Einstein Metrics and Stability

for Flat Connections

on Compact Hermitian Manifolds,

and a Correspondence

with Higgs Operators in the Surface Case

M. Lübke

Received: February 19, 1999 Revised: October 12, 1999

Communicated by Thomas Peternell

Abstract. A flat complex vector bundle (E, D) on a compact Riemannian manifold (X, g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the Dinvariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E, D) in this sense is equivalent to the existence of a so-called harmonic metric in E. In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X, g). We propose new notions of g-(poly-)stability of such bundles, and of g-Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a Kähler metric, but are different in general. Our main result is that the g-polystability in our sense is equivalent to the existence of a g-Hermitian-Einstein metric. Our notion of a g-Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

1991 Mathematics Subject Classification: 53C07

1 Introduction.

Let X be an n-dimensional compact complex manifold. If X admits a Kähler metric g, then it is known by work of in particular Simpson [S1],[S2],[S3] that there exists an canonical identification of the moduli space of polystable (or semisimple) flat bundles on X with the moduli space of g-polystable Higgsbundles with vanishing Chern classes on X. This identification has been used in showing that certain groups are not fundamental groups of compact Kähler manifolds. The construction uses the existence of canonical metrics, called g-harmonic in the case of flat bundles, and g-Einstein in the case of Higgs bundles. For flat bundles, the equivalence of semisimplicity and the existence of a gharmonic metric holds on compact Riemannian manifolds [C]. Furthermore, the equivalence of g-polystability and the existence of a g-Einstein metrics for Higgs bundles should generalize to the case of Hermitian manifolds as in the case of holomorphic vector bundles, using Gauduchon metrics. Nevertheless, an identification as above cannot be expected for general compact Hermitian manifolds, since it should imply restrictions on the fundamental group, but every finitely presented group is the fundamental group of a 3-dimensional compact complex manifold by a theorem of Taubes [T]. In the case of compact complex surfaces, however, things are different. We show that for an integrable Higgs bundle (E, d′′) with vanishing real Chern numbers and of g-degree 0 with g-Einstein metric h on a compact complex surface X with Hermitian metric g, there is an canonically associated flat connection D in E, again of g-degree 0, such that h is what we call a g-Einstein metric for (E, D), and that the converse is also true. Furthermore, this correspondence preserves isomorphism types and hence descends to a bijection between moduli spaces. The notion of a g-Einstein metric in a flat bundle makes sense in higher dimension, too, is equivalent to g-harmonicity in the case of a Kähler metric, but different in general, and we show that the existence of such a metric in a flat bundle (E, D) is equivalent to the g-polystability of this bundle in the sense that E is the direct sum of D-invariant g-stable flat subbundles. Here we call a flat bundle (E, D) g-stable if every D-invariant subbundle has g-slope larger(!) than the g-slope of (E, D). g-stability of a flat bundle is equivalent to its stability (in the sense of Corlette) in the Kähler case, but a weaker condition in general: A stable bundle is always g-stable, but the tangent bundles of certain Inoue surfaces are examples of g-stable bundles which are not stable. We expect that for a non-Kähler surface with Hermitian metric g, there is a natural bijection between the moduli space of g-polystable Higgs bundles, with vanishing Chern numbers and g-degree, and the moduli space of g-polystable flat bundles with vanishing g-degree. In the last section we consider the special case of line bundles on surfaces. Here the stability is trivial, and the existence of Einstein metrics is easy to show, so we get indeed the expected natural bijection between moduli spaces of line bundles of degree 0. We further show how this can be extended (in a non-natural way) to the moduli spaces of line bundles of arbitrary degree; this extension argument works in fact for bundles of arbitrary rank once the correspondence for degree 0 has been established.

Einstein Metrics and Stability for Flat Connections 489

Acknowledgments. The author wishes to thank A. Teleman for the suggestion to study the relation between flat connections and Higgs operators on Hermitian manifolds, and for several useful hints. Discussions with him and Ch. Okonek have been of considerable help in preparing this paper. The author was supported in part by EC-HCM project AGE ”Algebraic Geometry in Europe”, contract no. ERBCHRXCT 940557.

2 Preliminaries.

Let X be a compact n-dimensional complex manifold, and E ⟶ X a differentiable ℂ^{r}-vector bundle on X. We fix the following Notations: A^{p}(X) (resp. A^{p,q}(X)) is the space of differentiable p-forms (forms of type (p, q)) on X. A^{p}(E), A^{p,q}(E) are the spaces of differential forms with values in E. A(E) is the space of linear connections D in E. For a connection D ∈ A(E) we write D = D′ + D′′, where D′ is of type (1,0) and D′′ of type (0,1). A(E, h) ⊂ A(E) is the subspace of h-unitary connections d in E, where h is a Hermitian metric in E. We write d = ∂ + ¯∂, where ∂ is of type (1,0) and ¯∂ of type (0,1). A_{f}(E) := { D ∈ A(E) | D^{2} = 0 } is the subset of flat connections. Ā(E) is the space of semiconnections ¯∂ of type (0,1) in E (i.e. ¯∂ is the (0,1)- part of some D ∈ A(E) ). H(E) := { ¯∂ ∈ Ā(E) | ¯∂^{2} = 0 } is the subset of integrable semiconnections or holomorphic structures in E. A′′(E) := Ā(E) ⊕ A^{1,0}(EndE) = { d′′ = ¯∂ + θ | ¯∂ ∈ Ā(E), θ ∈ A^{1,0}(EndE) } is the space of Higgs operators in E. H′′(E) := { d′′ ∈ A′′(E) | (d′′)^{2} = 0 } is the subset of integrable Higgs operators. Often the same symbol is used for a connection, semiconnection, Higgs operator etc. in E and the induced operator in EndE. Two connections D_{1}, D_{2} ∈ A(E) are isomorphic, D_{1} ≅ D_{2}, if there exists a differentiable automorphism f of E such that f∘D_{1} = D_{2}∘f, which is equivalent to D(f) = 0, where D is the connection in EndE induced by D_{1} and D_{2}, i.e. D(f) = D_{2} ∘ f − f ∘ D_{1}. In the same way the isomorphy of semiconnections resp. Higgs operators is defined. If a Hermitian metric h in E is given, then a superscript ^{∗} means adjoint with respect to h.

For D = D′ + D′′ there are unique semiconnections δ′_{h},δ′′_{h} of type (1,0), (0,1) respectively such that D′ + δ′′_{h} and δ′_{h} + D′′ are h-unitary connections. Define δ_{h} := δ′_{h} + δ′′_{h}; then d_{h} := \frac{1}{2}(D + δ_{h}) is h-unitary, and Θ_{h} := D − d_{h} = \frac{1}{2}(D − δ_{h}) is a h-selfadjoint 1-form with values in EndE. Let d_{h} = ∂_{h} + ¯∂_{h} be the decomposition in the parts of type (1,0) and (0,1), and let θ_{h} be the (1, 0)-part of Θ_{h}; then it holds

D = d_{h} + Θ_{h} = ∂_{h} + ¯∂_{h} + θ_{h} + θ_{h}^{∗}.

The map

I_{h} : A(E) ⟶ A′′(E), I_{h}(D) := d′′_{h} := ¯∂_{h} + θ_{h} ∈ A′′(E)

is bijective; the inverse is given as follows. For d′′ = ¯∂ + θ ∈ A′′(E) let ∂_{h} be the unique semiconnection of type (1, 0) such that the connection d_{h} := ∂_{h} + ¯∂ is h-unitary, and define Θ := θ + θ^{∗}. Then

I_{h}^{−1}(d′′) = D_{h} := d_{h} + Θ ∈ A(E).

Remark 2.1 i) In general, if D_{1}, D_{2} ∈ A(E) are isomorphic, then I_{h}(D_{1}) and I_{h}(D_{2}) are not isomorphic, and vice versa. ii) D_{h} = d_{h}+θ+θ^{∗} is not h-unitary unless θ = 0, but the connections d_{h}−θ+θ^{∗} and d_{h} + θ − θ^{∗} are. iii) Any metric h′ in E is of the form h′ = f · h , i.e. h′(s, t) = h(f(s), t), where f is a h-selfadjoint and positive definite. For a connection D it is easy to show that the operator δ_{h·f} associated to D and f · h is given by δ_{h·f} = f^{−1} ∘ δ_{h} ∘ f = δ_{h} + f^{−1} ∘ δ_{h}(f), so it holds

d′′_{f·h} = d′′_{h} + \frac{1}{2}f^{−1} ∘ δ′′_{h}(f) − f^{−1} ∘ δ′_{h}(f)
= d′′_{h} + \frac{1}{2}f^{−1} ∘ ¯∂_{h}(f) − \frac{1}{2}f^{−1} ∘ θ_{h}^{∗}(f) − \frac{1}{2}f^{−1} ∘ ∂_{h}(f) + \frac{1}{2}f^{−1} ∘ θ_{h}(f).

Conversely, for a given Higgs operator d′′ one verifies

D_{f·h} = D_{h} + f^{−1} ∘ ∂_{h}(f) + f^{−1} ∘ θ(f).

In particular, if f is constant then the two maps I_{h} and I_{f·h} coincide.

Definition 2.2 i) G_{h} := (d′′_{h})^{2} is called the pseudocurvature of D with respect to h. ii) F_{h} := D_{h}^{2} is called the curvature of d′′ with respect to h.

