j. differential geometry

55 (2000) 229-324

COMPACT SELF-DUAL MANIFOLDS WITH TORUS ACTIONS

AKIRA FUJIKI

Abstract

We show that a compact self-dual four-manifold with a smooth action of a two-torus and with non-zero Euler characterestic is necessarily diffeomorphic to a connected sum of copies of complex projective planes, and furthermore the self-dual structure is isomorphic to one of those constructed by Joyce in [11]. This settles a conjecture of Joyce [11] affirmatively. Our method of proof is to show, by complex geometric techniques, that the associated twistor space, which is a compact complex threefold with the induced holomorphic action of algebraic two-torus, has a very special structure and is indeed determined by a certain invariant which is eventually identified with the invariant associated with the Joyce’s construction of his self-dual manifolds.

1. Introduction

Let m be a non-negative integer and M = mP^{2} the connected sum of m copies of complex projective plane P^{2}. Suppose that we are given a smooth effective action of the real two torus K on M. In [11] Joyce constructed a series of examples of conformal self-dual metrics on M which are invariant under the given K-action. Further he raised the following conjecture [11, 3.3.4]:

Let M be a compact connected oriented simply connected four-manifold with a smooth and effective K-action. Suppose that M admits a K-invariant conformal self-dual metric. Then M must be diffeomorphic to a connected sum of

Received January 12, 2000. Work parially supported by Grants-in-Aid for Scientific Research no. 10440020 and 10874014 and also by Sonderforschungsbereich 237 of the Deutsche Forschungsgemeinschaft.

complex projective planes, and the conformal self-dual metric coincides with one of his self-dual metrics above.

The purpose of this paper is to give an affirmative answer to this conjecture in the following stronger form:

Theorem 1.1. Let M be a compact connected oriented four-manifold with smooth and effective K-action. Suppose that its Euler-Poincare characterestic χ(M) is nonzero. Then if M admits a K-invariant selfdual metric, M is diffeomorphic to a connected sum of copies of complex projective planes and the conformal self-dual metric coincides with those constructed in [11].

Together with the results of Poon [22] this leads to a classification of self-dual conformal structures with two-torus actions in general.

Corollary 1.2. Let M be a compact connected oriented four-mani-fold with smooth and effective K-action. If M admits a K-invariant self-dual structure, then either M is conformally flat or M is isomorphic to one of Joyce’s examples.

As is shown in [22, Th. A], when M is locally conformally flat with K-action, it is one of the following standard examples: 1) a manifold covered by a real four-torus, 2) a four-sphere S^{4}, and 3) a Hopf surface. Note also that compact self-dual manifolds which admit a group of conformal automorphisms of dimension greater than two have been classified by Poon [21].

Now the method of Joyce [11] for constructing the K-invariant selfdual metric is to reduce, by using the K-symmetry, the self-duality equation on M to a certain system of linear partial differential equations on the quotient N := M/K, which is identified with the unit disc in the complex plane. Once the underlying smooth K-action on M is fixed, his solutions of the latter equation, i.e., his self-dual metrics, depend on a parameter which is a circular sequence of k points on the boundary of the unit disc considered modulo the action of the conformal automorphism group of the unit disc and the cyclic permutations, where k = m + 2.

With this last fact in mind, we start with a general compact orientated self-dual four-manifold M with K-action, and study in detail the complex analytic structure of the associated twistor space Z with the induced action of the complexfication G of K, which is an algebraic two-torus.

Under the assumption of Theorem 1.1 the singular set Σ(Z) of the induced K-action on Z, i.e., the points where the stabilizer group is nontrivial, turns out to be a disjoint union of cycles of nonsingular rational curves. We then determine the local G-action at each point of Σ(Z). Using these results, we further show that the closures of general G-orbits which intersect a sufficiently small neighborhood of Σ(Z) is necessarily analytic and smooth in Z. Together with some global consideration using the Douady space of Z, we then conclude that every G-orbit has an analytic closure and there exists a canonical meromorphic quotient \overline{f} : Z → P of the G-action on Z, where P is a nonsingular rational curve.

The meromorphic map \overline{f} becomes a holomorphic map f : Ẑ → P after blowing up Z with center a distinguished cycle C of nonsingular rational curves contained in Σ(Z). This is in fact the main object of study in this paper. Among other things we show that the map \overline{f} is given by a subsystem of a G-invariant elements of the fundamental system |− \frac{1}{2}K| of Z, which forms a pencil, and that the general members of this system are mutually isomorphic projective smooth toric surfaces whose isomorphism class is explicitly determined by the invariant of the given K-action on M introduced by Orlik and Raymond [18]. At this stage we can already show that M is diffeomorphic to mP^{2} and that Z is Moishezon, where m is the second betti number of M. Σ(M) consists then of a single cycle of two-spheres and the quotient N = M/K is diffeomorphic to a closed two disc by [18] as in the case of Joyce.

The real structure σ on Z induces ones on Ẑ and P, making f σequivariant. The fixed point set R of σ on P is diffeomorphic to S^{1} and is considered as the boundary of either of the closed two-discs D which are connected components of P − R. It turns out that there exist exactly k singular fibers for f and the corresponding points a_{1}, …, a_{k} on the base P all lie in the real part R, where k = m + 2 as above. Suppose that a_{1}, …, a_{k} are arragned cyclically on R. Then we consider the sequence of points (a_{1}, …, a_{k}) (modulo the action of the conformal automorphism group of D and cyclic permutations) as a basic invariant of our twistor space Z (endowed with G-action, the real structure σ and the twistor fibration t : Z → M), and hence, of the original self-dual manifold M with K-action via the inverse twistor correspondence.

We then proceed to show the effectivity of this invariant in the following sense: Suppose that we are given two self-dual conformal structures [g] and [g′] on M = mP^{2} which is invariant under the given K-action on M. Perform the construction above for the corresponding twistor spaces Z and Z′ and obtain fiber spaces f : Ẑ → P and f′ : Ẑ′ → P′ respectively. Suppose further that the sets of points on the bases corresponding to the singular fibers are identical with respect to a suitable identification of P and P′. Then there exists a biholomorphic map j : Z → Z′ which is compatible with the real structures and G-actions and which maps twistor lines to twistor lines. We shall show this, first by studying the local structure of the fiber space f : Ẑ → P along each singular fiber, and then by studying its global structure on the basis of this local consideration.

Finally, for the purpose of identifying our invariant (a_{1}, …, a_{k}) with the Joyce’s “invariant” attached to his examples, we first interpret Joyce’s invariant as a special case of an invariant which can be defined for any simply connected compact conformal four-manifold with K-action. We then show that in the case of a self-dual manifold with Kaction, this conformal invariant and the invariant above arising from the associated twistor space are naturally identified via a certain twistorlike correspondence between the quotient N and one of the discs D in P with boundary R. Combined with the effectivity of the invariant mentioned above this immediately yields Theorem 1.1.

This article is arranged as follows. In §2 we summarize what we need on toric surfaces which will appear in the fibers of the morphism f : Ẑ → P above. In §3 first we recall some of the results of Orlik-Raymond [18] on K-actions on a compact smooth four-manifold M. Furthermore when a K-invariant conformal structure on M is given, we introduce an invariant of this conformal K-action as a circular sequence of points on the boundary of the unit disc on a complex plane modulo certain equivalence.

From §4 on we start our study of a self-dual manifold M with Kaction. After summarizing the basic construction and properties of the associated twistor space we determine the singular set Σ = Σ(Z) of the K-action on Z and the isotropic representation of the stabilizer group at each point of Σ. By using this in §5 we determine the local behaviour of G-orbits near the singular set Σ . From this we deduce that any G-orbit which intersects with a sufficiently small neighborhood of Σ has, up to a finite number of exceptions, a smooth analytic closure in Z. Combined with global consideration using the structure of the space of divisors on Z we obtain in §6 the meromorphic quotient Z → P and the associated holomorphic fiber space f : Ẑ → P mentioned above. In this section we further study the structure of this fiber space in detail. Especially, we show that there exist exactly k singular fibers S_{a_{i}}, a_{i} ∈ P, for f and that a_{i} are real points with respect to the induced real structure on P.

We then proceed to show that f is determined uniquely by the location of the points a_{i}. For this purpose we first show in §7 that this is the case at least locally along each singular fiber. Here the deformation theoretic method and the toric method give supplmentary descriptions of the local structure of this fiber space. Then in §8 we prove the global uniqueness of the fiber space f with the given invariant (a_{1}, …, a_{k}). Finally in §9 we prove the coincidence of this invariant with the conformal invariant defined in §3.

Notation and Convention. Unless otherwise is mentioned, G and K denote respectively the real and algebraic two-torus so that G = C^{∗}×C^{∗} and K ≅ S^{1} × S^{1}. In this paper we often encounter the sentences which contain ± more than once such as: “A^{±} is contained in B^{±}”, which should always read as: “A^{+} is contained in B^{+} and A^{−} is contained in B^{−}.”

Acknowlegdement. The author expresses gratitude to A. Huckleberry for giving him the oppotunity for writing up the manusucript of the paper in the nice atomosphere of Bochum University. Thanks are also due to N. Honda for stimulating discussions at the early stage of this investigation.

2. Toric surfaces

(2.1) We recall some basic definitions and results on toric surfaces [17, 13]. Let G ≅ C^{∗} × C^{∗} be an algebraic torus of dimension two. A compact smooth toric surface is a compact connected nonsingular projective algebraic surface S on which G acts effectively and algebraically and with a unique open orbit U.

The complement C := S − U then forms a cycle of nonsingular rational curves in the sense that it has the irreducible decomposition C = C_{1}∪⋯∪C_{s}, where each C_{i} is a nonsingular rational curve, C_{i}∩C_{j} = ∅ if |i − j| ≥ 2, and C_{i} and C_{i+1} intersect transversally at exactly one point, with C_{s+1} := C_{1}. Moreover C with reduced structure belongs to the anti-canonical system | − K_{S}| of S and it is called the anti-canonical cycle of the toric surface S. We also refer to the curves C_{i} as boundary curves of S.

Let a_{i} = C_{i}^{2} be the self-intersection number of C_{i}. Then we may associate with S a weighted circular graph with vertices v_{i}, one for each C_{i}, with weights a_{i}. The isomorphism class of a toric surface is completely determined by the associated circular weighted graph modulo cyclic permutations and reversing the order (cf. [17, Cor. 1.29]).

We also note that by [24, 7.6] the anti-Kodaira dimension κ^{−1}(S) of S always satisfies

κ^{−1}(S) = 2. (1)

(2.2) A toric surface is completely described by the associated combinatorial data [17, 1.1,1.2]. Let N be the set of one parameter subgroups of G, which is naturally a free abelian group of rank 2. We may write G = C^{∗} ⊗Z N and any isomorphism N ≅ Z^{2} gives rise to an isomorphism G ≅ C^{∗} × C^{∗}. Let NR = N ⊗Z R ≅ R^{2}.

A complete fan on N, or a complete rational polyhedral decomposition of N, is by definition a collection

△ = {{0}, τ_{j}, σ_{i}}, 1 ≤ i, j ≤ s

of one-dimensional simplices τ_{i} and two-dimensional simplices σ_{j} in NR, both defined over the rationals, such that NR = ∪σ_{i} and that the faces of σ_{i} consist of τ_{i} and τ_{i+1}, where τ_{s+1} = τ_{1}, σ_{s+1} = σ_{1} by convention and τ_{1}, …, τ_{n} are numbered so that it is arranged cyclically around the origin in NR. For any τ_{i} there exists a unique primitive element ρ_{i} ∈ τ ∩N such that τ_{i} = R_{≥0}ρ_{i}. The fan △ is then completely recovered from the circular sequence {ρ_{1}, …, ρ_{s}}. Conversely, any such sequcence gives rise to a complete fan, if ρ_{i} are not contained in a fixed halfplane in NR, by the above correspondence.

(2.3) Compact smooth toric surfaces are in natural bijecitive correspondence with complete fans in NR such that

(2) for every i, ρ_{i} and ρ_{i+1} form a Z-basis of N.

If we fix an isomorphism N ≅ Z^{2}, and hence, an isomorphism G ≅ C^{∗} × C^{∗}, we may write ρ_{i} = (m_{i}, n_{i}) for coprime integers m_{i} and n_{i} so that ρ_{i} is the one parameter subgroup given by ρ_{i}(t) = (t^{m_{i}}, t^{n_{i}}), t ∈ C^{∗}. We assume that ρ_{i} are arranged counterclockwise in NR ≅ R^{2}. Then the condition (2) is equivalent to the condition

| \begin{matrix}m_{i} & n_{i} \\ m_{i+1} & n_{i+1} \end{matrix} | = 1. (3)

The correspondence between toric surfaces and fans is determined by the following condition: If a compact smooth toric surface S corresponds to a fan △, then the one-parameter subgroup ρ_{i} : C^{∗}(t) → G fixes each point of C_{i} and for any point x ∈ U we have

lim_{t→0} ρ_{i}(t)(x) ∈ C_{i}. (4)

We also note the relation [17, p.43 (∗)]

ρ_{i−1} + ρ_{i+1} = −a_{i}ρ_{i}. (5)

(2.4) Let Aut(S, C) be the group of automorphisms of S which preserves C, and Aut_{0}(S, C) its identity component. Moreover, denote by Aut(S, {C_{i}}) the subgroup of Aut(S, C) consisting of those elements which preserves each irreducible component of C.

Lemma 2.1. G = Aut_{0}(S, C) = Aut(S, {C_{i}}).

Proof. We have the natural inclusions

G ⊆ Aut_{0}(S, C) ⊆ Aut(S, {C_{i}}).

The Lie algebra of Aut_{0}(S, C) is naturally identified with the sheaf Θ(− log C) of holomorphic vector fields on S which are tangent to C, while the latter is isomorphic to O_{S} ⊗Z N (cf. [17, Prop. 3.1]). Thus we have

dim Aut_{0}(S, C) = dim H^{0}(S, Θ(− log C)) = 2

and hence the equality G = Aut_{0}(S, C).

It remains to see that the latter inclusion is an equality. Let g ∈ Aut(S, {C_{i}}) be an arbitrary element. It induces an automorphism a(g) : t → gtg^{−1} of G, t ∈ G. But since g leaves each boundary component fixed, it also fixes the corresponding one parameter subgroup. This implies that the automorphism of N induced by a(g) is trivial, and hence, a(g) itself is trivial. Thus g commutes with every element of G. Let x be an arbitrary element of U and t the unique element of G which sends g(x) to x. Then tg has a fixed point x on U. The commutativity with elements of G implies then that tg fixes any point of U. Thus tg is the identity. Hence g ∈ G as desired. q.e.d.

(2.5) We are interested in a special class of toric surfaces.

Definition. A compact smooth toric surface S is called symmetric if the number s of boundary curves is even, say s = 2k, with k ≥ 2, and the weithts a_{i} of the weighted circlular graph associated to S has the property that a_{i} = a_{i+k} for all i, 1 ≤ i ≤ k. In this case we call a_{1}, …, a_{k} the reduced weights of S.

For instance if S_{0} := P^{1} × P^{1} with standard toric structure, it is symmetric with k = 2 and a_{1} = a_{2} = 0.

Suppose that S is a symmetric toric surface with s = 2k. Denote by p_{j} the intersection point of C_{j} and C_{j+1}, 1 ≤ j ≤ 2k. We call a pair of points admissible if it is of the form (p_{i}, p_{i+k}) for 1 ≤ i ≤ k. We call a blowing up S̃ → S admissible if its center is an admissible pair of points on S. Then any toric surface obtained from a symmetric toric surface by an admissible blowing-up is again symmetric.

Lemma 2.2. Every symmetric toric surface is obtained from S_{0} = P^{1} × P^{1} by a finite succession of admissible blowing-ups. In particular if k ≥ 3, a_{i} = C_{i}^{2} < 0 for all i.

Proof. We proceed by induction on k ≥ 2. When k = 2, the possible weights for a toric surface S are a, 0, −a, 0 for some non-negative integer a [17, Cor. 1.29]. Thus S is symmetric if and only if a = 0, i.e., S = S_{0}.

Let S be a toric surface with s = 2k ≥ 6. Suppose that the lemma is verified for symmetric toric surfaces with smaller s. Since S is obtained from a minimal toric surface by a finite succession of blowing-ups at some intersection points of boundary divisors [17, Th. 1.28], there is a (−1)-curve, say C_{i}, on S. By symmetry, C_{i+k} also is a (−1)-curve. Then by blowing down these two (−1)-curves we obtain a symmetric toric surface with s = 2k − 2. By induction it is obtained from S_{0} by a finite succession of admissible blowing-ups. The lemma follows. q.e.d.

For later use we also give a result on the existence of (−1)-curves among the boundary curves considered in the above proof. We call a pair of (−1)-curves of the form (C_{j}, C_{j+k}) admissible.

Lemma 2.3. Let S be a symmetric toric surface with s = 2k ≥ 6. Fix a pair of admissible points (p_{i}, p_{i+k}) on S. Then there eixsts an admissible pair of (−1)-curves on S which do not pass through either of p_{i} and p_{i+k}.

Proof. We proceed by induction on k ≥ 3. If k = 3, S is obtained from S_{0} by an admissible blowing-up. Then all the weights are −1 and the conclusion of the lemma is true.

In the general case let S be obtained from a symmetric torus surface \overline{S} by an admissible blowing-up. Let (C_{j}, C_{j+k}) be the admissible pair of (−1)-curves which are the exceptional curves of this blowing up. If they do not pass through either of p_{i} and p_{i+k}, the proof is done. Suppose not. Then p_{i} and p_{i+k} are mapped to the center of this admissible blowing-up. By induction there exists on \overline{S} an admissible pair of (−1)-curves which do not pass through these points. Then the proper transforms of these (−1)-curves to S give rise to a desired admissible pair of (−1)-curves.

q.e.d.

(2.6) We give characterizations of symmetric toric surfaces.

Proposition 2.4. Let S be a toric surface such that the number s of boundary curves is an even number 2k(≥ 4). Then the following conditions are equivalent:

1) S is symmetric.

2) The fan defining S satisfies ρ_{i+k} = −ρ_{i} for all i, 1 ≤ i ≤ k.

3) There exists a holomorphic involution τ of S which interchanges C_{i} and C_{i+k}, 1 ≤ i ≤ k, and which satisfies

τgτ^{−1} = g^{−1}, g ∈ G. (6)

Proof. 3) ⟹ 2): Applying τ to (4) and taking into account the relation (6) we obtain

C_{i+k} = τ(C_{i}) ∋ lim_{t→0} τ(ρ_{i}(t)x) = lim_{t→0} ρ_{i}(t)^{−1}(τ(x))

This implies that −ρ_{i} = ρ_{i+k}. 2) follows.

2) ⟹ 3): If 2) is true, the multiplication by −1 is an isomorphism of the fan of S. Therefore it defines an involution τ of S which maps C_{i} to C_{i+k} satisfying (6) (cf. [17, Th. 1.13]).

3) ⟹ 1): Clear.

1) ⟹ 3): We proceed by induction on k. When k = 2, the involution τ of S_{0} = P^{1} × P^{1} defined by (z, w) → (1/z, 1/w) has the desired property. In the general case let S be obtained from a symmetric toric surface \overline{S} by an admissible blowing-up. Then by induction there exists an involution \overline{τ} of \overline{S} satisfying the desired properties. Since \overline{τ} interchanges the blown-up points p_{i} and p_{i+k}, it lifts to an involution τ on S inheriting the desired properties from \overline{τ}. q.e.d.

Let τ be as in 3) above. Let Ĝ be the semidirect product Ĝ := G·⟨τ⟩ with respect to the action (6) of τ on G. We have Ĝ = G ∐ τG and any element of τG is an involution and can play the role of τ above. The above lemma shows that for a symmetric toric surface S, Aut(S, C) always contains a subgroup isomorphic to Ĝ and any admissible blowingdown to S_{0} is Ĝ-equivariant.

(2.7) Given k = m + 2 ≥ 2, up to isomorphisms there exist at most a finite number n(k) of symmetric toric surfaces; in fact we have n(2) = n(3) = n(4) = n(5) = 1 and n(6) = 3. More precisely, we obtain the following table:

k (−a_{1}, …, −a_{k}) n(k)  S
2  (0,0)  1  Del Pezzo
3  (1,1,1)  1  Del Pezzo
4  (1,2,1,2)  1 generalized Del Pezzo
5  (2,1,3,1,2)  1

(1,2,3,1,2,3)

6 (1,3,1,3,1,3)  3
(2,1,4,1,2,2)  of maximal type

In general for every k there exists (up to equivalence) a unique weighted circular graph with 2k vertices which has −m as a reduced weight; then the whole reduced weights are given (up to cyclic permutations and the inversion of the order) by

(−a_{1}, …, −a_{k}) = (2, …, 2, 1, m, 1, 2, …, 2)

unless k ≤ 4. In this case we call the corresponding toric surface of maximal type. In this sense the surfaces for 2 ≤ k ≤ 5 are all of maximal type although it is not indicated in the table. Moreover the number n(k) can be determined explicitly for any k (cf. Proposition 3.4 below).

(2.8) We shall determine equivariant morphisms of a symmetric toric surface onto curves.

Lemma 2.5. Let S be a symmetric toric surface with boundary components C_{i}, 1 ≤ i ≤ 2k. Then for each i, 1 ≤ i ≤ k, there exists a natural equivariant morphism ν_{i} : S → P_{i} onto a nonsingular rational curve P_{i} with the following properties:

1) Any proper surjective morphism of S onto a curve coincides with one of ν_{i}. More generally, any nonsingular rational curve L in S with self-intersection number L^{2} = 0 is a fiber of ν_{i} for a unique i.

2) C_{i} and C_{i+k} are mapped isomorphically onto P_{i} by ν_{i}.

3) With respect to a suitable choice of an affine coordinate z of P_{i} the singular fibers of ν_{i} appear precisely over 0 and ∞ such that:

a) The singular fiber over 0 is a positive linear combination of C_{j}, i + 1 ≤ j ≤ i + k − 1, while the fiber over ∞ is one of C_{j} with i + k + 1 ≤ j ≤ i − 1 (index considered modulo 2k) where the coefficients of C_{j} and C_{j+k} coincide for any j.

b) The smooth fibers are orbit closures of the one parameter group ρ_{i}.

4) For any i, ν_{i} × ν_{i+1} : S → P^{1} × P^{1} is a birational morphism which contracts all the components C_{j} other than C_{l} and C_{l+k}, l = i, i + 1.

Proof. We give the definition of ν_{i} in two ways.

1) Geometric definition: Fix i. By Lemma 2.3 there exists an admissible blowing-down u : S → S_{0} in which all the components C_{j} other than C_{l}, C_{l+1}, l = i, i + 1, are contracted to an admissible pair of points on S_{0} and which is otherwise isomorphic. Then the compositions S → P^{1} of u with the two projections S_{0} → P^{1} yields holomorphic surjection ν_{i} and ν_{i+1}. The properties 2), 3) and 4) are all immediately checked as well as the independency of the definition of the choices of the points p_{i} or p_{i−1} used in Lemma 2.3.

2) Combinatorial definition: For every 1 ≤ i ≤ k we consider the quotient homomorphism χ_{i} : N → \overline{N}_{i} := N/Zρ_{i}. Since ρ_{i} is primitive, \overline{N}_{i} is a free abelian group of rank one and χ_{i} induces a natural map of fans (N, △) → (\overline{N}_{i}, \overline{△}), where if ξ is a genarator of \overline{N}_{i}, \overline{△} = \{R_{≥0}ξ, 0, −R_{≤0}ξ\}.

Corresponding to χ_{i} we get a G-equivariant morphism S → P_{i} onto the toric projective line P_{i}, which is easily identified with the above ν_{i}. The map ν_{i} × ν_{i+1} : S → S_{0} is the equivariant birational morphism corresponding to the “forgetting map” of the fans, where the fan of S_{0} is obtained by deleting all the primitive vectors ρ_{j} other than ρ_{l}, ρ_{l+1}, l = i, i + 1.

Proof of the property 1). It suffices to show the latter statement. If L^{2} = 0, the linear system defined by L gives a morphism f : S → P^{1} onto the complex projective line, which is necessarily equivariant with an induced G-action on P^{1} [23, Satz 1.3]. On the other hand, any equivariant morphism of toric varieties corresponds to a map of fans (cf. [17, Th. 1.13]); however, from the shape of our fan it is immediate that the possible such equivariant morphisms are those given by χ_{i}.

q.e.d.

(2.9) Let S be a symmetric toric surface. Let g → g^{∗} be the unique anti-holomorphic involution of G with fixed point set the maximal compact subgroup K ≅ S^{1}×S^{1}. Suppose that S admits an anti-holomorphic fixed point free involution satisfying

σgσ^{−1} = g^{∗}, g ∈ G. (7)

Then the following holds:

Lemma 2.6. The equivariant morphisms ν_{i} : S → P_{i} of Lemma 2.5 are all σ-equivariant with respect to an induced action of σ on P_{i}. If there exists a real fiber, then the fixed point set R_{i} of σ on P_{i} is diffeomorphic to S^{1} and consists of a single K-orbit.

Proof. Applying σ to (4) and taking into account (7) we obtain

σ(C_{i}) ∋ lim_{t→0} σ(ρ_{i}(t)(x)) = lim_{t→0}(ρ_{i}(t)^{∗})(σ(x)) ∈ C_{i+k}.

This implies that

σ(C_{i}) = C_{i+k} (8)

for any i. Thus the σ-equivariancy of ν_{i} follows from 3) b) of Lemma 2.5. The second assertion is obvious. q.e.d.

Lemma 2.7. There exist up to isomorphisms at most four antiholomorphic involutions on S satisfying the condition (7).

Proof. On S_{0} = P^{1} × P^{1} we have four natural anti-holomorphic involutions; with respect to affine coordinates (z, w) of S_{0} they are (z, w) → (±1/\overline{z}, ±1/\overline{w}). Note that these anti-holomorphic involutions are expressed as σs for some element s of order ≤ 2 of K. It is easy to see that these all satisfy the property (7). For any succession of admissible blowing-ups starting from S_{0} these involutions lift to the resulting symmetric toric surface. By Lemma 2.2 the existence of four anti-holomorphic involutions which satisfy (7) is thus verified in general.

Next we show that any such anti-holomorphic involution is isomorphic to one of them. If there exist two anti-holomorphic involutions σ and σ′ of S satisfying (7), then in view of (8) h := σ^{−1}σ′ is a holomorphic automorphism of S which preserves all the irreducible components of C. Then by Lemma 2.1 we have h ∈ G. Since σ′ is an involution, we must have σhσh = id; thus σhσh(x) = h^{∗}hx = x for any x ∈ S, or equivalently, with respect to a suitable isomorphism G ≅ C^{∗} × C^{∗} we may write h = (t_{1}, t_{2}) with t_{i} ∈ R. If we set |h| = (|t_{1}|, |t_{2}|), we may write h = s|h| with s = (±1, ±1). Then

a^{−1}σa(x) = σa^{2}(x) = σ|h|(x) for a = (\sqrt{|t_{1}|}, \sqrt{|t_{2}|}).

Thus, σh = σ|h|s = a^{−1}(σs)a. Namely, σh is conjugate to σs, and up to isomorphisms there exist at most four non-isomorphic anti-holomorphic involutions. q.e.d.

