Deformations of holomorphic Poisson manifolds

TL;DR

This paper proves an unobstructedness theorem for deformations of compact holomorphic Poisson manifolds and explores specific examples like rational surfaces and Hilbert schemes, showing how their deformations are characterized.

Contribution

It establishes an unobstructedness result for deformations of holomorphic Poisson manifolds and analyzes the deformation parameters of Hilbert schemes of points on Poisson surfaces.

Findings

Deformations of certain Poisson manifolds are unobstructed.

Hilbert schemes of points on Poisson surfaces are characterized by specific parameters.

A generic deformation of the projective plane's Hilbert scheme depends on an elliptic curve and a translation.

Abstract

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deformation is determined by two parameters -- an elliptic curve and a translation on it.