Remark 2.3 i) Obviously it holds: I_{h}(D) is an integrable Higgs operator if and only if G_{h} = 0, and I_{h}^{−1}(d′′) is a flat connection if and only if F_{h} = 0. ii) For i = 1, 2, let E_{i} be a differentiable complex vector bundle on X with Hermitian metric h_{i} and connection D_{i}. Let h be the induced metric and D the induced connection in Hom(E_{1}, E_{2}). Denote by G_{i,h} resp. G_{h} the pseudocurvature of D_{i} resp. D with respect to h_{i} resp. h. Then for f ∈ A^{0}(Hom(E_{1}, E_{2})) it holds G_{h}(f) = G_{2,h} ∘ f − f ∘ G_{1,h}. Similarly, the curvature F_{h} of the Higgs operator induced in Hom(E_{1}, E_{2}) by Higgs operators d′′_{i} in the E_{i} is given by F_{h}(f) = F_{2,h} ∘ f − f ∘ F_{1,h}. iii) If D is a connection, then D^{2} is the curvature of d′′_{h} with respect to h, and if d′′ is a Higgs operator, then (d′′)^{2} is the pseudocurvature of D_{h} with respect to h. This trivially follows from the bijectivity of I_{h}.

Lemma 2.4 i) For D ∈ A(E) let D = d_{h} + Θ_{h} = ∂_{h} + ¯∂_{h} + θ_{h} + θ_{h}^{∗} be the decomposition induced by h as above. If D is flat, then it holds δ_{h}^{2} = 0, d_{h}(Θ_{h}) = 0, i.e. ∂_{h}(θ_{h}) = ¯∂_{h}(θ_{h}^{∗}) = ∂_{h}(θ_{h}^{∗}) + ¯∂_{h}(θ_{h}) = 0, and furthermore d_{h}^{2} = −Θ_{h} ∧ Θ_{h}. ii) For d′′ = ¯∂ + θ ∈ A′′(E) let ∂_{h}, d_{h} and D_{h} be as above, and write d′_{h} := ∂_{h} + θ^{∗}. If d′′ is integrable, then it holds (d′_{h})^{2} = 0 , i.e. ∂_{h}^{2} = ∂_{h}(θ^{∗}) = θ^{∗} ∧ θ^{∗} = 0 , d_{h}^{2} = [∂_{h}, ¯∂], and hence F_{h} = d_{h}^{2} + [θ, θ^{∗}] + ∂_{h}(θ) + ¯∂(θ^{∗}).

Proof: i) For D = D′ + D′′ ∈ A_{f}(E) it holds

0 = ∂∂h(s, t)
= h((D′)^{2}(s), t) − h(D′(s), δ′′_{h}(t)) + h(D′(s), δ′′_{h}(t)) + h(s, (δ′′_{h})^{2}(t))
= h(s, (δ′′_{h})^{2}(t))

for all s, t ∈ A^{0}(E), i.e. (δ′′_{h})^{2} = 0. Similarly one sees (δ′_{h})^{2} = 0 = δ′_{h}δ′′_{h} + δ′′_{h}δ′_{h}, yielding δ_{h}^{2} = 0. We conclude

d_{h}(Θ_{h}) = \frac{1}{4}[D + δ_{h}, D − δ_{h}] = 0,

and

0 = D^{2} = (d_{h} + Θ_{h})^{2} = d_{h}^{2} + d_{h}(Θ_{h}) + Θ_{h} ∧ Θ_{h} = d_{h}^{2} + Θ_{h} ∧ Θ_{h}.

ii) For d′′ = ¯∂ + θ ∈ H′′(E) and d_{h} = ∂_{h} + ¯∂ it is well known that ∂_{h}^{2} = 0, and hence d_{h}^{2} = [∂_{h}, ¯∂]. Furthermore, for all s, t ∈ A^{0}(E) it holds

h(∂_{h}(θ^{∗})(s), t)
= h(∂_{h} ∘ θ^{∗}(s), t) + h(θ^{∗} ∘ ∂_{h}(s), t)
= ∂h(θ^{∗}(s), t) + h(θ^{∗}(s), ¯∂(t)) − h(∂_{h}(s), θ(t))
= ∂h(s, θ(t)) + h(s, θ ∘ ¯∂(t)) − h(∂_{h}(s), θ(t))
= h(∂_{h}(s), θ(t)) + h(s, ¯∂ ∘ θ(t)) + h(s, θ ∘ ¯∂(t)) − h(∂_{h}(s), θ(t))
= h(s, ¯∂(θ)(t)) = 0,

and

h(θ^{∗} ∧ θ^{∗}(s), t) = −h(s, θ ∧ θ(t)) = 0;

this shows ∂_{h}(θ^{∗}) = 0 = θ^{∗} ∧ θ^{∗}. Now let g be a Hermitian metric in X, and denote by ω_{g} the associated (1, 1)- form on X, by Λ_{g} the contraction by ω_{g}, and by ∗_{g} the associated Hodge-∗- operator. Recall that in the conformal class of g there exists a Gauduchon metric g̃, i.e. a metric satisfying ¯∂∂(ωn−1 ; g̃) = 0; g̃ is unique up to a constant positive factor if n ≥ 2 ([G] p. 502, [LT] Theorem 1.2.4).

There is a natural way to define a map

deg_{g} : H(E) ⟶ ℝ,

called g-degree, with the following properties (see [LT] sections 1.3 and 1.4):

- If g is a Gauduchon metric, and ¯∂ ∈ H(E) is a holomorphic structure, then deg_{g}(¯∂) is given as follows: Choose any Hermitian metric h in E, and let d be the Chern connection in (E, ∂) induced by h, i.e. the unique h-unitary connection in E with (0, 1)-part ¯∂. Then

deg_{g}(¯∂) := \frac{i}{2π} ∫_{X} tr(d^{2}) ∧ ω_{g}^{n−1} = \frac{i}{2nπ} ∫_{X} trΛ_{g}d^{2} · ω_{g}^{n} = \frac{i}{2nπ} ∫_{X} trΛ_{g}[¯∂, ∂] · ω_{g}^{n}.

- If g is arbitrary, then there is a unique Gauduchon metric g̃ in the conformal class of g such that deg_{g} = degg̃ . The g-slope of ¯∂ is

μ_{g}(¯∂) := \frac{deg_{g}(¯∂)}{r},

where r is the rank of E. If D = D′ + D′′ is a flat connection, then it holds (D′′)^{2} = 0, so D′′ is a holomorphic structure. We define the g-degree and g-slope of D as

deg_{g}(D) := deg_{g}(D′′), μ_{g}(D) := μ_{g}(D′′).

Similarly, for an integrable Higgs operator d′′ = ¯∂ + θ it holds ¯∂^{2} = 0, and we define

deg_{g}(d′′) := deg_{g}(¯∂), μ_{g}(d′′) := μ_{g}(¯∂).

Observe that in all three cases the g-degrees (resp. slopes) of isomorphic operators are the same.

Remark 2.5 Suppose that g is a Kähler metric, i.e. d(ω_{g}) = 0. Then the gdegree is a topological invariant of the bundle E, completely determined by the first real Chern class c_{1}(E)_{ℝ} ∈ H^{2}(X, ℝ). In particular, since all real Chern classes of a flat bundle vanish, it holds deg_{g}(D) = 0 for every flat connection D in E. On the other hand, if e.g. X is a surface admitting no Kähler metric and g is Gauduchon, then every real number is the g-degree of a flat line bundle on X ([LT] Proposition 1.3.13).

Lemma 2.6 If g is a Gauduchon metric, then for any metric h in E it holds:

i) If D is a flat connection, then

deg_{g}(D) = −\frac{i}{nπ} ∫_{X} trΛ_{g}G_{h} · ω_{g}^{n},

where G_{h} is the pseudocurvature of d′′ with respect to h. ii) If d′′ is an integrable Higgs operator, then

deg_{g}(d′′) = \frac{i}{2nπ} ∫_{X} trΛ_{g}F_{h} · ω_{g}^{n},

where F_{h} is the curvature of d′′ with respect to h.

Proof: i) Observe that Λ_{g}G_{h} = Λ_{g} ¯∂_{h}(θ_{h}) . The Chern connection in (E, D′′) induced by h is D′′ + ∂_{h} − θ_{h} = D − 2θ_{h}, and it holds

trΛ_{g}(D − 2θ_{h})^{2} = −2trΛ_{g}((¯∂ + θ^{∗})(θ) = −2trΛ_{g}(G_{h} + [θ, θ^{∗}]) = −2trΛ_{g}(G_{h}),

so the claim follows by integration. ii) Lemma 2.4 implies trΛ_{g}F_{h} = trΛ_{g}d_{h}^{2}; again the claim follows by integration.

3 Einstein metrics and stability for flat bundles.

We fix a Hermitian metric g in X; the associated volume form is vol_{g} := \frac{1}{n!}ω_{g}^{n}, and the g-volume of X is Vol_{g}(X) := ∫_{X} vol_{g}. We further fix a Hermitian metric

h in E, and denote by |. | the pointwise norm on forms with values in E (and associated bundles) defined by h and g. Let D ∈ A_{f}(E) be a flat connection in E, and write D = d+Θ = ∂ + ¯∂+θ +θ^{∗} as in section 1. Let d′′_{h} = I_{h}(D) = ¯∂ + θ ∈ A′′(E) be the Higgs operator associated to D, and G_{h} = (d′′_{h})^{2} its pseudocurvature. From Λ_{g}G_{h} = Λ_{g} ¯∂_{h}(θ_{h}) and Lemma 2.4 we deduce

(iΛ_{g}G_{h})^{∗} = −iΛ_{g}((¯∂(θ))^{∗}) = −iΛ_{g}∂(θ^{∗}) = iΛ_{g} ¯∂(θ) = iΛ_{g}G_{h},

so iΛ_{g}G_{h} is selfadjoint with respect to h.

