1. Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids
- Author
-
Maxence Mayrand
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Holomorphic function ,Kähler manifold ,01 natural sciences ,Section (fiber bundle) ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic manifold ,Mathematics::Complex Variables ,Applied Mathematics ,010102 general mathematics ,Zero (complex analysis) ,53D17, 53C26, 53C28, 32G05 ,Submanifold ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,Symplectic Geometry (math.SG) ,Cotangent bundle ,Mathematics::Differential Geometry ,010307 mathematical physics ,Symplectic geometry - 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., 20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)
- Published
- 2021