Poisson structures and generalized Kahler structures

TL;DR

This paper explores how holomorphic Poisson structures on compact Kähler manifolds induce deformations of generalized Kähler structures, leading to new examples and insights into their geometric properties.

Contribution

It demonstrates the existence of deformations of generalized Kähler structures from Poisson structures and shows Poisson submanifolds are also generalized Kähler submanifolds.

Findings

Existence of deformations of generalized Kähler structures from Poisson structures

Poisson submanifolds are generalized Kähler submanifolds

Construction of unobstructed deformations of bi-Hermitian structures

Abstract

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler structures and provide non-trivial examples of generalized Kahler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bi-Hermitian structures constructed from Poisson structures.