(2.10) We may “divide into halves” a symmetric toric surface along each of its k “diagonals” \overline{p_{i}p}_{i+k}. Let S be a symmetric toric surface with reduced weights (a_{1}, …, a_{k}). Then for each i, 1 ≤ i ≤ k, we define a pair of (not necessarily symmetric) projecitve smooth toric surfaces S^{+} and S^{−} as follows. Let ρ = {ρ_{i}, 1 ≤ i ≤ 2k} be the set of primitive elements giving the fan of S = S(ρ). Then define for each i

S_{i}^{±}(ρ) (9)

to be the toric surfaces determined by a sequence of primitive elements

±(ρ_{1}, …, ρ_{i}, −ρ_{i} + ρ_{i+1}, −ρ_{i+1}, …, −ρ_{k}) (10)

which generates 1-simplices of a complete fan. (Here and in what follows we adopt the following convention. If a sentence contains the notation ± more than once, it actually consist of two sentences, one for + only and the other for − only. See Convention and Notation in the Introduction.) It is clear that any two adjacent elements in (10) form a Z-basis so that the resulting toric surface is smooth and compact. Both surfaces have k + 1 boundary components, which we shall denote by (C_{1}^{±}, …, C_{i}^{±}, L_{i}^{±}, C_{i+1}^{±}, …, C_{k}^{±}) respectively. In view of (5), from

(11)  ρ_{i} + (−ρ_{i+1})  = (−1)(−ρ_{i} + ρ_{i+1})
(12)  ρ_{i−1} + (−ρ_{i} + ρ_{i+1}) = −a_{i}ρ_{i} − ρ_{i} = −(a_{i} + 1)ρ_{i}

the self-intersection numbers are computed as

L_{i}^{±2} = 1 and C_{l}^{±2} = a_{l} + 1, l = i, i + 1

respectively. Therefore the corresponding weighted cirular graphs for S_{i}^{±}(ρ) are the same and is given by

(a_{1}, … a_{i−1}, a_{i} + 1, 1, a_{i+1} + 1, a_{i+2}, …, a_{k}) (13)

modulo cyclic permutations; in particular S^{+}(ρ) and S^{−}(ρ) are mutually isomorphic. We shall note one important property of surfaces S_{i}^{±}(ρ).

Lemma 2.8. Let S be either of S_{i}^{±}(ρ) of (9). Then there exists a toric birational morphism f : S → P^{2} which blows down all the boundary curves other than L and the two irreducible components of C which are adjacent to L.

Proof. We proceed by induction on k ≥ 2. When k = 2, by (13) and Lemma 2.2 S = P^{2} and there is nothing to prove. Suppose that k > 2. Then by Lemma 2.3 there exists an admissible pair of (−1)-curves on S(ρ) which do not pass through neither p_{i} nor p_{i+k}. Contracting these (−1)-curves we obtain a symmetric toric surface S(ρ′) for a suitable ρ′ with (k − 1) boundary components. One of these (−1)-curves may be naturally identified with a (−1)-curve in S and contracting this curve in S we obtain a surface S′ which is identified with either of S_{i′}^{+}(ρ′) for some i′. So we may apply the induction hypothesis to finish the proof. (One may also note that the fan generated by the primitive vectors ρ_{i}, ρ_{i+1} and −ρ_{i} + ρ_{i+1} is exactly the fan defining a complex projective plane.) q.e.d.

We identify G-equivariantly S_{i}^{+} = S_{i}^{+}(ρ) and S_{i}^{−} = S_{i}^{−}(ρ) transversally along L. We shall call the resulting surface S_{i} := S_{i}^{+} ∪_{L} S_{i}^{+} with natural G-action a degenerate symmetric toric surface. The Gequivariancy implies that C_{i}^{+}, L and C_{i+1}^{−} (resp. C_{i}^{−}, L and C_{i+1}^{+}) intersect (transversally) at one point p_{i}^{+} (resp. p_{i}^{−}). We set b_{j}^{±} = C_{j}^{±} ∩ C_{j+1}^{±}, where the indices j are considered cyclically modulo k. We shall call any blowing up of S_{i} with center a pair of smooth points {b_{j}^{+}, b_{j}^{−}}, j ≠ i, an admissible blowing up. Then the resulting surface is again a degenerate symmetric toric surface naturally. In particular if k = 2, S_{i} are all isomorphic and is a union of two copies of the toric projective planes. Denote this surface by \overline{S}. Then from Lemma 2.8 we deduce immediately the following:

Lemma 2.9. Any degenerate symmetric toric surface S_{i} as above is obtained from a finite succession of admissible blowing-ups from \overline{S}.

We also use the following:

Lemma 2.10. Let S = S_{i}^{±}(ρ), L = L_{i} and C_{j} be as above. Then the complete linear system |−K_{S} −2L| is non-empty if and only if either of C_{i} and C_{i+1} has the self-intersection number 3 − k = 1 − m.

Proof. When k = 2, S is a toric projective plane P^{2} and the anticanonical cycle L_{0} + L_{1} + L_{2} consists of three lines L_{j}, one of which is our L. Thus | − K_{S} − 2L| = |L| on P^{2}, which is non-empty and L_{j}^{2} = 1 = 3 − 2. In the general case let f : S → P^{2} be as in Lemma 2.8 and suppose that L is the proper transform of, say L_{0}. If |−K_{S}−2L| ≠ ∅, then the successive blowing-ups, of which f is the composition, is done always on the proper transform of one of L_{i} for a fixed i(= 1 or 2). Then the proper transform of this L_{i} is either C_{i} or C_{i+1} and it has the desired self-intersection number 3 − k. It is also easy to see that conversely, if the latter holds, f is obtained precisely as described above. q.e.d.

3. Torus actions and conformal invariants

(3.1) Let M be a connected compact oritented smooth 4-manifold. Suppose that a real two-torus K ≅ S^{1}×S^{1} acts smoothly and effectively on M. This situation has been extensively studied by Orlik-Raymond [18], to which we refer for the details.

We begin with a simple and well-known remark on the fixed point set of our K-action.

Lemma 3.1. Let M with a K-action be as above. Then the Euler-Poincare characterestic χ(M) of M is nonnegative, and it is positive if and only if K has a fixed point. In particular the latter is the case when M is simply connected.

Proof. We first note that the fixed point set F of the action is isolated if it is not empty. Indeed, let x ∈ F be any point. Then the isotropic representation of K in the normal space N_{x} at x of the submanifold F in M is faithful. But this is possible only when dim N_{x} = 4, i.e., x is isolated.

Now if we take a general S^{1}-subgroup S of K, its fixed point set also coincides with F. Hence we have χ(M) = χ(F) + χ(M − F) with χ(M − F) = 0, S acting on M − F locally freely. Thus, with b_{i} the i-th betti number of M, we get

0 ≤ #F = χ(F) = χ(M) = 2 − 2b_{1} + b_{2}, (14)

where #F denotes the cardinality of F. The lemma follows. q.e.d.

(3.2) We call a union

B := B_{1} ⋃ ⋯ ⋃ B_{k}.

of 2-spheres B_{i} on M a cycle of two-spheres if it has the property that B_{i} ⋂ B_{j} = ∅ if |i − j| ≥ 2, and B_{i} and B_{i+1}, 1 ≤ i ≤ k, intersect transversally at a single point x_{i}, B_{i} ∩ B_{i+1} = {x_{i}}, 1 ≤ i ≤ k, where we consider the indices i and j modulo k so that B_{k+1} = B_{1}.

Let Σ(M) be the singular set of the K-action, i.e., the set of points whose stabilizer group is nontrivial.

Lemma 3.2. Suppose that χ(M) ≠ 0. Then Σ(M) is a disjoint union of cycles of two-spheres, each of which is of the form B := B_{1} ∪ ⋯ ∪ B_{k} as above such that:

1) Each x_{i} := B_{i} ∩ B_{i+1}, 1 ≤ i ≤ k, is a fixed point.

2) At each point x of B′_{i} := B_{i} − {x_{i−1}, x_{i}} the stabilizer group is a S^{1}-subgroup K_{i} of K which is independent of the choice of the point x, where x_{0} = x_{k}.

Moreover, if M is simply connected, B is unique and coincides with Σ(M).

Proof. By Lemma 3.1 χ(M) > 0 and there exists a fixed point x. Then the isotropic representation of K at x is a direct sum of two representations, in each of which the representation has a fixed S^{1}-subgroup as the kernel. The fixed point set of these S^{1}-subgroups consists of smooth surfaces embedded in M and intersect transversally at x. These surfaces must be diffeomorphic either to S^{2} or real projective plane RP^{2} since it admits an S^{1}-action with fixed points.

We claim that RP^{2} does not occur. Indeed, let R be one of the surfaces passing through x and K_{1} ≅ S^{1} the stabilizer group at a general point of R. We have the effective action of K/K_{1} on R. Suppose now that R is a real projective plane RP^{2}. Then the element −1 ∈ K/K_{1} ≅ S^{1} admits a component H of a fixed point set on R which is homeomorphic to S^{1}. We may lift −1 to an element s of order two on K. Then K_{1} × ⟨s⟩(≅ S^{1} × Z_{2}) acts effectively on the two dimensional normal vector space N_{x} at each point x of H, which is impossible. Thus R is never RP^{2}. Then for any fixed point x we have two embedded twospheres intersecting transversally at x. Take one of it, say B_{1}. On B_{1} we have exactly one more fixed point, say x_{2}, other than x_{1}. Repeat the same consideration at x_{2} and obtain another embedded two-sphere B_{2} intersecting transversally with B_{1} at x_{2} and is pointwise fixed by some S^{1}-subgroup of K. Continuing in this way we finally obtain a sequence of two-spheres B_{1}, …, B_{k} each of which is pointwise fixed by some S^{1}- subgroup of K, where B_{k} and B_{0} intersect transversally at a single point x_{k}. In this way we obtain a cycle of two-spheres B := B_{1} ∪⋯∪B_{k} with the desired property.

2) is well-known (cf. [18]) and the final assertion is proved in [18, §5]. q.e.d.

(3.3) Fix an identification K = S^{1} × S^{1}. We identify S^{1} with the multiplicative group of complex numbers of modulus one. For any pair (m, n) of coprime integers we introduce the S^{1}-subgroup K(m, n) of K by

K(m, n) := {(s, t) ∈ K = S^{1} × S^{1}; s^{m}t^{n} = 1}.

These groups are mutually different except that we have K(m, n) = K(−m, −n).

Now we consider a general K-action on M with fixed points as above. Let B = ∪B_{i} be one of the cycles of S^{2} contained in the singular set of the action. Then the stabilizer group K_{i} of a point x belonging to the set B′_{i} is independent of x and is written as K_{i} = K(m_{i}, n_{i}) for a pair (m_{i}, n_{i}) of coprime integers m_{i} and n_{i} which are determined up to (simultaneous) inversion of signs. Moreover, these pairs of coprime integers are subject to the condition:

| \begin{matrix}m_{i} & n_{i} \\ m_{i+1} & n_{i+1} \end{matrix} | = ±1 (15)

coming from the effectivity of the action. Thus, associated to the cycle B one obtains a circular sequence

{±(m_{1}, n_{1}), …, ±(m_{k}, n_{k})} (16)

of pairs of coprime integers considered up to signs as above. We shall call this sequence (16) the Orlik-Raymond invariant of the cycle B (or of M when M is simply connected.)

(3.4) When M is simply connected, also the structure of the quotient is determined (cf. [18, §5]). In this case, as follows from (14), the number m := k − 2 equals the second Betti number of M and the set {x_{1}, …, x_{k}} are precisely the set of fixed points of K The quotient N := M/K is naturally homeomorphic to the closed two-disc and in this way N admits a structure of a C^{∞} manifold with boundary. Let π : M → N be the quotient map and N_{0} the interior of N. Then M_{0} := π^{−1}(N_{0}) = M − B is the maximal open subset of M on which K acts freely.

Further, if bN denotes the boundary of N, we have π^{−1}(bN) = B, and bN = B/K. Moreover, the images

y_{i} := π(x_{i}), 1 ≤ i ≤ k,

of x_{i} are arranged cyclically on the circle bN with respect to an orientation of bN and then B_{i}/K is identified with the closed arc on bN connecting y_{i−1} and y_{i}. In this case the Orlik-Raymond invariant, considered modulo cyclic permutations and reversing the orders, completely determines the isomorphism class of the K-action on M [18].

(3.5) We shall give a supplement to a result of Joyce [11, 3.1]. Namely we show that when M is diffeomorphic to the connected sum mP^{2} of m-copies of the complex projective plane P^{2}, the number ι(m) of isomorphism classes of K-actions on M can be explicitly determined.

Proposition 3.3. Set j(m) = (2m)!/m!(m + 1)! for m ≥ 0, where 0! = 1. Then ι(m) for m > 0 is given by the formula:

ι(m) = j(m)/2(m + 2) + j((m − 1)/3)/3 + h(m),

where h(m) = 3j(m/2)/4 if m is even and = j((m − 1)/2)/2 if m is odd, with j(l) = 0 if l is not an integer.

Proof. By [11, Prop. 3.1.1] the Orlik-Raymond invariant

{±(m_{i}, n_{i}); 1 ≤ i ≤ k = m + 2}

can be normalized as follows: There exists a unique element h of SL(2, Z) such that, after changing the identification K = S^{1} × S^{1} by the automorphism of K induced by h if necessary, one may choose the signs of ±(m_{i}, n_{i}) so that it is arranged couterclockwise in this order in R^{2} such that

(m_{1}, n_{1}) = (1, 0) and (m_{k}, n_{k}) = (0, 1). (17)

In this case we automatically have m_{i} > 0 if i < k and

m_{i}n_{i+1} − m_{i+1}n_{i} = 1, 1 ≤ i < k. (18)

Moreover, by [11, Prop. 3.1.2] given an integer k ≥ 2 the number of the sequences (16) satisfying the above conditions (17) and (18) is equal to j(m). In his proof this number is in fact identified with the number of subdivisions of a regular k-gon into triangles.

On the other hand, the Orlik-Raymond invariant is determined up to the cyclic permutations and reversing the orders. In the above correspondence counting the number of possible Orlik-Raymond invariants modulo this equivalence relation exactly corresponds to counting the number of the above subdivisions modulo rotations and reflections of the k-gons. This latter numer is indeed computed to be equal to be the number given in the statement of the proposition by Moon and Moser [16] (cf. the reference of Brown’s article listed in [11]). Since the Orlik-Raymond invariant determines the action completely, this yields the proposition. q.e.d.

The number ι(m) also coincides with the number of the isomorphism classes of symmetric toric surfaces.

Proposition 3.4. For k ≥ 2 the number ι(k − 2) coincides with the number n(k) of isomorphism classes of symmetric toric surfaces with 2k boundary components.

Proof. If {±(m_{i}, n_{i}); 1 ≤ i ≤ k} is an Orlik-Raymond invariant, then with respect to a suitable choice of signs (which is unique up to simultaneous inversion of signs) the vectors

(n_{1}, −m_{1}), …, (n_{k}, −m_{k}), (−n_{1}, m_{1}), …, (−n_{k}, m_{k})

form a cyclic sequence of primitive vectors arranged counterclockwise in Z^{2}. These give rise to a fan defining a symmetric toric surface by setting ρ_{i} = (n_{i}, −m_{i}) and ρ_{i+k} = (−n_{i}, m_{i}) for 1 ≤ i ≤ k (cf. Lemma 4.12 below). This sets a bijective correspondence of the set of Orlik-Raymond invariants and symmetric fans when both are considered modulo the action of GL(2, Z). q.e.d.

Remark. One can establish a geometric correspondence between symmetric toric surfaces and mP^{2} with K-actions.

(3.6) In the rest of this section we shall introduce an invariant for a simply connected conformal four-manifold with K-action. So suppose now that M is simply connected and is endowed with a K-invariant conformal structure [g].

Let △ be the open unit disc {|z| < 1} in the complex plane C = C(z) with the unit circle S = {|z| = 1} as the boundary. We denote by A the set of (ordered) sequences (q_{1}, …, q_{k}) of distinct points q_{i} of S arranged cyclically on S with respect to an orientation of S and considered modulo cyclic permutations. Let H be the group of conformal automorphisms of △, any of which extends to a diffeomorphism of the closure \overline{Δ}. H thus acts on the set A and the quotient \overline{A} := A/H is the set in which our invariant for the action belongs. Note that two sequences (q_{1}, …, q_{k}) and (q′_{1}, …, q′_{k}) defines the same element of \overline{A} if and only if there exist a conformal automorphism h of △ and an integer l with 1 ≤ l < k such that h(q_{i}) = q_{i+l} (resp. q_{i−l}) for all i.

We use freely the notation of (3.4). We have a natural conformal structure [h] on N_{0}. In fact, for any point y ∈ N_{0} the corresponding conformal structure on the tangent space T_{y}N_{0} is described as follows. Take a point x of π^{−1}(y). Let Kx be the orbit of x and T_{x}^{⊥} the orthogonal complement of the tangent space of Kx in the tangent space of M at x with respect to the given conformal structure. Then via the natural isomorphism T_{x}^{⊥} →^{∼} T_{y}N we get an induced conformal structure, which is independent of the choice of the point x ∈ π^{−1}(y).

Since N_{0} is of real dimensional two, the conformal structure defines a unique complex structure up to complex conjugation. Hence by the Riemann’s mapping theorem there exists a conformal isomorphism of N_{0} onto the open unit disc △. (From the argument below it follows that N_{0} is not isomorphic to C.) Let α : N_{0} → △ be such a map. We shall prove below the following:

Lemma 3.5. α extends to a unique homeomorphism \overline{α} of N onto the closed disc \overline{△}.

Then our invariant for a conformal torus action is defined as follows.

Definition. Let x_{1}, …, x_{k}, k ≥ 2, be the fixed points of the Kaction and y_{i} = π(x_{i}) ∈ bN, 1 ≤ i ≤ k, be as in (3.4). Then we set

q_{i} = \overline{α}(y_{i}), 1 ≤ i ≤ k.

Since any other choice of \overline{α} is obtained by composing an element of H with α, the image [q_{1}, …, q_{k}] of the cyclic sequence (q_{1}, …, q_{k}) in \overline{A} is an invariant of the given K-action.

Example. In the construction of self-dual metrics on mP^{2} in [11] Joyce first puts a conformal structure on the quotient N_{0} = M_{0}/K by identifying N with the closed unit disc \overline{△} and then specifies k points {y_{1}, …, y_{k}} on the boundary bN, which is to be the parameter of his self-dual metrics. By the very construction of his, this conformal structure on N_{0} is precisely the one coming from the conformal structure of M as explained above and the points y_{i} are exactly the images by π : M → N of the fixed points of K on M. Thus the parameter of Joyce is naturally identified with the conformal invariant of the K-action on M defined above.

(3.7) For the proof of Lemma 3.5 we use the following classical result due to Lindelöf [2, p.86]:

Lemma 3.6. Let D be a simply connected bounded domain in the complex plane C whose boundary is a Jordan curve. Let α : D → △ be a biholomorphic map onto the unit disc, whose existence is guaranteed by the Riemann’s mapping theorem. Then α extends to a homeomorphism up to the boundary of both domains.

In order to apply this result we prove the following:

Proposition 3.7. N is realized as a closed subdomain of a simply connected manifold N̂ to which the conformal structure [h] on N_{0} extends smoothly.

Proof of Lemma 3.5 from Proposition 3.7. By Proposition 3.7 and the Riemann’s mapping theorem, there exists a conformal isomorphism of N̂ (with the extended conformal structure) onto the unit disc △ in C, after restricting N̂ if necessary. The restriction of this map to N realizes N_{0} as a bounded domain in C with smooth boundary so that we can apply Lemma 3.6 to get a desired homeomorphic extension of ϕ. q.e.d.

(3.8) Before going into the proof of Proposition 3.7 we recall some of the general facts in the construction of the quotient N = M/K in the smooth category.

To get N̂ we consider N as a C^{∞} manifold with corners rather than as a manifold with boundary. Here, in our two dimensional case a manifold with corners means a manifold structure which is modelled on open subsets of the first quadrant R_{≥0}^{2} := {(x, y); x, y ≥ 0} and C^{∞} maps i.e., the maps which extend to C^{∞} maps of open neighborhoods in R^{2}. Such a structure on N is determined roughly as follows.

For each point y ∈ N take a point x ∈ π^{−1}(y) and a local slice S at x for the action; S is a locally closed subset of M passing through x, which is mapped homeomorphically by π onto a neighborhood, say P, of y of N, and which is diffeomorphic to an open subset Q of R_{≥0}^{2}. Then the homeomorphism P → S → Q defines a coordinate neighborhood of N for the structure of a manifold with corners on N. In our case we prove further:

Lemma 3.8. A slice S as above can be taken to be an open subset of a locally closed submanifold Ŝ of M such that there exists a diffeomorphism of Ŝ onto an open subset of R^{2} which maps S onto an open subset of R_{≥0}^{2}.

Proof. The construction of the slices at x ∈ B is as follows. We take a K_{x}-invariant (complex) coordinate neighborhood W of x in M in which the action of K_{x} is complex linearlized. Let z_{1}, z_{2} be (smooth) complex coordinates around x and write z_{i} = u_{i} + \sqrt{−1}v_{i}, u_{i}, v_{i} ∈ R, i = 1, 2. We have two cases:

Case 1. x ≠ x_{i}.

In this case, by a suitable choice of z_{i} the K_{x}-action takes the form:

(z_{1}, z_{2}) → (z_{1}, sz_{2}), s ∈ K_{x} ≅ S^{1}

where the orbits of the induced action of K/K_{x} is defined by u_{1} = const. and z_{2} = const. We then define a local slice S (resp. its extension Ŝ) to be the intersection of W with the halfplane (resp. plane) {v_{1} = v_{2} = 0, u_{2} ≥ 0} (resp. {v_{1} = v_{2} = 0}).

Case 2. x = x_{i}.

With respect to a suitable isomorphism K ≅ S^{1}(s) × S^{1}(t), the K-action takes the form

(z_{1}, z_{2}) → (sz_{1}, tz_{2}).

We define a local slice S (resp. its extension Ŝ) as before to be the intersection of W with the quadrant (resp. the plane) {u_{i} ≥ 0, v_{i} = 0, i = 1, 2} (resp. {v_{1} = v_{2} = 0}).

Clearly in either case the pair (S, Ŝ) meets the requirement of the lemma. q.e.d.

(3.9) The main point in the proof of the existence of a smooth structure on the quotient is that if we take another slice S′ at the same point x defined in the same way as above, starting from another K_{x}- linearlized coordinates, we can construct naturally a diffeomorphism between S and S′.

To understand this, it is convenient to perform the real blowing up ν : M̃ → M along B = ∪B_{i}. Then M̃ is a smooth 4-manifold with corners to which the K-action naturally lifts. The main advantage in passing to M̃ is that the action becomes free there. Moreover, since each fiber of ν is in the same K-orbit, we have N = M̃/K. A slice on M is then nothing but a diffeomorphic image of a slice for the free K-action on M̃, and if S̃ and S̃′ are such slices on M̃ we have a corresponding deffeomorphism ũ : S̃ → S̃′ in the form ũ(s̃) = g̃(s̃)s̃ for a unique smooth map g̃: S̃ → K. Since ν is K-equivariant, this induces a diffeomorphism u : S → S′ with u(s) = g(s)s, where the smooth map g : S → K is naturally induced by g̃.

Now we show

Lemma 3.9. The above diffeomorphism u : S → S′ extends to a diffeomorphism û : Ŝ → Ŝ′ between their respective extensions Ŝ and Ŝ′ as defined in Lemma 3.8.

Proof. In Case 1 (resp. Case 2), by the construction of Ŝ in the proof of Lemma 3.8 there exists an element k (resp. k_{1}, k_{2}) of K_{x} of order two such that

Ŝ = S ∪ k(S) (resp. Ŝ = S ∪ k_{1}(S) ∪ k_{2}(S) ∪ (k_{1}k_{2})(S));

in fact in the local expression there we may take

k = −1 (resp. k_{1} = (−1, 0), k_{2} = (0, −1)).

Then for x ∈ k(S) we have

g(k^{−1}x)(x) = (kg(k^{−1}x)k^{−1})(x) = kg(k^{−1}x)(k^{−1}x) ∈ k(S′) ⊆ Ŝ′

where k = k_{i}, i = 1, 2 in Case 2. Thus the map g : S → K extends naturally to a continuous map ĝ : Ŝ → K such that x → ĝ(x)x gives a homeomorphism of Ŝ onto Ŝ′; in fact it is characterized by the equation

(19)  ĝ(x) = g(kx), x ∈ Ŝ − S and k as above.

It remains to show that the extended map is smooth on Ŝ. For this purpose we may consider the map g also as a smooth map g : P → K since S → P is diffeomorphic. On the other hand, we may consider P also as the quotient Ŝ/⟨k⟩ (resp. Ŝ/⟨k_{1}, k_{2}⟩). Then by (19) our map ĝ is nothing but the pull-back of the map g : P → K to Ŝ by the quotient map, and hence is smooth. q.e.d.

Proof of Proposition 3.7. Using Lemma 3.9 we can patch together the extended slices Ŝ in Lemma 3.8 and obtain a desired smooth manifold N̂ which contains N as a compact subdomain with corners. Restricting N̂ if necessary, we may assume that N̂ is simply connected.

It remains to show that the conformal structure on N is extendible to N̂. For this purpose we have only to show the extendibility locally across any boundary point of N; indeed, then we can use a partition of unity to obtain a global extension to N̂.

Let T_{π} be the relative tangent bundle for the projection π : M_{0} → N_{0}, which is a subbundle on M_{0} of the tangent bundle T M of M. Let S be any slice and Ŝ its extention as in Lemma 3.8. We claim that the restriction T_{π}|S ∩ M_{0} extends to a subbundle T′ of T M|Ŝ. Indeed, in either of Cases 1 and 2 we have only to define T′ to be the subbundle of T M|Ŝ generated by the two independent sections ∂/∂v_{1}, ∂/∂v_{2}, for T_{π} is generated on S ∩ M_{0} by the two sections ∂/∂v_{1} and u_{2}∂/∂v_{2} in Case 1 and by u_{1}∂/∂v_{1} and u_{2}∂/∂v_{2} in Case 2.

The natural conformal structure on the orthogonal complement T^{′⊥} of T′ in T M|Ŝ defines a desired local extension of the conformal structure on the interior of P = π(S) to P̂ := π(Ŝ). q.e.d.

4. Self-dual manifolds with two-torus action

(4.1) In the first few subsections we summarize basic facts about selfdual manifolds and the associated twistor spaces (cf. [1]). Let (M, g) be a compact connected oriented four-dimensional Riemannian manifold. (M, g) is called self-dual if the anti-self-dual part W_{−} of the Weyl conformal curvature tensor of (M, g) vanishes identitically. Since W_{−} depends only on the conformal class [g] of g, the notion of self-duality is defined actually for the conformal manifold (M, [g]).

For any self-dual conformal manifold (M, [g]), one constructs naturally a compact complex manifold Z of dimension three, called the twistor space of (M, [g]), with the following properties.

1) Z admits a structure of a C^{∞} fiber bundle t : Z → M over M.

2) Each fiber L_{x} := t^{−1}(x), x ∈ M, of t is a complex submanifold of Z and is isomorphic to a complex projective line P^{1}. Moreover, its holomorphic normal bundle N_{x} in Z is isomorphic to the direct sum O(1) ⊕ O(1), where O(1) is the line bundle of degree one on P^{1}.

3) There exists an anti-holomorphic and fixed point free involution σ on Z which leaves each fiber L_{x} invariant. We call t : Z → M the twistor fibration, each L_{x} a twistor line and σ the real structure of Z.

(4.2) We need some information on the construction of Z from (M, [g]). Let T be an oriented real four-dimensional vector space with a positive definite inner product g. We denote by C(T) = C(T, g, ι) the set of isometric complex structures J on T which defines an orientation that is opposite to the given orientation ι of T. Namely, J is an endmorphism of T such that J^{2} = −identity, g(Ju, Jv) = g(u, v) for any u, v ∈ T and for any complex basis e, f of the complex vector space (T, J), the resulting real basis e, Je, f, Jf of T defines the orientation opposite to the original one on T. C(T) is naturally identified with a complex projective line and each twistor line L_{x} is naturally identified with the set C(T_{x}) = C(T_{x}, g_{x}, ι_{x}) for any representative g of [g], where T_{x} = T_{x}M is the tangent space of M at x.

To explain these statements we take an orientation preserving isometry u : T → H of real vector spaces, where H is the space of real quaternions with the oriented orthonormal basis 1, i, j, k. Denote by Sp(1) the group of unit quaternions. Let

Sp(1)_{+} × Sp(1)_{−} (20)

act on H by left and right multiplications;

h → qhq′^{−1}, h ∈ H, (q, q′) ∈ Sp(1)_{+} × Sp(1)_{−} (21)

where Sp(1)_{±} are copies of Sp(1). If we identity H with R^{4} naturally, this action gives a natural isomorphism of groups

(Sp(1)_{+} × Sp(1)_{−})/⟨(−1, −1)⟩ ≅ SO(4). (22)

Now we consider the set

C : = {q ∈ Sp(1)_{−}; q^{2} = −1}
(23)  = {ai + bj + ck; a^{2} + b^{2} + c^{2} = 1, a, b, c ∈ R}
≅ S^{2}.

