Abstract:

An affine special pseudo-Kähler (ASK) structure is a geometric structure on the base of an algebraic integrable system away from the discriminant locus. This can locally be realised as a certain immersion into a complex symplectic vector space with real structure. Moreover thanks to certain constructions coming from the literature on supergravity, the tangent bundle of an ASK manifold carries a natural pseudo-hyperkähler structure. Inspired by recent work of Tom Bridgeland and Ian Strachan on complex hyperkähler structures on the tangent bundle of a holomorphic symplectic manifold equipped with a certain flat Ehresmann connection, we generalise the above picture coming from the rigid c-map to give two new constructions of real pseudo-hyperkähler manifolds. The first one takes as input an Ehresmann flat holomorphic symplectic manifold, along with some extra data, and produces a real pseudo-hyperkähler structure on the same manifold, while the second takes as input the Ehresmann analogue of an ASK manifold (which may be realised as an immersion into an Ehresmann flat holomorphic symplectic manifold) and produces a real pseudo-hyperkähler structure on the tangent bundle. These real pseudo-hyperkähler manifolds are related to the complex hyperkähler manifolds of Bridgeland and Strachan through submersions and immersions respectively. This talk will be based on upcoming joint work with Vicente Cortés, Alejandro Gil García, and Iván Tulli.