We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k<=0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k < 0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

# Past Geometry and Analysis Seminar

While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been

proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett)

or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface' to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric', i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric': i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context (joint work with Ivan Smith).

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds.

In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.

A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this

talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).

We describe a general program for the classification of flat connections on topological manifolds. This is motivated by the classification of locally homogeneous geometric structures on manifolds, in the spirit of Ehresmann and Thurston. This leads to interesting dynamical systems arising from mapping class group actions on character varieties. The mapping class group action is a discrete version of a continuous object, namely the extension of the Teichmueller flow to a unversal character variety over over the tangent bundle of Teichmuller space. We give several examples of this construction

and discuss joint work with Giovanni Forni on a mixing property of this suspended flow.

The theory of connections on curves and Hitchin systems is something like a “global theory of Lie groups”, where one works over a Riemann surface rather than just at a point. We’ll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of “Dynkin diagrams” as a step towards classifying some examples of such objects.

I shall discuss Zauner's conjecture about the existence of n^2 mutually equidistant points in complex projective space CP^{n-1} with its standard Fubini-Study metric. This is the so-called SIC-POVM problem, and is related to properties of the moment mapping that embeds CP^{n-1} into the Lie algebra su(n). In the case n=3, there is an obvious 1-parameter family of such sets of 9 points under the action of SU(3) and we shall sketch a proof that there are no others. This is joint work with Lane Hughston.

The moduli space M(G) of Higgs bundles for a complex reductive group G on a compact Riemann surface carries a natural hyperkahler structure and it comes equipped with an algebraically completely integrable system through a flat projective morphism called the Hitchin map. Motivated by mirror symmetry, I will discuss certain complex Lagrangians (BAA-branes) in M(G) coming from real forms of G and give a proposal for the mirror (BBB-brane) in the moduli space of Higgs bundles for the Langlands dual group of G. In this talk, I will focus on the real groups SU^*(2m), SO^*(4m) and Sp(m,m). The image under the Hitchin map of Higgs bundles for these groups is completely contained in the discriminant locus of the base and our analysis is carried out by describing the whole

(singular) fibres they intersect. These turn out to be certain subvarieties of the moduli space of rank 1 torsion-free sheaves on a non-reduced curve. If time permits we will also discuss another class of complex Lagrangians in M(G) which can be constructed from symplectic representations of G.

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real

Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane. In this talk, I will explain how to compute the mod 2 Betti numbers of M_R using Morse theory. A key role in this calculation is played by the Abel-Jacobi map from symmetric products of X to the Jacobian of X.

Character varieties have been studied largely by means of their correspondence to the moduli space of Higgs bundles. In this talk we will report on a method to study their Hodge structure, in particular to compute their E- polynomials. Moreover, we will explain some applications of the given method such as, the study of the topology of the moduli space of doubly periodic instantons. This is joint work with A. González, V.Muñoz and P. Newstead.