Remark 3.1 It also holds iΛ_{g}G_{h} = \frac{i}{2}Λ_{g}(¯∂(Θ) − ∂(Θ)) , which in the case of a Kähler metric g equals \frac{1}{2}d^{∗}(Θ), where d^{∗} is the L^{2}-adjoint of d = ∂ + ¯∂.

Definition 3.2 h is called a g-Einstein metric in (E, D) if iΛ_{g}G_{h} = c · id_{E} with a real constant c, which is called the Einstein constant.

Lemma 3.3 Let h be a g-Einstein metric in (E, D), and g̃ = φ · g conformally equivalent to g. Then there exists a g̃-Einstein metric h̃ in (E, D) which is conformally equivalent to h.

Proof: g̃ = φ · g implies Λg̃ = \frac{1}{φ} · Λ_{g}. From Remark 2.1 iii) it follows that for f ∈ C^{∞}(X, ℝ) it holds G_{e^{f} ·h} = G_{h} − \frac{1}{4} ¯∂∂(f) · id_{E}. Hence the condition iΛ_{g}G_{h} = c · id_{E} implies iΛg̃G_{e^{f} ·h} = (\frac{c}{φ} − \frac{1}{4}P (f)) · id_{E}, where P := iΛg̃ ¯∂∂. Since C^{∞}(X, ℝ) = imP ⊕ ℝ ([LT] Corollary 2.9), there exists an f such that \frac{c}{φ} − \frac{1}{4}P (f) is constant.

Lemma 3.4 If iΛ_{g}G_{h} = c · id_{E} with c ∈ ℝ, then it holds:

i) c = −\frac{π}{(n−1)!·Vol_{g}(X)} · μ_{g}(D) if g is Gauduchon. ii) deg_{g}(D) = 0 if and only if c = 0.

Proof: i) is an immediate consequence of Lemma 2.6. ii) If g is Gauduchon, then this follows from i). If g is arbitrary, then let g̃ = φ · g be the Gauduchon metric in its conformal class such that deg_{g} = degg̃ . Now we have

iΛ_{g}G_{h} = 0 ⟺ iΛg̃G_{h} = 0 ⟺ degg̃(D) = 0 ⟺ deg_{g}(D) = 0.

Remark 3.5 i) If two flat connections D_{1},D_{2} are isomorphic via the automorphism f of E, i.e. if D_{2} ∘ f − f ∘ D_{1} = 0, and if h is a g-Einstein metric in (E, D_{1}), then f_{∗}h is a g-Einstein metric in (E, D_{2}) with the same Einstein constant. ii) By Remark 2.3, a necessary condition for d′′_{h} = I_{h}(D) to be an integrable Higgs operator is that h is a g-Einstein metric for D with Einstein constant c = 0, so in particular deg_{g}(D) = 0. On the other hand it holds d^{2} = −Θ ∧ Θ (Lemma 2.4), and, if d′′_{h} is integrable, θ ∧ θ = 0 implying θ^{∗} ∧ θ^{∗} = 0 . This gives tr(d^{2}) = −tr[θ, θ^{∗}] = 0, which implies deg_{g}(d′′_{h}) = 0. iii) For complex vector bundles on compact Riemannian manifolds (X, g), Corlette defines a g-harmonic metric for a flat connection by the condition d^{∗}(Θ) = 0 ([C]). If X is complex and g is a Kähler metric, then the g-degree of any flat connection vanishes, so in this context g-harmonic is the same as g-Einstein (see Remarks 2.5 and 3.1), but in general the two notions are different.

Now we prove a useful Vanishing Theorem.

Proposition 3.6 Let D be a flat connection in E, and h a g-Einstein metric in (E, D) with Einstein constant c. If c > 0, then the only section s ∈ A^{0}(E) with D(s) = 0 is s = 0. If c = 0, then for every section s ∈ A^{0}(E) with D(s) = 0 it holds ¯∂(s) = θ(s) = 0 and ∂(s) = θ^{∗}(s) = 0, so in particular d′′_{h}(s) = 0 .

Proof: D(s) = 0 is equivalent to

∂(s) = −θ(s), ¯∂(s) = −θ^{∗}(s);  (1)

this implies

¯∂∂h(s, s) = −h(¯∂ ∘ θ(s), s) − h(θ(s), θ(s)) + h(¯∂(s), ¯∂(s)) − h(s, ∂ ∘ θ^{∗}(s)). (2)

The assumption that h is g-Einstein means iΛ_{g} ¯∂(θ) = iΛ_{g}G_{h} = c·id_{E}, which is equivalent to iΛ_{g}∂(θ^{∗}) = −c·id_{E} since (iΛ_{g} ¯∂(θ))^{∗} = −iΛ_{g}(¯∂(θ)^{∗}) = −iΛ_{g}∂(θ^{∗}); these relations can be rewritten as

iΛ_{g} ¯∂ ∘ θ = −iΛ_{g}θ ∘ ¯∂ + c · id_{E}, iΛ_{g}∂ ∘ θ^{∗} = −iΛ_{g}θ^{∗} ∘ ∂ − c · id_{E}. (3)

Using (1) and (3) we get

iΛ_{g}h(¯∂ ∘ θ(s), s) = −iΛ_{g}h(θ ∘ ¯∂(s), s) + c · |s|^{2} = iΛ_{g}h(¯∂(s), θ^{∗}(s)) + c · |s|^{2}
= −iΛ_{g}h(¯∂(s), ¯∂(s)) + c · |s|^{2} = |¯∂(s)|^{2} + c · |s|^{2},

and similarly

iΛ_{g}h(s, ∂ ∘ θ^{∗}(s)) = |θ(s)|^{2} + c · |s|^{2},

so (2) implies

iΛ_{g} ¯∂∂h(s, s) = −2 (|¯∂(s)|^{2} + |θ(s)|^{2} + c · |s|^{2}) .

Since the image of the operator iΛ_{g} ¯∂∂ on real functions contains no non-zero functions of constant sign ([LT] Lemma 7.2.7), this gives s = 0 in the case c > 0, and if c = 0 we get ¯∂(s) = θ(s) = 0, implying ∂(s) = θ^{∗}(s) = 0 because of (1). The following corollary will be used later in the context of moduli spaces.

Corollary 3.7 For i = 1, 2 let D_{i} ∈ A_{f}(E) be a flat connection, h_{i} a g-Einstein metric in (E, D_{i}), and d′′_{i} := I_{h_{i}}(D_{i}) ∈ A′′(E) the associated Higgs operator. If D_{1} and D_{2} are isomorphic via the automorphism f of E, then d′′_{1} and d′′_{2} are isomorphic via f, too.

Proof: Let h be the metric in EndE = E^{∗} ⊗ E induced by the dual metric of h_{1} in E^{∗} and h_{2} in E, and D the connection in EndE defined by D(f) = D_{2} ∘ f − f ∘ D_{1} for all f ∈ A^{0}(EndE). Then D is flat of g-degree 0 since D_{1} and D_{2} are flat of equal degree, and h is a g-Einstein metric in (EndE, D) with Einstein constant c = 0 (compare Remark 2.3). Furthermore, the Higgs operator d′′ in EndE defined by d′′(f) = d′′_{2} ∘ f − f ∘ d′′_{1} equals I_{h}(D). Hence Proposition 3.6 implies that an automorphism f of E with D(f) = 0 also satisfies d′′(f) = 0. If F ⊂ E is a D-invariant subbundle of E, then it is obvious that flatness of D implies flatness of D|_{F}, and hence the following definition makes sense.

Definition 3.8 A flat connection D in E is called g-(semi)stable iff for every proper D-invariant subbundle 0 ≠ F ⊂ E it holds μ_{g}(D|_{F}) > μ_{g}(D) (μ_{g}(D|_{F}) ≥ μ_{g}(D)). D is called g-polystable iff E = E_{1} ⊕ E_{2} ⊕ … ⊕ E_{k} is a direct sum of D-invariant and g-stable subbundles E_{i} with μ_{g}(D|_{E_{i}}) = μ_{g}(D) for i = 1, 2, …, k.

Remark 3.9 i) Let D be a flat connection in E, and 0 ≠ F ⊂ E a proper D-invariant subbundle. Then g-stability of D implies μ_{g}(D|_{F}) > μ_{g}(D) and hence the g-instability of the holomorphic structure D′′ in E (in the sense of e.g. [LT]) since F is a D′′-holomorphic subbundle of E. ii) Suppose that g is a Kähler metric; then deg_{g}(D) = 0 for every flat connection D (Remark 2.5). Hence a flat connection D in E is - always g-semistable,

- g-stable if and only if E has no proper non-trivial D-invariant subbundle,

- g-polystable if E is a direct sum of D-invariant g-stable subbundles. This means that g-(poly-)stability on a Kähler manifold coincides with (poly- )stability in the sense of Corlette [C]. iii) It is obvious that stability in the sense of Corlette always implies g-stability, but at the end of this section we will give an example of a g-stable bundle which is not stable in the sense of Corlette.

Definition 3.10 A flat connection D in E is simple if the only D-parallel endomorphisms f, i.e. those with D_{End}(f) = D ∘ f − f ∘ D = 0, are the homotheties f = a · id_{E}, a ∈ ℂ.

