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Description
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, the situation is much more flexible. Nevertheless, (compact) non-Kähler complex symplectic manifolds have so far received relatively little attention in the literature.
In my talk, I will briefly recall some of the history of the subject and then discuss some results on cohomological and deformation-theoretic properties of compact non-Kähler complex symplectic manifolds, obtained jointly with G. Bazzoni, A. Latorre and N. Tardini. Afterwards, I will describe a Gibbons–Hawking-type ansatz for four-dimensional (non-compact) complex symplectic manifolds endowed with a Hamiltonian S^1-action and explain how one of the associated monopole equations admits a natural sheaf-cohomological reformulation. The latter results are joint work with A. Gil-García.