Any element q of this set defines by right multiplications

h → hq^{−1} (24)

an isometric complex structure on H which defines an orientation that is opposite to the given one on H, and in this way one identifies C with C(H) = C(T_{x}) naturally.

The identification of this set with a complex projective line is seen as follows. Denote by V_{+} (resp. V_{−}) the complex two dimensional vector space (H, ×i) (resp. (H, i×)), where ×i and i× denote respectively the complex structures given by the right and the left multiplications by i ∈ H. Note that the natural action of Sp(1)_{±} on H defines a complex representation of these groups on V_{±} respectively.

Let π : V_{−} − 0 → C be defined by π(h) = h^{−1}ih. Then π gives a smooth fibration whose fiber over h^{−1}ih is precisely the complex line Ch of V_{−} with the origin deleted. Thus we obtain a natural identification

C = (V_{−} − 0)/C^{∗} =: P (V_{−}) (25)

and the map π is Sp(1)_{−}-equivariant if we let Sp(1)_{−} act on C by q → gqg^{−1}, g ∈ Sp(1)_{−}. Note that the latter action induces an action of Sp(1)_{−}/⟨−1⟩ ≅ SO(3) on C, and therefore by (22), of SO(4), where the action of Sp(1)_{+}-factor is trivial.

(4.3) Now for simplicity fix any Riemannian metric g in the given conformal class [g] on M although the construction below actually depends only on the conformal class [g]. Let α : P → M be the associated principal bundle of oriented orthonormal frames of M. Then Z is nothing but the fiber bundle over M with typical fiber P (V_{−}) = C associated to α via the above action of SO(4) on P (V_{−})

Z := P ×_{SO(4)} P (V_{−}) = P ×_{SO(4)} C.

In this way we get a natural interpretation of each fiber L_{x} as C(T_{x}) above. Further, the left multiplication by j on H descends to an antiholomorphic and fixed point free involution on P (V_{−}) which commutes with the action of SO(4); therefore it globalizes and gives the desired real structure σ on Z.

The complex structure of Z is defined as follows. First of all, the Levi-Civita connection of g defines at any point z of Z a horizontal subspace H_{z} of the real tangent space T_{z} of Z, which is mapped isomorphically by the differential t_{∗} onto the tangent space T_{x}, where x = t(z). We have thus a natural direct sum decomposition

T_{z} = H_{z} ⊕ T_{z}L_{x} = T_{x} ⊕ T_{z}L_{x} (26)

of real vector spaces. Let J_{z} be the complex structure on T_{x} corresponding to z ∈ L_{x} = C(T_{x}) and J′_{z} the natural complex structure of T_{z}L_{x} coming from the complex structure of L_{x}. Then the complex structure on Z is by definition the direct sum of J_{z} and J′_{z} with respect to (26) at each point of Z.

With respect to the identification L_{x} = P (V_{−}) = Sp(1)_{−}/S^{1} the normal bundle N_{x} of L_{x} in Z is described as the complex homogeneous vector bundle by

N_{x} = Sp(1)_{−} ×_{S^{1}} V_{+},

where S^{1} ⊆ Sp(1)_{−} is identified with {a + bi; a^{2} + b^{2} = 1, a, b ∈ R} acting on V_{+} = (H, ×i) by right multiplications. The map

Sp(1)_{−} ×_{S^{1}} V_{+} → V_{+}, (g, v) → gv (27)

gives rise to a canonical C^{∞} trivialization

N_{x} ≅ C × V_{+}. (28)

(4.4) Let (M, [g]) be a compact connected self-dual manifold with the associated twistor fibration t : Z → M. Suppose now that a real two-torus K ≅ S^{1} × S^{1} acts on (M, [g]) by conformal transformations. Then the K-action on M lifts naturally to a biholomorphic action on the twistor space Z, which in turn extends to a biholomorphic action of the complexfication G ≅ C^{∗} × C^{∗} of K. The lifted K-action clearly sends a real twistor line onto a real twistor line, commutes with the real structure σ and is compatible with the direct sum decomposition (26) at various points of Z. G-action satisfies (7).

We shall use the notation of §2 for the underlying topological action of K on M. In what follows we assume that χ(M) ≠ 0 and take and fix once and for all a cycle B of two-spheres which is a connected component of the singular set Σ(M) of the action.

(4.5) First we shall determine the singular set Σ = Σ(B) of the K-action on Z along t^{−1}(B), i.e., the set of points of t^{−1}(B) with nontrivial stabilizer groups.

For this purpose we study for each point x of B the induced action on the twistor line L_{x} of the stabilizer group K_{x} at x.

It is convenient to distinguish two cases:

Case A. x ≠ x_{i} for any i, 1 ≤ i ≤ k, so that x ∈ B′_{i} for some i. In this case K_{x} is the S^{1}-subgroup K_{i} := K(m_{i}, n_{i}) of K.

Case B. x = x_{i} for some i, 1 ≤ i ≤ k. In this case K_{x} = K.

First we show the following:

Lemma 4.1. For any point x ∈ B there exists on L_{x} exactly two fixed points of the stabilizer group K_{x}, which are σ-conjugate to each other. If x ∈ B′, the stabilizer group K_{z} at any other point z of L_{x} is trivial.

Proof. There exists a K_{x}-invariant orthogonal decomposition of T_{x} into two inequivalent two-dimensional K_{x}-invariant subspaces

T_{x} = T_{1} ⊕ T_{2}. (29)

In fact, in Case A we may take T_{1} to be the maximal subspace on which K_{x} acts trivially, i.e., the tangent space of B_{i} at x, and T_{2} to be its orthogonal complement, while in Case B, (29) is nothing but the isotropic representation of K_{x} = K = K_{i} × K_{i+1}, where e.g., K_{i} acts faithfully on T_{1} and trivially on T_{2}, so that T_{2} (resp. T_{1}) is identified with the tangent space of B_{i} (resp. B_{i+1}) at x.

On the other hand, a point z ∈ L_{x} is a fixed point of K_{x} if and only if the isotropic action of K_{x} on T_{x} commutes with the corresponding complex structure J_{z} on T_{x}. Thus, if z is a fixed point of K_{x}, then the complex structure J_{z} should preserve the direct sum decomposition (29). In fact, among J_{z}, z ∈ L_{x}, precisely two complex structures leave invariant the decomposition, and if one is denoted by J = J_{z_{0}}, z_{0} ∈ L_{x}, the other is given by −J = J_{σ(z_{0})}. Hence, we have proved the existence of precisely two σ-conjugate fixed points on L_{x}. This proves the first assertion.

For the second assertion, we note that in Case A if K_{z} is nontrivial at some point of z ∈ L_{x}, the above argument still works with K_{x} replaced by K_{z}, so that J_{z} still must preserve the decomposition (29). Thus it must be either of ±J_{z_{0}} above and z must coincide with one of the fixed points of K_{x}. q.e.d.

Denote these two K_{x}-fixed points on L_{x} by z^{±}, and when x = x_{i} we write them z_{i}^{±}. Since a point is fixed by K if and only if it is fixed by G, the above lemma immediately implies the following:

Corollary 4.2. The fixed point set of G on t^{−1}(B) consists of 2k isolated points z_{i}^{±}, 1 ≤ i ≤ k. The two points z_{i}^{±} are σ-conjugate to each other. When M is simply connected, they exhaust all the fixed points of G on Z.

From Lemma 4.1 we conclude:

Lemma 4.3. There exist precisely two irreducible nonsingular rational curves C_{i}^{±}, 1 ≤ i ≤ k, contained in Σ which are mapped diffeomorphically onto B_{i} by t. Moreover we have C_{i}^{+} ∩ C_{i}^{−} = ∅ and σ(C_{i}^{±}) = C_{i}^{∓}.

Proof. The fixed point set F_{i} of K_{i} on t^{−1}(B_{i}) is a σ-invariant complex submanifold of Z. From Lemma 4.1 we see readily that it is a nonsingular curve and is an unramified double covering of B_{i}. Since B_{i} is simply connected, F_{i} is a disjoint union of two nonsingular rational curves C_{i}^{±} which are σ-conjugate to each other. q.e.d.

We call twistor lines L_{x} over points x of B special and the twistor lines

L_{i} := t^{−1}(x_{i}), 1 ≤ i ≤ k

very special. We may assume that z_{i}^{±} ∈ C_{i}^{±}.

Proposition 4.4. The singular set Σ(B) of the K-action along t^{−1}(B) is written as a union of nonsingular rational curves as

Σ(B) = (∪_{i=1}^{k}L_{i}) ⋃(∪_{i=1}^{k}(C_{i}^{+} ∪ C_{i}^{−})).

Moreover, the stabilizer group of each point of C_{i}^{′±} := C_{i}^{±} − {z_{i−1}^{±}, z_{i}^{±}} coincides with K_{i}. Σ(B) is connected.

Proof. Since x_{i} is a fixed point, K acts naturally on L_{i}, but since the two-torus cannot act on a projective line effectively, L_{i} is contained in Σ(B). The rest follows immediately from Lemmas 4.1 and 4.3. q.e.d.

Remark. The singular set of K-action on Z is thus a disjoint union of Σ(B′) for various components B′ of Σ(M). The singular set of Gaction on Z is more difficult to identify because of the possible stabilizer group which is isomorphic to the additive group C. It finally turns out that the singular set of the G-action also coincides with that of K ; our proof for this, however, will be indirect.

(4.6) We shall give some notational remark. Let C denote the union of C_{i}^{±}, 1 ≤ i ≤ k;

C = ⋃_{i=1}^{k}(C_{i}^{+} ∪ C_{i}^{−}).

We have two possibilities:

a) C is connected, or

b) C has exactly two connected components which are σ-conjugate to each other.

In Case b) the ±-signs of C_{i}^{±} can be chosen in such a way that

C_{i}^{±} ∩ C_{i+1}^{±} = {z_{i}^{±}}, 1 ≤ i ≤ k, k + 1 = 1

and there exist no other intersections among any of the irreducible components of C. Thus there exist two mutually disjoint and σ-conjugate cycles of nonsingular rational curves C_{1}^{±} ∪ ⋯ ∪ C_{k}^{±} in C.

On the other hand, in Case a) the notation ± is not quite reasonable and it is more convenient to denote the curves C_{i}^{±} by C_{i} and C_{i+k}, and the points z_{i}^{±} by z_{i} and z_{i+k} in such a way that

C_{i} ∩ C_{i+1} = {z_{i}}, 1 ≤ i ≤ 2k, 2k + 1 = 1

and there exist no other intersections among any two irreducible components of C. C itself forms a cycle of nonsingular rational curves.

However, when we want a universal notation which is valid in both cases we make the following identification of the components

C_{i} = C_{i}^{+}, C_{k+i} = C_{i}^{−}, z_{i} = z_{i}^{+}, z_{k+i} = z_{i}^{−}, 1 ≤ i ≤ k.

For k < i ≤ 2k it is sometimes convenient to put L_{i} = L_{i−k} and K_{i} = K_{i−k}.

In these notations the intersections with the very special twistor lines are given by

L_{i} ∩ C_{i} ∩ C_{i+1} = {z_{i}}, 1 ≤ i ≤ 2k

in Case a), where the intersections are transversal. In Case b) for i = k and 2k we need obvious modifications.

(4.7) With the above notational conventions K_{i}, 1 ≤ i ≤ 2k, is the stabilizer group in K of points of C′_{i} := C_{i} − {z_{i−1}, z_{i}}. Let G_{i}(≅ C^{∗}) be the complexfication of K_{i}(≅ S^{1}).

Lemma 4.5. For any point z of C′_{i}, 1 ≤ i ≤ 2k, the stabilizer group G_{z} in G coincides with G_{i}.

Proof. Clearly we have the inclusion G_{i} ⊆ G_{z} with dim G_{z} = 1. If G_{z} ≠ G_{i}, G_{z}/G_{i} is a non-trivial closed subgroup of G/G_{i} ≅ C^{∗}. Since G_{z} ∩ K = K_{i}, it is easy to show that G_{z}/G_{i} is an infinite group. Since G_{z} is discrete in G, G_{z}/G_{i} is an infinite cyclic subgroup of G/G_{i}. Then the C^{∗}-orbit C′_{i} ≅ G/G_{z} must be a complex torus of dimension one, which is absurd. q.e.d.

Let z be any point of C. We next compute the isotropic representation of the stabilizer group G_{z} on the holomorphic tangent space T_{z}Z of Z at z.

Proposition 4.6.

1) Let z ∈ C′_{i}, 1 ≤ i ≤ 2k. Then with respect to suitable C-linear coordinates u, v, w of T_{z}Z and an isomorphism G_{z} ≅ C^{∗}(t), G_{z} acts on T_{z}Z by

(u, v, w) → (u, tv, tw),

where the tangent space of C′_{i} at z is given by v = w = 0.

2) Let z = z_{i} for some i, 1 ≤ i ≤ 2k. Then with respect to suitable Clinear coordinates u, v, w of T_{z}Z and an isomorphism G_{z} = G ≅ C^{∗}(s) × C^{∗}(t), G_{z} acts on T_{z}Z by

(u, v, w) → (stu, sv, tw),

where the tangent space of L_{i} (resp. C_{i}, resp. C_{i+1}) at z is given by v = w = 0 (resp. u = v = 0, resp. u = w = 0).

Proof. By what we have explained in (4.4) we see that the complex isotropic representation of K_{z} on T_{z}Z is a direct sum of the two complex representations, one on (T_{x}, J_{z}) and the other on T_{z}L_{x}, where x = t(z) and T_{z}L_{x} is the holomorphic tangent space of L_{x}.

A) First we normalize the representation of K_{x} on (T_{x}, J_{z}). The computation will be done by using a suitable model. Take an orientation preserving isometric C-linear isomorphism

(T_{x}, J_{z}) ≅ (H, ×i) (30)

which sends the decomposition (29) into H = C ⊕ jC, and we consider the corresponding action of K_{x} on the latter. Write an element of H = C ⊕ jC ≅ C^{2} as

(z, w) = z + jw, z, w ∈ C.

It is clear from (30) that in Case A, with respect to a suitable isomorphism K_{x} ≅ S^{1}(t), the action of K_{x} on H is given by

(z, w) → (z, tw), (31)

and in Case B with respect to the isomorphism K_{i}×K_{i+1} ≅ K composed with suitable isomorphisms K_{i} ≅ S^{1}(s) and K_{i+1} ≅ S^{1}(t) the action of K on H is given by

(z, w) → (sz, tw). (32)

In both cases the stabilizer group K_{x} is realized as (a subgroup of) the distinguished maximal torus T = S^{1} × S^{1} in U(2) ⊆ SO(4).

B) In order to understand the action of K_{x} on C, it is convenient to consider K_{x} as the image of a subgroup of the distinguished subtorus T̃ := S^{1}(u) × S^{1}(v) of Sp(1)_{+} × Sp(1)_{−} of (20), where S^{1} is considered as the subgroup {a + bi ∈ Sp(1); a, b ∈ R} of Sp(1). The action of Sp(1)_{+} × Sp(1)_{−} on H is given by (21) and so the induced action of (u, v) ∈ T̃ is given by

(z, w) → (uv^{−1}z, (uv)^{−1}w). (33)

In view of (31) and (32) this implies that the action of K_{x} is realized in Case A by the action of the subgroup {(u, u)} of T̃ with

t = u^{−2}, (34)

and in Case B by the action of the whole group T̃ via the double covering map (u, v) → (s, t) = (uv^{−1}, (uv)^{−1}), giving the isomorphism T̃/⟨(−1, −1)⟩ ≅ T = K. In particular we get in this case

st = v^{−2}. (35)

C) Now we detect the action of K_{x} on T_{z}L_{x}. By the isomorphism (30), the twistor line L_{x}, considered as a set of isometric complex structures on T_{x}, is mapped bijectively to the set C defined by (23), in which z ∈ L_{x} is mapped to −i ∈ C, i.e., J_{z} corresponds to the right multiplication by i. (Recall that C operates on H by (24).)

We next identify the tangent space T_{−i}C of C at −i and the induced tangential representation of T̃. Note first that by (25) the holomorphic tangent space of C = P (V_{−}) at −i is identified with the set of C-linear maps Hom(Cj, C) since −i = j^{−1}ij and (V_{−}/Cj) ≅ Ci = C. Since the action of Sp(1)_{−} on C is induced by the right multiplications on V_{−}, if f ∈ Hom(Cj, C), the action of v ∈ S^{1}(v) on f is given by

f(wj) → f(wjv)v^{−1} = v^{−1}f(v^{−1}wj) = v^{−2}f(h), wj ∈ Cj,

i.e., the action takes the form

f → v^{−2}f.

Because of (34) and (35) we get that the action of K_{x} on T_{−i}C ≅ T_{z}L_{x} is by multiplication by t in Case A and by st in Case B (cf. (4.2)).

Combining this with (31) and (32) we get the assertion of the proposition after passing to the complexfication G_{x} of K_{x}. q.e.d.

(4.8) Example. When M = S^{4} with standard conformal structure, the associated twistor space is the complex projective 3-space P^{3}(z_{0} : z_{1} : z_{2} : z_{3}) and the twistor fibration t : P^{3} → P^{1}H = S^{4} is given by

(z_{0} : z_{1} : z_{2} : z_{3}) → (z_{0} + jz_{3} : z_{1} + jz_{2}),

where S^{4} is identified with the right quaternionic projective line P^{1}H. Introduce the complex affine coordinates (u, v) on S^{4} = C^{2} ∪ {∞} so that (u + jv)(z_{0} + jz_{3}) = z_{1} + jz_{2}. Define the K-action on P^{3} and on S^{4} respectively by

(z_{0} : z_{1} : z_{2} : z_{3}) → (z_{0} : sz_{1} : tz_{2} : stz_{3}) and (u, v) → (su, tv)

for (s, t) ∈ K = S^{1} × S^{1}. Then we see readily that π is K-equivariant, in compatible with Proposition 4.6.

Remark. The restriction of t over C^{2} ⊆ S^{4} is nothing but the twistor fibration for C^{2} = R^{4} with flat metric. This has the following general significance. For a general self-dual manifold M and any point x ∈ M let N_{x} be the normal bundle of L_{x} in Z. Then the differential t_{∗} induces a surjection t_{x} : N_{x} → T_{x} giving a linear isomorphism of each fiber of N_{x} onto T_{x}, which is identified with (27). This map t_{x} may be considered as giving a twistor fibration of T_{x} with the flat metric g_{x}, and moreover, as such, it is isomorphic to the above fibration t^{−1}(R^{4}) → R^{4}. In this way we may elaborate on the above example so that it gives another proof of Proposition 4.6.

(4.9) By Lemma 4.5 for each i, 1 ≤ i ≤ 2k, the stabilizer group G_{i} of G at points of C′_{i} is independent of the choice of the points, is a complexfication of K_{i} and is isomorphic to C^{∗}. In particular G_{i} = G_{i+k}.

However, in view of the structure of the isotropic representation in Proposition 4.6 we can specify for each i not only the subgroup G_{i} of G, but also a one parameter subgroup ρ_{i} : C^{∗} → G with image G_{i}, which amounts to specifying one of the two possible isomorphisms C^{∗}(t) → G_{i}, by the following rule: For any point z in a sufficiently small neighborhood of a point of C_{i} we have

lim_{t→0} ρ_{i}(t)z ∈ C_{i}. (36)

Let N be the free abelian group of rank two of one parameter subgroups of G as in (2.2).

Lemma 4.7. For any i, 1 ≤ i ≤ k, we have ρ_{i} = −ρ_{i+k}.

Proof. Take any point x of B′_{i} and consider the induced action of G_{i} on the twistor line L_{x}. For any z ∈ L′_{x} = L_{x}−{z^{±}} if lim_{t→0} ρ_{i}(t)z = z^{±}, then lim_{t→∞} ρ_{i}(t)z = z^{∓}. From this the lemma follows readily. q.e.d.

5. Analyticity of orbit closures

(5.1) We study first the G-action in a neighborhood of C in Z. Let z be any point on C. Then there exist

1) a K_{z}-invariant neighborhood U of the origin o in the holomorphic tangent space T_{z} = T_{z}Z of Z at z,

2) a K_{z}-invariant neighborhood V of z in Z, and

3) a K_{z}-equivariant isomorphism ϕ : U → V, namely,

ϕ(gx) = gϕ(x), g ∈ K_{z}, x ∈ U (37)

(cf. [12, Satz 4.4]). We may use linear coordinates u, v, w of T_{z} as local coordinates of Z at z via ϕ. This local linear description of the K_{z}-action extends partly to that of the G_{z}-action.

Lemma 5.1. Let U_{1} be a connected open neighborhood of o which is relatively compact in U and let V_{1} = ϕ(U_{1}). Define an open subset A_{0} of G_{z} × U_{1} by A_{0} := {(g, x) ∈ G_{z} × U_{1}; gx ∈ U} and let A be the connected component of A_{0} containing K_{z} × U_{1}. Then for any element (g, x) of A, gϕ(x) belongs to V and the equivariancy (37) holds true on A.

Proof. Let A′ be the connected component of the open subset {(g, x) ∈ A; gϕ(x) ∈ V } of A containing K_{z} × U_{1}. The two holomorphic maps ϕ(gx) and gϕ(x) from A′ to Z are defined and coincide identically on K_{z} ×U_{1}. Since G_{z} is a complexfication of K_{z}, they must also coincide identically also on A′. We show that A′ is also closed in A. In fact, let (g_{0}, x_{0}) ∈ A be on the boundary of A′. Take a sequence (g_{n}, x_{n}) ∈ A′ converging to (g_{0}, x_{0}). We know that g_{n}ϕ(x_{n}) = ϕ(g_{n}x_{n}) ∈ U. Then we have

g_{0}ϕ(x_{0}) = lim_{n→∞} g_{n}ϕ(x_{n}) = lim_{n→∞} ϕ(g_{n}x_{n}) = ϕ(g_{0}x_{0})

with limit taken in Z; since g_{0}x_{0} ∈ U, we have g_{0}ϕ(x_{0}) ∈ V, i.e., (g_{0}, x_{0}) ∈ A′. Since A is connected, A = A′. q.e.d.

(5.2) Let z be any point of C. We describe a local action of G in a neighborhood of z.

We distinguish two cases:

Case A: z ∈ C′ := C − {z_{i}} and Case B: z = z_{i} for some i.

In each case by using Proposition 4.6 and Lemma 5.1 we see that there exists a neighborhood W of z with the following properties.

Case A. Suppose that z ∈ C′_{i} := C_{i} − {z_{i−1}, z_{i}} for some i. The corresponding one parameter group ρ_{i} : C^{∗}(s) → G has the image G_{i}, and G_{i} admits a complement G′ = C^{∗}(t) (e.g., G′ = G_{i−1} or G_{i+1}) and we obtain an isomorphism G = G_{i} × G′ = C^{∗}(s) × C^{∗}(t). There exists a coordinate neighborhood W of z in Z with local coordinates u, v, w such that

W = {(u, v, w); |u − 1| < ε, |v| < 1, |w| < 1}, ε > 0, z = (1, 0, 0), (38)

v = w = 0 is a defining equation of C in W and the action of G_{i} = G_{i}×1 takes the form

(u, v, w) → (u, sv, sw), |s| ≤ 1. (39)

Moreover, the action of t ∈ G′ is of the form (u, v, w) → (tu, v, w) where |tu − 1| < ε.

Case B. z = z_{i} for some i:

In this case the natural isomorphism

ρ_{i} × ρ_{i+1} : C^{∗}(s) × C^{∗}(t) → G

gives natural coordinates of G. There exists a coordinate neighborhood

W = {(u, v, w); |u| < 1, |v| < 1, |w| < 1}, (40)

of z in Z with local coordinates u, v, w around z such that 1) u = v = 0 and u = w = 0 are defining equations in W of C_{i} and C_{i+1} respectively, and v = w = 0 is one of L_{i}, and 2) the action of

(s, t) ∈ G ≅ C^{∗}(s) × C^{∗}(t)

takes the form

(u, v, w) → (stu, sv, tw), |s|, |t| ≤ 1. (41)

We shall call any of the above neighborhoods W of points of C simply an admissible neighborhood.

(5.3) Let W be an admissible neighborhood and set

W′ := W − (C ∪ L),

where L = ∪L_{j}. Define subsets W′_{ab} and W′′_{01} of W′ as follows. Case A. For any (a, b) ≠ (0, 0)

W′_{ab} := {(u, v, w) ∈ W′; av = bw}.

Case B. For any (a, b) with a ≠ 0

W′_{ab} := {(u, v, w) ∈ W′; au = bvw}

and

W′_{01} := {(u, v, w) ∈ W′; v = 0}, and W′′_{01} := {(u, v, w) ∈ W′; w = 0}.

From the description of G-action on W in (5.2) the following lemma is clear.

Lemma 5.2. The subsets defined above are all connected and closed submanifolds of W′ and are contained in a single G-orbit. Moreover, the closure of W′_{ab} contains C′ ∩ W in Case A.

Lemma 5.3. For any point x ∈ W′ the stabilizer G_{x} at x reduces to the identity.

Proof. We may assume that x belongs to an admissible neighborhood W of Case A. (If x belongs to some W of Case B, we may find a suitable admissible neighborhood of Case A containing x and contained in W.)

If g ∈ G_{x}, g fixes any points of the G-orbit of x. In particular by Lemma 5.2 g fixes any point of the subset W′_{ab} containing x, and hence g also fixes the points of C′_{i} ∩ W which is contained in the closure W′_{ab}. However, by Lemma 4.5 the stabilizer group of a point of C′_{i} is G_{i} itself; so g must belong to G_{i}. On the other hand, as follows from (39) and (41), no element of G_{i} other than the identity e fixes any point of W′. Thus g = e. q.e.d.

By Lemma 5.3 the G-action defines a foliation on W′ whose leaves we call local leaves on W′. Local leaves are precisely the subsets W′_{ab} and W′′_{01} of W defined above before Lemma 5.2

Lemma 5.4. The closure of each local leaf in W is an analytic submanifold of W.

Proof. The closures are defined in W by the same equations and their locus are again nonsingular. q.e.d.

(5.4) We next consider the behavior of G-orbits on Z along C. Let W and W′ = W − (C ∪ L) be as above. Let O be a G-orbit on Z which intersects with W′. Then O ∩ W′ consists of a union of local leaves on W′. We shall show that actually O ∩ W′ consists of a unique local leaf modulo finite number of exceptions. In this connection we shall call the two local leaves W′_{01} and W′′_{01} special leaves in Case B.

Lemma 5.5. Let F and F′ be distinct local leaves in W′. Suppose that neither F nor F′ is special in Case B. Then they are never contained in the same G-orbit.

Proof. Suppose that they are contained in one and the same G-orbit. We use the notation of (5.3) and consider the two cases separately.

Case A. Let z_{0} := (1, 0, 0) ∈ C be the center of the admissible neighborhood. There exist points z ∈ F and z′ ∈ F′ with the same u-coordinate 1. By assumption there exists an element g ∈ G with gz = z′. Then we have

gsz = sgz = sz′ (42)

for any s = (s, 1) ∈ G_{i} ⊆ G, and then letting s tend to zero, we have

gz_{0} = z_{0}. (43)

This implies that g ∈ G_{i} by Lemma 4.5. This, however, is impossible because any element s of G_{i} with |s| ≤ 1 preserves each local leaf. (If g = (s_{g}, 1) satisfies |s_{g}| > 1, consider g^{−1} instead, interchanging the roles of z and z′.)

Case B. The proof is same as in Case A. Since neither F nor F′ is special, we can find two points z ∈ F and z′ ∈ F′ with the same w-coordinate, w_{0} ≠ 0 say. Take an element g ∈ G with gz = z′. Then again we have (42) and then have (43) with z_{0} = (0, 0, w_{0}), which implies that g ∈ G_{i}. This is, however, impossible exactly by the same reasoning as above. q.e.d.

We call a G-orbit O on Z special if O contains a special local leaf at some z_{i}. Clearly special orbits are finite in number (≤ 4k). Now we take and fix a small neighborhood U of C in Z which is covered by admissible neighborhoods. We set U′ = U − (C ∪ L).

