j. differential geometry

61 (2002) 341-364

Â-GENUS ON NON-SPIN MANIFOLDS WITH S^{1} ACTIONS AND THE CLASSIFICATION OF

POSITIVE QUATERNION-KÄHLER 12-MANIFOLDS

HAYDEÉ HERRERA & RAFAEL HERRERA

Abstract

We prove that the Â-genus vanishes on certain non-spin manifolds. Namely, Â(M) vanishes on any oriented, compact, connected, smooth manifold M with finite second homotopy group and endowed with non-trivial (isometric) smooth S^{1} actions. This result extends that of Atiyah and Hirzebruch on spin manifolds endowed with smooth S^{1} actions [1] to manifolds which are not necessarily spin.

We prove such vanishing by means of the elliptic genus defined by Ochanine [23, 24], showing that it also has the special property of being “rigid under S^{1} actions” on these (not necessarily spin) manifolds.

We conclude with a non-trivial application of this new vanishing theorem by classifying the positive quaternion-Kähler 12-manifolds. Namely, we prove that every quaternion-Kähler 12-manifold with a complete metric of positive scalar curvature must be a symmetric space.

Introduction

This article is divided into two parts. The first part is devoted to proving the following vanishing theorem:

Theorem 1. Let M be a 2n-dimensional, oriented, compact, connected, smooth manifold with finite second homotopy group and endowed with a smooth S^{1} action. Then,

Â(M) = 0.

The second author was partially supported by a Guggenheim Fellowship and NSF grant DMS-0204002.

Received 04/24/2002.

We prove this theorem by means of the “rigidity under S^{1} actions” of the elliptic genus [23, 24, 6, 14, 18, 31, 32] on these manifolds. This, in fact, consists of showing the rigidity of certain elliptic operators (see Theorem 3). This vanishing is new since it does not follow from results on spin (spin^{c} or spin^{h}) manifolds. The manifolds under consideration are not necessarily spin (nor spin^{c}, nor spin^{h}). Hence, Theorem 1 extends that of Atiyah and Hirzebruch on spin manifolds endowed with smooth S^{1} actions [1].

The second part of the paper is devoted to a classification problem in Riemannian Geometry. Namely, the classification of the 12-dimensional quaternion-Kähler manifolds with a complete metric of positive scalar curvature. Complete quaternion-Kähler manifolds with positive scalar curvature have only been classified in 4 dimensions by Hitchin [16], and in 8 dimensions by Poon and Salamon [27]. In this paper, we show that all such manifolds belong to the list of 12-dimensional symmetric spaces given by Wolf in [33], which is accomplished by applying the vanishing Theorem 1 at a crucial point.

Theorem 2. A complete 12-dimensional quaternion-Kähler manifold with positive scalar curvature is isometric to one of the following symmetric spaces:

1. The quaternionic projective space ℍℙ^{3}.

2. The complex Grassmannian 𝔾r_{2}(ℂ^{5}).

3. The real Grassmannian 𝔾r_{4}(ℝ^{7}).

Acknowledgements. The first named author wishes to thank Loring Tu for his encouragement and valuable discussions. The second named author wishes to thank S. Salamon, R. Lee, R. Szczarba, Y.S. Poon, B. Wong and P. Baier for valuable conversations and encouragement, as well as Yale University and IHÉS for their hospitality, financial support and stimulating research environments during the preparation of this work.

Part I

1. Â and elliptic genera on non-spin manifolds

Â-genus on non-spin manifolds 343

The main theorem of this part, Theorem 1, states the vanishing of the characteristic number Â(M) on certain non-spin manifolds. Such manifolds admit no spin structure and, therefore, have neither spin bundle, nor spinors, nor Dirac operator. This means that the characteristic number Â(M) is, a priori, a rational number and that we cannot estimate it in the usual index-theoretical way.

Instead, by noticing that the elliptic genus defined by Ochanine [23, 24] has two standard representations as Laurent series, one involving indices of (well-defined) signature operators on the manifold and the other twisted Â-genera, we study it in our context and prove a rigidity result under circle actions which eventually leads to the vanishing theorem.

The elliptic genus has been well studied on spin manifolds. Witten gave an interpretation of the elliptic genus [31, 32] as the localized value of the equivariant signature (to the fixed point set of a circle action) on the loop space of the manifold, and predicted various rigidity theorems for elliptic operators on the original spin manifold. These theorems were proved by Taubes [30], Bott and Taubes [6], and Liu [21], and have been generalized further by others [13, 17, 9].

In the same vein of ideas, we shall prove a rigidity theorem for certain elliptic operators on π_{2}-finite manifolds with circle actions (see Theorem 3), a case that has not been studied so far. The proof is carried out along the lines of [6] and the key point is to prove a couple of spin-like properties of these manifolds in Lemmas 1 and 2 by using [7].

This part is organized as follows: In §§1.1 we review the definition of the elliptic genus and state the rigidity theorem. In §§1.2-1.3 we prove the rigidity theorem, and in §§1.4 we prove our main result, Theorem 1.

1.1 Elliptic genus and rigidity

Let D: Γ(E) ⟶ Γ(F) be an elliptic operator acting on sections of the vector bundles E and F over a compact manifold M. The index of D is the virtual vector space

ind(D) = ker(D) − coker(D).

If M admits a circle action preserving D, i.e. such that S^{1} acts on E and F, and commutes with D, ind(D) admits a Fourier decomposition into complex 1-dimensional irreducible representations of S^{1}

ind(D) = ∑ a_{m} L^{m},

where a_{m} ∈ ℤ and L^{m} is the representation of S^{1} on ℂ given by e^{iθ} ↦ e^{imθ}.

Definition 1. The elliptic operator D is called rigid if a_{m} = 0 for all m ≠ 0, i.e., ind(D) consists of the trivial representation with multiplicity a_{0}.

The elliptic operator D is called universally rigid if it is rigid under any S^{1} action on M by isometries.

We shall be concerned with the elliptic operators associated with the signature operator. Let ⋀_{c}^{±} be the even and odd complex differential forms on an oriented, compact, 2n-dimensional, smooth manifold M under the Hodge ∗-operator, respectively. The signature operator

d_{s}: ⋀_{c}^{+} ⟶ ⋀_{c}^{−}

is elliptic and the virtual dimension of its index equals the signature of M, τ(M). If W is a complex vector bundle on M endowed with a connection, we can twist the signature operator to forms with values in W

d_{s} ⊗ W: ⋀_{c}^{+}(W) ⟶ ⋀_{c}^{−}(W).

This operator is also elliptic and the virtual dimension of its index is denoted by τ(M, W).

Definition 2. Let M be an oriented, compact, 2n-dimensional smooth manifold, and T = T M ⊗ ℂ its tangent bundle. Let R_{i} be the sequence of bundles defined by the formal series

R(q, T) = ∑_{i=0}^{∞} q^{i} R_{i} = ⨂_{i=1}^{∞} ⋀_{q^{i}}T ⊗ ⨂_{j=1}^{∞} S_{q^{j}}T,

where S_{t}T = ∑_{k=0}^{∞} t^{k} S^{k}T, ⋀_{t}T = ∑_{k=0}^{∞} t^{k} ⋀^{k}T, and S^{k}T, ⋀^{k}T denote the k-th symmetric and exterior tensor powers of T, respectively. The elliptic genus of M is defined as

τ_{q}(M) = τ(M, R(q, T)) = ∑_{i=0}^{∞} q^{i} · τ(M, R_{i}). (1)

The first few terms of the sequence R(q, T) are

R_{0} = 1, R_{1} = 2T R_{2} = 2(T^{⊗2} + T), …

so that, in particular, the constant term of τ_{q}(M) is τ(M).

Theorem 3 (Rigidity Theorem). Let M be an oriented, compact, connected, smooth 2n-manifold endowed with smooth S^{1} actions. In addition, assume that π_{2}(M) is finite. Then, each of the operators d_{s} ⊗ R_{i} is universally rigid.

1.2 Proof of the rigidity theorem

The manifolds under consideration posses a very special property induced by the π_{2}-finiteness, which is proved in [7] and which is used at a key point in the proof to replace the spin condition. We recall the notation from [6] and their main line of argument for the convenience of the reader and to give appropriate context to Lemmas 1 and 2 below.

Applying the Atiyah-Segal G-signature theorem [2], we have

τ_{q}(M) = ∑_{{P }} μ(P)

where P runs over the connected components of the fixed point set of the S^{1} action (cf. [6, p. 155]). The contribution μ(P) of P to τ_{q}(M) is given by the index of the signature operator on P twisted by an appropriate power series of vector bundles on P (see [6] for details).

The contributions μ(P) are meromorphic functions on the 2-dimen-sional torus 𝕋_{q^{2}} = ℂ^{∗}/q^{2} (the quotient of the multiplicative group of non-zero complex numbers ℂ^{∗} by the subgroup generated by the element q^{2} ≠ 0). The proof of the rigidity theorem is equivalent to showing that τ_{q}(M) = ∑_{{P }} μ(P) has no poles at all on 𝕋_{q^{2}}.

Define the translation t_{a}τ_{q}(M) of τ_{q}(M) by a ∈ ℂ^{∗}, to be given by the map at the character level λ ↦ aλ. The rigidity theorem for τ_{q}(M) will follow from showing that none of the translations t_{a}τ_{q}(M), by points a ∈ 𝕋_{q^{2}} of finite order, has a pole on the circle |λ| = 1. It is enough to consider k ranging over ℕ and a ranging over the roots of the form

a = α^{s}, α^{k} = q,

with k and s relatively prime. The translations t_{α^{s}}τ_{q}(M) can be expressed as twists of τ_{q} on the connected components M_{k} of the fixed point submanifold of the subgroup ℤ_{k} ⊂ S^{1}, generated by e^{2πi/k} ∈ S^{1}, which do contain fixed points of the S^{1} action. In fact, the translation t_{α^{s}}τ_{q}(M) is the index of the signature operator on M′_{k} twisted by an appropriate power series in certain bundles T_{r}, where M′_{k} is the submanifold M_{k} with a specific orientation. We explain the meaning of r and T_{r}, and the choice of orientation of M_{k} below (cf. [6, (8.13)]).

Remark. The submanifolds M_{k} are intermediate steps between M and the fixed point set {P } of the S^{1} action. Therefore, we shall be interested only in those connected components of M_{k} that contain connected components P of fixed points of S^{1} (cf. [6, p. 153]). Moreover, the submanifolds M_{k} must be orientable in order to define a signature operator and the translations t_{a}τ_{q}(M). We ensure the orientability of these components by the following lemma whose proof is postponed until Subsection 1.3:

Lemma 1. Let M be an oriented, 2n-dimensional, smooth manifold endowed with a smooth S^{1} action. Consider ℤ_{k} ⊂ S^{1} and its corresponding action on M. If k is odd then the fixed point set M_{k} of the ℤ_{k} action is orientable. If k is even and M_{k} contains a fixed point of the S^{1} action, M_{k} is also orientable.

The translations t_{α^{s}}τ_{q}(M) converge on some annulus containing the unit circle |λ| = 1 to the Laurent series of a meromorphic function on 𝕋_{q^{2}} which has no poles on the unit circle.

The bundles T_{r}

The subgroup ℤ_{k} acts on the normal bundle of M_{k} in M so that T = T M splits over M_{k} as

T |_{M_{k}} = T M_{k} ⊕ T_{1}^{#} ⊕ … ⊕ T_{k/2−1}^{#} ⊕ T_{[k/2]}^{#}, (2)

where each T_{r}^{#} is an irreducible real representation of ℤ_{k}, and [k/2] is the greatest integer smaller than or equal to k/2. The S^{1} action on M induces an S^{1} action on M_{k}, whose differential induces an action on T |_{M_{k}}, preserving the decomposition, and making each T_{r}^{#} an S^{1} bundle over M_{k}, for r = 1, …, [(k − 1)/2]. Each T_{r}^{#}, with r ≠ k/2 if k is even, is endowed with a complex structure such that λ ∈ S^{1} acts by λ^{r}, for r = 1, …, [(k − 1)/2]. Hence, T_{r}^{#} comes from a complex vector bundle T_{r}. For k even, the action on T_{k/2}^{#} = T_{k/2} is multiplication by −1, and it does not necessarily come from a complex vector bundle.

The T_{r}^{#} inherit an orientation from the complex structure on T_{r}, for r = 1, …, [(k − 1)/2]. Hence, if k is odd, T M_{k} has an induced orientation. If k is even, however, we only know that T M_{k} ⊕ T_{k/2} is orientable. On the other hand, Lemma 1 guarantees that M_{k} is also orientable. Let us, therefore, choose an orientation. In this way, T_{k/2} inherits an orientation from M and M_{k}.

The orientation of M′_{k}

Observe that M is oriented and that the submanifolds P and M_{k} are orientable. The orientation of M_{k} should be chosen to be compatible with the orientations of the components P as follows: Let P be a connected component of the fixed point set of the S^{1} action. Along P

T M|_{P} = T P ⊕ ⨁_{i=1}^{l} E_{i}^{#},

where E_{i}^{#} denotes the canonical underlying real bundle of the complex bundle E_{i} on which S^{1} acts by sending λ to λ^{m_{i}} (|m_{i}| ≠ |m_{j}| unless i = j). Let d_{i} = rank_{ℂ}(E_{i}).

When k is odd, the decomposition in (2) determines an orientation on T M_{k} denoted by +1. If P ⊂ M_{k}, choose the exponents along P so that each m_{j} ≢ 0 (mod k) is congruent to some r ∈ {1, …, (k − 1)/2}. Choose the orientation of T P and the sign of those m_{j} ≡ 0 (mod k) so that the induced orientation on T M|_{P} is the given one. The induced orientation on T M_{k}|_{P} will be the +1 orientation. For each m_{j}, let (l_{j}, ω_{j}) ∈ ℤ × {1, …, k − 1} be such that

s · m_{j} = l_{j} · k + ω_{j}, (3)

and define

ε(P) = ∑_{j} d_{j} · l_{j}. (4)

The orientation for M′_{k} is now defined as +1 · (−1)^{ε(P)}, if M_{k} ⊇ P. Lemma 2 below ensures that this orientation is well defined.

When k is even, T M|_{M_{k}} decomposes according to (2). Since ⨁_{r=1}^{k/2−1} T_{r}^{#} is naturally oriented, T M_{k} ⊕ T_{k/2} inherits an orientation. Let us choose an orientation for T M_{k} and call it +1, which induces an orientation on T_{k/2}. If P ⊂ M_{k}, select the exponents at P as follows: If m_{j} ≢ 0, k/2 (mod k), make the choice as before so that (m_{j})_{mod k} ∈ {1, … k/2−1}. Choose the signs for those m_{j} ≡ 0, k/2 (mod k) and the orientation of T P to make the induced orientation of (T M_{k} ⊕ T_{k/2})|_{P} correct. This ensures that the induced orientation of T M|_{P} is correct. The induced orientation of T M_{k}|_{P}, however, may not be the correct one (+1). Let ε_{0} = 0, 1, with ε_{0} = 0 if the induced orientation on T M_{k}|_{P} is correct, and ε_{0} = +1 if the induced orientation on T M_{k}|_{P} is incorrect. For each m_{j}, define (l_{j}, ω_{j}) by (3) and set

ε(P) = ε_{0} + ∑_{j} d_{j} · l_{j}. (5)

The orientation of M′_{k} is again +1·(−1)^{ε(P)}, if M_{k} ⊃ P. This orientation is well defined by the following lemma, whose proof is postponed until Subsection 1.3:

Lemma 2. Let M be an oriented, compact, 2n-dimensional, smooth manifold with π_{2}(M) finite and endowed with a smooth S^{1} action. Let k ∈ ℕ and M_{k} be a connected component of the fixed point set of ℤ_{k} ⊂ S^{1}. Let s ∈ ℤ be relatively prime to k, and P, P′ ⊂ M_{k} be connected fixed point submanifolds of the S^{1} action. Use the prescription above,

(4) or (5) respectively, to define the numbers ε(P) and ε(P′). Then (−1)^{ε(P)} = (−1)^{ε(P′)}.

Provided with all the conditions above, the function of q

t_{α^{s}}τ_{q}(M),

is regular on an annulus containing the unit circle for all k ∈ ℕ, so that t_{α^{s}}τ_{q}(M) has no poles on the unit circle |λ| = 1. Hence, τ_{q}(M) has no poles at all on 𝕋_{q^{2}}, and must be constant.

1.3 The lemmas

Remark. Notice that Lemma 1 is valid on any compact smooth manifold with a circle action. It does not require any special condition on neither the fundamental group, nor (co)homology, nor homotopy type.

