Almost-K¨ahler Geometry John Armstrong i Title: Almost-K¨ahlerGeometry Name: John Armstrong Society: Wadham College Degree: D. Phil. Year: 1998 Term: Trinity Abstract The central theme of this thesis is almost-K¨ahler,Einstein 4-manifolds. We shall show how to answer the question of when a given Riemannian met- ric admits a compatible almost-K¨ahlerstructure. We shall illustrate this by proving that hyperbolic space of any dimension does not admit a compati- ble almost-K¨ahlerstructure. In the case of four dimensional manifolds, we shall show further that no anti-self-dual, Einstein manifold admits a non- K¨ahler,almost-K¨ahlerstructure. Indeed we shall prove that any Einstein, weakly ∗-Einstein, almost-K¨ahler4-manifold is given by a special case of the Gibbons{Hawking ansatz. We shall consider a number of other curvature conditions one can impose on the curvature of an almost-K¨ahler4-manifold. In particular we shall show that a compact, Einstein, almost-K¨ahler4-manifold whose fundamental two form is a root of the Weyl tensor is necessarily K¨ahler.We shall also show that a compact, almost-K¨ahler,Einstein 4-manifold with constant ∗-scalar curvature is necessarily K¨ahler. We shall prove, using the Seiberg{Witten invariants, that rational surfaces cannot admit a non-K¨ahleralmost-K¨ahler,Einstein structure. We shall also briefly consider the related topic of Hermitian, Einstein 4- manifolds. We find a new proof of the relationship between Ricci-flat Her- mitian manifolds given in [PB87] and the SU(1)-Toda field equation and obtain an analogous result for Hermitian, Einstein manifolds with non-zero scalar curvature. Introduction The initial motivation for this thesis comes from the following conjecture due to Goldberg: Conjecture 1 [Gol69] A compact, almost-K¨ahler,Einstein manifold is nec- essarily K¨ahler. An almost-K¨ahler manifold is an almost-Hermitian manifold whose funda- mental two form is closed. In other words, an almost-K¨ahlermanifold is a symplectic manifold equipped with a compatible metric. Broadly speak- ing, this thesis is about what the possibilities are for the curvature of an almost-K¨ahlermanifold. Goldberg's conjecture is still far from resolved | and we shall not resolve it. The conjecture will, however, serve as a guide throughout this thesis. One natural approach to Goldberg's conjecture is to impose additional cur- vature conditions and see whether or not one can prove that almost-K¨ahler manifolds which satisfy these additional conditions are necessarily K¨ahler. Of course, the motivation for studying such problems is two-fold: one may obtain some insight into how to prove non-existence results for almost- K¨ahler,Einstein metrics; alternatively one may obtain some insight into how to construct such metrics. Experience suggests that the Einstein met- rics which are easiest to find are the ones which satisfy the most curvature conditions. We recall that the curvature decomposes into three pieces under SO(2n) | namely the trace free Ricci tensor, the Weyl tensor and the scalar curvature. When n ≥ 3 all of these components are irreducible. However, when n = 2 (i.e. when the manifold is four dimensional), the Weyl tensor decomposes ii iii into the self-dual and anti-self-dual parts of the Weyl tensor. On an Einstein manifold, the trace free part of the Ricci tensor vanishes and the scalar curvature is constant. Thus the only interesting part of the curvature tensor is the Weyl tensor. Thus any additional curvature conditions we wish to impose will be conditions on the Weyl tensor of our manifold. Of course, this immediately makes the four-dimensional case stand out. Most of this thesis will be devoted to the four-dimensional case. The self-dual part of the Weyl tensor decomposes under U(2) into three pieces, one of which is a scalar. The anti-self-dual part of the Weyl ten- sor remains irreducible under U(2). So the curvature tensor of an Einstein, almost-Hermitian 4-manifold has 4 interesting components. Correspond- ingly there are 4 special types of almost-Hermitian, Einstein 4-manifold each imposing one additional condition on the Weyl tensor: 1. weakly ∗-Einstein, 2. constant ∗-scalar curvature, + 3. W00 ≡ 0, 4. self-dual. The names are, admittedly, not memorable. In this introduction we shall refer to these conditions simply as conditions (1), (2), (3) and (4). Manifolds with some, or all, of the above curvature properties constitute the most natural types of special almost-Hermitian, Einstein 4-manifold. Even if we cannot prove the Goldberg conjecture, we would like to prove that compact, almost-K¨ahler,Einstein manifolds with some of these additional properties are necessarily K¨ahler. We should point out that all four conditions are defined for non-Einstein almost-Hermitian manifolds. However, in this case conditions (1) and (2) are no