Corollary 5.6. For any non-special G-orbit O in Z with O∩U′ ≠ ∅, the closure of O in U is analytic and smooth.

Proof. By Lemma 5.5 O ∩ W′ is a single local leaf in W′ and the closure of the latter in W is analytic and smooth by Lemma 5.4. q.e.d.

Remark. Actually, with a little more preparation we can show that the above lemma, and hence the corollary also, is actually true without exceptions by the same method.

(5.5) We turn to considering the closures of G-orbits in the whole Z. We first prove the following:

Lemma 5.7. Any non-special G-orbit O on Z with O ∩ U′ ≠ ∅ intersects with any admissible neighborhood of points of C.

We use the following:

Lemma 5.8. Let z be any point of C′_{i} := C_{i} − {z_{i−1}, z_{i}} (resp. L′_{i} := L_{i} − {z_{i}, z_{i+k}}) and W an admissible neighborhood of z_{i−1} or z_{i} (resp. z_{i}^{+} or z_{i}^{−}). Then there exist a neighborhood V of z in Z and an element g of G such that g(V) ⊆ W.

Proof. First we consider the case z ∈ C′_{i}. Take an admissible neighborhood of z in Z with coordinates u, v, w as in (5.2). Then the equation u = 0 defines a G_{z}-invariant 2-dimensional local slice D for the local G-action at z. We consider the induced action of G′

G′ × D → Z

which is locally biholomorphic onto an open neighborhood of C′_{i} = G′z in Z. Since C′_{i} ∩ W ≠ ∅, the lemma follows.

The proof for L′_{i} is similar. In this case its stabilizer group H_{i} is the image of the one parameter group ρ_{i}ρ_{i+1}^{−1} : C^{∗}(s) → G and it admits a complement G′ = C^{∗}(t) (e.g., = G_{i} or G_{i+k}) so that we have an isomorphism G ≅ H_{i} × G′ = C^{∗}(s) × C^{∗}(t). There exists a coordinate neighborhood W of z in Z with local coordinates u, v, w around z as in (38) such that v = w = 0 is a defining equation of L_{i} in W, where the action of H_{i} takes the form

(u, v, w) → (u, sv, s^{−1}w), |s| ≤ 1. (44)

Moreover, for t ∈ G′ the action is of the form (u, v, w) → (tu, v, w) where |tu − 1| < ε. So the same argument applies also in this case.

q.e.d.

Proof of Lemma 5.7. Let W_{1} be any admissible neighborhood and z_{1} an arbitrary point of C ∩ W_{1}. Then any non-special local leaf in W_{1} intersects with any neighborhood of z_{1}, as follows from the description of local leaves.

Now let W be an arbitrary admissible neighborhood of a point z of C. Take an admissible neighborhood W_{0} of a point z_{0} of C which intersects with O. Suppose that z_{0} ∈ C_{i} and z ∈ C_{j}. Since Σ = C ∪ L is connected by Proposition 4.4, we have a finite chain {Y_{1}, …, Y_{n}} of irreducible components of Σ such that Y_{i} ∩ Y_{i+1} = {y_{i}} ≠ ∅ for 1 ≤ i ≤ n − 1, Y_{1} = C_{i} and Y_{n} = C_{j}. Then since O is nonspecial, in view of the first remark, by applying succesively the previous lemma we can transform a point of W_{0} by elements of G into a point of W. This shows that O also intersects with W. q.e.d.

(5.6) Let O be any non-special G-orbit with O ∩ U′ ≠ ∅. By Corollary 5.6 the closure Ô of O ∩ U′ in U is analytic and smooth. We next consider the closure of O in the whole Z. Our basic observation is that Ô looks like a neighborhood of the anti-canonical cycle of a compact smooth toric surface. We shall determine this toric surface first. For each i, 1 ≤ i ≤ 2k, we specified in (4.9) a one parameter subgroup ρ_{i} : C^{∗}(t) → G determined by the following two conditions

1) ρ_{i}(t) fixes any element of C_{i}, and

2) for any sufficiently small neighborhood V_{i} of C_{i} in Z we have

lim_{t→0} ρ_{i}(t)z ∈ C_{i}, z ∈ V_{i}.

Also recall by Lemma 4.7 that we have ρ_{i+k} = −ρ_{i} as elements of N.

Lemma 5.9. For any element ρ of N which is in the open simplex spanned by ρ_{i} and ρ_{i+1} the following holds true: Let z be any point of an admissible neighborhood of the point z_{i}. Then we have lim_{t→0} ρ(t)z = z_{i}. Here the range of indices are 1 ≤ i ≤ 2k, 2k + 1 = 1, in Case a) of (4.6) and 1 ≤ i ≤ k − 1, k + 1 ≤ i ≤ 2k − 1, in Case b) of (4.6) respectively. Moreover, in Case b) the same statment holds true if we replace in the above statements (ρ_{i}, ρ_{i+1}) and z_{i} = C_{i} ∩ C_{i+1} by (ρ_{k}, ρ_{1}) and z_{k} = C_{k} ∩ C_{1} respectively.

Proof. We only consider the first statement, the supplementary case being treated in the same way. By assumption ρ is written as ρ = mρ_{i} + nρ_{i+1} for some positive integers m and n. By Proposition 4.6, in an admissible neighborhood of z_{i}, with respect to the isomorphism ρ_{i}×ρ_{i+1} : C^{∗}(t_{1})×C^{∗}(t_{2}) → G the action of G takes the form (u, v, w) = (t_{1}t_{2}u, t_{1}v, t_{2}w), and therefore the induced action of ρ takes the form ρ(t) = (t^{m+n}x, t^{m}y, t^{n}z). The assertion is clear from this. q.e.d.

The following lemma is crucial for the construction of our toric surface.

Lemma 5.10. For 1 ≤ i ≤ 2k, 2k + 1 = 1, in Case a) (resp. for 1 ≤ i ≤ k − 1, k + 1 ≤ i ≤ 2k − 1 in Case b)) of (4.6), there exists no ρ_{j}, j ≠ i, i + 1, in the closed simplex spanned by ρ_{i} and ρ_{i+1}. In Case b) the same statement holds true if we replace (ρ_{i}, ρ_{i+1}) and j ≠ i, i + 1 in the above statements by (ρ_{k}, ρ_{1}) and j ≠ k, 1 respectively.

Proof. We only show the first statement. Suppose that some ρ_{j}, j ≠ i, i + 1, is between ρ_{i} and ρ_{i+1}. Take a G-orbit O with O ∩ U′ ≠ ∅ which is not special. By Lemma 5.7 we can find a point z (resp. y) of O ∩ U′ which is in an admissible neighborhood of a general point of C′_{j} (resp. in an admissible neighborhood W of z_{i}). We can find an element g ∈ G which maps z to a point y.

Suppose first that ρ_{j} ≠ ρ_{i}, ρ_{i+1}. By Lemma 5.9 our assumption implies that lim_{t→0} ρ_{j}(t)y = z_{i}, while we have z_{0} := lim_{t→0} ρ_{j}(t)z ∈ C′_{j}. Since G is commutative, we thus get

z_{i} = lim_{t→0} ρ_{j}(t)y = lim_{t→0} ρ_{j}(t)gz = g lim_{t→0} ρ_{j}(t)z = gz_{0}.

But this is impossible since z_{i} is a fixed point and z_{0} ∈ C′_{j}.

Suppose next that ρ_{j} = ρ_{i}, say. In the above proof we have only to replace the equality lim_{t→0} ρ_{j}(t)y = z_{i} by the following one

v := lim_{t→0} ρ_{j}(t)y = lim_{t→0} ρ_{i}(t)y ∈ C′_{i},

where the derivation of the contradiction is similar, noting the assumption that j ≠ i, i + 1 and the fact that C′_{i} and C′_{j} are different G-orbits.

q.e.d.

As an important conclusion of the lemma we obtain

Corollary 5.11. C is connected.

Proof. Suppose that C is not connected, i.e., we are in Case b). The lemma says that there exists no other ρ_{j}’s between ρ_{i} and ρ_{i+1} for any 1 ≤ i ≤ k, k + 1 = 1. This would lead to the non-existence of ρ_{k+j}, 1 ≤ j ≤ k, which is absurd. q.e.d.

Remark. A. Huckleberry pointed out that once Lemma 5.3 is proved, then the connectedness of C and the analyticity of the closure of the general orbits follows by topological argument using the fact that G has only one end. This could certainly simplify the exposition above and below.

(5.7) By Lemma 5.10 we conclude that

ρ_{1}, …, ρ_{2k}, ρ_{i+k} = −ρ_{i},

are arranged on N couterclockwise or clockwise in this order around the origin. To fix an idea let us assume that they are arranged couterclockwise, reversing the numbering if necessary. If we fix an identification K = S^{1} × S^{1}, we have the corresponding identifications G = C^{∗} × C^{∗} and N = Z^{2}.

Lemma 5.12. Let {±(m_{i}, n_{i})} be the Orlik-Raymond invariant of the underlying K-action on M, and (k_{i}, l_{i}) ∈ Z^{2} the coordinates of ρ_{i}. Then these satisfy the following equalities for any i; (k_{i}, l_{i}) = ±(−n_{i}, m_{i}), (−l_{i+k}, k_{i+k}) = −(−l_{i}, k_{i}) and

| \begin{matrix}k_{i} & l_{i} \\ k_{i+1} & l_{i+1} \end{matrix} | = 1. (45)

Proof. The assertions follows readily from the fact that the image G_{i} of ρ_{i} : C^{∗} → G is the complexfication of K_{i} = K(m_{i}, n_{i}), noting the dual description of K_{i} and G_{i} via characters and one parameter groups respectively. q.e.d.

Clearly, the one parameter groups ρ_{i}, 1 ≤ i ≤ 2k, define a complete fan △(ρ) on N and we get a (compact smooth) symmetric toric surface

S = S(ρ) (46)

with the anti-canonical cycle D = D_{1}∪⋯∪D_{2k}, D_{i} corresponding to ρ_{i}. We identify the open orbit of S with G once and for all (by fixing a point of the open orbit) and consider S as a distinguished compactification of G = C^{∗} × C^{∗} associated to our G-action on Z.

Lemma 5.13. Let O be a non-special G-orbit with O∩U′ ≠ ∅. Then its closure \overline{O} in Z is analytic and smooth. \overline{O} is obtained by adding C to O, is G-equivariantly isomorphic to the toric surface S = S(ρ) of (46), and has C as the anti-canonical cycle. In particular such closures are all isomorphic to each other.

Proof. Let F = O ∩ U and N = \overline{O} − O. Clearly O = GF. Take any point o of N and a sequence {x_{n}} of points of O which converges to o. Fix any point z ∈ F and write z_{n} = g_{n}z for some g_{n} ∈ G which is unique by Lemma 5.3. Then after passing to a subsequence if necessary, we can assume that g_{n} converge to a point g_{0} ∈ S with respect to the compactification G ⊆ S above. If g_{0} ∈ G, then o = lim x_{n} = lim g_{n}z = g_{0}z and o is in the orbit O, which contradicts our choice of o.

Thus g_{0} ∉ G so that g_{0} ∈ D_{i} ⊆ S for some i. Suppose first that g_{0} is not a G-fixed point. With respect to the isomorphism

ρ_{i} × ρ_{i+1} : C^{∗}(s) × C^{∗}(t) ≅ G,

g_{0} admits an affine neighborhood in S of the form C(s)×C^{∗}(t) containing G, where s = 0 is a defining equation of D_{i}. In this neighborhood we may identify g_{0} with a point (0, t_{0}) with t_{0} ≠ 0 and g_{n} takes the form (s_{n}, t_{n}), where s_{n} → 0 and t_{n} → t_{0} when n → ∞. Thus we may write g_{n} = t_{n}s_{n} with s_{n} = (s_{n}, 1) ∈ G and t_{n} = (1, t_{n}) by abuse of notations.

Now coming back to the action on Z, take an admissible neighborhood W of a point of C′_{i}. Since z ∈ F ⊆ O and F ∩W ≠ ∅ by Lemma 5.7, there exists an element h of G such that hz ∈ W. Then we have

g_{n}z = t_{n}s_{n}h^{−1}(hz) = h^{−1}(t_{n}s_{n}hz). (47)

Note that s_{n}hz converge to a point, say b, on C_{i}. Hence we have

o = lim g_{n}z = h^{−1} lim t_{n}(s_{n}hz) = h^{−1}t_{0}b ∈ C′_{i} (48)

since C′_{i} is G-invariant.

Next suppose that g_{0} is a fixed point so that g_{0} = D_{i} ∩ D_{i+1} for some i. With respect to the isomorphism ρ_{i} ×ρ_{i+1} : C^{∗}(s)×C^{∗}(t) ≅ G, then g_{0} admits an affine neighborhood in S of the form C(s) × C(t) containing G, where st = 0 is a defining equation of D_{i} ∪ D_{i+1} and g_{0} is identified with the origin (0, 0). Further, g_{n} takes the form (s_{n}, t_{n}) with s_{n}, t_{n} → 0 when n → ∞. Write g_{n} = t_{n}s_{n} as before.

Now coming back again to the action on Z take an admissible neighborhood W of z_{i}. As in the previous case we may find an element h of G such that hz ∈ W. Then we again have (47), where in this case t_{n}s_{n}hz converge to z_{i}. Hence as in (48) we obtain

o = lim_{n→∞} g_{n}z = h^{−1}t_{0}z_{i} = z_{i} (49)

since z_{i} is fixed by G.

By (48) and (49) we get that N ⊆ C and by Corollary 5.6 the closure \overline{O} of O in Z is analytic and smooth since \overline{O} = [O ∩ (Z − U)] ∪ Ô, where Ô is the closure of O ∩ U′ in U. Note also that since F intersecs with any admissible neighborhood we actually have N = C.

Finally, the compact nonsingular G-surface \overline{O} is a toric surface with all the data of the boundary divisors same with those of S, D_{i} corresponding to C_{i} with the same one parameter group ρ_{i}. This implies that \overline{O} is isomorphic to S = S(ρ) as a toric surface. q.e.d.

Since the special orbits are finite in number, summarizing what we have obtained we get the following:

Proposition 5.14. There exist uncountably many G-orbits on Z such that their closures in Z are analytic and smooth. Moreover, they are projecitve nonsingular toric surfaces which are isomorphic to S(ρ) and which contain C as the anti-canonical cycles. In particular, the closures are obtained by adding C to the G-orbits.

So far we have fixed throughout a connected component B of the singular set Σ(M) of the K-action on M and have been dealing exclusively with the G-action along C ⊆ t^{−1}(B). However, we shall see in the next section that B is actually the unique connected component of Σ(M).

6. Associated fiber space and invariants

(6.1) In order to show that all the orbit closures are analytic, instead of studying the individual orbit structure more in detail, we apply the method of meromorphic quotient. In fact for the application of the latter, Proposition 5.14 is sufficient. To explain this we begin with the following general setting.

Let Z be a compact connected complex manifold of dimension n. Then there exists a natural structure of a reduced complex space on the set D of effective divisors on Z [5]. The following property of this space proved in [8] is important for our purpose.

Lemma 6.1. Any connected component of D is compact and projective.

We note that the essential ingredient of the proof is the result of [6] to the effect that, up to finite number of exceptions, every irreducible divisor E on Z is of the form h^{∗}\overline{E}, where h : Z → Y is an algebraic reduction of Z and \overline{E} is a divisor on the projective variety Y. See [8] for the detail.

(6.2) Suppose now that a connected complex Lie group G acts biholomorphically on Z such that general orbits are of codimension one, namely there exists a nonempty Zariski open subset U of Z such that every G-orbit of a point of U is of codimension one. Let D^{G} be the fixed point set of the induced G-action on D. For any d ∈ D let Y_{d} be the corresponding effective divisor on Z. Denote by D_{0}^{G} the union of those irreducible components F_{α} of D^{G} such that for a general point d of F_{α} the corresponding divisor Y_{d} is reduced and irreducible. Then take an irreducible component F of D_{0}^{G} of positive dimension (if any). F is projective by Lemma 6.1. We consider the universal family of divisors on Z parametrized by F:

(50)  [figure: Y; p; ⊆; F × Z; p_{F}; F]

Lemma 6.2. An irreducible component F of D_{0}^{G} as above is unique (if any) and is of dimension one. The natural projection q : Y → Z is bimeromorphic.

Proof. Y is compact of dimension n − 1 + dim F and the natural projection q : Y → Z is surjective. For every d ∈ F, p^{−1}(d) is identified with the corresponding divisor Y_{d} of Z and is G-invariant. There exists a nonempty Zariski open subset V of F such that for any d ∈ V, Y_{d} is reduced and irreducible and Y_{d} ∩ U ≠ ∅, and hence, Y_{d} is a closure of a G-orbit. Since different orbits do not intersect each other, q^{−1}(U) ∩ p^{−1}(V) → Z is injective; it follows that q is bimeromorphic and F is of dimension one. Also by the construction we see that F is the unique irreducible component of D_{0}^{G} of positive dimension. q.e.d.

We call the commutative diagram

[figure: Y; q; p; Z; pq^{−1}; F] (51)

or simply, the meromorphic map pq^{−1} : Z → F, the meromorphic quotient of Z by G, which is canonically associated with the given action of G on Z. In this case Y is nothing but the graph of the meromorphic map pq^{−1} and there exists a natural G-action on the diagram (51), where the action on F is trivial.

(6.3) On the existence of the meromorphic quotient we have the following criterion:

Proposition 6.3. Let Z be a compact connected complex manifold. Suppose that a connected complex Lie group G acts effectively on Z such that its general orbits are of codimension one and that the closures of uncountably many orbits of codimension one are analytic. Then the meromorphic quotient (51) of Z by G exists. Moreover, every orbit of codimension one has an analytic closure in Z and is identified with an irreducible component of a fiber of p.

Proof. Let O_{t}, t ∈ T, be a family of G-orbits of codimension one whose closure \overline{O}_{t} in Z is analytic, where the index set T is uncountable. Let d_{t} be the point of D corresponding to the irreducible divisor \overline{O}_{t} of Z. Since \overline{O}_{t} is G-invariant, d_{t} is a point of D_{0}^{G}. Moreover, since D_{0}^{G} as well as D has at most countably many irreducible components, there must exist an irreducible component F of D_{0}^{G} containing infinitely many d_{t}’s. This F must be of positive dimension and the meromorphic quotient exists.

Since Z is nonsingular, there exists an analytic subset A of Z of codimension ≥ 2 such that q is isomorphic over q^{−1}(Z − A). Then by the equivariancy of the diagram (51) every one-codimensional orbit on Z is uniquely lifted to one on Y, which then should be mapped to a point of F. Thus the latter is Zariski open in an irreducible component of a fiber of f; in particular its closure is analytic, and then, as its biholomorphic image in Z, the closure of the original G-orbit in Z also is analytic. q.e.d.

(6.4) We now apply the above consideration to our G-action on the twistor space Z, where G is an algebraic two-torus.

Proposition 6.4. Let Z be the twistor space with G-action as in (4.4).

1) There exists the meromorphic quotient of Z by G :

\begin{aligned}& Ẑ & \underrightarrow{μ} & Z \\ f & ↓ \\ & P \end{aligned} (52)

where P is a nonsingular rational curve.

2) Every two-dimensional orbit has an analytic closure in Z and is identified with an irreducible component of a fiber of f.

3) The action of σ on Z naturally lifts to one on the diagram (52). In particular, P admits a naturally induced real structure.

Proof. In our case the general orbit has codimension one and the assumption of Proposition 6.3 is fulfilled by Proposition 5.14. Therefore by that proposition we obtain 1) and 2) except the identification of P.

Since a general twistor line is mapped holomorphically and surjectively onto P, P is rational. This implies that for general a ∈ P the divisors S_{a} := μ(f^{−1}(a)) are all linearly equivalent. Let Q be the complete linear system containing such S_{a}’s. Then G acts (projective) linearly on Q and it is easily seen that P coincides with a connected component of the fixed point set of this action. Thus it must be a complex projective line in Q since G is an algebraic torus.

The anti-holomorphic involution on D induced by σ preserves P because of (7). 3) follows. q.e.d.

(6.5) We now study the structure of the diagram (52) more in detail. Note first that the meromorphic map fμ^{−1} : Z → P is obtained by the linear pencil V whose members are S_{a} := μ(Ẑ_{a}), a ∈ P, where Ẑ_{a} := f^{−1}(a). (For simplicity of notations, however, in what follows we often identify S_{a} with Ẑ_{a} by μ and use the two notations interchangeably.)

Proposition 6.5.

1) μ is obtained by the blowing-up of Z with center C. E := μ^{−1}(C) is isomorphic to C × P and f|E is identified with the projection to the second factor.

2) General members of V are smooth and every smooth member is a projecitve toric surface isomorphic to S(ρ) of (46) and has C as its anti-canonical cycle.

Proof. By Proposition 5.14 there exist uncountably many members of the system V which are closures of G-orbits obtained by adding C to the orbits, and are smooth projecitve toric surfaces isomorphic to S(ρ). Now the meromorphic map fμ^{−1} is defined by the pencil V and Ẑ ⊆ Z × P is its graph. Thus μ is obtained by the blowing up of the intersection of two general members S_{1} and S_{2} of this system. In view of the above remark we may take them to be smooth orbit closures containing C and intersecting each other only along C.

Locally their intersection property can be read off in each admissible neighborhood W of a point of C; in Case A the intersections of S_{i} with W are given by smooth hyperplanes au = bv, (a, b) ≠ (0, 0) and S_{1} and S_{2} intersect transversally along C ∩ W; on the other hand, in Case B the intersections of S_{i} with W are given by equations of the form auv = bw, a ≠ 0. Hence we conclude that the base locus of the pencil V is precisely the curve C with multiplicity one, and therefore that μ coincides with the blowing up of Z with center C. The assertion about the structure of E is then clear. In particular every fiber of f contains C naturally; for a ∈ P, Ẑ_{a} ∩ E = a × C = C. This proves 1).

On the other hand, up to finite exceptions every fiber is smooth and it is obtained as a deformations of (projecitve) toric surfaces. Since it is known and easy to show that a deformation of a rational surface is again a rational surface, every smooth fiber of f must be a projecitve toric surface in view of its admitting G-action with open orbit. As it contains C in the complement of the open orbit as a connected component, C must coincide with their anti-canonical cycle. Since the data for the isotropic one parameter subgroups along each component of C, which determines a toric surface, is invariant under deformations, they are indeed isomorphic to each other. q.e.d.

(6.6) On the singularities of Ẑ we prove:

Lemma 6.6. Ẑ has 2k ordinary double points r_{i}, 1 ≤ i ≤ 2k, such that μ(r_{i}) = z_{i}, 1 ≤ i ≤ 2k, and σ(r_{i}) = r_{i+k}, 1 ≤ i ≤ k. Ẑ is smooth outside these points. Moreover, f(r_{i}) = f(r_{i+k}) for any i.

Proof. By Proposition 6.5 μ is the blowing up with center C. Thus Ẑ is smooth over the smooth points of C. At the singular points z_{i} of C, the defining equation of C in an admissible neighborhood W of (40) is of the form w = uv = 0. Thus Ẑ is described in a neighborhood of μ^{−1}(z_{i}) as a hypersurface wξ_{1} = uvξ_{0} in W × P^{1}, with the unique singular point r_{i} := {(0, 0), (1 : 0)}, where ξ_{i} are homogeneous coordinates of P^{1}. These ordinary double points r_{i} are thus the only singular points of Ẑ.

The twistor line L_{i} is defined in W by u = v = 0, and therefore its proper transform L̂_{i} in Ẑ passes through r_{i}; similarly, it passes through r_{i+k}. Since L̂_{i} is an orbit closure, it must be mapped to a point by f. Therefore

f(r_{i}) = f(L_{i}) = f(r_{i+k}), 1 ≤ i ≤ k.

Moreover, the relation σ(z_{i}) = z_{i+k} lifts to Ẑ to give σ(r_{i}) = r_{i+k}. q.e.d.

Let

a_{i} := f(r_{i}) = f(r_{i+k}), 1 ≤ i ≤ k.

In view of the σ-equivariancy of f and the above lemma, a_{i} must be a real point of P. In particular σ has a fixed point on P. The fixed point set R is then diffeomorphic to S^{1} and P − R consists of two connected components. We may choose a suitable inhomogeneous coordinate z of P so that

P = C(z) ∪ {∞} (53)

and σ takes the form z → 1/\overline{z} in this coordinate. Then R is given by |z| = 1 and a_{i} ∈ R.

(6.7) Recall that every effective divisor S on Z has a positive intersection number S · L with any twistor line L. We call this number the degree of S. S is called an elementary divisor if its degree is one, and a quadratic divisor when it is two.

Lemma 6.7. Let S be any member of the pencil V. Then S is a quadratic divisor on Z in the sense defined above.

Proof. Take any smooth member S of V. We show that S · L_{i} = 2. Note that L_{i} − {z_{i}, z_{i+k}} is a one-dimensional G-orbit, while one dimensional orbits in S are one of C′_{j}, 1 ≤ j ≤ 2k. Therefore S intersects L_{i} only at two points z_{i} and z_{i+k}. In an admissible neighborhood (40) of either of these two points, a defining equation of L_{i} is u = v = 0 and one of S is w = cuv for some nonzero c. Thus the intersection of L_{i} and S is transversal there. Hence we get L_{i} · S = 2. q.e.d.

Since V contains a real member, it contains also a real smooth member S_{a} for a general point a of R. Thus by a theorem of Pedersen-Poon [19, Theorem 1] we have the following

Corollary 6.8. M is diffeomorphic to mP^{2}, where m is the second betti number of M. In particular M is simply connected.

Proposition 6.9. Z is Moishezon, i.e., bimeromorphic to a complex projective manifold.

Proof. Let r : Z̃ → Ẑ be any resolution of Ẑ. Consider the resulting holomorphic map f̃: Z̃ → P. It suffices to show that Z̃ is Moishezon. Any smooth fiber Z̃_{a} is isomorphic to the toric surface S = S(ρ); in particular its anti-Kodaira dimension κ^{−1}(Z̃_{a}) = 2 by (1). From this follows the Moishezon property of Ẑ immediately by applying [27, Prop. 12.2] to the case where D is the anti-canonical divisor of Z in the notation there. q.e.d.

Let −\frac{1}{2}K be the canonical square-root of the anti-canonical line bundle −K = −K_{Z} of Z defined on any twistor space (cf. [10]).

Proposition 6.10. The linear system V coincides with the unique positive dimensional linear subsystem |− \frac{1}{2}K|^{G} of the half-anti-canonical system | − \frac{1}{2}K| consisting only of G-invariant elements.

Proof. Since both V and −\frac{1}{2}K are real, and −\frac{1}{2}K·L = 2, Lemma 6.7 shows that −\frac{1}{2}K and the line bundle A := [S_{a}] determined by the members of V coincide in the chern class level (cf. [10, §3]). On the other hand, we have 2 dim H^{1}(O_{Z}) = b_{1}(Z) for the Moishezon manifold Z, where b_{1} denotes the first betti number. Thus, in view of b_{1}(Z) = b_{1}(M) Corollary 6.8 gives H^{1}(O_{Z}) = 0. It follows that −\frac{1}{2}K and A are isomorphic as line bundles.

The members of V are clearly G-invariant members of | − \frac{1}{2}K|. On the other hand, by Proposition 6.4 G-invariant divisors which are not members of V are identified with union of some of the irreducible components of fibers of f and they are discrete. From this the final assertion follows. q.e.d.

(6.8) We shall next detect the singular fibers of f precisely. We use the results of Poon [20] and Pedersen-Poon [19], which we quote as:

Lemma 6.11.

1) ([19, Lemma 2.1]) Any real and irreducible quadratic divisor on Z is smooth.

2) ([20, Lemma 1.9]) Every elementary divisor S on Z is smooth and contains a unique twistor line L. Moreover, if we set \overline{S} := σ(S), \overline{S} is another elementary divisor in Z such that S ∩ \overline{S} = L, where the intersection is transversal.

Following our notational convention (cf. (6.5)) we denote the fiber Ẑ_{a} of f by S_{a} in what follows. Similarly, L_{i} also denotes its proper transform in Ẑ.

Proposition 6.12. The fiber S_{a}, a ∈ P, of f is smooth if and only if a ≠ a_{i} for any i, 1 ≤ i ≤ k. The points a_{i} are all distinct and S_{i} := S_{a_{i}} is a union of two nonsingular projecitve toric surfaces S_{i}^{+} and S_{i}^{−} which intersect transversally along L_{i} and are mutually σ-conjugate. S_{i}^{+} and S_{i}^{−} are isomorphic to the toric surfaces S_{i}^{+}(ρ) and S_{i}^{−}(ρ) of (9) respectively.

Proof. By Lemma 6.7 every member S of V is quadratic. Since every effective divisor on Z has a positive intersection number with twistor lines, either S is irreducible or has two irreducible components which are elementary divisors.

Suppose first that S is irreducible. Since every member of V intersects with C, it is a closure of a two dimensional orbit O with O∩U′ ≠ ∅. If O is non-special, by Lemma 5.13 its closure is irreducible and is isomorphic to the smooth toric surface S(ρ).

On the other hand, if O is special, then there exist some i, 1 ≤ i ≤ 2k, and an admissible neighborhood W of z_{i} as in (40) such that S ∩ U′ contains either of the plane u = 0 or v = 0. However, since u = v = 0 defines the twistor line L_{i} and L_{i} is mapped to a_{i} ∈ P, we must have S = S_{i} and both the planes u = 0 and v = 0 are mapped to the same point a_{i} by f. Thus, S must contain both the planes u = 0 and v = 0 since S is assumed to be irreducible. In particular S is not smooth. Now since a_{i} is a real point, S is σ-invariant. Thus S must be smooth by 1) of Lemma 6.11. This is a contradiction and we conclude that O cannot be special.

Next suppose that S is reducible so that S is a union of two irreducible components S^{+} and S^{−}. This is the case for S = S_{i} as the previous argument shows. By 2) of Lemma 6.11 S^{±} are smooth and if we set \overline{S}^{+} = σ(S), \overline{S}^{+} also is an elementary divisor and L := S^{+} ∩ \overline{S}^{+} is the unique twistor line contained in S. Now \overline{S}^{+} is G-invariant as well as S^{+}, and therefore L also is G-invariant. But the only G-invariant twistor lines in Z are very special twistor lines L_{i}. Thus L coincides with L_{i} for some i, and then both S^{+} and \overline{S}^{+} must be mapped to a_{i} by f; hence S = S_{i}. This implies that S is real and \overline{S}^{+} = S^{−}. The transversality of the intersection of S^{+} and S^{−} follows from Lemma 6.11 or directly from the local description (40).

Moreover, S_{i} never intersects with other twistor lines L_{j}, j ≠ i, at its general point since if otherwise S_{i} would contain L_{j} since L_{j} is Ginvariant, which is impossible by Lemma 6.11, 2). It follows that a_{i} ≠ a_{j} if i ≠ j.

It remains to identify the irreducible components S_{i}^{±} of S_{i}, 1 ≤ i ≤ k. First of all, since they contain an open orbit of G, they are themselves smooth toric surfaces. (These surfaces are projective since Z is Moishezon.) L_{i} is the unique common irreducible component of the anti-canonical cycles of both surfaces. L_{i} is mapped to itself by σ with z_{i} and z_{i+k} interchanged on it. Further, σ interchanges the two surfaces, their anti-canonical cycles, and interchanges C_{j} and C_{j+k} for any j. Thus, if C_{j} is in S^{+}, then C_{j+k} are in S^{−} and vice versa. It follows that if C_{i} is adjacent to L_{i} in the anti-canonical cycle of S^{+}, then C_{i+k} is adjacent in S^{−} to L_{i}.

From these observations we can readily conclude that the anti-canoni-cal cycle of S_{i}^{+} consists of curves L_{i}, C_{i+k+1}, C_{i+k+2}, …, C_{2k}, C_{1}, …, C_{i} in this order and that of S_{i}^{−} consists of L_{i}, C_{i+1}, C_{i+2}, …, C_{i+k} in this order. So in order to identify these surfaces we have only to identify the one parameter subgroup corresponding to each irreducible component of the anti-canonical cycles. For C_{i}, this is again ρ_{i} as is clear from the description (39) of the action along C_{i}. For L_{i}, because of the local description (41) of the action at z_{i} we see readily that it is ∓(ρ_{i} − ρ_{i+1}) on S_{i}^{±}. It follows that S_{i}^{±} have the associated weighted circular sequence (13) of S_{i}^{±}(ρ). q.e.d.

(6.9) By Proposition 6.10 and the adjunction formula it follows that the normal bundle of a smooth member S of V in Z is isomorphic to the anti-canonical bundle −K_{S} of S, while for the irreducible components S_{i}^{±} of the singular members S_{i} we have

Lemma 6.13. The normal bundle [S_{i}^{±}]|S_{i}^{±} of S_{i}^{±} in Z is isomorphic to −K_{S_{i}^{±}} − 2L_{i} on S_{i}^{±} (written additively).

Proof. We have

[S]|S_{i}^{±} = [S_{i}^{+} + S_{i}^{−}]|S_{i}^{±} = [C]|S_{i}^{±},

where [C] is the line bundle defined by the Cartier divisor C on S_{i}. On the other hand, we have [S_{i}^{±}]|S_{i}^{∓} = [L_{i}] on S_{i}^{∓}. From this we have

[S_{i}^{±}]|S_{i}^{±} = [C ∩ S_{i}^{±}] − [L_{i}] = (−K_{S_{i}^{±}} − L_{i}) − L_{i} = −K_{S_{i}^{±}} − 2L_{i}

since −K_{S_{i}^{±}} = L_{i} + C ∩ S_{i}^{±}. q.e.d.

Using the above lemma we shall give a characterization of LeBrun’s twistor spaces [14] among our twistor spaces; we shall also determine the dimension of the linear system | − \frac{1}{2}K|.

Proposition 6.14. For a twistor space Z with G-action as above the following three conditions are equivalent.

1) The discrepancy | − \frac{1}{2}K|^{G} ≠ | − \frac{1}{2}K| occurs.

2) The general member S of V is of maximal type in the sense defined in (2.7).

3) Z coincides with one of the twistor spaces of LeBrun [14] associated to an S^{1}-invariant self-dual metric on mP^{2}.

Moreover, in this case we have

dim | − \frac{1}{2}K| = 9 − 2m for m ≤ 3 and = 3 for m ≥ 3.

Proof. 1) ⇔ 2): From the short exact sequence

0 → O_{Z} → −\frac{1}{2}K → −K_{S} → 0

we obtain the cohomology exact sequence

0 → H^{0}(Z, O_{Z}) → H^{0}(Z, −\frac{1}{2}K) → H^{0}(S, −K_{S}) → H^{1}(Z, O_{Z}) = 0.

Similarly, from the short exact sequence

0 → O_{S} → −K_{S} → −K_{S}|C → 0

we get the cohomology exact sequence

0 → H^{0}(S, O) → H^{0}(S, −K) → H^{0}(C, −K_{S}|C) → H^{1}(S, O) = 0. (54)

Thus we get dim | − \frac{1}{2}K| = dim | − K_{S}| + 1 = dim | − K_{S}|C| + 2. Denote in general a symmetric toric surface with b_{2} = 2m + 2 by S_{m}. S_{m} is up to isomorphisms unique up to m = 3 (cf. (2.7)). Further, S_{m+1} is obtained by blowing up a pair of σ-conjugate fixed points on the anticanonical cycle C of a certain S_{m}. Moreover, we compute easily that dim | − K_{S}| = 8 for m = 0 and know that | − K_{S}| is base point free for m = 0, 1, 2. Since | − K_{S_{m+1}}| is considered as the set of elements of | − K_{S_{m}}| which pass through the points of blowing up, if the blown-up points are not contained in the base locus, we get dim | − K_{S_{m+1}}| = dim | − K_{S_{m}}| − 2, while if the points are in the base locus the dimension does not change. On the other hand, using (54) we can verify that a point p on the anti-canonical cycle C is in the base locus of −K_{S} if and only if p is on an irreducible component of C with self-intersection number ≤ −3. From all these assertions it is not difficult to check the above statements.

2) ⇔ 3): By Poon [20, Th. 3.1] 3) holds if and only if there exists an elementary divisor D on Z such that dim |D| ≥ 1 on Z. Since G acts on the complete linear system |D| with fixed point, we may assume that D is G-invariant. Then by Propositions 6.4 and 6.10 D is one of S_{i}^{±}. By Lemma 6.13 and the fact that H^{1}(Z, O_{Z}) = 0, dim|S_{i}^{±}| ≥ 1 if and only if the complete linear system | − K_{S_{i}^{±}} − 2L_{i}| on S_{i}^{±} is non-empty. By Lemma 2.10 this is precisely the case where one of the irreducible components of the anti-canonical cycle of S_{i}^{±} which is adjacent to L_{i} has self-intersection number 1 − m. In view of (13) this is true for some i if and only if the general member S of the system V is of maximal type. q.e.d.

Remark. The equivalent conditions of the proposition depends only on the underlying smooth K-action on M. In fact, the condition 2) depends only on the Orlik-Raymond invariant of the K-action on M.

(6.10) Recall that a twistor line L is called special if t(L) ∈ B, or equivalently L ∩ C ≠ ∅. Otherwise we call L general. A general twistor line L will be identified with its proper transform in Ẑ, and will be denoted by the same letter L.

Proposition 6.15. Let L be a twistor line.

1) If L is general, then the restriction f|L : L → P is a double covering with precisely two mutually σ-conjugate branch points on P. Neither of the branch points is on R.

2) If L is special, L is contained in a unique member S_{a} of V with a ∈ R, and it is never tangent to other members S_{a′}, a′ ≠ a.

Proof. 1) By Lemma 6.7 the intersection number of L with every fiber of f equals two. Thus f makes L a double covering of P with two branch points. Suppose that one of it, say a, is on R. Since L is tangent to S_{a} at some point z ∈ S_{a} and both L and S_{a} are real, L is also tangent to S_{a} at σ(z) ≠ z, which implies that the intersection number L · S_{a} ≥ 4. This is a contradiction. Thus the last assertion is proved. Since L is σ-invariant and f|L is σ-equivariant, the two branch points are (non-real and) σ-conjugate to each other.

2) In this case L is in the closure of a G-orbit and hence its proper transform L̂ in Ẑ is mapped to a point a of P by f, f being G-invariant. Since L is real, a must also be real, i.e., a ∈ R. Finally, if L is not very special, L intersects C transversally at two points other than z_{i}, at which S_{a} and S_{a′} intersect transversally for any a′ ≠ a. This proves the last assertion for such special twistor lines. If L is very special, we have a = a_{i} and the transversality of the intersection is already shown in the proof of Lemma 6.7. q.e.d.

(6.11) Recall that N denotes the quotient M/K and bN its boundary (cf. (3.4)).

Proposition 6.16. There exists a natural homeomorphism ϕ : bN → R which sends y_{i} to a_{i}, and hence, the arcs y_{i}y_{i+1} to a_{i}a_{i+1}, 1 ≤ i ≤ k.

For the continuity part we use the following easy:

Lemma 6.17. Let X, Y and W be topological spaces, and h : W → X and g : W → Y continuous maps with h proper and surjective. Suppose that for every point x ∈ X, g(h^{−1}(x)) is a single point y of Y. Then u : X → Y, u(x) := y, is a continuous map. q.e.d.

Proof of Proposition 6.16. Let y ∈ bN and take an arbitrary point x ∈ B over y. Then by Proposition 6.15 the special twistor line L = L_{x} is contained in S_{a} for a unique a = a(x) ∈ R. Moreover, since S_{a} is K-invariant, this point a(x) is invariant if we replace x by tx for any t ∈ K. Hence a(x) depends only on y and we obtain a map ϕ : bN → R with ϕ(y) = a(x).

More precisely, if x = x_{i} for some i, then L_{x} = L_{i} and is mapped to a_{i} and if x ≠ x_{i} for any i, a(x) never coincides with any a_{i}; in fact since σ(S_{i}^{±}) = S_{i}^{∓}, the only twistor line contained in S_{i} is L_{i}. Thus we have

ϕ(y_{i}) = a_{i}, and if y ∈ bN, y ≠ y_{j}, then ϕ(y) ≠ a_{i}, 1 ≤ i, j ≤ k. (55)

We proceed to show the bijectivity of ϕ. In view of (55), it suffices to show the bijectivity of bN − {y_{i}} and R − {a_{i}}.

Injectivity. Take any point x of B_{i} − {x_{i−1}, x_{i}} and set a = a(x). By the definition of ϕ it suffices to show that the twistor lines contained in S_{a} are K-translates of L = L_{x}.

On S_{a} we consider the equivariant morphism ν_{i} : S_{a} → P_{i} of Lemma 2.5. Let G_{i} be the subgroup of G corresponding to ±ρ_{i}, which is the complexfication of K_{i}. Then every G_{i}-orbit on S_{a} is contained in a fiber of ν_{i}. The twistor line L = L_{x} is invariant by K_{i} and hence by G_{i}; thus it is contained in a fiber of ν_{i}. Since ν_{i} is G-equivariant and L is not G-invariant, its image is a point of an open orbit on P_{i}. In particular L is itself a fiber of ν_{i}. Then any other twistor line contained in S_{a} is in a fiber of ν_{i} since it cannot intersect with L.

The linear system which defines ν_{i} is σ-invariant by Lemma 2.6, and has a real member L. Thus by that lemma the set of real points R_{i} on P_{i} consists of a single K-orbit.

Surjectivity. Take any point a ∈ R−{a_{i}}. S_{a} is then a real smooth quadratic divisor on Z. By the definition of ϕ it suffices to show that S_{a} contains a special twistor line. Suppose otherwise. Since no twistor line is tangent to S_{a} unless it is contained in S_{a} by Proposition 6.15, the twistor fibration induces an unramified double covering t|S_{a} : S_{a} → M, which is absurd since M is simply connected by Corollary 6.8. Thus S_{a} contains a twistor line L. Clearly, L is not general, but not very special. Therefore we may write L = L_{x} for some x ∈ B − {x_{i}}; this means that a = a(x) is an image of a point of bN − {y_{i}} by ϕ.

Finally, the continuity of ϕ follows from Lemma 6.17 by putting X = bN, Y = R and W = (tμ)^{−1}(B) ∩ f^{−1}(R) there. q.e.d.

Remark. The fact that S_{a}, a ∈ R − {a_{i}}, contains twistor lines and these are obtained as fibers over real points of a σ-equivariant holomorphic map S_{a} → P follows at once from [20, Lemma 1.3]. Here we would like to emphasize the additional information that this map onto P is given exactly by ν_{i} as long as a is on the arc a_{i}a_{i+1}.

(6.12) Proposition 6.16 shows in particular that the points a_{1}, …, a_{k} are arranged cyclically in this order on R ≅ S^{1} with respect to an orientation of R.

Definition. With respect to the coordinate (53) the sequence (a_{1}, …, a_{k}) naturally defines an element [a_{1}, …, a_{k}] of \overline{A} which is independent of the choices of the affine coordinate (53), where \overline{A} is as in (3.6). By construction this invariant depends only on the isomorphism classes of the self-dual manifold (M, [g]) with which we have started. We call [a_{1}, …, a_{k}] the twistorial invariant of (M, [g]).

Let m be a nonnegative integer and M = mP^{2}, where 0P^{2} = S^{4}. Fix an effective and smooth K-action on M. Then denote by C the set of K-equivariant isomorphism classes of K-invariant self-dual conformal structures on M. Then we have the natural map

I_{t} : C → \overline{A} (56)

obtained by associating the twistorial invariant to a K-invariant selfdual structure on M.

We shall show in the next two sections that this map is actually injective and show in the last section that this map coincides with the other map

I_{c} : C → \overline{A}

given by the conformal invariant defined in (3.6).

(6.13) By a theorem of Hitchin [10] we know that Z is not projecitve when k − 2 = m ≥ 2. In our case we have a direct proof of this fact.

Proposition 6.18. Z is non-projective for k > 3.

Proof. Suppose that Z is projective. Then as in [10] we conclude that −\frac{1}{2}K must be ample as it is the unique real line bundle up to multiples. On the other hand, the general orbit closures are members of this system and therefore the intersection of two of them is ample on each member. By Proposition 6.5 such orbit closures are toric surfaces and the intersection is the anti-canonical cycle on each surface. However, such an anti-canonical cycle is ample on a toric surface if and only if k = 2 or 3 (cf. [17, Prop. 2.21]). q.e.d.

On the other hand, we show in general that Z is not merely a Moishezon manifold, but has a structure of a complete algebraic variety of special type. For each p_{i}, 1 ≤ i ≤ 2k, the local action of the one parameter subgroup ρ_{i} + ρ_{i+1} with respect to the coordinates (41) at p_{i} takes the form

(x, y, z) → (sx, sy, s^{2}z), s ∈ C^{∗}, |s| ≤ 1.

We then set

U_{i} := {z ∈ Z; lim_{s→0} sz = p_{i}}.

We have p_{i}, p_{i+k} ∈ S_{i} and we shall name the two irreducible components S_{i}^{±} of S_{i} such that p_{i} ∈ S_{i}^{+} and p_{i+k} ∈ S_{i}^{−}. Then by our previous description we check easily the following:

U_{i} = Z − ⋃_{j≠i} S_{j}^{−}, 1 ≤ i ≤ k, and U_{i} = Z − ⋃_{j≠i−k} S_{j}^{+}, k + 1 ≤ i ≤ 2k.

Moreover, by the result of [3] we see that for each i there exists an isomorphism of U_{i} onto the complex affine space C^{3} which extends to a bimeromorphic map of Z to the natural compactification P^{3} of C^{3}. (In [3] the result is proved for a compact Kähler manifold, but the proof applies equally well for a Moishezon manifold in general.) Thus we obtain the following:

Proposition 6.19. Z has a natural structure of a rational complete algebraic variety which is covered by 2k copies of C^{3}, the complement of each of which is a union of k − 1 smooth toric surfaces in Z.

For k = 2, Z = P^{3} is covered by 4 copies of C^{3}, the complement of each of which is isomorphic to P^{2}. For k > 2 the complementary divisor are reducible and are not of normal crossings type. (The author is grateful to M. Pontecorvo for asking him about the rationality of Z.)

7. Local structures along the fibers

(7.1) Let m be a nonnegative integer. Let M = mP^{2} be the connected sum of m copies complex projective planes for m > 0 and set 0P^{2} = S^{4}. Fix an effective and smooth K-action on M. Suppose that we are given two K-invariant self-dual conformal structures (M, [g]) and (M, [g′]) on M. Let Z and Z′ be the corresponding twistor spaces with the associated fiber spaces (cf. Propositions 6.4 and 6.5):

f : Ẑ → P and f′ : Ẑ′ → P, (57)

where P is the complex projective line with the coordinate (53) and with real structure σ. Then supposing that the points a_{1}, …, a_{k} of R corresponding to the singular fibers of f and f′ coincide, we shall show that Ẑ and Ẑ′ are isomorphic over P.

By Propositions 6.5 and 6.12 the fibers of f and f′ over the same point a of P are isomorphic to each other. In this section, as a first step we prove that f and f′ are locally isomorphic along each fiber. Namely we prove:

Theorem 7.1. Let the notations and assumptions be as above. For every point a ∈ P there exists a neighborhood U of a in P such that f^{−1}(U) and f^{′−1}(U) are isomorphic as a complex space over U.

The result is well-known if a ≠ a_{i} for any i, since in this case both f and f′ are isomorphic over U as above to the product U × S(ρ) if U is taken sufficiently small. Therefore we assume that a = a_{i} in what follows.

There exist two ways for treating the problem, one by a deformationtheoretic method and the other by using the theory of torus embeddings. We first follow the former method and prove the above theorem as a consequence of Propositions 7.10 and 7.11 below. Then by using the latter method we give an explicit description of f (or f′) along the fiber over a_{i} in terms of torus embeddings. Also we shall indicate how the latter method gives another proof of the local uniqueness.

(7.2) We consider the fiber space f : Ẑ → P. For any a_{i} ∈ P the fiber S_{i} contains a cycle of nonsingular rational curves C = a_{i} × C, and consider deformations of the pair (S_{i}, C) which induces a trivial deformation of C. Our first purpose is to construct explicitly a Kuranishi family of this deformation theory.

For the notational convenience, however, fixing i, 1 ≤ i ≤ k, we write S = S_{i} and L = L_{i} in what follows. Accordingly we set S^{±} = S_{i}^{±} so that S = S^{+} ∪ S^{−} and L = S^{+} ∩ S^{−}. Let U^{±} be the open orbits in S^{±} so that C^{±} := S^{±} − U^{±} forms the anti-canonical cycle of S^{±}. Using the notation of (2.10) we write the irreducible components of C^{±} as (C_{1}^{±}, …, C_{i}^{±}, L_{i}^{±}, C_{i+1}^{±}, …, C_{k}^{±}). Then B := L ∩ C consists of two points p = L ∩ C_{i}^{+} = L ∩ C_{i+1}^{−} and q = L ∩ C_{i+1}^{+} = L ∩ C_{i}^{−}. Thus in this notation the one parameter group ρ_{i} fixes each point of C_{i}^{+} and C_{i}^{−}.

For holomorphic maps h_{i} : Y_{i} → T, i = 1, 2 over the same base space T by a (relative) isomorphism of Y_{1} to Y_{2} over T we shall mean an isomorphism u : Y_{1} → Y_{2} of complex spaces with h_{2}u = h_{1}. We consider deformations of the pair (S, C) which induce trivial deformations of C, i.e., a flat morphism f : S → T over an analytic germ T = (T, o) together with a relative divisor C of S over T such that (S_{o}, C_{o}) = (S, C) (fixed isomorphism) and there exists an isomorphism ϕ : C × T → C over T, where (S_{o}, C_{o}) denotes the fiber over o (Thanks are due to K. Ohno for asking the precise meaning of the deformations here, which has eventually led to the correction of the original argument.) We call such a deformation simply a deformation of the pair (S, C) and denote it by f : (S, C) → T. If f′ : (S′, C′) → T is another deformation over the same space T, the two deformations f and f′ are said to be isomorphic, if there exists an isomorphism u : S → S′ over T which sends C over C′ (in compatible with the given identifications of the fibers over o with (S, C)). In what follows we shall call the corresponding semiuniversal deformation briefly a Kuranishi family of (S, C).

Lemma 7.2. A Kuranishi family of (S, C) exists.

Proof. Let u : (S̃, C̃) → T̃ be a Kuranishi family of (S, C) in the ordinary sense (for which C̃ is not necessarily trivial) (cf. [7, Th.8.5] applied to the case of an embedding) and v : Ĉ → P a Kuranishi family of C. We have an induced versal map b : T̃ → P, i.e., the map induced by the versality of the family v. Then the restriction of u to the inverse image b^{−1}(o) is clearly a Kuranishi family for the deformations of (S, C) in our sense here, where o is the base point of P. q.e.d.

(7.3) We shall consutruct explicitly a deformation of (S, C) which turns out to be a Kuranishi family of (S, C). First we introduce some terminology. Let f : (S, C) → T be a deformation of the pair (S, C). Then C is a union of divisors C_{j}^{±} which are (trivial) deformations of C_{j}^{±}, 1 ≤ j ≤ k. Then for any j ≠ i, b_{j}^{±} := C_{j}^{±} ∩ C_{j+1}^{±} (with j considered cyclically modulo k) defines a holomorphic section of f which does not intersect L ⊆ S_{o} = S. We call any blowing up η : (S̃, C̃) → (S, C) of (S, C) with center b_{j}^{+} ∐ b_{j}^{−}, j ≠ i, an admissible blowing up of (S, C).

Proposition 7.3. There exists a natural deformation f : (S, C) → C of (S, C) parametrized by C such that (S, C) admits a natural relative G-action over C and that for t ≠ 0 the fibers S_{t} are mutually isomorphic nonsingular symmetric toric surface with anti-canonical cycle C_{t} with respect to the induced G-action. Moreover, S is nonsingular except at the two points p and q of S = S_{0}, where S has ordinary double points.

Proof. First consider the case k = 2. In this case S^{±} are complex projective planes and S is realized as a reducible quadric Q := {ξ_{1}ξ_{2} = 0} in P^{3} = P^{3}(ξ_{1} : ξ_{2} : ξ_{3} : ξ_{4}) with irreducible components Q^{±} := {ξ_{i} = 0}, i = 1, 2, where C =: A is the union of four lines A_{ij} := {ξ_{i} = ξ_{j} = 0}, i = 1, 2, j = 3, 4. Then we consider the one parameter deformation ξ_{1}ξ_{2} = tξ_{3}ξ_{4}, t ∈ C of Q, in which each member Q_{t}, t ≠ 0, is a smooth quadric isomorphic to P^{1}×P^{1} and which contains all the above four lines. Therefore we may consider this deformation, denoted especially by q : (Q, A) → C, as a deformation of the pairs (Q, A) in the sense defined above. With respect to the G-action

(ξ_{1} : ξ_{2} : ξ_{3} : ξ_{4}) → (ξ_{1} : s_{1}ξ_{2} : s_{2}ξ_{3} : s_{1}s_{2}ξ_{4}), (s_{1}, s_{2}) ∈ G = C^{∗2}

each member Q_{t}, t ≠ 0, becomes a symmetric torus surface as in the proposition. It is also immediate to check the final assertion concerning the singularity of Q.

Next we consider the general case. In this case the desired deformation will be obtained by successive blowing-ups of the above family q. By Lemma 2.9 our S is obtained from the above Q for k = 2, identified with \overline{S} in that lemma, by a succession of admissible blowing ups. Since our deformation is trivial for C, we can extend the succession of admissible blowing ups sideways to a succession of admissible blowing ups of the families so that starting from the deformation (Q, A) → C for the case k = 2 constructed above, we obtain a deformation of (S, C) for any pairs under consideration with the same parameter space C (where the triviality of C over T is obvious). Moreover, the relative G-action on the family defined above for the case k = 2 lifts canonically to any of the deformation obtained in the above way. The final assertion follows from that for the case k = 2 since all the blowing-ups occur on the smooth parts of the varieties under consideration. q.e.d.

(7.4) Our purpose is to show the following:

Proposition 7.4. Let (S, C) be as above. Then the deformation f : (S, C) → C of (S, C) consutructed in Proposition 7.3 is a Kuranishi family of (S, C).

In view of the construction of f in the proof of Proposition 7.3, we immediately obtain the following:

Corollary 7.5. Any Kuranishi family u : (S, C) → C of (S, C) is obtained from a Kuranishi family of the pair (Q, A) with k = 2 by a finite succession of admissible blowing-ups.

As for the proof of Proposition 7.4, since the irreducible components of C which intersect with L are not Cartier divisors on S, the cohomological description of the tangent space for our deformation problem seems a bit complicated. So we shall follow some round-about way in showing the proposition. Namely we consider a modified deformation problem for which the cohomological description is easier. For this purpose we prepare some more notations. We define

D^{±} = C_{i}^{±} ⋃ C_{i+1}^{±} and C^{′±} = ⋃_{j≠i,i+1} C_{j}^{±}.

Then we set

C′ = C^{′+} ⋃ C^{′−} and D = D^{+} ⋃ D^{−}.

Note that D is a Cartier divisor in S as the inclusion D ⊆ S looks locally at p or q like

D = {xy = z = 0} ⊆ S = {xy = 0} ⊆ C^{3}.

By a D-deformation of the pair (S, C) we mean a deformation f : (S, C) → T of (S, C) as above together with an isomorphism ϕ : D × T → D, where D = ∪_{±}(C_{i}^{±} ∪ C_{i+1}^{±}). We shall study the tangent space of this modified deformation theory.

First we recall some general notation and terminology. Let X be a complex space and Y a divisor with normal crossings contained in the smooth locus of X. We shall denote by Ω_{X}^{1} the sheaf of germs of holomorphic 1-forms on X. Then its dual Θ_{X} := Hom_{O_{X}} (Ω_{X}^{1}, O_{X}) is the holomorphic tangent sheaf of X. Ω_{X}^{1}(log Y) will denote the sheaf of germs of meromorphic 1-forms with at most logarithmic poles along Y and then its dual Θ_{X}(− log Y) is the sheaf of germs of holomorphic vector fields which are tangent to Y. If D is a Cartier divisor on X, the notations (D) and (−D) are used to indicate ⊗[D] and ⊗[D]^{−1} respectively, where [D] is the line bundle associated to D, e.g.,

Ω_{X}^{1}(log Y)(D) = Ω_{X}^{1}(log Y) ⊗ [D].