Remark. The content of the Lemma 2 essentially says that a compact smooth non-spin manifold with π_{2} finite behaves under circle actions in the same way as (or better than) spin manifolds.

Proof of Lemma 1. We begin by recalling an argument from [30, pp. 488–489]. First, note that the case when k is odd follows from (2).

Let k = 2 and denote the S^{1} action on M by φ: S^{1} × M ⟶ M. Let T_{0} = T M_{2} and T_{1} be the normal bundle of M_{2} (with respect to an appropriate metric on the fibers). Since φ(π, ·) is the identity on M_{2}, φ(π, ·)_{∗}: T_{0} ⟶ T_{0} is the identity map, and φ(π, ·)_{∗}: T_{1} ⟶ T_{1} is the involution induced by the ℤ_{2} action. Given an S^{1} invariant metric on M, there is a metric on T_{1} with respect to which φ(π, ·)_{∗} = −I, where I is the identity automorphism.

Suppose that T_{0} is not orientable. Then there exists a loop in M_{2}, f: S^{1} ⟶ M_{2}, on which T_{0} is not trivial, i.e. w_{1}(f^{∗}T_{0}) ≠ 0 and f^{∗}T_{0} ≅ (S^{1} ×_{ℤ_{2}} ℝ) × ℝ^{l−1}, where l = dim M_{2}. Similarly, since w_{1}(T_{0}) = w_{1}(T_{1}) f^{∗}T_{1} ≅ (S^{1} ×_{ℤ_{2}} ℝ) × ℝ^{m−1}, where m = rank(T_{1}).

With the help of φ and f we define another map

f_{1} : S^{1} × S^{1} ⟶ M_{2},

(t, s) ↦ f_{1}(t, s) = φ(t/2, f(s)),

which represents a 2-torus mapped into M_{2} ⊂ M, by letting the original loop follow its orbit under the S^{1} action. For fixed s, f_{1}(·, f(s))^{∗}T_{0} = f(s)^{∗}φ_{∗}T_{0}, so that

f_{1}^{∗}T_{0} ≅ S^{1} × (S^{1} ×_{ℤ_{2}} ℝ) × ℝ^{l−1}. (6)

For the bundle T_{1}, f_{1}(·, f(s))^{∗}T_{1} = f(s)^{∗}φ_{∗}T_{1}, so that

f_{1}^{∗}T_{1} ≅ S^{1} ×_{ℤ_{2}} ((S^{1} ×_{ℤ_{2}} ℝ) × ℝ^{m−1}) , (7)

where ℤ_{2} acts on ((S^{1} ×_{ℤ_{2}} ℝ) × ℝ^{m−1}) as multiplication by ±1.

The ℤ_{2} cohomology of S^{1} × S^{1} has generators z_{1}, z_{2} which restrict trivially to the second S^{1} and to the first S^{1}, respectively. The total Stiefel-Whitney class of f_{1}^{∗}T_{0} can be obtained from (6)

w(f_{1}^{∗}T_{0}) = 1 + z_{2}, (8)

and that of f_{1}^{∗}T_{1} from (7)

w(f_{1}^{∗}T_{1}) = 1 + m_{mod(2)} · z_{1} + z_{2} + (m − 1)_{mod(2)} · z_{1} ∧ z_{2}. (9)

From (8) and (9), the total Stiefel-Whitney class of f_{1}^{∗}(T M)

w(f_{1}^{∗}T M) = 1 + m_{mod(2)} · z_{1} + (m_{mod(2)} + (m − 1)_{mod(2)}) · z_{1} ∧ z_{2}, (10)

which implies

w_{2}(f_{1}^{∗}T M) ≠ 0. (11)

On the other hand, the 2-torus is homotopic to a sphere with S^{1} acting on it by rotations. Indeed, let p ∈ M_{2} be a fixed point of the S^{1} action, and let p′ be a point in the original loop f(S^{1}). Consider a path γ in M_{2} joining p and p′. Let S^{1} act on γ to generate a disk, whose “central” point is p. This disk is attached to the 2-torus by a circle (its border) which is an orbit of the S^{1} action. We can think of it as the image of an equivariant map from the 2-sphere to M, where S^{1} acts on S^{2} by rotations with its north and south poles as fixed points being mapped to the same point p. Let us denote this map by f_{2} : S^{2} ⟶ M_{2}, and note that

w_{2}(f_{2}^{∗}T M) = w_{2}(f_{1}^{∗}T M) ≠ 0 (12)

by (11).

Observe that w_{2}(f_{2}^{∗}T M) is now the reduction mod 2 of the Chern class c_{1}(f_{2}^{∗}T M) on S^{2} (cf. [6, Lemmas 9.1]). Furthermore, c_{1}(f_{2}^{∗}T M) is even by [6, Lemmas 9.2], since the north and south pole of our sphere are mapped to the same point. This contradicts (12) (and (11)).

For even k > 2, use the same arguments together with the fact that c_{1}(E) = 0 for any bundle E ⟶ S^{1} × S^{1} to which the rotations around an S^{1} lift. q.e.d.

Sketch of proof of Lemma 2. Since Lemma 2 is the analogue of [6, Lemma 8.1] in our set-up, we shall only point out the relevant change in the proof.

Let k ∈ ℕ, P and P′ be two distinct connected fixed point submanifolds of S^{1} in M_{k}. Let p ∈ P and p′ ∈ P′. Consider a path joining p and p′ which avoids other fixed points of S^{1}. Let S^{1} act on it to generate a sphere with “north” and “south” poles p and p′ respectively. Let the sets of integers {m_{i}} and {m′_{i}} denote the exponents of the S^{1} action on T_{p}M and T_{p′}M respectively.

