Theorem of existence and completeness for holomorphic Poisson structures

TL;DR

This paper extends classical theorems of existence and completeness to the realm of holomorphic Poisson structures, providing foundational results for their deformation theory.

Contribution

It introduces a deformation framework for compact holomorphic Poisson manifolds and proves analogues of classical theorems within this context.

Findings

Established an integrability condition for holomorphic Poisson deformations.

Proved an existence theorem for holomorphic Poisson structures.

Demonstrated a completeness theorem analogous to classical complex structures.

Abstract

In this paper, we define a concept of a family of compact holomorphic Poisson manifolds on the basis of Kodaira-Spencer's deformation theory and deduce the integrability condition. We prove an analogue of their `Theorem of existence for complex analytic structures' under some analytic assumption, and establish an analogue of their `Theorem of completeness for complex analytic structures' in the context of holomorphic Poisson deformations.