Let D be a flat connection in E, 0 ≠ F ⊂ E a D-invariant subbundle, and Q := E/F the quotient with natural projection π : E ⟶ Q. Then D induces a flat connection D_{Q} in Q such that D_{Q} ∘π = π∘D. In particular, F is a holomorphic subbundle of (E, D′′), and D′′_{Q} is the induced holomorphic structure in Q. Since the g-degree of a flat connection D by definition equals the g-degree of the associated holomorphic structure D′′, it follow deg_{g}(D) = deg_{g}(D_{1}) + deg_{g}(D_{Q}). Hence as in the case of holomorphic bundles one verifies (compare [K] Chapter V)

Proposition 3.11 i) A flat connection D in E is g-(semi)stable if and only if for every D-invariant proper subbundle 0 ≠ F ⊂ E with quotient Q = E/F it holds μ_{g}(D_{Q}) < μ_{g}(D) (resp. μ_{g}(D_{Q}) ≤ μ_{g}(D).) ii) Let (E_{1}, D_{1}) and (E_{2}, D_{2}) be g-stable flat bundles over X with μ_{g}(D_{1}) = μ_{g}(D_{2}). If f ∈ A^{0}(Hom(E_{1}, E_{2})) satisfies D_{2} ∘ f = f ∘ D_{1}, then either f = 0 or f is an isomorphism. iii) A g-stable flat connection D in E is simple.

Next we prove the first half of the main result of this section.

Proposition 3.12 Let D be a flat connection in E, and h a g-Einstein metric in (E, D) with Einstein constant c; then D is g-semistable. If D is not g-stable, then D is g-polystable; more precisely, E = E_{1}⊕E_{2}⊕…⊕E_{k} is a h-orthogonal direct sum of D-invariant g-stable subbundles such that μ_{g}(D|_{E_{i}}) = μ_{g}(D) for i = 1, 2, …, k. Furthermore, h|_{E_{i}} is a g-Einstein metric in (E_{i}, D|_{E_{i}}) with Einstein constant c for all i, and the direct sum is invariant with respect to the Higgs operator d′′_{h} = I_{h}(D).

Proof: First we consider the case when g is a Gauduchon metric. Let 0 ≠ F ⊂ E be a D-invariant proper subbundle of rank s; then E = F ⊕ F^{⊥}, where F^{⊥} is the h-orthogonal complement of F. With respect to this decomposition, we write operators as 2 × 2 matrices, so D has the form

D = ( \begin{matrix}D_{1} & A \\ 0 & D_{2} \end{matrix} ) ,

where D_{1} = D|_{F} and D_{2} is a flat connection in F^{⊥}. We use notations as in section 2; it is easy to see that the operator δ associated to D by h has the form

δ = ( \begin{matrix}δ_{1} & 0 \\ A^{∗} & δ_{2} \end{matrix} ) ,

where the δ_{i} are the operators associated to the D_{i} by h. Similarly it holds

¯∂ = \frac{1}{2}(D′′ + δ′′) = \frac{1}{2} ( \begin{matrix}D′′_{1} + δ′′_{1} & A′′ \\ A′^{∗} & D′′_{2} + δ′′_{2} \end{matrix} ) = ( \begin{matrix}¯∂_{1} & \frac{1}{2}A′′ \\ \frac{1}{2}A′^{∗} & ¯∂_{2} \end{matrix} ) ,

and

θ = \frac{1}{2}(D′ − δ′) = ( \begin{matrix}D′_{1} − δ′_{1} & A′ \\ −A′′^{∗} & D′_{2} − δ′_{2} \end{matrix} ) = ( \begin{matrix}θ_{1} & \frac{1}{2}A′ \\ −\frac{1}{2}A′′^{∗} & θ_{2} \end{matrix} ) ,

where A′ resp. A′′ is the part of A of type (1, 0) resp. (0, 1). This implies

¯∂(θ) = [¯∂, θ]
= ( \begin{matrix}¯∂_{1}(θ_{1}) + \frac{1}{4}(A′ ∧ A′^{∗} − A′′ ∧ A′′^{∗})  ∗ \\ ∗  ¯∂_{2}(θ_{2}) + \frac{1}{4}(A′^{∗} ∧ A′ − A′′^{∗} ∧ A′′) \end{matrix} ) ,

hence

c · id_{E} = iΛ_{g}G_{h}
= ( \begin{matrix}iΛ_{g}G_{1,h} + \frac{i}{4}Λ_{g}(A′ ∧ A′^{∗} − A′′ ∧ A′′^{∗})  ∗ \\ ∗  iΛ_{g}G_{2,h} + \frac{i}{4}Λ_{g}(A′^{∗} ∧ A′ − A′′^{∗} ∧ A′′) \end{matrix} ) ,(4)

and thus

sc = tr(iΛ_{g}G_{1,h} + \frac{i}{4}Λ_{g}(A′ ∧ A′^{∗} − A′′ ∧ A′′^{∗})) = itrΛ_{g}G_{1,h} + \frac{1}{4}|A|^{2} .

Using Lemma 2.6 and Lemma 3.4 we conclude

μ_{g}(D_{1}) = −\frac{i}{snπ} ∫_{X} trΛ_{g}G_{1,h} · ω_{g}^{n} ≥ −\frac{c(n − 1)!}{π}Vol_{g}(X) = μ_{g}(D); (5)

this prove that D is g-semistable. If D is not g-stable, then there exists a subbundle F as above such that equality holds in (5), which implies A = 0. This means not only that F^{⊥} is D-invariant, too, with D|_{F^{⊥}} = D_{2}, but also that

iΛ_{g}G_{1,h} = c · id_{F} , iΛ_{g}G_{2,h} = c · id_{F^{⊥}}

by (4). Hence the restriction of h to F resp. F^{⊥} is g-Einstein for D_{1} resp. D_{2}, and it holds μ_{g}(D_{1}) = μ_{g}(D) = μ_{g}(D_{2}) by Lemma 3.4. Furthermore, the D-invariance of F means that the inclusion i : F ↪ E is parallel with respect to the flat connection in Hom(F, E) induced by D_{1} and D. Using Remark 2.3 and Proposition 3.6 as in the proof of Corollary 3.7, we conclude that i is also parallel with respect to the associated Higgs operator, i.e. that F is d′′_{h}- invariant; the same argument works for F^{⊥}. If D_{1} and D_{2} are stable, then we are done; otherwise the proof is finished by induction on the rank. Now let g be arbitrary, let g̃ be the Gauduchon metric in its conformal class with deg_{g} = degg̃, and let h̃ be a g̃-Einstein metric in the conformal class of h, which exists by Lemma 3.3; then the theorem holds for g̃ and h̃. Since g and g̃ define the same degree and slope, and hence stability, it follows that D is g̃-semistable. If D is not g-stable, then there exists a D-invariant proper subbundle F as above with μg̃(D_{1}) = μ_{g}(D_{1}) = μ_{g}(D) = μg̃(D). Note that the h-orthogonal complement F^{⊥} of F is also the h̃-orthogonal complement, since h and h̃ are conformally equivalent. Hence, using g̃ and h̃ we conclude as above

that D = ( \begin{matrix}D_{1} & 0 \\ 0 & D_{2} \end{matrix} ) with respect the decomposition E = F ⊕ F^{⊥}; now we

can proceed as in the Gauduchon case.

Another consequence of Proposition 3.6 is

Proposition 3.13 Let D be a simple flat connection in E. If a g-Einstein metric in (E, D) exists, then it is unique up to a positive scalar.

Proof: Let h_{1},h_{2} be g-Einstein metrics in (E, D), and c ∈ ℝ the Einstein constant. There are differentiable automorphisms f and k of E, selfadjoint with respect to both h_{1} and h_{2}, such that f = k^{2} and h_{2}(s, t) = h_{1}(f(s), t) = h_{1}(k(s), k(t)) for all s, t ∈ A^{0}(E). Since D is simple it suffices to show D(f) = 0. We define a new flat connection D̃ := k ∘ D ∘ k^{−1}. In what follows, operators δ, d, Θ etc. with a subscript i are associated to D by the metric h_{i}, without a subscript they are associated to D̃ by h_{1}. One verifies

δ_{2} = f^{−1} ∘ δ_{1} ∘ f, δ = k^{−1} ∘ δ_{1} ∘ k = k ∘ δ_{2} ∘ k^{−1},

implying

d = \frac{1}{2}(D̃ + δ) = k ∘ d_{2} ∘ k^{−1}, Θ = \frac{1}{2}(D̃ − δ) = k ∘ Θ_{2} ∘ k^{−1}

and hence

iΛ_{g}G_{h_{1}} = iΛ_{g} ¯∂(θ) = ik ∘ Λ_{g} ¯∂_{2}(θ_{2}) ∘ k^{−1} = ik ∘ Λ_{g}G_{2,h_{2}} ∘ k^{−1} = c · id_{E},

so h_{1} is a g-Einstein metric in (E, D̃). It follows that h_{1} induces a g-Einstein metric with Einstein constant 0 for the flat connection D̃_{End}(.) = .∘D −D̃ ∘. in EndE. By definition it holds D̃_{End}(k) = 0, so Proposition 3.6 implies d̃_{End}(k) = 0. Since δ̃_{End} = 2d̃_{End} − D̃_{End}, it follows

0 = δ̃_{End}(k) = k ∘ δ_{1} − δ ∘ k = k ∘ δ_{1} − k^{−1} ∘ δ_{1} ∘ k^{2} = k^{−1} ∘ (f ∘ δ_{1} − δ_{1} ∘ f) , implying δ_{1,End}(f) = 0, where δ_{1,End} is the operator on EndE induced by D and h_{1}. But this is equivalent to δ′_{1,End}(f) = 0 and δ′′_{1,End}(f) = 0 , and taking adjoints with respect to h_{1} we get

0 = (δ′_{1,End}(f))^{∗} = D′′_{End}(f), 0 = (δ′′_{1,End}(f))^{∗} = D′_{End}(f),

i.e. D_{End}(f) = 0.