By [6, Lemma 9.1], the number

ε(P) − ε(P′) ≡ c · (∑_{i}(m_{i} − m′_{i})) (mod 2),

where c is a constant. In other words, we must check that the parity of this sum of exponents is even. This is an immediate consequence of the following theorem by G. Bredon:

Theorem 4 ([7]). Let N be a smooth manifold with a smooth S^{1} action and assume that π_{2i}(N) is finite for 1 ≤ i ≤ r − 1. Let x and y be two fixed points of the S^{1} action. Then T_{x}N − T_{y}N is divisible by (1 − L)^{r} in the representation ring R(S^{1}), where L is the standard representation of S^{1}.

In our set up π_{2}(M) is finite, then the virtual representation T_{p}M − T_{p′}M decomposes as follows:

T_{p}M − T_{p′}M = (1 − L)^{2} ⊗ (∑_{j} b_{j}L^{j}) ,

where L ≅ ℂ is the standard representation of S^{1} and {b_{j} ∈ ℤ} is a finite set of integers. This means that

∑_{i}(m_{i} − m′_{i}) = ∑_{j} b_{j} · j − 2 ∑_{j} b_{j} · (j + 1) + ∑_{j} b_{j} · (j + 2) = 0,

which proves the assertion. q.e.d.

Let us contrast the content of the previous paragraph with the following example of a manifold with smooth S^{1} actions, but with infinite π_{2}:

Example. Given our interest in 12-dimensional manifolds in Part II, let us consider the complex S^{1} representation V = L^{0} ⊕ L^{1} ⊕ L^{2} ⊕ L^{3} ⊕ L^{4}, with the obvious S^{1} action on each complex line. The complex Grassmannian 𝔾r_{2}(V) is non-spin and has an induced S^{1} action with fixed points. We have, for instance, the following fixed points with their corresponding exponents:

• The complex 2-plane p_{1} = L^{0} ⊕ L^{1} with exponents (1, 2, 2, 3, 3, 4), which add up to 15.

• The complex 2-plane p_{2} = L^{0}⊕L^{2} with exponents (−1, 1, 1, 2, 3, 4), which add up to 10.

The difference of the sums of the exponents is 5 ≢ 0 (mod 2), failing to satisfy Lemma 2. This is due to the fact that 𝔾r_{2}(V) is neither spin nor has finite π_{2}. In fact, π_{2}(𝔾r_{2}(V)) = ℤ. Furthermore, one can check directly that 𝔾r_{2}(V) does not satisfy Theorem 1

Â(𝔾r_{2}(V)) ≠ 0.

1.4 Vanishing of the Â-genus

As a consequence of the rigidity theorem, we get the vanishing Theorem 1.

Remark. The manifold M may be neither spin, nor spin^{c}, nor spin^{h}. In such a case, Â(M) is only defined as a characteristic number and does not represent the index of an elliptic operator. Thus, Â(M) does not even have to be an integer. The prescribed homotopy type, however, yields this number as zero.

Proof of Theorem 1. We can assume that dim(M) = 4n. Given that S^{1} acts on M, the equivariant genus τ_{q}(M)_{g} is defined for any g ∈ S^{1} as

τ_{q}(M)_{g} = ∑ τ(M, R_{i})_{g} · q^{i},

where τ(M, R_{i})_{g} = tr|_{g} ker(d_{s}⊗R_{i})−tr|_{g}coker(d_{s}⊗R_{i}). The coefficients of its q-development are now equivariant twisted signatures. Thus, according to Theorem 3, the value of τ_{q}(M)_{g} does not depend on g.

Applying the Atiyah-Segal G-signature theorem [2], τ_{q}(M)_{g} can be expressed in terms of the fixed point set M^{g} of g and the action of g on the normal bundle of M^{g} ⊂ M. In particular, let g be the orientation preserving involution in ℤ_{2} ⊂ S^{1}. Note that the submanifold M^{g} = M_{2} may not be connected. We denote the transversal self-intersection of M_{2} by M_{2} ∘ M_{2}. In [15, p. 315], Hirzebruch and Slodowy showed that

τ_{q}(M)_{g} = τ_{q}(M_{2} ∘ M_{2}).

On the other hand, applying the rigidity theorem (Theorem 3), τ_{q}(M) = τ_{q}(M)_{g}, i.e.

τ_{q}(M) = τ_{q}(M_{2} ∘ M_{2}). (13)

The codimension of M_{2} is positive and even, so that the elliptic genus τ_{q}(M) can be computed from the elliptic genera of submanifolds of M of codimension at least 4.

Let us now recall the expansion of τ_{q}(M) at the other cusp [14]

τ̃_{q}(M) = \frac{1}{q^{n/2}} ∑_{j=0}^{∞} Â(M, R′_{j}) · q^{j},

where R′_{j} is the sequence of virtual tensor bundles given by

R′(q, T) = ⨂_{k=2m+1} ⋀_{−q^{k}}T ⊗ ⨂_{k=2m+2} S_{q^{k}}T,

and Â(M, R′_{j}) = ⟨Â(M) · ch(R′_{j}), [M]⟩ may only be defined as a characteristic number (since M will not, in general, be spin). The first few terms of the sequence are

R′_{0} = 1, R′_{1} = −T, R′_{2} = ⋀^{2}T + T, …

This expansion is obtained by considering q = e^{πit} and changing the t coordinate in (1) by t → −1/t, and then by t → 2t (cf. [14]). This expansion has, a priori, a pole of order n/2.

