Deformations of compact holomorphic Poisson submanifolds

TL;DR

This paper investigates the deformation theory of compact holomorphic Poisson submanifolds, extending classical results and exploring simultaneous deformations of Poisson structures and submanifolds, with algebraic and geometric perspectives.

Contribution

It extends Kodaira's deformation theory to holomorphic Poisson submanifolds, including algebraic formulations and analysis of first-order deformations and obstructions.

Findings

Identifies first-order deformations of Poisson submanifolds

Analyzes obstructions to deformations

Establishes conditions for stability and semi-regularity

Abstract

In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifolds in the context of holomorphic Poisson deformations. We also study simultaneous deformations of holomorphic Poisson structures and holomorphic Poisson submanifolds on a fixed underlying compact complex manifold. In appendices, we present deformations of Poisson closed subschemes in the language of functors of Artin rings which is the algebraic version of deformations of holomorphic Poisson submanifolds. We identify first-order deformations and obstructions.