SR2

I shall describe joint work with Gritsenko and Hulek in which we study the moduli spaces of polarised holomorphic symplectic manifolds via their periods. There are strong similarities with moduli spaces of K3 surfaces, but also some important differences, notably that global Torelli fails. I shall explain (conjecturally) why and show how the techniques used to obtain general type results for K3 moduli can be modified to give similar, and quite strong, results in this case. Mainly I shall concentrate on the case of deformations of Hilbert schemes of K3 surfaces.

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

Given K a separably closed field of finite ( > 1) degree of imperfection, and semiabelian variety A over K, we study the maximal divisible subgroup A^{sharp} of A(K). We show that the {\sharp} functor does not preserve exact sequences and also give an example where A^{\sharp} does not have relative Morley rank. (Joint work with F. Benoist and E. Bouscaren)

The Nielsen realisation problem asks when a collection of diffeomorphisms, which form a group up to isotopy, is isotopic to a collection of diffeomorphisms which form a group on the nose. For surfaces this problem is well-studied, I'll talk about this problem in the context of K3 surfaces.