longer conditions on the Weyl tensor alone. Before the author began this thesis, a certain amount of work had been done on such questions. One important result was the proof ([Sek85] and [Sek87]) that, if one makes the additional assumption that the manifold has positive scalar curvature, then Goldberg's conjecture is true. Nevertheless, little was known about the case of negative scalar curvature. Indeed the following question was still unanswered: iv Question 1 Can a constant-curvature manifold admit a compatible strictly almost-K¨ahlerstructure? This question was answered by Olzsak in [Ols78] for manifolds of dimensions 8 and above, but the problem was still unresolved in dimensions 4 and 6 | although Sekigawa and Oguro proved in [SO94] that the result is true if one looks for a global almost-K¨ahlerstructure on a complete hyperbolic manifold. Our first result result towards answering this type of question is given in Chapter 2: we prove that a compact, almost-K¨ahler,Einstein 4-manifold which satisfying (2) is necessarily K¨ahler.This tells us that compact anti- self-dual, Einstein, almost-K¨ahler four manifolds are necessarily K¨ahler. However, in terms of what it tells us about the constant-curvature case, this result is unsatisfactory. The condition that the manifold is constant- curvature is about as stringent a condition as one could impose on a man- ifold's curvature. It tells us that the manifold must be locally isometric to either a sphere or a hyperbolic space or it must be flat. One feels that one should surely be able to prove, using an entirely local argument, that constant-curvature manifolds cannot admit compatible strictly almost- K¨ahlerstructures. Our problem is that we have no method of determining when a given Riemannian metric admits a compatible almost-K¨ahlerstruc- ture. We shall devise a strategy for answering this question which we shall apply to prove that constant-curvature manifolds cannot admit a compati- ble strictly almost-K¨ahlerstructure. The strategy is rather complex in that it requires examining a surprisingly large number of derivatives of the cur- vature. So we shall have to build up a good body of knowledge about the curvature, and derivatives of the curvature, of almost-Hermitian manifolds. However, once we have done this, we shall be able to use our strategy to generalise our result. It will be a relatively simple matter to prove that an almost-K¨ahler,Einstein 4-manifold satisfying either both (1) and (2) or both (1) and (3) must be K¨ahler| even locally. This includes anti-self-dual, Einstein manifolds. Another line of enquiry one might consider in studying Goldberg's conjec- ture is whether or not almost-K¨ahler,Einstein metrics exist locally. Once again this question was unanswered when the author began his thesis. The author attempted to tackle the problem by applying Cartan–K¨ahlertheory to obtain an abstract existence result. Indeed, he believed he had succeeded v in doing so. Nevertheless a gap in the proof was pointed out to the au- thor which seems too difficult to fix. Notice that experts on Cartan–K¨ahler theory had believed that it would be easy to prove the existence of almost- K¨ahler,Einstein metrics using Cartan–K¨ahlertheory. Thus the difficulty one experiences in attempting to do this is surprising. Between the time of producing the \proof" and the mistake being pointed out, Nurowski and Przanowski ([PN]) found an explicit example of an almost- K¨ahler,Einstein metric. Motivated by their example, Tod ([Tod97a]) went on to find a family of examples all based on the Gibbons{Hawking ansatz. We shall refer to this method of producing almost-K¨ahler,Einstein metrics as Tod's construction. Since explicit examples now exist, the motivation for providing an abstract proof of the existence of almost-K¨ahlerEinstein metrics has rather dimin- ished. Nevertheless, Tod's examples raise as many questions as they answer. For example: do there exist almost-K¨ahler Einstein metrics in 4-dimensions which are not given by Tod's construction? Additional motivation for ask- ing such a question is given by the fact that a comparable result is true for Hermitian, Einstein manifolds. Specifically one has the Riemannian Goldberg{Sachs theorem which states that any Hermitian Einstein manifold automatically satisfies curvature condition (1). There is a sense in which Cartan–K¨ahlertheory should allow one to an- swer such questions as \Is there any unexpected condition the curvature of an almost-K¨ahler,Einstein manifold must satisfy?". If one could pro- vide a proof using Cartan–K¨ahlertheory for the existence of almost-K¨ahler, Einstein metrics then one would be able to answer such a question as an immediate corollary. Thus, it is still natural to attempt to apply Cartan–K¨ahlertheory to our problem and so we shall do this in chapter 4.