On the other hand, by (13) we also have the following:

τ̃_{q}(M) = τ̃_{q}(M_{2} ∘ M_{2}), (14)

whose right hand side has a pole of order at most (n/2 − 1/2), since the dimension of any connected component of M_{2} ∘ M_{2} is at most 4n − 4. Therefore (14) implies that the first coefficient on the left hand side vanishes, i.e.,

Â(M) = 0.

q.e.d.

Example. It is known that for every m ∈ ℕ, the real 8m + 4dimensional Grassmannians 𝔾r_{4}(ℝ^{2m+5}) of 4-planes in ℝ^{2m+5} are nonspin and π_{2}(𝔾r_{4}(ℝ^{2m+5})) = ℤ_{2}. The Â-genus can be computed explicitly

Â(𝔾r_{4}(ℝ^{2m+5})) = 0,

in accordance with our theorem. In fact, not only the Â-genus vanishes on these manifolds, but the entire elliptic genus [12]

τ_{q}(𝔾r_{4}(ℝ^{2m+5})) = 0.

Example. Borel and Hirzebruch proved in [5, Theorm 23.3.(iii)] the following result: Let G be a compact connected Lie group and U a closed connected subgroup of G containing a maximal torus. If the second Betti number of G/U is zero, then

Â(G/U) = 0.

Notice that the space G/U is not necessarily spin. Hence, it remains an interesting problem to determine whether the other coefficients of the elliptic genus have any integrality properties on the non-spin spaces.

Furthermore, given the subject of interest in Part II we have the following corollary:

Corollary 1. Let M be an oriented, compact, connected, smooth 12dimensional manifold with π_{2}(M) finite and endowed with a smooth S^{1} action. Suppose that M_{2}, the fixed point manifold of ℤ_{2} ⊂ S^{1}, contains no connected component of dimension 8. Then,

τ_{q}(M) ≡ 0,

and, in particular,

τ(M) = 0.

Proof. This follows from (14) since the Laurent expansion on the right hand side has no half-integral powers of q at all, which means that the genus must vanish identically. q.e.d.

The theorems in this first part can be generalized further. For instance, one can consider other versions of the elliptic genus twisted with more general vector bundles on the manifolds and also prove rigidity theorems with respect to smooth S^{1} actions. Furthermore, it is clear that the more information one obtains at the level of fixed point sets, the more vanishing theorems one may deduce. Along these lines, there is room for an analysis similar to that of Hirzebruch and Slodowy [15] for homogeneous spaces with vanishing second Betti number.

We shall explore these possibilities in the near future. In the meantime, let us pass to Part II to give a non-trivial application of the vanishing Theorem 1.

Part II

2. Quaternion-Kähler 12-manifolds

Let us consider ℝ^{4n} ≅ ℍ^{n} as a right module over the quaternions ℍ, whose elements are column vectors with entries in ℍ. Let A ∈ Sp(n), q ∈ Sp(1) and

μ: Sp(n) × Sp(1) ⟶ GL(n, ℍ) ⊆ GL(4n, ℝ)

be the representation defined by μ(A, q)(X) = AXq^{−1}, for X ∈ ℍ^{n}. Since ker(μ) = ℤ_{2}, we define Sp(n)Sp(1) = Sp(n) ×_{ℤ_{2}} Sp(1), where we identify Sp(n) and Sp(1) with subgroups of GL(4n, ℝ) so that

Sp(n)Sp(1) ⊂ SO(4n) ⊂ GL(4n, ℝ).

An oriented, connected, irreducible, Riemannian 4n-manifold M is called a quaternion-Kähler manifold, n ≥ 2, if its linear holonomy is contained in Sp(n)Sp(1). We shall call M positive if its metric is complete and has positive scalar curvature. When n = 1 we add the condition that the manifold M must be Einstein and self-dual, since Sp(1)Sp(1) = SO(4).

A quaternion-Kähler manifold is Einstein, which renders three different types according to the scalar curvature being positive, negative or zero. The quaternion-Kähler manifolds with zero scalar curvature happen to be locally hyperkähler.

In the case of positive scalar curvature, Wolf showed in [33] that each compact centerless Lie group G is the isometry group of a quaternion-Kähler symmetric space given as the conjugacy class of a copy of Sp(1) in G, determined by a highest root of G. They are called “Wolf spaces” and are the only known examples with complete metrics of positive scalar curvature. Moreover, we know that there are only finitely many positive quaternion-Kähler manifolds for each n (cf. [20]). These facts have given some support to the following conjecture:

Conjecture 1. Every positive quaternion-Kähler manifold is isometric to a (symmetric) Wolf space.

Two decades ago, Hitchin proved in [16] that this conjecture is true in 4 dimensions (n = 1), namely that the 4-manifold must be isometric to S^{4} or ℂℙ^{2}. A decade ago, Poon and Salamon proved it in 8 dimensions (n = 2, see [27]). They proved that a positive 8-dimensional quaternion-Kähler manifold M is isometric to the quaternionic projective space ℍℙ^{2}, or the complex Grassmannian 𝔾r_{2}(ℂ^{4}), or the exceptional space G_{2}/SO(4). They carried out a careful study of the standard twistor space Z of M as a polarized algebraic variety, and were able to pin down the few candidates of polarized varieties that can occur as twistor spaces. In the last few years, more supporting evidence for the conjecture has been produced, such as the results in [20, 4, 3, 26].

By using Theorem 1, we are now able to produce a proof of the conjecture in 12 dimensions (n = 3). Theorem 2 states that a positive quaternionic Kähler 12-manifold must be a Wolf space.

Our approach is very different from those in [16, 27], since the main ingredients turn out to be topological and differential geometric, rather than algebraic geometric. Moreover, the study is carried out almost entirely on the (Riemannian) quaternion-Kähler manifolds, not on their twistor spaces.