Let (E, D), (Ẽ, D̃) be flat bundles with g-Einstein metrics h, h̃. Let E = ⨁_{i=1}^{k} E_{i}

and Ẽ = ⨁_{i=1}^{l} Ẽ_{i} be the orthogonal, invariant splittings given by Proposi-

tion 3.12. We write D_{i} := D|_{E_{i}}, D̃_{i} := D̃|Ẽ_{i}, h_{i} := h|_{E_{i}}, h̃_{i} := h̃|Ẽ_{i}. Using Propositions 3.11 and 3.13 one verifies

Corollary 3.14 If there exists an isomorphism f ∈ A^{0}(Hom(E, Ẽ)) satisfying f ∘ D = D̃ ∘ f, then it holds k = l, and, after renumbering of the summands if necessary, there are isomorphisms f_{i} ∈ A^{0}(Hom(E_{i}, Ẽ_{i})) such that f_{i} ∘ D_{i} = D̃_{i} ∘ f and f_{∗}(h_{i}) = h̃_{i}.

The following result is the converse of Proposition 3.12.

Proposition 3.15 Let (E, D) a g-stable flat bundle over X. Then there exists a g-Einstein metric for (E, D).

Sketch of proof: The proof is very similar to the one for the existence of a g-Hermitian Einstein metric in a g-stable holomorphic vector bundle as given in Chapter 3 of [LT]. Therefore we will be brief, leaving it to the reader to fill in the necessary details. First observe that by Lemma 3.3 we may assume that g is a Gauduchon metric. For any metric h in E it holds

G_{h} = ¯∂_{h}(θ_{h}) = \frac{1}{4}[D′′ + δ′′_{h}, D′ − δ′_{h}] = −\frac{1}{4}[D′′, δ′_{h}] + \frac{1}{4}[D′, δ′′_{h}]

since D^{2} = δ_{h}^{2} = 0. Observe that [D′′, δ′_{h}] resp. [D′, δ′′_{h}] is the curvature of the h-unitary connection D′′ + δ′_{h} resp. D′ + δ′′_{h}. Fix a metric h_{0} in E, and let δ = δ′ +δ′′, d = ∂ + ¯∂, Θ = θ+θ^{∗} be the operators associated to D = D′+D′′ and h_{0} as in section 2. Consider for an h_{0}-selfadjoint positive definite endomorphism f of E and ε ∈ [0, 1] the differential equation

L_{ε}(f) := K^{0} − \frac{i}{4}Λ_{g}D′′(f^{−1} ∘ δ′(f)) + \frac{i}{4}Λ_{g}D′(f^{−1} ∘ δ′′(f)) − ε · log(f) = 0, (6)

where K^{0} := iΛ_{g} ¯∂(θ) − c · id_{E} = −\frac{i}{4}Λ_{g}([D′′, δ′] − [D′, δ′′]) − c · id_{E}, and c is the constant associated to a possible g-Einstein metric for (E, D). The metric f · h_{0}, defined by f · h_{0}(s, t) := h_{0}(f(s), t) for sections s, t in E, is g-Einstein if and only if L_{0}(f) = 0.

The term T_{1} := iΛ_{g}D′′(f^{−1} ∘ δ′(f)) (associated to the unitary connection d_{1} := δ′ + D′′ ) in equation (6) is of precisely the same type as the term T_{0} := iΛ_{g} ¯∂(f^{−1} ∘ ∂_{0}(f)) (associated to the unitary connection d_{0} = ∂_{0} + ¯∂ ) in equation (∗∗) on page 62 in [LT], and the term T_{2} := −iΛ_{g}D′(f^{−1} ∘ δ′′(f)) (associated to the unitary connection d_{2} := D′ + δ′′ ) is almost of this type; e.g. the trace of all three terms equals iΛ_{g} ¯∂∂(tr(log f), and the symbols of the differential operators \frac{d}{df} T̂_{i}, where T̂_{i}(f) := f ∘ T_{i}(f), are equal, too. Therefore most of the arguments in [LT] can easily be adapted to show first that for a simple flat connection D equation (6) has solutions f_{ε} for all ε ∈ (0, 1], which satisfy det f_{ε} ≡ 1, and which converge to a solution f of L_{0}(f) = 0 if the L^{2}-norms of the f_{ε} are uniformly bounded. (There are two places where one has to argue in a slightly different way: In the proof of the analogue of [LT] Lemma 3.3.1, one uses the Laplacian Δ_{D} = D^{∗} ∘ D instead of Δ ¯∂, and in the proof of the analogue of [LT] Proposition 3.3.5 the sum Δ_{d_{1}} + Δ_{d_{2}} of the two Laplacians associated to d_{1} and d_{2} instead of just one.) Then, under the assumptions that rkE ≥ 2 and that the L^{2}-norms of the f_{ε} are unbounded, one shows that for suitable ε_{i} ⟶ 0, ρ(ε_{i}) ⟶ 0, the limit

π := id_{E} − lim_{σ⟶0} ( lim_{i⟶∞} ρ(ε_{i}) · f_{ε_{i}})^{σ}

exists weakly in L_{1}^{2}, and satisfies in L^{1} π = π^{∗} = π^{2} and

(id_{E} − π) ∘ D(π) = 0.  (7)

This implies (id_{E}−π)∘D′′(π) = 0, so π defines a weakly holomorphic subbundle F of the holomorphic bundle (E, D′′) by a theorem of Uhlenbeck and Yau (see [UY], [LT] Theorem 3.4.3). F is a coherent subsheaf of (E, D′′), a holomorphic subbundle outside an analytic subset S ⊂ X of codimension at least 2, and π is smooth on X \ S. Therefore (7) implies that F|_{X\S} is in fact a D-invariant subbundle of E|_{X\S}, which extends to a D-invariant subbundle F of E by the Lemma below. Again using arguments as is [LT], one finally shows that F violates the stability condition for (E, D).

Lemma 3.16 Let X be a differentiable manifold, E a differentiable vector bundle over X, and D a flat connection in E. Let S ⊂ X be a subset such that X \ S is open and dense in X, and with the following property: For every point x ∈ S and every open neighborhood U of x in X there exists an open neighborhood x ∈ U′ ⊂ U such that U′ \ S is path-connected. Then every D-invariant subbundle F of E|_{X\S} extends to a D-invariant subbundle F of E.

Proof: For every x ∈ S choose an open neighborhood x ∈ U ⊂ X such that U \ S is path connected and (E|_{U}, D) ≅ (U × V, d), where V is a vector space and d the trivial flat connection. Since F is D-invariant and U \ S is path connected, it holds

(F|_{U\S}, D) ≅ ((U \ S) × W, d),

where W ⊂ V is a constant subspace. Define F over U by F |_{U} :≅ U × W; then the topological condition on S implies that this is well defined on S, and hence gives a D-invariant extension F of F over X. The following main result of this section is a direct consequence of Propositions 3.12 and 3.15.

Theorem 3.17 A flat connection D in E admits a g-Einstein metric if and only if it is g-polystable.

As for stable vector bundles and Hermitian-Einstein metrics, the gauge theoretic interpretation of our results is as follows. The group

G^{ℂ} := A^{0}(GL(E))

of differentiable automorphisms of E acts on A(E) by D · f = f^{−1} ∘ D ∘ f, so

A(E)/G^{ℂ}

is the moduli space of isomorphism classes of connections in E. Observe that flatness, simplicity and g-stability are preserved under this action. Fix a metric h in E; then it holds:

Corollary 3.18 The following two statements for a flat connection D are equivalent:

i) D is g-stable. ii) D is simple, and there is a connection D_{0} in the G^{ℂ}-orbit through D such that h is g-Einstein for D_{0}.

The essential uniqueness of a g-Einstein metric (Proposition 3.13) implies that the connection D_{0} in ii) is unique up to the action of the subgroup

G := A^{0}(U(E, h)) ⊂ G^{ℂ}

of h-unitary automorphisms. This means that the moduli space

M_{f}^{st}(E) = { D ∈ A^{f}(E) | D is g − stable }/G^{ℂ}

of isomorphism classes of g-stable flat connections in E coincides with the quotient

{ D ∈ A^{f}(E) | D is simple and h is g − Einstein for D }/G.

Example: We now give the promised example of a flat bundle which is gstable, but not stable in the sense of Corlette.

An Inoue surface of type S_{N}^{±} is the quotient of ℍ×ℂ by an affine transformation group G generated by

g_{0}(w, z) := (αw, ±z + t),
g_{i}(w, z) := (w + a_{i}, z + b_{i}w + c_{i}), i = 1, 2,
g_{3}(w, z) := (w, z + c_{3}),

with certain constants α, a_{i}, b_{i}, c_{3} ∈ ℝ, c_{1}, c_{2} ∈ ℂ (see [P] p. 160). Since the second Betti number of S_{N}^{±} vanishes, the degree map

deg_{g} : Pic(S_{N}^{±}) ⟶ ℝ

associated to a Gauduchon metric g is, up to a positive factor, independent of the chosen metric g. In particular, all Hermitian metrics g define the same notion of g-stability ([LT] Remark 1.4.4 iii)). The trivial flat connection d on ℍ × ℂ induces a flat connection D in the tangent bundle E := T_{S_{N}^{±}}. A D-invariant sub-line bundle of E is in particular a holomorphic subbundle, so it defines a holomorphic foliation of S_{N}^{±}. According to [B] Théorème 2, there is precisely one such foliation, namely the one induced by the G-invariant vertical foliation (i.e. with leaves {w} × ℂ) of ℍ × ℂ. The corresponding trivial line bundle L_{0} on ℍ × ℂ is d-invariant, so it descends to a unique D-invariant subbundle L of E; this shows that E is not stable in the sense of Corlette. Observe that L has factors of automorphy χ(g_{i}) = ±1, i = 0, 1, 2, 3, so the standard flat metric in L_{0} defines a metric h in L such that the associated Chern connection in (L, D′′|_{L}) is flat; this implies μ_{g}(D|_{L}) = deg_{g}(D|_{L}) = 0. On the other hand, the g-degree, and hence the gslope, of E is negative by [P] Proposition 4.7; this implies the g-stability of E since L is the only D-invariant proper subbundle of E.

