Deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes

TL;DR

This thesis explores the deformation theory of compact holomorphic Poisson manifolds and algebraic Poisson schemes using both analytic and algebraic frameworks, advancing understanding of their structural variations.

Contribution

It provides a comprehensive study of deformations in the context of Poisson geometry, integrating Kodaira-Spencer and Grothendieck theories for the first time in this setting.

Findings

Established deformation criteria for compact holomorphic Poisson manifolds.

Extended algebraic deformation theory to Poisson schemes.

Identified conditions under which deformations preserve Poisson structures.

Abstract

In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.