A PROBLEM LIST FOR COMPACT HYPERKÄHLER MANIFOLDS

Abstract. In his thesis, Caldararu described twisted Fourier-Mukai transforms for elliptic fibrations. In this talk I will describe how certain holomorphic symplectic manifolds can be deformed to integrable sys-tems, i.e. fibrations by abelian varieties. These are higher dimensional analogues of elliptic K3 surfaces, and twisted Fourier-Mukai transforms once again arise. 1. Title of talk: Twisted Fourier-Mukai transforms for holomorphic symplectic manifolds 2. A problem list for compact hyperkähler manifolds In the 50s Berger classified the possible holonomy groups of Riemannian manifolds. Examples of Riemannian manifolds with non-generic holonomy include Calabi-Yau, G2, and Spin(7) manifolds; for these manifolds, we now know of hundreds of compact examples (for instance, see Joyce’s book [11]). In contrast, hyperkähler manifolds also have non-generic holonomy, but we know of very few compact examples. Currently the list of compact irre