4 Einstein metrics and stability for Higgs bundles.

Again we fix Hermitian metrics g in X and h in E. Let d′′ = ¯∂ + θ ∈ A′′_{i}(E) be an integrable Higgs operator,

D_{h} = I_{h}^{−1}(d′′) = d + Θ = ∂ + ¯∂ + θ + θ^{∗} ∈ A(E)

the connection associated to d′′ as in section 2, and F_{h} = D_{h}^{2} its curvature.

Definition 4.1 h is called a g-Einstein metric in (E, d′′) if and only if K_{h} := iΛ_{g}F_{h} = c · id_{E} with a real constant c, the Einstein constant.

Lemma 4.2 Let h be a g-Einstein metric in (E, d′′), and g̃ = φ · g conformally equivalent to g. Then there exists a g̃-Einstein metric h̃ in (E, d′′) which is conformally equivalent to h.

Proof: From Remark 2.1 iii) it follows that for f ∈ C^{∞}(X, ℝ) it holds F_{e^{f} ·h} = F_{h} + ¯∂∂(f) · id_{E}. Using this, the proof is analogous to that of Lemma 3.3. Notice that since d′′ is integrable it holds (compare Lemma 2.4)

K_{h} = iΛ_{g}(d^{2} + [θ, θ^{∗}]) = iΛ_{g}([∂, ¯∂] + [θ, θ^{∗}])

where d = ∂ + ¯∂. An immediate consequence of Lemma 2.6 and Lemma 4.2 is (compare the proof of Lemma 3.4)

Lemma 4.3 If iΛ_{g}F_{h} = c · id_{E} with c ∈ ℝ, then it holds:

i) c = \frac{2π}{(n−1)!·Vol_{g}(X)} · μ_{g}(d′′) if g is Gauduchon. ii) deg_{g}(d′′) = 0 if and only if c = 0.

Remark 4.4 (compare Remark 3.5)

i) If two integrable Higgs operators d′′_{1},d′′_{2} are isomorphic via the automorphism f of E, i.e. if d′′_{2} ∘ f − f ∘ d′′_{1} = 0, and if h is a g-Einstein metric in (E, d′′_{1}), then f_{∗}h is a g-Einstein metric in (E, d′′_{2}), and the associated Einstein constants are equal. ii) By Remark 2.3, a necessary condition for D_{h} = I_{h}(d′′) to be a flat connection is h to be Einstein with Einstein constant c = 0, so in particular deg_{g}(d′′) = 0. On the other hand, the Chern connection in (E, D′′_{h}) is ∂ − θ + ¯∂+ θ^{∗}, so the gdegree of D_{h} is obtained by integrating trΛ_{g}[¯∂+θ^{∗}, ∂ −θ] which equals trΛ_{g}[¯∂, ∂] since d′′ is integrable (Lemma 2.4 ii)). If D_{h} is flat, we furthermore have d^{2} = −Θ ∧ Θ (Lemma 2.4 i)), implying tr[¯∂, ∂] = 0 and hence deg_{g}(D_{h}) = 0.

In analogy with the case of Hermitian-Einstein metrics in holomorphic vector bundles, the following vanishing theorem holds.

Proposition 4.5 Let h be a g-Einstein metric in (E, d′′) with Einstein constant c. If c < 0, then the only section s ∈ A^{0}(E) with d′′(s) = 0 is s = 0. If c = 0, then for every section s ∈ A^{0}(E) with d′′(s) = 0 it holds D_{h}(s) = 0.

Proof: For s ∈ A^{0}(E), d′′(s) = 0 is equivalent to ¯∂(s) = 0 = θ(s). This implies

c · |s|^{2} = c · h(s, s) = h(K_{h}(s), s) = iΛ_{g} (h(¯∂∂(s), s) + h(θ^{∗}(s), θ^{∗}(s))) . (8)

We have

iΛ_{g} ¯∂∂h(s, s) = iΛ_{g} (h(¯∂∂(s), s) − h(∂(s), ∂(s)))

since ¯∂(s) = 0, and using (8) we get

iΛ_{g} ¯∂∂h(s, s) = c · |s|^{2} − |∂(s)|^{2} − |θ^{∗}(s)|^{2}.

Now the claim follows as in the proof of Proposition 3.6. The proof of the following corollary is analogous to that of Corollary 3.7.

Corollary 4.6 For i = 1, 2 let d′′_{i} ∈ A′′_{i}(E) be an integrable Higgs operators, h_{i} a g-Einstein metric in (E, d′′_{i}), and D_{i} := I_{h_{i}}^{−1}(d′′_{i}) ∈ A(E) the associated connection. If d′′_{1} and d′′_{2} are isomorphic via the automorphism f of E, then D_{1} and D_{2} are isomorphic via f, too.

Let d′′ = ¯∂ + θ be an integrable Higgs operator in E. A coherent subsheaf F of the holomorphic bundle (E, ¯∂) is called a Higgs-subsheaf of (E, d′′) iff it is d′′-invariant. For the definition of the g-degree and g-slope of a coherent sheaf see [LT].

Definition 4.7 An integrable Higgs operator d′′ in E is called g-(semi)stable iff for every coherent Higgs-subsheaf F of (E, d′′) with 0 < rkF < rkE it holds μ_{g}(F) < μ_{g}(E) ( μ_{g}(F) ≤ μ_{g}(E) ). d′′ is called g-polystable iff E is a direct sum E = E_{1} ⊕ E_{2} ⊕ … ⊕ E_{k} of d′′-invariant and g-stable subbundles E_{i} with μ_{g}(d′′|_{E_{i}} = μ_{g}(d′′) for i = 1, 2, …, k.

Definition 4.8 An integrable Higgs operator d′′ in E is called simple iff for every f ∈ A^{0}(EndE) with d′′ ∘ f = f ∘ d′′ it holds f = a · id_{E} with a ∈ ℂ.

As in the case of stable vector bundles or flat connections, (semi)-stability can equivalently be defined using quotients of E; again it follows

Lemma 4.9 i) A g-stable integrable Higgs operator in E is simple. ii) Let d′′_{1}, d′′_{2} be g-stable integrable Higgs operators in bundles E_{1}, E_{2} on X such that μ_{g}(d′′_{1}) = μ_{g}(d′′_{2}) . If f ∈ A^{0}(Hom(E_{1}, E_{2})) satisfies d′′_{2} ∘ f = f ∘ d′′_{1}, then either f = 0 or f is an isomorphism.

Furthermore, using arguments similar to those in the proof of Proposition 3.13, we get the following consequence of Proposition 4.5.

Proposition 4.10 Let d′′ be a simple integrable Higgs operator in E. If a g-Einstein metric in (E, d′′) exists, then it is unique up to a positive scalar.

The proof of the next result is a straightforward generalization of that in the Kähler case [S2] (just as for the proof of the corresponding statement for Hermite-Einstein metrics in vector bundles, see [LT]).

Proposition 4.11 Let d′′ be an integrable Higgs operator in E, and h a g-Einstein metric in (E, d′′) with Einstein constant c; then d′′ is gsemistable. If d′′ is not g-stable, then d′′ is g-polystable; more precisely, E = E_{1} ⊕ E_{2} ⊕ … ⊕ E_{k} is an h-orthogonal direct sum of d′′-invariant and gstable subbundles such that μ_{g}(d′′|_{E_{i}}) = μ_{g}(d′′) for i = 1, 2, …, k . Furthermore, h|_{E_{i}} is a g-Einstein metric in (E_{i}, d′′|_{E_{i}}) with Einstein constant c for all i, and the direct sum is invariant with respect to the connection D_{h} = I_{h}^{−1}(d′′).

Let d′′, d̃′′ be integrable Higgs operators in bundles E, Ẽ with g-Einstein met-

rics h, h̃. Let E = ⨁_{i=1}^{k} E_{i} and Ẽ = ⨁_{i=1}^{l} Ẽ_{i} be the orthogonal, invariant split-

tings given by Proposition 4.11. We write d′′_{i} := d′′|_{E_{i}} , d̃′′_{i} := d̃′′|Ẽ_{i}, h_{i} := h|_{E_{i}}, h̃_{i} := h̃|Ẽ_{i}. As in the previous section (but now using Lemma 4.9 and Proposition 4.10) we deduce

Corollary 4.12 Suppose that there exists an isomorphism f ∈ A^{0}(Hom(E, Ẽ)) satisfying f ∘ d′′ = d̃′′ ∘ f. Then it holds k = l, and, after renumbering of the summands if necessary, there are isomorphisms f_{i} ∈ A^{0}(Hom(E_{i}, Ẽ_{i})) such that f_{i} ∘ d′′_{i} = d̃′′_{i} ∘ f and f_{∗}(h_{i}) = h̃_{i}.

