Symplectic realizations of holomorphic Poisson manifolds

TL;DR

This paper provides an explicit construction of holomorphic symplectic realizations for any holomorphic Poisson manifold, solving a classical problem by extending symplectic structures to neighborhoods in the cotangent bundle.

Contribution

It introduces a method to explicitly construct holomorphic symplectic structures on neighborhoods of the zero section in the cotangent bundle of any holomorphic Poisson manifold.

Findings

Existence of holomorphic symplectic structures near the zero section

Construction of explicit holomorphic symplectic forms

Zero section as a holomorphic Lagrangian submanifold

Abstract

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, \pi)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on $Y \subseteq T^*X$.