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*COMPACT RIEMANNIAN 7-MANIFOLDS WITH HOLONOMY . I COMPACT RIEMANNIAN 7-MANIFOLDS WITH HOLONOMY G2. I*

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COMPACT RIEMANNIAN

7-MANIFOLDS WITH HOLONOMY G2. I

DOMINIC D. JOYCE

published as J. Differential Geometry 43 (1996), 291–328.

1. Introduction

The list of possible holonomy groups of Riemannian manifolds given by Berger [3] includes three intriguing special cases, the holonomy groups G2, Spin(7) and Spin(9) in dimensions 7, 8 and 16 respectively. Subsequently [1] it was shown that Spin(9) does not occur as a non-symmetric holonomy group, but Bryant [5] showed that both G2 and Spin(7) do occur as non-symmetric holonomy groups. Bryant’s proof is a local one, in that it proves the existence of many metrics of holonomy G2 and Spin(7) on small balls in R7 and R8 respectively. He also gives some explicit examples of such metrics. In a subsequent paper [6], Bryant and Salamon construct complete metrics of holonomy G2.

This is the first of two papers in which we shall construct examples of compact Riemannian 7-manifolds with holonomy G2. These are, to our knowledge, the first such examples known. We believe that they are the first nontrivial examples of odd-dimensional, compact, Ricci-flat Riemannian manifolds. The author has also used similar methods to construct compact 8-manifolds with holonomy Spin(7), and is preparing a paper [12] giving the proof.

The goal of this first paper is to study a single example in depth. We shall describe a certain simply-connected, compact 7-manifold M , and construct a family of metrics on it with holonomy G2. The 7-manifold M was chosen because it is the simplest example that we know of. The content of the paper is mostly introductory material, and proofs using a lot of analysis. The second paper will describe many different compact 7-manifolds admitting metrics of holonomy G2, and will have a more topological emphasis. It will also contain much more discussion of the results, and some interesting questions.

The method we shall use to construct Riemannian 7-manifolds with holonomy G2 is modelled on the Kummer construction for K3 surfaces (described in §1.3), which is a way of approximating some of the family of metrics of holonomy SU(2) on the K3 surface, that exist by Yau’s proof of the Calabi conjecture [19]. We begin with a flat Riemannian 7-torus T 7, divide by the action of a finite group Γ, and then resolve the resulting singularities in a certain special way to get a compact, nonsingular 7-manifold M . The finite group preserves a flat G2- structure on T 7, and a family of metrics are defined on M modelled on the flat G2- structure on T 7/Γ, that are close to having holonomy G2 in a suitable sense. It is then shown using analysis that these metrics can be deformed to metrics that do have holonomy G2.

Typeset by AMS-TEX 1

The paper is divided into three chapters. This first chapter contains introductory sections §1.1 on the geometry of the holonomy group G2, §1.2 on the analytic tools and results that will be needed later, and §1.3 on hyperkähler 4-manifolds and K3 surfaces. Chapter 2 defines a compact 7-manifold M and a family of G2- structures ϕt on M depending on a parameter t. The main results of the paper are then stated in §2.3. They are divided into three theorems, Theorems A-C. From Theorems A- C we deduce that M has a smooth family of metrics of holonomy G2, and this is the goal of the paper. The third chapter is then devoted wholly to the proofs of Theorems A-C of §2.3.

Now a G2- structure on a 7-manifold M defines a metric g and hence a Levi- Civita connection ∇. There is a geometric invariant of the G2- structure called the torsion, which measures how far the G2- structure is from being preserved by ∇. The G2- structure comes from a metric g with holonomy contained in G2 if and only if the torsion is zero, and the G2- structure is then called torsion-free. Theorem A of §2.3 shows that under certain conditions, a G2- structure ϕ with small torsion on a compact 7-manifold M may be deformed to a torsion-free G2- structure ϕ̃. Theorem B shows that the conditions of Theorem A apply to the G2- structures ϕt on M of §2.2, for small enough t. Theorem C shows that a torsion-free G2- structure on a compact 7-manifold is part of a smooth family of diffeomorphism classes of torsion-free G2- structures, with dimension b3(M).

The most important issue behind the proofs of Theorems A and B is the fol- lowing. The G2- structures of Chapter 2 are defined by smoothing off a singular metric (from the singular manifold T 7/Γ) to make it nonsingular, and they depend on a parameter t, which we may think of as the degree of smoothing. The torsion of the G2- structures is the error introduced by the smoothing process, so that the torsion is small when t is small. But when t is small, the metric is quite close to being singular in some sense, and this means that the curvature is large and the injectivity radius is small.

Our aim is to prove that a small deformation to a torsion-free G2- structure exists. One would hope to prove a theorem stating that if the torsion is smaller than some a priori bound, then such a deformation exists. However, one would expect this a priori bound to depend on geometric information such as the curvature and the injectivity radius. Therefore there is a problem in proving that the G2- structures of Chapter 2 can be deformed to zero torsion, because although one can get the torsion of the initial structure very small by choosing t very small, the a priori bound that the torsion must satisfy is also small when t is small.

