Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids

TL;DR

This paper generalizes the Feix-Kaledin theorem to construct hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids, linking deformations of holomorphic symplectic and Poisson structures to hyperkahler geometry.

Contribution

It extends the Feix-Kaledin construction to neighborhoods of Lagrangian submanifolds and holomorphic symplectic groupoids, connecting deformations of structures to hyperkahler metrics.

Findings

Hyperkahler structures exist near Lagrangian submanifolds in holomorphic symplectic manifolds.

Holomorphic symplectic groupoids over Poisson surfaces admit hyperkahler structures near their identity.

Deformations of holomorphic Poisson structures are key to constructing hyperkahler metrics.

Abstract

The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem.