Complex symplectic structures arise naturally in hyperkähler geometry, and in complex algebraic geometry the terms “complex symplectic” and “hyperkähler” are often used interchangeably, since the two notions are essentially equivalent in the presence of a compatible Kähler metric. Without the Kähler assumption, however, the situation is much more flexible. In fact, in the early 1990s Guan constructed the first higher-dimensional non-hyperkähler examples on nilmanifolds and solvmanifolds.

In this talk, I will present some general construction methods for non-hyperkähler complex symplectic structures on nilmanifolds and solvmanifolds, and discuss certain cohomological and deformation-theoretic properties of compact complex symplectic manifolds. If time permits, I will also explain the relation to semi-integrable almost hyperhermitian structures introduced very recently by Gentili and Moroianu and describe a Gibbons–Hawking-type ansatz for four-dimensional S^1-invariant complex symplectic manifolds, where one of the associated monopole equations admits a natural sheaf-cohomological formulation.

The talk is based on joint work with G. Bazzoni, A. Latorre, N. Tardini, B. Meinke, and A. Gil-García.

Cultural coffee by Renan Berto Cuevas