Thus it is not clear that the a priori bound on the torsion will be satisfied for any t. The way this is solved is by writing inequalities on the torsion, curvature, and so on explicitly in terms of functions of t, and then proving that the powers of t come out in such a way that when t is small, the necessary conditions hold and the deformation to a torsion-free G2- structure exists. Heuristically speaking, Theorems A and B show that if the torsion is O(t4), the curvature is O(t−2), and the injectivity radius is at least O(t), then for t sufficiently small one may deform to a G2- structure with zero torsion.

1.1. The holonomy group G2. We begin with some necessary facts about the structure group G2, which can be found in [16], Chapter 11. Let R7 be equipped with an orientation and its standard metric g, and let y1, . . . , y7 be an oriented orthonormal basis of (R7)∗. Define a 3-form ϕ on R7 by

2

ϕ =y1 ∧ y2 ∧ y7 + y1 ∧ y3 ∧ y6 + y1 ∧ y4 ∧ y5 + y2 ∧ y3 ∧ y5 − y2 ∧ y4 ∧ y6 + y3 ∧ y4 ∧ y7 + y5 ∧ y6 ∧ y7.

(1)

Let GL+(7,R) be the subgroup of GL(7,R) preserving the orientation of R7. The subgroup of GL+(7,R) preserving ϕ is the exceptional Lie group G2, which is a compact, semisimple, 14-dimensional Lie group. It is a subgroup of SO(7), so that g can be reconstructed from ϕ. Applying the Hodge star ∗ of g we get the 4-form

∗ϕ =y1 ∧ y2 ∧ y3 ∧ y4 + y1 ∧ y2 ∧ y5 ∧ y6 − y1 ∧ y3 ∧ y5 ∧ y7 + y1 ∧ y4 ∧ y6 ∧ y7 + y2 ∧ y3 ∧ y6 ∧ y7 + y2 ∧ y4 ∧ y5 ∧ y7 + y3 ∧ y4 ∧ y5 ∧ y6.

(2) The subgroup of GL+(7,R) preserving ∗ϕ is also G2.

Let M be an oriented 7-manifold, and define Λ3+M, Λ 4 +M to be the subsets of

Λ3T ∗M and Λ4T ∗M respectively of forms admitting oriented isomorphisms with the forms ϕ and ∗ϕ defined by (1) and (2). Then Λ3+M and Λ4+M are both canon- ically isomorphic to the bundle of oriented G2- structures on M , and so have fibre GL+(7,R)/G2. Let Θ : Λ3+M → Λ4+M be the (nonlinear) natural identification. A dimension count reveals that Λ3+M and Λ4+M are open subbundles of Λ3T ∗M and Λ4T ∗M respectively.

Let ϕ be a smooth section of Λ3+M . Then ϕ is a smooth 3-form on M , and defines a G2- structure on M . By abuse of notation, we will usually identify a G2- structure on M with its 3-form ϕ. A G2- structure ϕ induces a metric g on M from the inclusion G2 ⊂ SO(7). With the Hodge star ∗ of g we may define the 4-form ∗ϕ, which by definition of Θ is equal to Θ(ϕ). Now the most basic invariant of a G- structure on a manifold is called the torsion of the G- structure, and is the obstruction to finding a torsion-free connection∇ on M preserving the G- structure. The condition for ϕ to be the G2- structure of a metric with holonomy contained in G2 is that the torsion of ϕ should be zero. Let ∇ be the Levi-Civita connection of g. Then the condition for ϕ to have zero torsion is that ∇ϕ = 0. By [16, Lemma 11.5], this is equivalent to the condition dϕ = d ∗ ϕ = 0.

When a metric g on M has holonomy group H, the Levi-Civita connection ∇ of g must preserve an H- structure on M , and this in turn implies that the Riemann curvature R of g lies in a bundle with fibre S2h, where h is the Lie algebra of H. Therefore a holonomy reduction imposes a linear restriction on the Riemann curvature. For the case of holonomy G2, [16, Lemma 11.8] shows that metrics with holonomy contained in G2 are Ricci-flat, which is one reason to study them.

The action of G2 on R7 gives an action of G2 on Λk(R7)∗, which splits Λk(R7)∗ into an orthogonal direct sum of irreducible representations of G2. Suppose that M is an oriented 7-manifold with a G2- structure, so that M has a 3-form ϕ and a metric g as above. Then in the same way, ΛkT ∗M splits into an orthogonal direct sum of subbundles with irreducible representations of G2 as fibres. In this section we shall describe these splittings, and some results associated with them. We shall use the notation Λkl for an irreducible representation of dimension l lying in ΛkT ∗M .

Proposition 1.1.1. Let M be an oriented 7-manifold with G2- structure, giving a 3-form ϕ and a metric g on M . Then ΛkT ∗M spli