Note that when dim X = 1, we have

Ω_{X}^{1}(log Y) = Ω_{X}^{1}(Y)

and

Θ(− log Y) = Θ(−Y).

Using these notation the tangent space of the modified deformation theory is identified with the Ext-group

Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S})

which we shall now compute. (Actually, this space parametrizes the infinitesimal deformations which induce locally analytically trivial deformations of C with globally trivialization of D. But for a cycle of nonsingular rational curves we know that every locally analytically trivial deformation is necessarily globally trivial, and hence, this is a right choice.)

(7.5) With the simplified notations

Ξ := Θ(− log C′)(−D) and τ := Ext^{1}(Ω_{S}^{1}(log(C′)(D), O_{S}).

the local-to-global spectral sequence of Ext functor gives us the following exact sequence of vector spaces

\begin{aligned}0 → H^{1}(S, Ξ) & → Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S}) \\ & → H^{0}(S, τ) → H^{2}(S, Ξ). \end{aligned} (58)

Then we prove the following:

Proposition 7.6.

1) There exist a natural isomorphism

H^{1}(S, Ξ) ≅ H^{0}(L, Θ_{L}(B))(≅ C)

and we have H^{2}(S, Ξ) = 0.

2) τ ≅ O_{L}.

Corollary 7.7. The exact sequence (58) reduces to

\begin{matrix}0 → H^{1}(S, Ξ) ≅ C → Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S}) \\ →^{β} H^{0}(S, τ) ≅ C → 0. \end{matrix} (59)

In particular dim Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S}) = 2.

Proof of Proposition 7.6. 1) In order to compute H^{i}(S, Ξ) we recall the following sheaf exact sequence [4, (5.4)]

0 → Θ_{S} → ⨁_{±} n_{∗}^{±}Θ_{S^{±}}(− log L) → Θ_{L} → 0, (60)

where n^{±} : S^{±} → S are the inclusions. Since C′ does not intersect with L and D is a Cartier divisor with D ∩ L = B := {p, q}, from (60) we get

0 → Ξ → ⨁_{±} n_{∗}^{±}Ξ^{±} → Θ_{L}(−B) → 0, (61)

where

Ξ^{±} = Θ_{S^{±}}(− log(C^{±} ⋃ L))(−D^{±}).

Note that Θ_{L}(−B) ≅ O_{L} so that H^{i}(L, Θ_{L}(−B)) = 0 for i = 1, 2 and ≅ C for i = 0. From the cohomology exact sequence associated to (60) we have the following long exact sequence of cohomology and an isomorphism

0 → H^{0}(S, Ξ) → ⨁_{±} H^{0}(S^{±}, Ξ^{±}) → H^{0}(L, Θ_{L}(−B))

→ H^{1}(S, Ξ) → ⨁_{±} H^{1}(S^{±}, Ξ^{±}) → 0 (62)

H^{2}(S, Ξ) ≅ ⨁_{±} H^{2}(S^{±}, Ξ^{±}). (63)

We next compute H^{i}(S^{±}, Ξ^{±}). Consider the natural short exact sequence of sheaves

0 → Ξ^{±} → Θ_{S^{±}}(− log C^{±}) → ⨁_{j=i,i+1} Θ_{C_{j}^{±}}(−B_{j}^{±}) → 0 (64)

where B_{j}^{±} consists of two intersection points of C_{j}^{±} with the other components of C^{±}. Since Θ_{C_{j}^{±}}(−B_{j}^{±}) are isomorphic to O_{C_{j}^{±}} we have

H^{i}(C_{j}^{±}, Θ_{C_{j}^{±}}(−B_{j}^{±})) = 0

for i = 1, 2 and ≅ C for i = 0. On the other hand, by the sheaf isomorphism ([17, Prop.3.1])

Θ_{S^{±}}(− log C^{±}) ≅ O_{S^{±}} ⊗Z N ≅ O_{S}^{⊕2},

we have H^{i}(S^{±}, Θ_{S^{±}}(− log C^{±})) = 0 for i = 1, 2 and ≅ C^{2} for i = 0. Then H^{0}(S^{±}, Ξ^{±}) consists of vector fields associated to the torus action on S^{±} which vanishes identically on C_{i}^{±} ∪ C_{i+1}^{±}, which must be identically zero on the whole S^{±}; namely H^{0}(S^{±}, Ξ^{±}) = 0. Thus from the cohomology exact sequence associated to (64) we have H^{2}(S^{±}, Ξ^{±}) = 0 and the following exact sequence of vector spaces

\begin{matrix}0 → H^{0}(S^{±}, Θ_{S^{±}}(− log C^{±})) ≅ C^{2} \\ → ⨁_{j=i,i+1} H^{0}(C_{j}^{±}, Θ_{C_{j}^{±}}(−B_{j}^{±})) ≅ & C^{2} → H^{1}(S^{±}, Ξ^{±}) → 0. \end{matrix} (65)

Therefore we also have dim H^{1}(S^{±}, Ξ^{±}) = 0. Substituting the results into (62) we get the desired result.

2) Since Ω_{S}^{1}(log C′)(D) is locally free outside L and D is a Cartier divisor on the whole S, τ has support in L and we have

τ ≅ Ext^{1}(Ω_{S}^{1}(D), O_{S}) ≅ Ext^{1}(Ω_{S}^{1}, O_{S})(−D).

On the other hand, as is well-known the last term is an O_{L}-module and we have

Ext^{1}(Ω_{S}^{1}, O_{S}) ≅ N_{L/S^{+}} ⊗ N_{L/S^{−}} = O_{L}(2)

where N_{L/S^{±}} denotes the normal bundle of L in S^{±}. Thus

τ ≅ O_{L}(2)(−B) ≅ O_{L}. q.e.d.

Remark. From the above argument it also follows that

Ext^{2}(Ω_{S}^{1}(log C′)(D), O_{S}) = 0.

(7.6) We shall construct a Kuranishi family for D-deformation of (S, C). In view the proof of 1) of Proposition 7.6 nonzero elements of H^{1}(S, Ξ) are described as follows. We set

V^{+} = S^{+} − C_{i+1}^{+}, W^{+} = S^{+} − C_{i}^{+}, V^{−} = S^{−} − C_{i}^{−}, W^{−} = S^{−} − C_{i+1}^{−}.

Then V := V^{+} ∪ V^{−} and W := W^{+} ∪ W^{−} are open subsets of S and V ∩ W = S − D. Fix any nonzero element v of H^{0}(L, Θ_{L}(−B)). Let χ = χ_{i} be a holomorphic vector field on S coming from the action of the one parameter group ρ_{i} which restricts to v on L. (The action of ρ_{i} is nontrivial on L.) Since ρ_{i} fixes each point of C_{1}^{+} and C_{k}^{−}, χ vanishes identically on C_{1}^{+} and C_{k}^{−}. Thus (χ, 0) defines an element of H^{0}(V^{+}, Ξ_{1}) ⨁ H^{0}(V^{−}, Ξ_{2}) which restricts to v on L − q. Similarly (0, −χ) defines an element of H^{0}(W^{+}, Ξ_{1}) ⨁ H^{0}(W^{−}, Ξ_{2}) which restricts to v on L − p. Then (χ, χ) = (χ, 0) − (0, −χ) is considered to be an element of H^{0}(V^{+} ∩ V^{−}, Ξ) which defines a nonzero element χ of H^{1}(S, Ξ) corresponding to to v ∈ H^{0}(L, Θ_{L}(−B)) via the isomorphism of Proposition 7.6.

The corresponding D-deformation parametrized by C^{∗} with the base point 1 ∈ C^{∗} is described as follows. Define a family of complex spaces S_{s} parametrized by s ∈ C^{∗} by patching V and W via ρ_{i}(s); S_{s} = V ∪_{ρ_{i}(s)} W. For s = 1 we obtain the original space S although clearly all S_{s} are isomorphic to S. As a D-deformation this family is isomorphic to the trivial deformation (S, C) × C^{∗} → C^{∗} of (S, C) together with the non-trivial trivialization ϕ : D × C^{∗} → D × C^{∗} given by the condition that ϕ_{s}|C_{i+1}^{−} = ρ_{i}(s)|C_{i+1}^{−} and ϕ_{s} is the identity on the other irreducible components of D. The infinitesimal deformation at s = 1 is then precisely the above element χ up to constants.

We can perform the same construction for the family f : (S, C) → C constructed in Proposition 7.3 using its relative G-action. Fixing a trivialization ϕ : D × T → D we may consider f as a D-deformation of (S, C). As in the absolute case we set

V^{+} = S − C_{i+1}^{+}, W^{+} = S − C_{i}^{+}, V^{−} = S − C_{i}^{−}, and W^{−} = S − C_{i+1}^{−}.

Then V := V^{+} ∪ V^{−} and W := W^{+} ∪ W^{−} are open subsets of S and V ∩ W = S − D. We patch together as before V and W via ρ_{i}(s), s ∈ C^{∗}. In this way we obtain a D-deformation f̂ : (Ŝ, Ĉ) → R of (S, C) parametrized by R := C × C^{∗} with the base point o := (0, 1), where the trivialization of D is obtained by perturbing the original one ϕ by using ρ_{i} as in the absolute case. By construction the following is clear.

Lemma 7.8. As a deformation of the pair (S, C) the family f̂ is isomorphic to the family induced from f : (S, C) → T via the natural projection p : R → T = C.

(7.7) Now we are ready to identify a Kuranishi family for D-defor-mations of (S, C).

Proposition 7.9. The map f̂ : (Ŝ, Ĉ) → R of (S, C) gives a Kuranishi family for the D-deformations of (S, C).

Proof. First we shall show that the tangent space of R at o is naturally identified with the vector space Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S}). Indeed, the tangent space of the subspace 0 × C^{∗} is, by the description of (7.6) in the absolute case, naturally identified with the subspace H^{1}(S, Ξ) in (39). On the other hand, if w : T_{0}C → Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S}) is the natural tangential map (Kodaira-Spencer map) for the original family f : (S, C) → C with ϕ chosed and fixed, where T_{0}C is the tangent space of T at 0, the composition of w with the quotient map β in (39) is necessarily surjective. In fact, the image of βw measures the infinitesimal variation of the singularity of S along L; however by Proposition 7.3 the total space S of the family f is smooth along L so that the composite map βw cannot be the zero map. Since H^{0}(S, τ) ≅ C, the assertion is verified.

The proposition is deduced from this. However, for the D-deforma-tions we have not found a suitable reference to quote for the existence of a Kuranishi family in general. So we shall proceed in an ad hoc way here. First of all it is not difficult to show that the D-deformation functor has a hull in the sense of [25]; indeed, the proof is basically the same as in the case of the ordinary deformation functor (cf. [28, Th.3.6], [25, (3.7)]) using the criterion of Schlessinger [25, Th.2.11]. Then, since the parameter space of our family is smooth, and the tangent space is identified with the tangent space of the functor, i.e., Ext^{1}(Ω_{S}^{1}(log C′)(D), O_{S}), by [28, Th.3.5] this family is formally semiuniversal in the sense that it is semiuniversal for any deformations whose parameter space is zerodimensional. Then by the same argument as in the proof of [28, Th.3.7] (in the case of ordinary deformation functor) using, e.g., [9, Lemma 2] we conclude that the family is complete (or versal), and hence is a Kuranishi family. q.e.d.

Proof of Proposition 7.4. Let f′ : (S′, C′) → T′ be any deformation of the pair (S, C). By taking a trivialization D × T′ → D we may consider f′ as a D-deformation of the pair (S, C). Then by the previous proposition we have a semiuniversal map δ : T′ → R. Compose this with the projection p : R → T. Then by Lemma 7.8 u is isomorphic as a deformation of (S, C) to the induced family from f via pu. This shows that f is a versal family. Moreover, if (S̃, C̃) → T̃ is a Kuranishi family of (S, C), the differential of the versal map T_{o}C → T_{o}T̃ is not the zero map since otherwise S will have singularity along L. Thus the Kodaira-Spencer map of f is injective and f is a semiuniversal deformation of (S, C) as desired. q.e.d.

We come back to our twistor space Z and consider the associated morphism f : Ẑ → P or more precisely the morphism f : (Ẑ, E) → P. Take any a_{i} ∈ R and consider the germ f_{i} of f along the fiber S_{i} = S_{i}^{+} ∪ S_{i}^{−} as a deformation of the pair (S_{i}, C) inducing the trivial deformation of C.

Proposition 7.10. The map f_{i} gives rise to a Kuranishi family of (S_{i}, C) at a_{i}.

Proof. Let f : (S, C) → (T, o) be the Kuranishi family of the pair (S_{i}, C) constructed in Proposition 7.4. We have a versal map τ : (P, a_{i}) → (T, o) which is induced by the versality of the family f. If τ has ramification at a_{i}, then Ẑ should have singularity along the intersection L_{i} = S_{i}^{+} ∩ S_{i}^{−}. This is a contradiction; so τ must be isomorphic. Thus f_{i} and f are isomorphic as a deformation of (S_{i}, C), and hence, f_{i} itself is a Kuranishi family of (S_{i}, C). q.e.d.

(7.8) In order to prove Theorem 7.1 we still have to prove the following:

Proposition 7.11. Let u : (S, C) → (T, o) and u′ : (S′, C′) → (T, o) be two Kuranishi families of the pair (S, C) with the same parameter space T. Then there exists an isomorphism j : (S, C) → (S′, C′) over T which maps each irreducible component C_{j} to the corresponding component C′_{j}. Moreover, if we are given a section s : T → S (resp. s′ : T → S′) of u (resp. u′) which do not pass through C (resp. C′), then j can be taken so that js = s′ on T.

Remark. The versality in general implies the existence of such an isomorphism over some automorphism α of T. Our claim is that, in our case, an arbitrary automorphism of the base lifts to an isomorphism of the two families. This would not be automatic in general.

Proof. We first consider the case k = 2. In this case S = S^{+} ∪ S^{−}, S^{±} ≅ P^{2}, and the intersection S^{+} ∩ S^{−} is a line L. Moreover, the general fiber of u is isomorphic to P^{1} × P^{1}. Our method is to perform some canonical bimeromorphic modifications to both the fiber spaces u and u′ and reduce them to the trivial fiber spaces for which the existence of the isomorphism is clear; and then we follow the inverse process to get a desired isomorphism between the original fiber spaces. Now the total space S has two singular points p and q, which are ordinary double points. We may perform a small resolution S̃ → S at p and q with the exceptional nonsingular rational curves N_{p} and N_{q} respectively. At each point there exist two choices of such small resolutions; our choice is that N_{p} and N_{q} are contained in one and the same component of S, say S^{+}. (In the presence of a section s we take the component which does not intersect with the image of s, noting that the image never intersects with L.) Then the proper transform S̃^{−} of S^{−} in S̃, which is still isomorphic to P^{2}, is exceptional in S̃; hence we can blow down S̃^{−} to a smooth point r of a complex manifold \overline{S}:

[figure: S; u; S̃; \overline{S}; \overline{u}; T] (66)

It turns out that the resulting morphism \overline{u} : \overline{S} → T is smooth and is a trivial bundle with fiber P^{1} × P^{1}. We denote by \overline{A}_{i}, i = 1, 2, the proper transforms in \overline{S} of the two irreducible components of the anticanonical cycles of S^{+} other than L, and by \overline{N}_{p} and \overline{N}_{q} those of N_{p} and N_{q} respectively. Then one sees that \overline{N}_{p} ∩ \overline{N}_{q} = \{r\}. Furthermore, the cycle of rational curves \overline{C}_{0} := \overline{N}_{p} + \overline{A}_{1} + \overline{A}_{2} + \overline{N}_{q} extends to the anti-canonical cycle \overline{C}_{t} on the general fiber \overline{S}_{t} so that \overline{u} is considered as a deformation of the pair (\overline{S}_{0}, \overline{C}_{0}).

We perform the same construction to the other family u′ : (S′, C′) → (T, o) and obtain a manifold S̃′ and a trivial family \overline{u}′ : \overline{S}′ → T considered as a deformation of the pair (\overline{S}′_{0}, \overline{C}′_{0}) defined in the same way as (\overline{S}_{0}, \overline{C}_{0}). Let r′ ∈ \overline{S}′ be the point corresponding to r.

Now take an isomorphism \overline{j} : (\overline{S}, \overline{C}) → (\overline{S}′, \overline{C}′) over T which sends \overline{A}_{i}, \overline{N}_{p}, \overline{N}_{q} to \overline{A}′_{i}, \overline{N}′_{p}, \overline{N}′_{q} respectively, and hence sends r to r′, which is always possible. Then j clearly lifts to an isomorphism j̃: S̃ → S̃′ of the blown-up manifolds which automatically sends N_{p} onto N′_{p} and N_{q} to N′_{q}. Hence, after contracting these curves, j̃ further descends to an isomorphism j between the original fiber spaces S → T and S′ → T which obviously sends each irreducible component of C to the corresponding components of C′. This is a desired isomorphism.

Finally suppose that we are given sections s and s′. Let \overline{s} and \overline{s}′ be the corresponding sections to \overline{u} and \overline{u}′. Since \overline{j}\overline{s}(t) and \overline{s}′(t) are in the open orbit of S′_{t} for any t ∈ T, there exists a unique element g(t) of G such that g(t)\overline{j}\overline{s}(t) = \overline{s}′(t). This g(t) defines a holomorphic map g : T → G. Then replacing \overline{j} by g\overline{j} in the above argument we get a desired isomorphism j.

Next we consider the general case. By Proposition 7.4 and Corollary 7.5 u : (S, C) → T is obtained by a succession ϕ := ϕ_{1} ⋯ ϕ_{m} of admissible blowing-ups ϕ_{d} : S_{d} → S_{d−1}, d = 1, …, m, over T such that S_{m} = S and that (S_{0}, C_{0}) is a Kuranishi family of the pair (Q, A) for the case k = 2 where m = k − 2 (cf. the proof of Proposition 7.3). The similar statement is true for u′, too. Let ϕ′_{d} : S′_{d} → S′_{d−1}, d = 1, …, m, over T be the corresponding admissible blowing ups. Moreover, these blowing-ups are determined by the induced admissible blowing-ups of the central fiber so that we may assume that at each step (S_{d}, C_{d}) and (S′_{d}, C′_{d}) are deformations of one and the same pair.

Now by the proof in the case k = 2 there exists an isomorphism j_{0} : S_{0} → S′_{0} over T which maps the irreducible components of C_{0} onto the corresponding components of C′_{0}, and in the presence of sections maps the image of the section s in S_{0} to that of the section s′ in S′_{0}. Then by the definition of admissible blowing-ups it is immediately seen that j_{0} lifts to a desired isomorphism j of S onto S′. q.e.d.

Proof of Theorem 7.1. Immediate from Propositions 7.10 and 7.11.

(7.9) Let f : X → Y be holomorphic maps of complex spaces. Then a relative automorphism of X over Y is an autmorphism h : X → X of complex spaces such that f = fh. We also need the following result in the next section.

Proposition 7.12. Let u : (S, C) → T be a Kuranishi family of (S, C). Let β : S → S be a relative automorphism over T which preserves each irreducible component C_{i} of C. Then there exists a unique holomorphic map h : T → G such that the automorphism β_{t} : S_{t} → S_{t}, t ∈ T, is induced by the action of the element h(t).

Proof. The existence of a unique holomorphic map h defined on T − {0} follows from Lemma 2.1 immediately. The problem is to show that it extends across the origin 0.

First consider the case k = 2. In this case in view of Proposition 7.4 and the proof of Proposition 7.3 we may assume that u is the family of quadrics in P^{3} constructed in the proof of Proposition 7.3. Thus S is naturally embedded in P^{3} × T over T and β is given by a holomorphic map h : T → P GL(3) which has values in G for t ≠ 0 with respect to the natural embedding G ↪ P GL(3) induced by the given action. Since G is closed in P GL(3), h(0) also belongs to G.

In the general case, as in the last part of the proof of the previous proposition we take a bimeromorphic morphism ϕ : S → S_{0} over T. Then β descends to an automorphism \overline{β} of S_{0} over T preserving the irreducible components of C_{0} and with \overline{β}(t), t ≠ 0, induced by the action of h(t). By what we have proved above for the case k = 2, h extends across the origin. But then the extended map h : T → G already meets the requirement of the lemma for the original S, too. q.e.d.

Remark. It is clear that G is the identity component of the group of extendible automorphisms of the fiber S_{0}. But a priori it may contain another components to which β_{o} belongs as in the case of degeneration of elliptic curves.

(7.10) We shall construct by the method of torus embedding a local model of the degeneration at a_{i} of the morphism f for any i, 1 ≤ i ≤ k.

Let N = Z^{3} be the additive group of one parameter subgroups of the algebraic three-torus C^{∗3}. Decompose N into the direct sum N = N_{2} ⊕ N_{1} = Z^{2} ⊕ Z. Let

ρ = {ρ_{1}, …, ρ_{2k}}, ρ_{j+k} = −ρ_{j}, 1 ≤ j ≤ k,

be the set of primitive elements in N_{2} = Z^{2} arranged counterclockwise in this order.

Fixing i take any elements η and η′ of N whose last coordinates are equal to 1 and which satisfy the condition

η − η′ = ρ_{i} − ρ_{i+1}. (67)

Define a fan △ in NR by the following prescription: The one-dimensional simplices are those generated by the primitive vectors η, η′ and ρ_{j}, 1 ≤ j ≤ 2k. The two-dimensional simplicies consist of

{η, η′}, {ρ_{j}, ρ_{j+1}}, 1 ≤ j ≤ 2k (2k + 1 = 1),
{η, ρ_{j}}, j = 1, …, i, i + k + 1, …, 2k, and
{η′, ρ_{j}}, j = i + 1, …, i + k,

while three-dimensional simplices are given by

{η, ρ_{1}, ρ_{2}}, … {η, ρ_{i−1}, ρ_{i}}, {η, −ρ_{i+1}, −ρ_{i+2}}, … {η, −ρ_{k}, −ρ_{1}}

and

{η′, −ρ_{1}, −ρ_{2}}, … {η′, −ρ_{i−1}, −ρ_{i}}, {η′, ρ_{i+1}, ρ_{i+2}}, … {η′, ρ_{k}, ρ_{1}}.

(Here, for example {η, ρ_{1}, ρ_{2}} denotes the simplex spanned by η, ρ_{1}, ρ_{2}.) Besides these, we take the following two three-dimensional quadrilateral cones

{η, η′, ρ_{i}, ρ_{i+1}} and {η, η′, −ρ_{i+1}, −ρ_{i}}

as constituents of our fan. It is immediate to see that these data actually define a fan △ whose support coincides with the half space N_{2R} ⊕ N_{1R_{≥0}}. Let X be the corresponding three dimensional toric variety.

Let △_{1} = {{0}, R_{≥0}} be a fan on N_{1R} = R, which gives the toric affine line C = C^{∗}(s) ∪ {0}. Then the projection N → N_{1} gives rise to a map of fans ψ : (N, △) → (N_{1}, △_{1}), which in turn induces a proper morphism of toric varieties h : X → C (cf. [17, Th. 1.13, Th. 1.15]).

Lemma 7.13. Given ρ, the fiber space h is, up to C^{∗3}-equivariant isomorphisms, independent of the choices of η and η′ above.

Proof. It suffices to show that the map of fans ψ is up to isomorphisms independent of the choice of η and η′. Consider the subgroup H ≅ Z^{2} ⊆ SL(3, Z) consisting of elements of the form

g = ( \begin{matrix}1 & 0 & α \\ 0 & 1 & β \\ 0 & 0 & 1 \end{matrix} ) , α, β ∈ Z.

The natural action of the group H on N leaves fixed any element of N_{2} ⊆ N and induces the identity on the quotient N_{1} = N/N_{2}. Thus if we apply g to our original fan △, the change occurs only on η and η′ with the difference η − η′ unchanged. On the other hand, the image of (0, 0, 1) becomes (α, β, 1). From this we easily conclude that any two fans constructed according to the above prescription are mutually mapped to each other by some element of H. q.e.d.

(7.11) We show that h gives a local model of f along S_{i}.

Proposition 7.14. We consider the morphism f : Ẑ → P as a germ along the fiber S_{i} = S_{a_{i}} over a_{i} and consider the morphism h : X → C as a germ along the fiber X_{0} over 0 ∈ C. Then f and h are isomorphic.

Proof. Let E_{j} be the irreducible divisor on X corresponding to ρ_{j} and S (resp. S′) the irreducible divisor corresponding to η (resp. η′). Because of (67) X has ordinary double points at the intersection point of the four divisors S, E_{i}, E_{i+1} and S′ (resp. S, E_{i+k}, E_{i+k+1} and S′) (cf. [13, pp. 38–39] [17, p. 37, Ex.]). Since the third coordinate of η (resp. η′) is equal to one, η, ρ_{j} and ρ_{j+1} form a Z-basis of N so that X is nonsingular at any other point.

Let △(ρ) be the fan in N_{2R} = R^{2} generated by ρ, which defines the toric surface S(ρ). Then we have the natural inclusion of fans (N_{2}, △(ρ)) ↪ (N, △), giving rise to an embedding of the toric surface S(ρ) onto the fiber of h over 1, and hence onto any fiber over s ≠ 0. Note that the divisors E_{j} are of the form E_{0j} × C on X, where E_{0j} = E_{j} ∩ h^{−1}(0).

The fiber over s = 0 consists of two toric surfaces S and S′, whose structures are easily read off from the fan △. For S, for instance, we set according to [17, Cor.1.7] N_{η} := N/Zη and

△_{η} := \{\overline{σ} = the image of σ in N_{ηR}; η is a face of σ\}.

Then S is isomorphic to the toric surface associated to the fan (N_{η}, △_{η}) and we have

△_{η} = \{\overline{η}′ and \overline{ρ}_{j}, j = 1, …, i, i + k + 1, …, 2k\},

where \overline{η}′ = \overline{η}′−\overline{η} = −\overline{ρ}_{i}+\overline{ρ}_{i+1} because of (67). Therefore S is isomorphic to S_{i}^{+}(ρ). Similarly, we get S′ ≅ S_{i}^{−}(ρ) of (9). Finally since they intersect transversally along the curve L := S ∩ S′ corresponding to the two-simplex {η, η′}, it follows that X_{0} ≅ S_{i} = S_{a_{i}}.

Thus, if we set E = ∪_{j}E_{j} with its fiber E_{0} = ∪_{j}E_{0j} over 0, the map h, regarded as a map (X, E) → C, is considered as giving a local deformation of the pair (X_{0}, E_{0}) which is isomorphic to (S_{i}, C) by the above consideration. Hence we have a versal map τ : (C, 0) → (P, a_{i}) of analytic germs, considering f : Ẑ → P as a Kuranishi family of the pair (S_{i}, C) by Proposition 7.10. However since X is smooth along the general points of L, τ cannot have ramifcation, i.e., τ is isomorphic. Thus the proposition follows. q.e.d.

In passing we also note the following:

Proposition 7.15. There exists a relative involution of X over C which interchanges the two irreducible components of the fiber X_{0} over the origin 0 ∈ C.

Proof. Lemma 7.13 shows that we may take η to be (0, 0, 1). Then if η′ = (α, β, 1), the element

h = ( \begin{matrix}−1 & 0 & α \\ 0 & −1 & β \\ 0 & 0 & 1 \end{matrix} ) ∈ SL(3, Z)

defines an involution of the fan (N, △) which interchanges the vectors (0, 0, 1) and (α, β, 1). This induces a desired relative automorphism.

q.e.d.

(7.12) Note that the existence of toric description of the Kuranishi family of (S_{i}, C) as above is actually a priori guaranteed by the following result of Mumford (cf. the proof of Theorem in [13, IV,§1]).

Lemma 7.16. Let f : A → B be a proper morphism of a complex algebraic variety A of dimension n + 1 onto a complex smooth algebraic curve B. Suppose that there exists an effective algebraic action of the algebraic torus C^{∗n} on A over B. Let b ∈ B be an arbitrary point. Then there exist a toric variety X of dimension n+1 and an equivariant morphism h : X → C such that the germ of f, considered along the fiber over b, and the germ of h, considered along the fiber over the origin 0 of C, are C^{∗n}-equivariantly and analytically isomorphic, where the action of C^{∗n} on X over C is induced by the given torus action (first n factors).

