Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids

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.

arXiv comment20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

References in corpus (7)

• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• Poisson sigma models and symplectic groupoids
• A hyperkahler structure on the cotangent bundle of a complex Lie group
• Integration of Holomorphic Lie Algebroids
• Constructions and classifications of projective Poisson varieties
• Canonical complex extensions of Kähler manifolds
• Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations

Cited by in corpus (1)

• Geometry of the Space of Sections of Twistor Spaces with Circle Action