Part II is organized as follows: In §§2.1 we review preliminaries of quaternionic Kähler geometry. In §§2.2 we review the index theory of certain elliptic operators on quaternion-Kähler manifolds. In §§2.3 we give the proof of Theorem 2, and some immediate applications.

2.1 Preliminaries on quaternion-Kähler geometry

The existence of the Sp(3)Sp(1)-structure induces an isomorphism

T^{∗}M ⊗ ℂ ≅ E ⊗ H,

where E and H denote the locally defined vector bundles over M associated to the standard (faithful) complex representations of Sp(3) and Sp(1) on E = ℂ^{6} and H = ℂ^{2}, respectively.

Consider the globally defined bundle Z = ℙ(H) ⟶^{π} M with fiber ℂℙ^{1}, usually called the standard twistor space of M. It parameterizes orthogonal almost-complex structures at each tangent space of M, and Z itself is a complex manifold. When the scalar curvature is positive, Z has a canonical Kähler structure.

The quaternionic structure of M is characterized by a 4-form u coming from the second Chern class of the quaternionic line bundle H, i.e., u = −c_{2}(H). The multiple 4u = −c_{2}(S^{2}H) is integral and nondegenerate [28], and we shall call it the quaternionic class. There is a complex line bundle L ⟶ Z such that locally π^{∗}H ≅ L^{1/2} ⊕ L^{−1/2} and l^{2} = 4π^{∗}u, where l = c_{1}(L). Such a class l is represented by a Kähler form on Z, which gives the following result to be used in Theorem 2, communicated to the authors by S. Salamon (see [22, 29] for similar results):

Lemma 3. Let M be a compact connected quaternion-Kähler 12manifold of positive scalar curvature. The symmetric bilinear form Q on H^{4}(M) defined by

Q(α, β) = ∫_{M} α ∧ β ∧ (4u),

where α, β ∈ H^{4}(M), is positive definite.

Proof. This result follows from the Hodge-Riemann bilinear relations [11] on the twistor space Z of M. Indeed, since Z is a ℂℙ^{1}-fibration over M

H^{4}(Z) ≅ π^{∗}(H^{4}(M)) ⊕ {l} ⊗ π^{∗}(H^{2}(M))

and, at the same time,

H^{4}(Z) = P^{2,2} ⊕ L^{1}(P^{1,1}) ⊕ {l^{2}}

is the Lefschetz decomposition into primitive subspaces, where L^{1} represents the map given by wedging with the Kähler form. Clearly, π^{∗}(H^{4}(M)) ≅ P^{2,2} ⊕ {l^{2}} so that by twistor transform

∫_{M} α ∧ β ∧ (4u) = \frac{1}{2} ∫_{Z} π^{∗}(α) ∧ π^{∗}(β) ∧ l^{3},

for α, β ∈ H^{4}(M), and the result follows from the fact that this bilinear form is positive definite on P^{2,2} ⊕ {l^{2}}. q.e.d.

Remark. Here, we have restricted the statement to dimension 12 although it is valid in higher degrees and dimensions.

Given the content of Conjecture 1, it is clear that the (real) dimension d of the isometry group G of M is a key differential geometric invariant to be determined. Indeed, in 8 dimensions, d played a central role in the classification and it was seen to be the only relevant characteristic number of M. Similarly, d also plays a central role in 12 dimensions, as well as the pairing

v(M) = ⟨(4u)^{3}, [M]⟩

which we shall call the quaternionic volume, and which now constitutes a separate parameter.

2.2 Quaternionic spinors

Let Δ be the 2^{6}-dimensional faithful spin representation of Spin(12). The representation Δ splits as

Δ = Δ_{+} ⊕ Δ_{−},

where Δ_{±} are two copies of the 2^{5}-dimensional irreducible representation of Spin(11) ⊂ Spin(12). Clifford multiplication of an element of Δ_{+} by an element of T = T^{∗}M gives an element of Δ_{−}.

There is an anti-linear mapping

σ: Δ ⟶ Δ with σ^{2} = (−1)^{3} = −1

which allows us to see Δ as a complex space underlying a quaternionic one (σ = j). Combining σ with an invariant Hermitian metric on Δ, we get an equivariant isomorphism Δ ≅ Δ^{∗}.

The irreducible representations of Sp(1) are the symmetric tensor powers S^{q}H of H = ℂ^{2}, with dim(S^{q}H) = q+1. The tensor products of these representations behave according to the Clebsch-Gordan formula

S^{j}H ⊗ S^{k}H ≅ ∑_{r=0}^{min(j,k)} S^{j+k−2r}H,

and to the K-theory formula

(H − 2)^{⊗m} = ∑_{j=0}^{m}(−1)^{j} \{\binom{2m}{j} − \binom{2m}{j − 2}\} S^{m−j}H, (17)

which can be easily proved by induction.

We can define the Dirac operator with coefficients in S^{q}H by (cf. [20])

D(S^{q}H): Γ(Δ_{+} ⊗ S^{q}H) ⟶ Γ(Δ_{−} ⊗ S^{q}H),

with index

f(q) = indD(S^{q}H) = ⟨Â(M) · ch(S^{q}H), [M]⟩, (18)

provided that q ≥ 0 and 3+q is even. The parity condition ensures that the corresponding coupled Dirac operator is globally defined. Moreover, f(q) is a polynomial in q.

Theorem 5 ([20, 28]). Let M be a 12-dimensional positive quaternion-Kähler manifold. Then

π_{1}(M) = 0,

π_{2}(M) = \{ \begin{matrix}0 & iff M is homothetic to ℍℙ^{3}, \\ ℤ & iff M is homothetic to 𝔾r_{2}(ℂ^{5}), \\ finite with 2-torsion & otherwise. \end{matrix}

and M is spin if and only if M is homothetic to ℍℙ^{3}.

The polynomial f(q) satisfies

f(−q − 2) = −f(q),

and

f(q) = \{ \begin{matrix}0 & if q = −3, −1, 1, \\ 1 & if q = 3, \\ d = dim(G) & if q = 5. \end{matrix}

2.3 Proof of the classification theorem

Note that Theorem 5 does not provide any relation between the parameters d and v. Thus, the strategy of the proof is to pin down the pairs (d, v) that can occur for actual positive quaternion-Kähler manifolds. Applying Theorem 5, we can assume that π_{2}(M) is finite, i.e., M ≅ 𝔾r_{2}(ℂ^{5}). We know that d ≥ 5 (cf. [28]), so that M admits smooth S^{1} actions.

We know that v is an integer and v ≥ 1 since 4u = −c_{2}(S^{2}H) is an integral cohomology class. Let us consider the (locally defined) virtual bundle H − 2 whose Chern character is

ch(H − 2) = u + \frac{1}{12}u^{2} + \frac{1}{360}u^{3},

since −u = c_{2}(H). Clearly,

⟨Â(M) · ch((H − 2)^{⊗3}), [M]⟩ = \frac{v}{64},

⟨Â(M) · ch((H − 2)^{⊗4}), [M]⟩ = 0, and

⟨Â(M) · ch((H − 2)^{⊗5}), [M]⟩ = 0.

Applying formula (17) to these identities, gives the following system of equations:

(19) \frac{v}{64} = f(3) − 6f(2) + 14f(1) − 14f(0),
0 = f(4) − 8f(3) + 27f(2) − 48f(1) + 42f(0),
0 = f(5) − 10f(4) + 44f(3) − 110f(2) + 165f(1) − 132f(0).

The key point in the proof is to find another zero of f(q).

Lemma 4. Let M be a positive quaternion-Kähler 12-dimensional manifold different from 𝔾r_{2}(ℂ^{5}). Then

f(0) = 0.

Proof. We know that π_{2}(M) is finite and that M admits smooth S^{1} actions. Thus, by Theorem 1

Â(M) = 0,

which means f(0) = 0. q.e.d.

Remark. Clearly, the conclusion of Lemma 4 also holds for any positive quaternion-Kähler manifold different from the complex Grassmannian and which admits a smooth S^{1} action.

The system (19) therefore reduces to

v = \frac{12}{5} d − \frac{112}{5}. (20)

This relation between d and v greatly reduces the problem of determining the possible values of (d, v). In particular, it implies that d ≥ 11, already hinting that the manifold must be homogeneous.

Hence, Equation (20) gives us the following list of integral pairs:

(i) (d, v) = (11, 4),

(ii) (d, v) = (16, 16),

(iii) (d, v) = (21, 28), (iv) (d, v) = (26, 40),

(v) (d, v) = (31, 52),

(vi) (d, v) = (36, 64).

The pair (vi) must give rise to the quaternionic projective space, since the isotropy group K at any point of M must have dimension at least 24 and is contained in Sp(3)Sp(1).

The pairs (i) and (ii) are ruled out since the quadratic form Q in Lemma 3 is positive definite, i.e.,

v^{2} − 64v − 16vd + 576 − 288d + 36d^{2} < 0.

The pairs (iv) and (v) are ruled out since there are no semi-simple Lie groups of the given dimension and rank less than or equal to 3, the upper bound found by Bielawski [4].

The pair (iii) is the only one in which we have to do more work. Since d = 21, the isotropy group K at any point must have dimension 9 ≤ dim(K) ≤ 21. By [28], the curvature tensor R of M splits as R = tR_{0} + R_{1}, where R_{0} is the curvature tensor of quaternionic projective space, t is the scalar curvature, and R_{1} is a section of S^{4}E. Therefore, in order to measure the covariant derivative ∇R it suffices to look at ∇R_{1} as a section of S^{5}E ⊗ H, which must be invariant under K.

Consider the map given by the composition of the isotropy representation and projection onto the second factor

Lie(K) ⟶ 𝔰𝔭(3) ⊕ 𝔰𝔭(1) ⟶ 𝔰𝔭(1).

There are two cases: Either the map is surjective or is identically zero. By checking the possible groups K, we see that if the map is surjective, there are no groups K with invariants in S^{5}E ⊗ H, so ∇R_{1} = 0. On the other hand, if the map is zero, the only candidate for K with possible invariants in S^{5} ⊗ H is Sp(2), so that the orbits of G on M have dimension 11. This is not possible due to the classification of cohomogeneity one quaternion-Kähler manifolds in [8, 25]. q.e.d.

As we have seen, the new vanishing Theorem 1 has a powerful application to geometry. The keys to the proof of the vanishing are the rigidity property under smooth S^{1} actions and the dimension of the fixed point manifolds of S^{1}. This situation provides strong evidence for Conjecture 1 since it is feasible to gain very detailed information on fixed point manifolds of S^{1} actions (see [3, 26]) in the quaternion-Kähler setup. It is therefore feasible to obtain further vanishing theorems on these manifolds, which will completely pin down their topology and geometry (see Corollary 1).

We conclude by simply stating immediate applications of our result to certain manifolds which fiber over positive quaternion-Kähler 12manifolds. By using [19, 10] we have the following corollaries:

Corollary 2. Let Z be a (compact) Fano contact manifold of complex dimension 7, which admits a Kähler-Einstein metric. Then Z is a homogeneous space, isometric to the twistor space of a Wolf space.

Corollary 3. Let S be a compact regular 3-Sasakian 15-manifold. Then S is homogeneous.

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