In fact by applying the lemma to our G-action on Ẑ over P at each point a_{i} ∈ R, we obtain a priori the exsitence of a toric description of f along the singular fiber S_{i}.

Conversely, we can replace the deformation theoretic argument in the first part of this section by the toric argument above in showing the local uniqueness of the fibering of f over a_{i}. In fact, from the structure of Ẑ along S_{i} we conclude that any of the toric models describing f along S_{i} must be isomorphic to h in Proposition 7.14. For instance, from the fact that Ẑ has ordinary double point at z_{i} or z_{i+k} we obtain the condition (67), from the fact that Ẑ is otherwise smooth along S_{i} we get that the last coordinates of η and η′ should be equal to one, and etc.

Remark. In order to apply Lemma 7.16 we have to check that a suitable neighborhood of S_{i} is algebraic; in this respect it is not difficult to show that if we set P_{i} = {a ∈ P; a ≠ a_{j}, j ≠ i}, h^{−1}(P_{i}) is quasiprojective over P_{i}.

8. Effectivity of invariants

(8.1) The purpose of this section to prove the following global isomorphism theorem which implies the effectivity of the twistorial invariants, i.e., the injectivity of the map I_{t} in (56).

Theorem 8.1. Let (M, [g]), (M, [g′]), Z, Z′, f : Ẑ → P and f′ : Ẑ′ → P be as in (7.1). Suppose that the points a_{1}, …, a_{k} on P corresponding to the singular fibers of f and f′ coincide. Then there exists an isomorphism j : Z → Z′ of complex manifolds which is Gequivariant, is compatible with real structures and maps twistor lines to twistor lines.

By the (equivariant) inverse twistor correspondence this implies the following

Corollary 8.2. The original self-dual manifolds (M, [g]) and (M, [g′]) are K-equivariantly isomorphic.

We first construct a global isomorphism between the two fiber spaces f : Ẑ → P and f′ : Ẑ′ → P forgetting the additonal structures for a while.

(8.2) As a preliminary result, we first deduce a consequence of Proposition 7.12 on the (global) automorphism group of Ẑ over P. Let E = μ^{−1}(C) ≅ C × P and E_{i}, 1 ≤ i ≤ 2k, its irreducible components as in Proposition 6.5.

Lemma 8.3. Let Aut_{P} (Ẑ, {E_{i}}) be the group of relative automorphisms of Ẑ over P which preserve any irreducible components E_{i} of E. Then the embedding G ↪ Aut_{P} (Ẑ, {E_{i}}) induced by the natural Gaction on Ẑ is an isomorphism. Moreover, the same conclusion is also true for any fiber space f_{1} : (Ẑ_{1}, {E′_{1i}}) → P_{1} obtained by pulling back f by a finite covering P_{1} → P which is unramified over all a_{i}, where P_{1} is any compact curve.

Proof. Let g ∈ Aut_{P} (Ẑ, {E_{i}}) be any element. Let P_{0} := P −{a_{i}, 1 ≤ i ≤ k}. By Lemma 2.1 for any a ∈ P_{0} there exists a unique element h(a) of G depending holomorphically on a such that g|Ẑ_{a} is given by the action of h(a). Moreover, by Proposition 7.12 the resulting holomorphic map h : P_{0} → G extends to a holomorphic map h̃ : P → G such that g|Y_{a} coincides with the action of the element h̃(a) for all a ∈ P. Then h̃ must be a constant map since G is Stein. This implies that g is an element of G. The proof is completely the same for f_{1}. q.e.d.

(8.3) In what follows for an auxiliary purpose we fix general twistor lines L and L′, one for each Ẑ and Ẑ′, with the same branch points a and \overline{a} := σ(a) on P (cf. (6.10)).

Take and fix a ramified double covering u : P̃ → P ramified precisely over a and \overline{a}. We set R̃ := u^{−1}(R), which is connected and is an unramified double covering of R. We write

u^{−1}(a_{i}) = {b_{i}, b_{i+k}},

where b_{1}, …, b_{2k} are arranged cyclically in this order on the circle R̃.

We consider the pull-back f̃: Z̃ → P̃ of f by u, i.e., Z̃ = Ẑ ×_{P} P̃ with f̃ the natural projection. Z̃ inherits a natural G-action over P̃ from Ẑ. Denote the inverse image of E in Z̃ by Ẽ(≅ C × P̃). Since L is a double coverings of P branched exactly over a and \overline{a}, the inverse image of L in Z̃ decomposes into two irreducible components which determine sections of f̃. We then choose and fix one of its irreducible components L̃. The branch locus of the double covering map p : Z̃ → Ẑ is precisely the two nonsingular fibers Ẑ_{a} := f^{−1}(a) and Ẑ\overline{a} := f^{−1}(\overline{a}). The morphism f̃ has singular fibers precisely over b_{i}. L̃ intersects neither with Ẽ nor with the inverse images of special twistor lines.

We perform the same construction to f′ : Ẑ′ → P and denote the corresponding objects by putting ′ on the same letter; for instance its pullback to P̃ is denoted by f̃′ : Z̃′ → P̃. By our assumption the general fibers of f and f′ are isomorphic to the same toric surface S(ρ). So the irreducible components of Ẽ and Ẽ′ may be numbered in such a way that Ẽ_{i} and Ẽ′_{i} correspond with respect to a fixed isomorphism of the general fibers. This in particular implies that the one parameter groups corresponding to Ẽ_{i} and Ẽ′_{i} are the same and is equal to ρ_{i}. We then show the following:

Lemma 8.4. There exists an isomorphism j̃: Z̃ → Z̃′ over P̃ which sends Ẽ_{i} to Ẽ′_{i}, 1 ≤ i ≤ 2k, and L̃ to L̃′.

Proof. Let I = IsomP̃((Z̃, {Ẽ_{i}}, L̃), (Z̃′, {Ẽ′_{i}}, L̃′)) be the analytic space over P̃ whose fiber I_{q}, q ∈ P̃, parametrizes all the isomorphisms of (Z̃_{q}, {Ẽ_{iq}}, L̃_{q}) and (Z̃′_{q}, {Ẽ′_{iq}}, L̃′_{q})), namely the isomorphisms of Z̃_{q} to Z̃′_{q} which sends each Ẽ_{iq} to Ẽ′_{iq} and L̃_{q} to L̃′_{q}, where L̃_{q} and L̃′_{q} consist of single points. See [26] and [9, 3.1]. We have natural identifications (Z̃_{q}, {Ẽ_{iq}}) = (S_{u(q}), {C_{iu(q)}}) and (Z̃′_{q}, {Ẽ′_{iq}}) = (S′_{u(q)}, {C′_{iu(q)}}).

Since we know that each smooth fiber of f and f′ are isomorphic as a toric surface, the set of such isomorphisms Isom((Z̃_{q}, {Ẽ}_{iq}), (Z̃′_{q}, {Ẽ}′_{iq})) is non-empty, and then by using the torus action we conclude that the fiber I_{q} = Isom((Z̃_{q}, {Ẽ}_{iq}, L̃_{q}), (Z̃′_{q}, {Ẽ}′_{iq}, L̃′_{q})) is non-empty either; in fact the latter consists of a single element in view of Lemma 2.1.

Thus the natural projection i : I → P̃ is an isomorphism over P̃_{0} := P̃ − {b_{i}} and hence the closure I′ of i^{−1}(P̃_{0}) in I is an irreducible component of I. On the other hand, at any point of P̃ we have by Proposition 7.11 a local section to I → P̃. This implies that I′ is mapped surjectively, and hence isomorphically, onto P̃. Thus we have obtained a section of I → P̃, which in turn gives rise to a desired isomorphism of two fiber spaces. q.e.d.

(8.4) We want the P̃-isomorphism j̃ : Z̃ → Z̃′ obtained in Lemma 8.4 to descend to a P -isomorphism between original fiber spaces f : Ẑ → P and f′ : Ẑ′ → P. For this it suffices to show that j is compatible with the covering transformations κ and κ′ of the double coverings Z̃ → Ẑ and Z̃′ → Ẑ′ respectively;

j̃κ = κ′j̃. (68)

For this purpose we first prove the following:

Lemma 8.5. The covering transformation κ is the unique holomorphic involution of Z̃ which covers the covering transformation of P̃ → P and which induces the identity on the two fibers Z̃_{b} and Z̃\overline{b}, where b = u^{−1}(a) and \overline{b} = u^{−1}(\overline{a}).

Proof. Suppose that there exist two such involutions, say κ and κ_{1}. Then g := κ^{−1}κ_{1} gives a relative automorphism of Z̃ over T which induces the identity on Z̃_{b} and Z̃\overline{b}. In particular g preserves each irreducible component of Ẽ. Then by Lemma 8.3 g is induced by the action of an element of G, which will be denoted by the same letter g. Then g must be the identity since g|Z̃_{b} is the identity; therefore κ = κ_{1} as desired. q.e.d.

From this we get the desired uniqueness immediately.

Lemma 8.6. The isomorphism j̃ : Z̃ → Z̃′ obtained in Lemma 8.4 descends to an isomorphism ĵ : Ẑ → Ẑ′ over P which sends each E_{i} to E′_{i}, where E_{i} and E′_{i} are the images of Ẽ_{i} and Ẽ′_{i} respectively.

Proof. The composite map j̃^{−1}κ′j̃ is a holomorphic involution on Z̃ with all the properties of Lemma 8.5. Hence, by Lemma 8.5 we have j̃^{−1}κ′j̃ = κ and (68) follows. The rest of the assertion is clear. q.e.d.

(8.5) So far we have shown that the two fiber spaces f and f′ are isomorphic as complex spaces over P. Next we show that such an isomorphism can be taken to commute with real structures. First we show the following:

Lemma 8.7. Let σ and σ′ be anti-holomorphic and fixed point free involutions of Ẑ which send the divisors E_{j} to E_{j+k} for every j, 1 ≤ j ≤ k, and which induce the given real structure on P. Then there exists an automorphism d of Ẑ such that σ′ = dσd^{−1}.

Proof. As usual if we set v = σ′σ^{−1}, v defines a holomorphic and relative automorphism of Ẑ over P. By our assumptions v preserves each irreducible component of E. Hence by Lemma 8.3 we conclude that v is induced by the action of some element g of G. Thus we may write σ′ = σg, g ∈ G.

From the fact that σg is an involution, by the same argument as in Lemma 2.7, after conjugation by a suitable element of G we may assume that g is either an element of order two of K or the unit element. We show, however, that if g ≠ e and k > 2, σg necessarily has a fixed point on Z and therefore is inappropriate.

Indeed, since g is an involution which is homotopic to the identity, and M is topologically mP^{2} with m > 0 by our assumption, it has always a two dimensional component in its fixed point set on M (cf. the proof of [15, Prop. 1]), which then must be one of B_{i}. Take a point x of B′_{i}. Then g has two fixed points on the twistor line L_{x}; take an isomorphism L_{x} → P which sends these two points to 0 and ∞. Take an affine coordinate z of P such that the action of g takes the form z → −z. Then the action on σ on L_{x} takes the form z → −a/\overline{z} with a > 0. It follows that σg admits a fixed point set |z| = \sqrt{a} on L_{x} ⊆ Z.

When k = 2, there exists a unique element g of order two with only isolated fixed points. However, in this case Z is the complex projective space and a direct computation shows easily that σg is conjugate to σ by some element of Aut Ẑ. The detail is left to the reader. q.e.d.

Lemma 8.8. The isomorphism ĵ : Ẑ → Ẑ′ over P obtained in Lemma 8.6 can be taken so that it is compatible with the real structures σ̂ on Ẑ and σ̂′ on Ẑ′.

Proof. We set σ̂_{1} = ĵ^{−1}σ̂ĵ. Then both σ̂ and σ̂_{1} are anti-holomorphic involutions satisfying the condition of Lemma 8.7. Hence by that lemma there exists an element d ∈ G such that σ̂ = dσ̂_{1}d^{−1}. Then if we replace the original ĵ by ĵd^{−1}, we get the desired compatibility. q.e.d.

(8.6) Next we verify the G-equivariance of the isomorphism.

Lemma 8.9. The isomorphism ĵ : Ẑ → Ẑ′ above is G-equivariant.

Proof. By Lemma 8.3 we have natural identifications

G = Aut(Ẑ, {E_{i}})

and

G = Aut(Ẑ′, {E′_{i}}).

On the other hand, the map

g → ĵgĵ^{−1}, g ∈ Aut(Ẑ, {E_{i}}),

induces a group isomorphism of Aut(Ẑ, {E_{i}}) and Aut(Ẑ′, {E′_{i}}), which in view of the above identification is considered as an automorphism ζ : G → G. We have to show that this is the identity; in fact since ĵ maps the irreducible components E_{i} to E′_{i}, the induced automorphism of the group N of one parameter groups reduces to the identity. Thus ζ itself must be the identity, i.e., ĵ is G-equivariant. q.e.d.

(8.7) By the above lemmas we have obtained a G-equivariant isomorphism ĵ : Ẑ → Ẑ′ over P which is compatible with the real structures of both spaces. Since it maps E onto E′, it descends to an isomorphism j : Z → Z′ of the twistor spaces which is compatible with the real structures of both spaces. Finally the following lemma finishes our proof of Theorem 8.1.

Lemma 8.10. The isomorphism j : Z → Z′ above sends any twistor line of Z to a twistor line of Z′.

Proof. Let D be the open subset of the Douady space of Z [5] consisting of points whose corresponding subspaces are nonsingular rational curves with normal bundle isomorphic to O(1) ⊕ O(1). Then the real structure σ of Z acts on D as an anti-holomorphic involution and M is considered as a connected component of the fixed point set of this action. Similar statement holds true also for Z′ and M′ and the corresponding objects will be denoted with the superscript ′.

Since j is (σ, σ′)-equivariant, the induced map δ : D → D′ sends the fixed point set of σ to the fixed point set of σ′. Now ĵ sends the fiber Ẑ_{a_{i}} isomorphically onto Ẑ′_{a_{i}} and hence the corresponding singular locus L̂_{i} to L̂′_{i}, which are the proper transforms of the twistor lines L_{i} and L′_{i} on Z and Z′ respectively. This implies that j sends the twistor line L_{i} onto L′_{i}, and hence δ(d_{i}) = d′_{i}, where d_{i} ∈ D and d′_{i} ∈ D′ are the points corresponding to L_{i} and L′_{i} respectively. By what we have remarked above, this already shows that σ maps M diffeomorphically onto M′. Namely, the twistor lines are mapped to twistor lines by j. q.e.d.

(8.8) For the reference in the next section we study further the structure of the real part of the fiber space f : Ẑ → P, i.e., we consider the structure of the induced fiber space f_{R} : Ẑ_{R} → R, where Ẑ_{R} = f^{−1}(R). Especially we are interested in the structure of

Q := (tμ)^{−1}(B) ∩ Ẑ_{R} (69)

and its complement in Ẑ_{R}

A := Ẑ_{R} − Q.

Q is the union of the proper transforms in Ẑ of all the special twistor lines on Z. Thus for any a ∈ R the fiber Q_{a} is the union of twistor lines contained in Ẑ_{a} = S_{a}. K acts smoothly on Ẑ_{R} over R and the action is free on A.

Lemma 8.11. Each fiber A_{a}, a ∈ R, of A has exactly two connected components.

Proof. 1) When a ≠ a_{i} and a is on the arc a_{i}a_{i+1}, consider the σ-equivariant morphism ν_{i} : S_{a} → P_{i} in Lemma 2.6 and write P_{i} − R_{i} = D_{i}^{+} ∐ D_{i}^{−} as two disjoint open discs D_{i}^{±} where R_{i} is the real part of P_{i}. Then

Q_{a} = S_{a} ∩ t^{−1}(B) = ν_{i}^{−1}(R_{i})

and

A_{a} = S_{a} − ν_{i}^{−1}(R_{i}) = ν_{i}^{−1}(D_{i}^{+}) ∐ ν_{i}^{−1}(D_{i}^{−});

thus A_{a} has two connected components.

2) When a = a_{i}, S_{a} = S_{i} and we have S_{i} ∩ t^{−1}(B) = L_{i}. Thus A_{a} = S_{i} − L_{i} = (S_{i}^{+} − L_{i}) ∪ (S_{i}^{−} − L_{i}) and A_{a} has two connected components. q.e.d.

Nevertheless we have the following:

Lemma 8.12. A is connected.

Proof. We consider the quotients \overline{Z}_{R} := Ẑ_{R}/K, \overline{Q} := Q/K and \overline{A} := A/K over R. We have

\overline{A} = \overline{Z}_{R} − \overline{Q}.

For each a ∈ R we set \overline{S}_{a} := \overline{Z}_{Ra} = S_{a}/K. For a ≠ a_{i} it is a 2k-gon, i.e., a polygon with 2k-edges (together with its interior) considered as a 2-manifold with corners (cf. [17, 1.3]). For a = a_{i} it is a union of two (k + 1)-gons with a unique common edge l_{i} which is the image of the very special twistor line L_{i}. We may “flatten” this union topologically along the edge l_{i} and consider it as a 2k-gon in this case too, regarding l_{i} as a special diagonal of it.

The boundary ∂\overline{S}_{a} of \overline{S}_{a} is naturally identified with the quotient \overline{C} := C/K and the union ⋃_{a} ∂\overline{S}_{a} is a trivial fiber bundle over R with fiber \overline{C}. Thus \overline{C}_{i} := C_{i}/K, 1 ≤ i ≤ 2k, form boundary edges of each 2k-gon \overline{S}_{a} and the images \overline{z}_{i} of z_{i}(= C_{i} ∩ C_{i+1}), 1 ≤ i ≤ 2k, form the vertices of the 2k-gon. For a = a_{i} the mentioned diagonal l_{i} connects exactly the vertices \overline{z}_{i} and \overline{z}_{i+k}. Also for a ≠ a_{i}, real twistor lines in Q_{a} are K-equivalent to each other and the quotient Q_{a}/K gives again a kind of diagonal l_{a} connecting some points \overline{z}_{ia} and \overline{z}_{i+k,a} on the edges \overline{C}_{i} and \overline{C}_{i+k} other than the vertices.

Thus, topologically we may consider \overline{Z}_{R} → R as the trivial bundle R × D → R where D is a closed two-disc, and \overline{Q} is then the union of diagonals ⋃_{a∈R} l_{a}, and form a bundle over R = S^{1} with closed interval as a fiber. Since its boundary \overline{C} is connected, \overline{Q} is homeomorphic to a Möbius band; indeed, when a moves from a_{i} to a_{i+1} on the edge a_{i}a_{i+1}, a diagonal starts from l_{i} and by way of l_{a} reaches to l_{i+1}, while both edges of the diagonal which are the points on \overline{C}_{i} (resp. \overline{C}_{i+k}) goes from \overline{z}_{i} (resp. \overline{z}_{i+k}) to \overline{z}_{i+1} (resp. \overline{z}_{i+k+1}) via \overline{z}_{ia} (resp. \overline{z}_{i+k,a}) for 1 ≤ i ≤ k, with the indices i considered cyclically modulo 2k. Therefore the complement \overline{A} of \overline{Q} in \overline{Z}_{R} is connected, and hence its inverse image A also is connected. q.e.d.

(8.9) Let L be a general twistor line, identified with its inverse image in Ẑ. By Proposition 6.15 the induced map f|L : L → P is a double covering branched exactly over σ-conjugate points, say a and \overline{a} ∈ P − R. We note that L ∩ Ẑ_{R} ⊆ A; otherwise it would intersect with some other twistor line in view of the definition of A.

As in (8.3) take a double covering u : P̃ → P branced exactly over the two points a and \overline{a} and let b_{1}, …, b_{2k} be points of R̃ := u^{−1}(R) which are the inverse images of a_{i}. We set

Ẑ_{0} = Ẑ − (E ⋃(∪L̂_{i}))

where L̂_{i} is the proper transform of L_{i} in Ẑ. Ẑ_{0} is a G-invariant set on which G acts freely. Then Z̃_{0} := Ẑ_{0} ×_{P} P̃ is a G-invariant Zariski open subset of the total space of the induced fiber space Z̃ := Ẑ ×_{P} P̃ → P̃ with the induced G-action. The inverse image of L in Z̃ decomposes into two irreducible components L^{±}. The following lemma is used in the proof of the injectivity of a certain map in the next section.

Lemma 8.13. There exist G-invariant Zariski open subsets V^{±} of Z̃ contained in Z̃_{0} such that Z̃_{0} = V^{+} ∪ V^{−}, V^{±} ≅ P̃ × G and L^{±} ⊆ V^{±} by a suitable choice of the signs ±.

Proof. By Lemma 8.12 A is connected, while its inverse image Ã:= A ×_{R} R̃ in Z̃ admits exactly two connected components Ã^{±}. In fact, the pull-back Q̃ := \overline{Q} ×_{R} R̃ → R̃ of the Möbius band \overline{Q} → R becomes a trivial bundle over R̃. Correspondingly Ã/K = \overline{A}×_{R} R̃ gets exactly two components with common boundary Q̃. This implies that Ã also has two connected components. Ã is contained in Z̃_{0} and Ã_{b_{i}}^{±} := Ã^{±} ∩ S̃_{b_{i}} is Zariski open in S̃_{b_{i}} := f̃^{−1}(b_{i}). We now define

V^{±} := Z̃_{0} − ⋃_{1≤i≤2k} Ã_{b_{i}}^{∓}

so that V^{±} ∩ Ã_{b_{i}} = Ã_{b_{i}}^{±}. V^{±} are clearly G-invariant Zariski open subsets of Z̃ contained in Z̃_{0} with Z̃_{0} = V^{+} ∪ V^{−}.

Since L± ; R̃ := L^{±} ∩ f̃^{−1}(R̃) is connected and contained in Ã, after changing the signs if necessary, we may assume that L± ; R̃ ⊆ Ã^{±}. Then we have L^{±} ⊆ V^{±}.

Finally the map L^{±} × G → V^{±} induced by the G-action is isomorphic since the G-action on Z̃_{0} is free, G-orbits on V^{±} are precisely the fibers of V^{±} → P̃, and that the intersection of L^{±} with each G-orbit is transversal at exactly one point. q.e.d.

9. Coincidence of invariants

(9.1) Let m be a non-negative integer and M = mP^{2} with an effective and smooth K-action. Suppose that we are given a K-invariant self-dual conformal structure [g] on M. Then the quotient N = M/K is a closed two-disc and we have a cyclic sequence (y_{1}, …, y_{k}) of points on the boundary bN which are the images of fixed points of the K-action on M. On the other hand, let Z be the twistor space associated to (M, [g]) and f : Ẑ → P the associated fiber space, where P is endowed with the affine coordinate z (53) with σ given by σ(z) = 1/\overline{z}. Let a_{1}, …, a_{k} be the points of the fixed point set R of σ corresponding to the singular fibers of f such that y_{i} is mapped to a_{i} by the natural homeomorphism ϕ : bN → R obtained in Proposition 6.16. We set

D := {z ∈ P; |z| ≤ 1}

so that R is the boundary of D. In this section we prove that ϕ extends naturally to a diffeomorphism ψ : N → D which preserves the geometric structures on the interiors D_{0} and N_{0} respectively: namely the induced complex structure on D_{0} and the induced conformal structure of N_{0} (cf. (3.6)). Recall that in real dimension two, giving a conformal structure is equivalent to giving a complex structure modulo the complex conjugate. The precise statement of our theorem is as follows.

Theorem 9.1. There exists a natural homeomorphism ψ : N → D with the following properties:

1) ψ restricts to ϕ on the boundary bN; in particular ψ sends y_{i} to a_{i} for each 1 ≤ i ≤ k, and

2) ψ induces a diffeomorphism N_{0} → D_{0} of the interiors, with respect to which the conformal structure on N_{0} and the holomorphic structure of D_{0} modulo its complex conjugate are compatible, i.e., ψ sends the conformal structure of N_{0} to one on D_{0} which coincides with that defined by the complex structure of D.

One may think of D_{0} as a kind of (mini-mini)-twistor space of N_{0}. We first deduce Theorem 1.1 from Theorem 9.1.

Proof of Theorem 1.1. By Corollary 6.8 we may assume that M = mP^{2}.

Let J be the subset of C of (6.12) consisting of the isomorphism classes of Joyce’s self-dual manifolds [11]. We have a diagram

\begin{matrix}J & →^{I_{c}} & A \\ ∩ & & ∥ \\ C & →^{I_{t}} & A \end{matrix}

where I_{c} and I_{t} are the maps given by the conformal invariants and the twistorial invariants respectively (cf. (6.12)). This diagram is commutative by Theorem 9.1 and I_{t} is injective by Theorem 8.1. Thus I_{c} on the top line is also injective and we have J = C. This is precisely the assertion of Theorem 1.1. q.e.d.

Proof of Corollary 1.2. Since M admits a K-action, χ(M) is nonnegative by Lemma 3.1. The case χ(M) > 0 is treated in Theorem 1.1. If χ(M) = 0, the K-action has no fixed points. Thus by [22, Th. 3.6] M must be conformally flat. q.e.d.

The rest of this section is devoted to the proof of Theorem 9.1.

(9.2) First we give the definition of ψ(y), y ∈ N, set-theoretically. When y ∈ bN, we just set ψ(y) = ϕ(y) with ϕ in (6.11). In case y ∈ N_{0}, we define ψ(y) ∈ D_{0} as follows. Let x ∈ M − B be an arbitrary point with π(x) = y and L = L_{x} the corresponding general twistor line. By Proposition 6.15 precisely one of the branch points of the double covering f|L : L → P is in D_{0}, which we shall denote by a(x). Since S_{a} are K-invariant, we have a(x) = a(t(x)), and hence, we may write a(x) = a(y). Thus we can define

ψ(y) = a(x).

We show the following:

Lemma 9.2. ψ|N_{0} : N_{0} → D_{0} is bijective.

Together with Proposition 6.16 this would imply

Corollary 9.3. ψ : N → D is bijective.

We need the following lemma for the injectivity.

Lemma 9.4. Let L be a general twistor line. If g(L) is again a twistor line for some g ∈ G, then g ∈ K.

Proof. If g(L) is a twistor line, we have (σgσ)L = g(L). On the other hand, since σgσ^{−1} = g^{∗} (cf. (2.9)) we get g^{−1}g^{∗}(L) = L. Namely, |g|^{2} := g^{−1}g^{∗} ∈ G stabilizes the point d of the Douady space D_{Z} of Z corresponding to the submanifold L; d = t(L) ∈ M_{0} ⊆ D_{Z}. Since Z is Moishezon and L is general, the stabilizer group of d is finite, and therefore it is contained in K; |g|^{2} ∈ K. Since the K-action on M_{0} is free, this implies that |g|^{2} = 1, i.e., g ∈ K. q.e.d.

Proof of Lemma 9.2. Surjectivity: Let a be any point of D_{0} and consider the corresponding non-real fiber S = S_{a} of f : Ẑ → P. By Lemma 6.7 t induces a double covering s : S → M, which cannot be unramified since M is simply connected. If z ∈ S is a ramification point, then the twistor line L_{x}, x = t(z), is tangent to S at z. Thus a = a(x) = ψ(π(x)).

Injectivity. Suppose that a = ψ(y) = ψ(y′) in D_{0} for y and y′ in N_{0}. Take any point x and x′ in M_{0} over y and y′ respectively. Then the twistor lines L_{x} and L_{x′} are both tangent to the fibers S_{a} and S\overline{a}. Take a double covering P̃ → P branced exactly at a and \overline{a}, and consider the pull-back f̃: Z̃ → P̃ of f as in (8.9). Then the inverse images of L_{x} and L_{x′} in Z̃ are decomposed into two irreducible components L_{x}^{±} and L_{x′}^{±} respectively and define sections of f̃. L_{x}^{+} and L_{x′}^{+} are entirely contained in the Zariski open subset V^{+} ≅ G × P̃ by Lemma 8.13. Since any compact curves in V^{+} is of the form g × P̃ for some g ∈ G, we see that L_{x}^{+} is a G-translate of L_{x′}^{+}, and then as their G-equivariant images L_{x} is a G-translate of L_{x′}. Thus by Lemma 9.4 L_{x} further is a K-translate of L_{x′}, which implies that y = y′. q.e.d.