Remark 4.13 We expect that the existence of a g-Einstein metric for a gstable Higgs operator d′′ can be proved by solving (again using the continuity method as in [LT]) the differential equation

K_{h} + iΛ_{g}d′′(f^{−1} ∘ d′(f)) = c · id_{E}

for a positive definite and h-selfadjoint endomorphism f of E, where h is a suitable fixed metric in E.

5 Surfaces.

In this section we consider the special case n = 2, i.e. where X is a compact complex surface; again we fix a Hermitian metric g in X. In this case, the real Chern numbers c_{1}^{2}(E), c_{2}(E) ∈ H^{4}(X, ℝ) ≅ ℝ can be calculated by integrating the corresponding Chern forms of any connection in E, independently of the chosen metric g. In particular, if E admits a flat connection, then these Chern numbers vanish.

Proposition 5.1 Suppose that D ∈ A_{f}(E) is a flat connection of g-degree 0, and that h is a g-Einstein metric in (E, D). Then it holds G_{h} = 0. In particular, the Higgs operator d′′_{h} associated to D and h is integrable with deg_{g}(d′′_{h}) = 0, and h is a g-Einstein metric for (E, d′′_{h}).

Proof: (see [S2]) For ϵ > 0 we define a new connection B_{ϵ} := d + \frac{1}{ϵ}θ + ϵθ^{∗}, and F_{ϵ} := B_{ϵ}^{2}. Observe that n = 2 implies F_{ϵ}^{2} = \frac{1}{ϵ^{2}} ∇_{ϵ}^{4}, where ∇_{ϵ} = d′′_{h} + ϵd′. The vanishing of the Chern numbers of E implies ∫_{X} trF_{ϵ}^{2} = 0, and hence

∫_{X} tr∇_{ϵ}^{4} = 0 for all ϵ > 0. Taking the limit ϵ → 0 it follows

∫_{X} trG_{h}^{2} = 0.  (9)

Write G_{h} = G_{1,1} + G_{2}, where G_{1,1} is the component of the 2-form G_{h} of type (1, 1). Then it holds

∗_{g}G_{1,1} = −G_{1,1}, ∗_{g}G_{2} = G_{2};  (10)

the first equation is a consequence of Λ_{g}G_{h} = 0, which follows from the assumption and Lemma 3.4. On the other hand, it holds G_{h} = ¯∂^{2} + ¯∂(θ) + θ ∧ θ, so Lemma 2.4 implies

G_{1,1} = ¯∂(θ) = ∂(θ^{∗})^{∗} = −¯∂(θ)^{∗} = −G_{1,1}^{∗},  (11)

and

G_{2} = ¯∂^{2} + θ ∧ θ = −θ^{∗} ∧ θ^{∗} − θ ∧ θ = (θ ∧ θ + θ^{∗} ∧ θ^{∗})^{∗} = G_{2}^{∗}.  (12)

(11) and (12) combined with (10) give ∗_{g}G_{h}^{∗} = G_{h}, so from (9) it follows

0 = ∫_{X} trG_{h}^{2} = ∫_{X} tr(G_{h} ∧ ∗_{g}G_{h}^{∗}) = ∫_{X} |G_{h}|^{2}vol_{g},

implying (d′′_{h})^{2} = G_{h} = 0. Hence d′′_{h} is integrable, deg_{g}(d′′_{h}) vanishes (Remark 3.5), and h is g-Einstein for (E, d′′)h) because the curvature of d′′_{h} with respect to h equals D^{2} = 0.

Proposition 5.2 Suppose that c_{1}^{2}(E) = c_{2}(E) = 0, that d′′ is an integrable Higgs operator of g-degree 0, and that h is a g-Einstein metric in (E, d′′). Then it holds F_{h} = 0. In particular, the connection D_{h} associated to d′′ and h is flat with deg_{g}(D_{h}) = 0, and h is a g-Einstein metric for (E, D_{h}).

Proof: Define F_{1,1} := d^{2} + [θ, θ^{∗}], F_{2} := ∂(θ) + ¯∂(θ^{∗}); then F_{h} = F_{1,1} + F_{2}. Observe that F_{1,1} is of type (1,1) because d is a unitary connection in the holomorphic bundle (E, ¯∂). Since deg_{g}(d′′) = 0, Lemma 4.3 implies 0 = Λ_{g}F_{h} = Λ_{g}F_{1,1}, hence it holds ∗_{g}F_{1,1} = −F_{1,1} and ∗_{g}F_{2} = F_{2}. On the other hand, it is easy to see that F_{1,1}^{∗} = −F_{1,1} and F_{2}^{∗} = F_{2} . Combining these relations we get ∗_{g}F_{h}^{∗} = F_{h}. Since c_{1}^{2}(E) resp. c_{2}(E) are obtained by integrating −\frac{1}{4π^{2}} (trF_{h})^{2} resp. −\frac{1}{8π^{2}} ((trF_{h})^{2} − tr(F_{h}^{2})), we get

0 = ∫_{X} tr(F_{h}^{2}) = ∫_{X} tr(F_{h} ∧ ∗_{g}F_{h}^{∗}) = ∥F_{h}∥^{2},

implying D_{h}^{2} = F_{h} = 0. Hence D_{h} is flat, deg_{g}(D_{h}) vanishes (Remark 4.4), and h is g-Einstein for (E, D_{h}) because the pseudocurvature of D_{h} with respect to h equals (d′′)^{2} = 0.

Remark 5.3 The above proposition implies in particular the following: Suppose that c_{1}^{2}(E) = c_{2}(E) = 0; if there exists an integrable Higgs operator d′′ in E with g-degree 0 admitting a g-Einstein metric, then the real Chern class c_{1}(E)_{ℝ} ∈ H^{2}(X, ℝ) vanishes, because there is a flat connection in E.

We define A_{f}(E)_{g}^{0} to be the space of D ∈ A_{f}(E) of g-degree 0 such that there exists a g-Einstein metric in (E, D), and A′′_{i}(E)_{g}^{0} to be the space of d′′ ∈ A′′_{i}(E) of g-degree 0 such that there exists a g-Einstein metric in (E, d′′). By Remark 3.5 and Remark 4.4, the two moduli sets

M_{f}(E)_{g}^{0} := A_{f}(E)_{g}^{0}/ isomorphy of connections

and

M′′(E)_{g}^{0} := A′′_{i}(E)_{g}^{0}/isomorphy of Higgs operators

are well defined. The main result of this section is

Theorem 5.4 There is a natural bijection

I : M_{f}(E)_{g}^{0} ⟶ M′′(E)_{g}^{0}.

Proof: First observe that we may assume that the real Chern classes of E vanish, since otherwise both spaces are empty (see Remark 5.3). Let D be a flat connection in E with g-degree 0, and h a g-Einstein metric in (E, D). By Proposition 5.1, the associated Higgs operator d′′_{h} = I_{h}(D) is integrable with g-degree 0, and h is a g-Einstein metric in (E, d′′_{h}). We will show that the map I defined by I([D]) := [d′′_{h}] is well defined and bijective. Suppose that D, D̃ ∈ A_{f}(E)_{g}^{0} are isomorphic via the automorphism f of E; then f_{∗}h is g-Einstein in (E, D̃) (Remark 3.5), the Higgs-operator d̃′′ associated to D̃ and f_{∗}h is isomorphic to d′′ via f (Corollary 3.7), and f_{∗}h is a g-Einstein metric in (E, d̃′′) (Remark 4.4). To prove that I is well defined it thus suffices to show that two different g-Einstein metrics h, h̃ for a fixed D ∈ A_{f}(E)_{g}^{0} produce isomorphic Higgs operators d′′_{h}, d′′ ; h̃. For this consider the

D-invariant and h- resp. h̃-orthogonal splittings E = ⨁_{i=1}^{k} E_{i} resp. E = ⨁_{i=1}^{l} Ẽ_{i}

associated to h resp. h̃ by Proposition 3.12. According to Corollary 3.14 (with E = Ẽ, D = D̃, f = id_{E} ) it holds k = l, and we may assume that there are isomorphisms f_{i} : (E_{i}, D_{i}, h_{i}) ⟶ (Ẽ_{i}, D̃_{i}, h̃_{i}) of flat bundles of g-degree 0 with g-Einstein metrics, where D_{i} := D|_{E_{i}}, D̃_{i} := D|Ẽ_{i}, h_{i} := h|_{E_{i}}, h̃_{i} := h̃|Ẽ_{i}. This means in particular that the Higgs operator d′′_{i} in E_{i} associated to D_{i} and h_{i} is isomorphic via f_{i} to the Higgs operator d̃′′_{i} in Ẽ_{i} associated to D̃_{i} and h̃_{i}. Hence d′′_{h} = d′′_{1} ⊕ … d′′_{k} is isomorphic to d′′ ; h̃ = d̃′′_{1} ⊕ … ⊕ d̃′′_{k} via the isomorphism f := f_{1} ⊕ … ⊕ f_{k}. In the same way, but using Proposition 5.2 and the results of section 4, one shows that there is a well defined map from M′′(E)_{g}^{0} to M_{f}(E)_{g}^{0}, associating to the class of an integrable Higgs operator d′′ with g-Einstein metric h the class of the connection D_{h} = I_{h}^{−1}(d′′); this obviously is an inverse of I.