(9.3) We next check the smoothness of ψ on N_{0}. Let M_{0} = M − B and Z_{0} = t^{−1}(M_{0}). We consider the subset

Γ := {z ∈ Z_{0}; L_{x} is tangent to S_{a}, x = t(z), a = a(x) = f(z)}. (70)

Lemma 9.5. Γ is a closed submanifold of real codimension two in Z_{0} and consists of two connected components Γ^{+} and Γ^{−} which are mapped diffeomorphically onto M_{0} by t and which are mutually σconjugate.

Proof. Take any point z_{0} ∈ Z_{0}. Since t is a submersion, we can find real local coordinates x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2} of Z_{0} at z_{0} such that t is described by the projection to the first four coordinates and that z := y_{1} + \sqrt{−1}y_{2} gives a holomorphic local coordinate on L_{x} for any x in a neighborhood of t(z_{0}). Then with respect to a suitable complex coordinate of P around f(z_{0}), f is locally considered as a complexvalued smooth function which is holomorphic when restricted to each L_{x}. Thus a point z in a neighborhood of z_{0} belongs to Γ if and only if (∂f/∂z)(z) = 0. Namely, g := ∂f/∂z = 0 is a defining equation of Γ in a neighborhood of z_{0}. In particular, Γ is closed in Z_{0}.

Since the ramification indices at branch points of L_{x} → P are always two, at any point z of Γ we have ∂g/∂z ≠ 0. Thus the implicit function theorem shows that a neighborhood of z of Γ is mapped diffeomorphically onto a neighborhood of t(z) in M. This shows that Γ is a submanifold of real codimension two in Z_{0} and t|Γ : Γ → M is locally diffeomorphic. Since each general twistor line has exactly two ramification points, we conclude that t|Γ is an unramified double covering.

Let F := μ(f^{−1}(R)). Then F is a closed subset of Z, and Z − F has two connected components

Z^{+} := μ(f^{−1}(D_{0}) − E)

and

Z^{−} := μ(f^{−1}(σ(D_{0})) − E);

namely Z is a disjoint union of F, Z^{+} and Z^{−} with Z^{+} and Z^{−} σconjugate to each other. By Proposition 6.15 Γ never intersects with F so that Γ is a disjoint union of Γ^{+} := Γ ∩ Z^{+} and Γ^{−} = Γ ∩ Z^{−} with σ(Γ^{±}) = Γ^{∓}. Since for each general twistor line the corresponding two ramification points are σ-conjugate to each other, each of Γ^{±} is mapped surjectively onto M_{0}. Thus t|Γ^{±} is a diffeomorphism onto M_{0}. q.e.d.

Corollary 9.6. The restricted map ψ|N_{0} : N_{0} → D_{0} is smooth.

Proof. The map πt|Γ^{+} : Γ^{+} → N_{0} is a submersion by Lemma 9.5; thus it admits a smooth section N_{0} → Γ^{+}. Composing this with the smooth map f|Γ^{+} : Γ^{+} → D_{0} we obtain ψ, which is smooth. q.e.d.

(9.4) Next we show the following:

Proposition 9.7. ψ : N → D is a homeomorphism.

Since N and D are compact and Hausdorff and ψ is bijective by Corollary 9.3, Proposition 9.7 follows from the next lemma.

Lemma 9.8. The map ψ : N → D is continuous.

Proof. We use Lemma 6.17. We set X = N and Y = D. Both are compact. Define

W := Q ∪ Γ^{+} ⊆ Ẑ

where Q is defined by (69) and Γ^{+} is a closed submanifold of Z_{0} in Lemma 9.5 and is identified with a subset of Ẑ via μ^{−1}. We may also consider Q as the proper transform of (πt)^{−1}(bN) = t^{−1}(B) in Ẑ, namely, it is the closure of μ^{−1}(t^{−1}(B) − C) in Ẑ. By the definition Q is clearly compact.

Putting on W the relative topology from Ẑ we have the continuous maps h := πtμ|W : W → X and g := f|W : W → Y with h surjective; moreover by our definition of the map ψ, it is clear that h and g satisfy the condition ψ(x) = g(h^{−1}(x)), x ∈ X. Thus, in view of Lemma 6.17, we have only to show that W is closed in Ẑ, and hence, is compact.

Let Γ̂^{+} be the closures of Γ^{+} in Ẑ. Clearly, Γ̂^{+} ∪ Q is closed. It suffices to show that Γ̂^{+} ∪ Q = W, which in turn would follow if we show that Γ̂^{+} − Γ^{+} ⊆ Q. Since Γ^{+} is closed in Ẑ − μ^{−1}(t^{−1}(B)), we must have

Γ̂^{+} − Γ^{+} ⊆ μ^{−1}(t^{−1}(B)) = Q ∪ μ^{−1}(C).

Now take an arbitrary point y ∈ Γ̂^{+} − Γ^{+}. Take a sequence of points y_{n} ∈ Γ^{+} converging to y. Then μ(y_{n}) converge to μ(y) =: ỹ in Z and x_{n} := tμ(y_{n}) converge to x := t(ỹ) in M. Thus ỹ ∈ L_{x} and y ∈ μ^{−1}(L_{x}). On the other hand, since y ∉ Γ^{+}, we have x ∈ B and L_{x} is a special twistor line. Thus μ^{−1}(L_{x}) has the form l_{x}^{+} ∪ L̂_{x} ∪ l_{x}^{−}, where L̂_{x} is the proper transform of L_{x} in Ẑ, and l_{x}^{±} = μ^{−1}(z^{±}), where L_{x} ∩ C = {z^{+}, z^{−}}.

We have to show that y ∈ L̂_{x} since L̂_{x} ⊆ Q. So supposing that y ∉ L̂_{x} and hence that y ∈ l_{x}^{+}−L̂_{x} say, we shall derive a contradiction. Since x_{n} converge to x, we have the convergence L_{x_{n}} → L_{x} as subspaces of Z. Thus by taking the inverse image by μ we see that in a neighborhood of y, L_{x_{n}} (identified with its inverse image μ^{−1}(L_{x_{n}}) in Ẑ) converge to l_{x}^{+}, and y_{n} ∈ L_{x_{n}}; in particular we get the convergence of the tangent spaces T_{y_{n}}L_{x_{n}} → T_{y}l_{x}^{+}. But l_{x}^{+} is transversal, at each of its point z, to the fiber f^{−1}(f(z)). Hence, for all sufficiently large n, L_{x_{n}} should also be transversal at y_{n} to the fiber of f passing through this point. This contradicts the fact y_{n} ∈ Γ^{+}. q.e.d.

(9.5) Finally, we study in detail the differential ψ_{∗} of ψ on N_{0}. Especially we show that it is isomorphic everywhere; this would follow from the submersiveness of fμ^{−1}|Γ : Γ → P, in view of the commutative diagram

Γ ⊆ Z ←^{μ} Ẑ
t ∥  t ↓
M_{0} ⊆ M  ↓ f
π ↓  π ↓
N_{0} ⊆ N →^{ψ} P.

For this purpose we pass to the complexfication of the situation. This has the advantage that we can work in a purely complex analytic setting. The desired submersiveness of ψ will be proved in Proposition 9.17 below in this complexfied setting.

We start with the complexfication of M. Let M^{′C} be the irreducible component of the Douady space of Z [5] which contains the points corresponding to twistor lines of Z, and MC the Zariski open subset of M^{′C} corresponding to nonsingular rational curves whose normal bundle in Z is of type O(1) ⊕ O(1). Then MC is a 4-dimensional complex manifold and M, as the parameter space of twistor lines, is considered as a totally real submanifold of MC. Indeed, σ induces an anti-holomorphic involution on MC and M is one of the connected components of its fixed point set. MC also admits a natural G-action. We have the universal family tC : ZC → MC over MC which fits into the commutative diagram

Z ⊆ ZC →^{α} Z
t ↓  ↓ tC
M ⊆ MC

where α : ZC → Z is the natural projection and α|Z : Z → Z is the identity. The G- and σ-actions lift naturally to ZC making α and tC equivariant. ZC is a 5-dimensional complex manifold and tC is a holomorphic submerion.

For each point x ∈ MC the fiber tC^{−1}(x) is identified with the corresponding subspace of Z and will be denoted again by L_{x}. We call such an L_{x} a complex twistor line in general.

We take and fix a G-invariant Zariski open subset MC_{0} of MC such that MC_{0} ∩M = M_{0} and L_{x} ⊆ Z −(C ∪L) for any x ∈ MC_{0} (cf. (5.3)) so that fμ^{−1}|L_{x} : L_{x} → P is holomorphic and is a double covering; e.g., we may take MC_{0} = MC − t(Z^{′C}), where Z^{′C} = {z ∈ ZC; dim G_{z} > 0}. Moreover, we have

Lemma 9.9. The G-action on MC_{0} is free.

Proof. Let x be any point of MC_{0}. Then the stabilizer G_{x} acts on L_{x}, while the G-action on Z is free at each point of L_{x} since L_{x} ⊆ Z−(C∪L). Thus the action of G_{x} is free on L_{x} ≅ P^{1}, which implies that G_{x} reduces to the identity. q.e.d.

(9.6) For any z ∈ Z the fiber F (z) := α^{−1}(z) ⊆ ZC of α is mapped isomorphically by tC onto its image MC(z) := tC(F (z)) in MC, which in turn is identified with the subspace of MC parametrizing complex twistor lines L_{x} with z ∈ L_{x}. Denote by N_{x} the normal bundle of L_{x} in Z.

Lemma 9.10. F (z) and MC(z) are smooth of dimension two. The map α : ZC → Z is a submersion.

Proof. Let N_{x}(−z) denote the sheaf of sections of N_{x} vanishing at z. Since H^{1}(L_{x}, N_{x}(−z)) = 0 for any x ∈ MC(z), by standard deformation theory we have that MC(z) is smooth of dimension two. The assertion on F (z), and hence the submersiveness of α also, follows from this. q.e.d.

The inclusions of the tangent spaces at x ∈ MC(z) are described cohomologically as follows:

\begin{aligned}T_{x}MC(z) & ⊆ & T_{x}MC \\ ∥ & & ∥ \\ H^{0}(L_{x}, N_{x}(−z)) & ⊆ & H^{0}(L_{x}, N_{x}). \end{aligned} (71)

When x ∈ M, T_{x}MC is identified with the complexfication of the (real) tangent space T_{x}M. If, further, x ∈ M_{0}, then since the K-action is free at x, the associated infinitesimal action on M determines a (real) two dimensional subspace E = E_{x} of T_{x}M, whose complexfication EC = E_{Cx} arises from the infinitesimal G-action on MC at x;

E ⊆ EC
∩  ∩
T_{x}M ⊆ T_{x}MC.

Lemma 9.11. For any z ∈ Γ and x ∈ MC(z) we have

dim(T_{x}MC(z) ∩ EC) = 1. (72)

Proof. Since z ∉ k(L_{x}) for any k ∈ K with k ≠ e, we have T_{x}MC(z)∩E = {0}. It thus suffices to show that T_{x}MC(z)∩EC ≠ {0} since for T_{x}MC(z) and EC are of dimension two. By the definition of Γ, L_{x} is tangent to S_{a} at z, where a = f(z). Since S_{a} coincides with the G-orbit of z at z, there exists an element v of the Lie algebra Lie G of G whose associated vector field \underline{v} on Z is tangent to L_{x} at z. Then the section w ∈ H^{0}(L_{x}, N_{x}) induced from \underline{v}|L_{x} via the natural projection T Z|L_{x} → N_{x} vanishes at z. However w is identified, with respect to the identification (71), with the element δ(v) of T_{x}MC where δ denotes the infinitesimal G-action on MC at x;

Lie G  →^{δ}  T_{x}MC
↓  a
H^{0}(L_{x}, T Z|L_{x}) → H^{0}(L_{x}, N_{x}).

Thus if we consider δ(v) ∈ T_{x}MC as an element of H^{0}(L_{x}, N_{x}), it vanishes at z ∈ L_{x}. Then δ(v) is an element of T_{x}MC(z) in view of (71), while it is obviously an element of EC. Note finally that δ(v) is nonzero because the G-action on MC is locally free at x by Lemma 9.9.

q.e.d.

(9.7) Set ZC_{0} = tC^{−1}(MC_{0}). We now define a complex analogue of Γ by

ΓC := {z ∈ ZC_{0}; L_{x} is tangent to S_{a}, where x = tC(z) and a = fα(z)}

which fits into the commutative diagram

Γ ⊆ ΓC ⊆ ZC_{0}
↓ t ↓ tC ↓ tC
M_{0} ⊆ MC_{0} = MC_{0}.

Clearly we have Γ = ΓC ∩ Z; moreover, Γ and ΓC are K- and Ginvariant respectively and the induced actions are free by Lemma 9.9; moreover they are σ-invariant. Analogously to Lemma 9.5 we get

Lemma 9.12. ΓC is a σ-invariant closed and smooth divisor in ZC_{0} and is an unramified double covering of MC_{0}.

Proof. The ramifcation locus of v = tC ×(fμ^{−1}α) : ZC_{0} → MC_{0} ×P coincides with ΓC. Thus it is a closed divisor in ZC_{0}. More precisely, for any point z_{0} ∈ ZC_{0} we may take holomorphic local coordinates of MC at x := tC(z_{0}) and (z_{1}, z_{2}, z_{3}, z_{4}, z) of ZC_{0} around z_{0} such that tC is given by tC(z_{1}, z_{2}, z_{3}, z_{4}, z) = (z_{1}, z_{2}, z_{3}, z_{4}). Then z|L_{x} is a local parameter on L_{x} and the Jacobian matrix for v is of the form Jac

v = ( \begin{matrix}I & 0 \\ ∗ & ∂f/∂z \end{matrix} ) ,

where I is the identity matrix of size four and we have written f = fμ^{−1}α for simplicity. Therefore

z ∈ ΓC ⇔ detJac v(z) = 0 ⇔ \frac{∂f}{∂z} (z) = 0

Since the defining function ∂f/∂z of ΓC restricted to L_{x} is locally of the form qz for some function q with q(z_{0}) ≠ 0, ΓC is smooth and its intersection with each fiber of tC is transversal.

The final assertion follows from the fact that L_{x} ⊆ Z − (C ∪ L) and L_{x} → P is a double covering. q.e.d.

(9.8) Let ẼC (resp. Ẽ) be the subspace of the tangent space T_{z}ΓC (resp. of T_{z}Γ) arising from the infinitesimal action of G on ΓC (resp. K on Γ).

Lemma 9.13. For any z ∈ Γ we have ẼC ∩ T_{z}F (z) = {0} and

dim(T_{z}F (z) ∩ T_{z}ΓC) = 1.

Proof. Let v be any element of ẼC ∩ T_{z}F (z). Since v ∈ T_{z}F (z), we have α_{∗}v = 0, while since α is G-equivariant and the G-action is free at z ∈ Z_{0}, the restriction α_{∗}|ẼC must be injective. Thus v = 0 and the first assertion is proved.

Since F (z) is of dimension two and ΓC is of codimension one, we have dim(T_{z}F (z) ∩ T_{z}ΓC) ≥ 1. So assuming that this dimension is equal to two, i.e., that T_{z}F (z) is contained in T_{z}ΓC, we shall derive a contradiction. In fact the natural isomorphism t_{C∗} : T_{z}ΓC ≅ T_{x}MC, x = t(z), induces isomorphisms ẼC ≅ EC and T_{z}F (z) ≅ T_{x}MC(z). Then (72) would imply that ẼC ∩ T_{z}F (z) ≠ {0}. This contradicts the first assertion. q.e.d.

Corollary 9.14. In a neighborhood of Γ the induced map α|ΓC : ΓC → Z is a holomorphic submersion; therefore so is the composite morphism fμ^{−1}α|ΓC : ΓC → P.

Proof. This follows from the above lemma and the fact that the kernel of the differential of α|ΓC is precisely T_{z}F (z) ∩ T_{z}ΓC for z ∈ Γ.

q.e.d.

(9.9) Let x be any point of M_{0}. T_{x}MC admits a holomorphic conformal structure which induces the given conformal structure on T_{x}M. Let E^{⊥} be the orthogonal complement of E in T_{x}M with respect to the given conformal structure. Then its complexfication EC^{⊥} is the orthogonal complement of EC with respect to the holomorphic conformal structure on T_{x}MC. We have the direct sum decomposition

T_{x}MC = EC ⊕ EC^{⊥} (73)

which is defined over the reals.

For the computation below it is convenient to work with a concrete model of the space T_{x}MC = H^{0}(L_{x}, N_{x}) as follows. Fix an oriented (conformal) isometry T_{x}M → H inducing the identification L_{x} = C, where 1, i, j, k is an oriented orthonormal basis of H, such that its complexfication

T_{x}MC → H ⊗ C (74) maps the direct sum decomposition (73) into the natural decomposition

H ⊗ C = (C ⊕ Ci) ⊕ (Cj ⊕ Ck), (75)

where one should not confuse i ∈ H with \sqrt{−1} ∈ C; in particular 1 and i are linearly independent over C.

Correspondingly, we take a (real) model of the normal bundle N = N_{x} as C × H as in (28). In this identification the fiber N_{q} over q ∈ C of N → C is identified with the complex vector space H with complex structure given by the right multiplication by q^{−1}. Thus, when one considers an element w of H ⊗ C as a section of N_{x} with respect to the identifications (71) and (74), at any point q ∈ C = L_{x} the value w(q) ∈ N_{q} ≅ H is obtained by replacing any \sqrt{−1} in the expression of w via (75) by q^{−1} = −q. Now we have the following:

Lemma 9.15. Let x ∈ M_{0} be as above. Then we have Γ∩L_{x} = {±i} with respect to our identifications L_{x} = C above.

Proof. Consider an element of EC as a holomorphic section of N_{x} via the isomorphisms (73) and (71). Let z ∈ Γ ∩ L_{x}. By Lemma 9.11 and (71) there exists a nonzero element β ∈ EC which vanishes at z. Let z correspond to q ∈ C with respect to our identification L_{x} = C. We may write β = u + \sqrt{−1}v for some u, v ∈ R + Ri. Now for a point q ∈ C, β vanishes at q as a section of N_{x} if and only if u − vq = 0, or equivalently, u − vsi − jvt = 0 if q = si + jt, s ∈ R, t ∈ C, namely, vt = 0 and u = vsi. This implies that t = 0; thus z = q = ±i. From this follows the assertion since Γ ∩ L_{x} consists of two points. q.e.d.

The method of the above proof also yields the following:

Lemma 9.16. Let x ∈ M_{0} and z ∈ Γ ∩ L_{x}. Then we have

dim(T_{x}MC(z) ∩ EC^{⊥}) = 1.

Proof. The element ±\sqrt{−1}j + k ∈ Cj + Ck = EC^{⊥} vanishes at ±i ∈ C when considered as a section of N_{x}. Thus by Lemma 9.15 either of ±\sqrt{−1}j + k belongs to T_{x}MC(z), and hence, T_{x}MC(z) ∩ EC^{⊥} ≠ ∅.

On the other hand, dim T_{x}MC ∩ EC^{⊥} ≤ 1 by Lemma 9.11. The lemma follows. q.e.d.

Remark. The method of the proof of Lemma 9.15 also shows the following: Let J be the complex structure of T_{x}M corresponding to z ∈ L_{x}. Let T_{z}^{±}MC be the ±\sqrt{−1}-eigenspaces of J on T_{x}MC. Then we have the natural identifications T^{+}MC = T_{x}MC(\overline{z}) and T^{−}MC = T_{x}M(z). Thus we have the identical direct sum decompositions

T_{x}MC = T_{x}MC(z) ⊕ T_{x}MC(\overline{z}) = T_{z}^{−}MC ⊕ T_{z}^{+}MC.

This in turn yields the following identification of the normal bundle N_{x}: (N_{x})_{z} = T_{x}M(\overline{z}) for each z ∈ L_{x}.

(9.10) Using Lemma 9.15 we now prove the anticipated result:

Proposition 9.17. For any point z of Γ the composite map fμ^{−1}|Γ : Γ → P is submersive at z. In fact, the differential (fμ^{−1})_{∗} induces a real isomorphism of Ẽ^{⊥} onto T R_{a}P, where Ẽ^{⊥} = (t|Γ)_{∗}^{−1}(E^{⊥}), a = fμ^{−1}(z) and T R_{a}P denotes the real tangent space of P at a.

Proof. Note first the following facts: α_{∗} : T_{z}Γ → T_{z}Z is injective, the kernel of (fμ^{−1})_{∗} at z is T_{z}S_{a} and α_{∗} maps ẼC isomorphically onto T_{z}S_{a}; indeed, first two assertions are clear and the last follows from the fact that α is G-equivariant and S_{a} is G-homogeneous locally at z. In particular ẼC is mapped into the kernel of f_{∗}. Thus, in view of fμ^{−1}|Γ = fμ^{−1}α|Γ, our task is to show that

α_{∗}T_{z}Γ ∩ T_{z}S_{a} = α_{∗}Ẽ.

Indeed, by dimensional consideration we then deduce that (fμ^{−1}α)_{∗} induces an isomorphism Ẽ^{⊥} → T R_{a}P.

Suppose now that α_{∗}T_{z}Γ ∩ T_{z}S_{a} is strictly larger than α_{∗}Ẽ, namely that there exists a (real) subspace Ẽ′ of T_{z}Γ of dimension ≥ 3 which contains Ẽ and is mapped into T_{z}S_{a} by α_{∗}. Since α_{∗}(ẼC) = T_{z}S_{a}, any element d of Ẽ′ is then written in T_{z}ΓC in the form

d = b + c, b ∈ ẼC, c ∈ Ker α_{∗} = T_{z}F (z). (76)

Passing to T_{x}MC, with respect to the natural isomorphisms (T_{z}ΓC, T_{z}Γ) ≅ (T_{x}MC, T_{x}M) and T_{z}F (z) ≅ T_{x}MC(z) we may consider (76) as a relation in T_{x}MC with d ∈ E′ := t_{C∗}(Ẽ′), b ∈ EC and c ∈ T_{x}MC(z). Since dim E′ ≥ 3, E′ ∩ E^{⊥} ≠ {0}. We consider (76) in the case 0 ≠ d ∈ E^{⊥}. We shall derive a contradiction from this.

For this purpose we use (74) and (75) to identify d as an element of Rj + Rk and b as an element of C + Ci. We interpret these elements as sections of N_{x} on L_{x} by the isomorphism (71). By Lemma 9.15 we have z = ±i under the identification L_{x} = C there. Then, considering d = b + c as a relation among sections, and evaluating at z = ±i, we get d(z) = b(z) as an element of (N_{x})_{z} = H. This is a contradiction because if b = u + \sqrt{−1}v, where u, v ∈ R + Ri, then b(z) = u − zv = u ∓ iv ∈ C + Ci, while we have 0 ≠ d(z) = d ∈ Rj + Rk by our choice of d. q.e.d.

Corollary 9.18. The differential ψ_{∗} of ψ is isomorphic at every point y of N_{0}. It is identified with the composite of natural isomorphisms

T_{y}N_{0} ≅ E^{⊥} ≅ Ẽ^{⊥} ≅ T R_{a}P.

Proof. For any point y ∈ N_{0}, we may canonically identify the tangent space of N_{0} at y with the orthogonal complement E^{⊥} of E in T_{x}M, where x is an arbitrary point over y. Then ψ_{∗} is identified with the real isomorphism E^{⊥} → T R_{a}P in Proposition 9.17. q.e.d.

(9.11) For the proof of Theorem 9.1 it remains to show that the conformal structure of N_{0} and the complex structure of D_{0} are compatible with respect to the diffeomorphism ψ. For this purpose we need the following two lemmas.

Lemma 9.19. For any z ∈ Γ the restriction to EC^{⊥} of the composite map

(fμ^{−1}α)_{∗}(tC|ΓC)_{∗}^{−1} : T_{x}MC → T_{z}ΓC → T_{a}P, x = t(z), a = fμ^{−1}(z)

is surjective and its kernel is EC^{⊥} ∩ T_{x}MC(z).

Proof. First we consider (fμ^{−1}α)_{∗} : T_{z}ΓC → T_{a}P. Let V be its kernel. By Corollary 9.14 it is three dimensional and contains both ẼC and T_{z}F (z) ∩ T_{z}ΓC. By Lemma 9.13 the latter is one dimensional and they have zero intersection. Hence we have

V = ẼC ⊕ (T_{z}F (z) ∩ T_{z}ΓC).

On the other hand, by the isomorphism t_{C∗} : T_{z}ΓC ≅ T_{x}MC, ẼC is mapped onto EC. The surjectivity of EC^{⊥} → T_{a}P already follows.

If we let I be the image of T_{z}F (z) ∩ T_{z}ΓC in T_{x}MC, the kernel of our composite map is a direct sum EC ⊕ I = EC ⊕ I′, where I′ denotes the orthogonal projection of I onto EC^{⊥} in (73). We know that

I ⊆ T_{x}MC(z), and as follows from Lemmas 9.11 and 9.16 T_{x}MC(z) is compatible with the direct sum decomposition (73); therefore I′ must coincide with EC^{⊥} ∩ T_{x}MC(z). The lemma follows. q.e.d.

Lemma 9.20. Let W be a 2-dimensional real vector space with (positive definite) inner product. Let W^{±} be the (non-trivial) isotropic subspaces of the complexfication W C with the complexified inner product. Then W^{±} are complex conjugate to each other and we have W C = W^{+} ⊕ W^{−}. Moreover, the projections W C → W^{±} induces a real isomorphism W → W^{±}, with respect to which the conformal structure on W and the complex structures on W^{±} are compatible, i.e., the conformal structure on W is hermitian with respect to the complex structures of W^{±}.

Proof. We take an orthonormal basis of W, with respect to which we identify W with R^{2} and W C with C^{2}. The complexified quadratic form takes the form C^{2} ∋ (z, w) → z^{2} + w^{2} and we may identify W^{±} = {z ± iw = 0}. This shows the first assertion.

Moreover, the projection W C → W^{±} restricted to W is identified with the map R^{2} ∋ (x, y) → (x ± iy, ±i(x ± iy)) ∈ W^{±} ⊆ C^{2}. Then the pull-back by this map of the hermitian structure z\overline{z} + w\overline{w} on C^{2} to R^{2} is just 2(x^{2} + y^{2}). Thus the latter is hermitian on W with respect to the complex structure induced from W^{±}. q.e.d.

(9.12) Proof of Theorem 9.1. Fix a Riemannian metric in the given conformal class [g]. Let y be any point of N_{0} and x a point of M_{0} over N_{0}. Let Q be the null-cone of the complex conformal structure of T_{x}MC = H^{0}(L_{x}, N_{x}) consisting of sections vanishing at some point. By Lemma 9.20 applied to W = E^{⊥}, Q ∩ EC^{⊥} is a union of two σ-conjugate

isotropic subspaces. One of this is EC^{⊥} ∩ T_{x}MC(z) by Lemma 9.16 and

the σ-conjugate of this space is EC^{⊥} ∩\overline{T}_{x}MC(z) = EC^{⊥} ∩T_{x}MC(\overline{z}). We consider the following commutative diagram describing the differential ψ_{∗}:

T_{y}N_{0} ≅  E^{⊥}  ≅ T R_{a}P
u ↓  ↓
EC^{⊥}/(T_{x}MC(z) ∩ EC^{⊥}) ≅ T_{a}P,

where the top and bottom horizontal isomorphisms are due to Proposition 9.17 and Lemma 9.19 respectively, while the vertical real isomorphisms are the natural ones. We have to compare the conformal structure on T_{y}N_{0} and the complex structure on T_{a}P and this is reduced to proving the compatibility of the corresponding structures on both hand sides of the map

u : E^{⊥} → EC^{⊥}/(T_{x}MC(z) ∩ EC^{⊥}).

This however follows from Lemma 9.20, where W = E^{⊥}, W = EC^{⊥},

W^{+} = T_{x}MC(z) ∩ EC^{⊥}

and

W^{−} = T_{x}MC(\overline{z}) ∩ EC^{⊥}.

q.e.d.

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Osaka University, Japan