6 Line bundles on non-Kähler surfaces.

Isomorphism classes of flat complex line bundles (L, D) on a manifold X are parametrized by H^{1}(X, ℂ^{∗}). On the other hand, an integrable Higgs operator d′′ = ¯∂ + θ in a complex line bundle L consists of a holomorphic structure ¯∂ in L and a holomorphic 1-form θ on X (the condition θ ∧ θ = 0 now is trivial). Furthermore, two integrable Higgs operators d′′_{1} and d′′_{2} in L are isomorphic if and only if the two holomorphic line bundles (L, ¯∂_{1}) and (L, ¯∂_{2}) are isomorphic and θ_{1} = θ_{2}. Hence, the space parametrizing isomorphism classes of integrable Higgs operators is H^{1}(X, O^{∗}) ⊕ H^{0}(X, Ω^{1}(X)) = Pic(X) ⊕ H^{1,0}(X). In particular, the moduli sets M_{f}(L)_{g}^{0} and M′′(L)_{g}^{0} defined in the previous section are subsets of H^{1}(X, ℂ^{∗}) resp. Pic(X) ⊕ H^{1,0}(X).

Lemma 6.1 Let L be a complex line bundle on X, and g a Hermitian metric in X. Then every flat connection in L and every integrable Higgs operator in L admits a g-Einstein metric.

Proof: Let h_{0} be fixed metric in L, then every metric is of the form h_{f} = e^{f} · h_{0} with f ∈ C^{∞}(X, ℝ). Let D be a flat connection in L; then h_{f} is a g-Einstein metric for D if and only if it is a solution of the equation iΛ_{g}G_{h_{0}} − \frac{i}{2}Λ_{g} ¯∂∂(f) = c with a real constant c. Such a solution exists by [LT] Corollary 7.2.9. A similar argument works for integrable Higgs operators. From now on let X be a surface, and g a fixed Hermitian metric in X. Then the map deg_{g} : Pic(X) ⟶ ℝ is a morphism of Lie groups ([LT] Proposition 1.3.7; recall that deg_{g} = degg̃ for some Gauduchon metric g̃). We define

H^{1}(X, ℂ^{∗})^{f} := { [(L, D)] ∈ H^{1}(X, ℂ^{∗}) | deg_{g}(D) = 0 },
Pic(X)^{T} := { [(L, ¯∂)] ∈ Pic(X) | c_{1}(L)_{ℝ} = 0 },

and

Pic(X)^{f} := ker(deg_{g} |_{Pic(X)^{T}}).

Observe that Pic(X)^{f} can be identified with the set of isomorphism classes of line bundles admitting a flat unitary connection ([LT] Proposition 1.3.13). Theorem 5.4 and Lemma 6.1 imply

Proposition 6.2 There is a natural bijection

I_{1} : H^{1}(X, ℂ^{∗})^{f} ⟶ Pic(X)^{f} × H^{1,0}(X).

If X admits a Kähler metric, i.e. if the first Betti number of b_{1}(X) is even, then deg_{g} is a topological invariant for every metric g ([LT] Corollary 1.3.12 i)). Hence in this case it holds H^{1}(X, ℂ^{∗})^{f} = H^{1}(X, ℂ^{∗}) and Pic(X)^{f} = Pic(X)^{T} , and I_{1} is the natural bijection from the moduli space of isomorphism classes of flat line bundles to the moduli space of integrable Higgs operators in line bundles with vanishing first real Chern class, which (e.g. by the work of Simpson) already is known to exist for a Kähler metric g.

So let us assume that b_{1}(X) is odd. Then deg_{g} |_{Pic^{0}(X)} : Pic^{0}(X) ⟶ ℝ is surjective, and it holds

Pic(X)^{T}/Pic(X)^{f} ≅ Pic^{0}(X)/Pic^{0}(X)^{f} ≅ ℝ

([LT] Corollary 1.3.12 and Proposition 1.3.13). We will show that I_{1} extends to a (non-natural) bijection from H^{1}(X, ℂ^{∗}) to Pic(X)^{T} ×H^{1,0}(X) in this case, too.

Lemma 6.3 There is a bijection i : Pic(X)^{T} ⟶ Pic(X)^{f} × ℝ such that the diagram

Pic(X)^{T}  ⟶^{deg_{g}} ℝ
i ↓  ∥
Pic(X)^{f} × ℝ ⟶^{proj.} ℝ

commutes.

Proof: deg_{g} |_{Pic^{0}(X)} is surjective, so we can choose L_{1} := [(L_{1}, ¯∂_{1})] ∈ Pic^{0}(X) with deg_{g}(L_{1}) = deg_{g}(¯∂_{1}) = 1, and a class α ∈ H^{1}(X, O) such that L_{1} = π(α) where π : H^{1}(X, O) ⟶ Pic^{0}(X) is the natural surjection. For λ ∈ ℝ define

L_{λ} := π(λ · α);

then deg_{g}(L_{λ}) = λ since deg_{g} ∘π : H^{1}(X, O) ⟶ ℝ is linear. Now define i by

i(L) := (L ⊗ L_{− deg_{g}(L)}, deg_{g}(L));

then it is obvious that the inverse of i is given by (L, λ) ↦ L ⊗ L_{λ}, and that the diagram above commutes. In the proof of a similar statement for H^{1}(X, ℂ^{∗}) we will use

Lemma 6.4 The natural map

l^{1} : H^{1}(X, ℂ^{∗}) ⟶ Pic(X)^{T} , l^{1}([(L, D)]) := [(L, D′′)].

is surjective, i.e. a holomorphic structure ¯∂ in a differentiable line bundle L on X is the (0,1)-part of a flat connection if and only if the real first Chern class c_{1}(L)_{ℝ} vanishes.

Proof: Pic(X)^{f} is naturally identified with H^{1}(X, U(1)), such that the inclusion Pic(X)^{f} ↪ Pic(X) becomes the injection k^{1} : H^{1}(X, U(1)) ↪ H^{1}(X, O^{∗}) ([LT] p. 38). Observe that k^{1} is the composition of the natural map H^{1}(X, U(1)) ⟶ H^{1}(X, ℂ^{∗}) and l^{1}, so it holds

Pic(X)^{f} = im(k^{1}) ⊂ im(l^{1}).

Each component of Pic(X)^{T} contains a component of Pic(X)^{f} ([LT] Remark 1.3.10), hence for each component

Pic^{c}(X) := { [(L, ¯∂)] ∈ Pic(X) | c_{1}(L)_{ℤ} = c } ⊂ Pic(X)^{T}

there exists a class [(L_{c}, D_{c})] ∈ H^{1}(X, ℂ^{∗}) such that l^{1}([(L_{c}, D_{c})]) ∈ Pic^{c}(X). Define H^{1}(X, ℂ^{∗})^{0} := { [(L, D)] ∈ H^{1}(X, ℂ^{∗}) | c_{1}(L)_{ℤ} = 0 }. The commutative diagram with exact rows

0 ⟶ ℤ ⟶ ℂ ⟶^{exp} ℂ^{∗} ⟶ 0
∥  ↓  ↓
0 ⟶ ℤ ⟶ O ⟶^{exp} O^{∗} ⟶ 0

induces the commutative diagram

H^{1}(X, ℂ) ⟶ H^{1}(X, ℂ^{∗})^{0}
h^{1} ↓  ↓ l^{1}
H^{1}(X, O) ⟶ Pic^{0}(X)

with surjective horizontal arrows. Since X is a surface, the left vertical arrow h^{1} is also surjective ([BPV] p. 117), hence l^{1} maps H^{1}(X, ℂ^{∗})^{0} surjectively onto Pic^{0}(X). Now it is easy to see that every element of Pic^{c}(X) ⊂ Pic(X)^{T} is of the form l^{1}([(L_{c} ⊗ L, D_{c} ⊗ D)]) for some [(L, D)] ∈ H^{1}(X, ℂ^{∗})^{0}.

Lemma 6.5 There is a bijection j : H^{1}(X, ℂ^{∗}) ⟶ H^{1}(X, ℂ^{∗})^{f} × ℝ such that the diagram

H^{1}(X, ℂ^{∗})  ⟶^{deg′_{g}} ℝ
j ↓  ∥
H^{1}(X, ℂ^{∗})^{f} × ℝ ⟶^{proj.} ℝ

commutes, where deg′_{g} := deg_{g} ∘l^{1} is the map associated to the g-degree of flat connections.

Proof: Choose L_{1} ∈ Pic^{0}(X), α ∈ H^{1}(X, O) as in the proof of Lemma 6.3, and a class β ∈ H^{1}(X, ℂ) with h^{1}(β) = α. Let π′ : H^{1}(X, ℂ) ⟶ H^{1}(X, ℂ^{∗}) be the map induced by exp : ℂ ⟶ ℂ^{∗}, and define L′_{1} := π′(β) ∈ H^{1}(X, ℂ^{∗}). Since the diagram

H^{1}(X, ℂ) ⟶^{π′} H^{1}(X, ℂ^{∗})
h^{1} ↓  ↓ l^{1}
H^{1}(X, O) ⟶ Pic(X)^{T}

commutes, it holds deg′_{g}(L′_{1}) = 1. The rest of the proof is as for Lemma 6.3. We conclude

Theorem 6.6 The composition

Ī: H^{1}(X, ℂ^{∗}) ⟶^{j} H^{1}(X, ℂ^{∗})^{f} × ℝ ⟶^{I_{1}×id_{ℝ}} H^{1,0}(X) × Pic(X)^{f} × ℝ

⟶^{id_{H^{1,0}(X)}×i^{−1}} H^{1,0}(X) × Pic(X)^{T}

is a bijective extension of the map I_{1}, and preserves the g-degree.

We finish with the obvious remark that the map l^{1} : H^{1}(X, ℂ^{∗}) ⟶ Pic(X)^{T} in general does not coincide with the composition of Ī and projection onto Pic(X)^{T}.

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M. Lübke Mathematical Institute Leiden University PO Box 9512 NL 2300 RA Leiden lubke@math.leidenuniv.nl