Generalized K\"ahler metrics from Hamiltonian deformations

TL;DR

This paper introduces a new way to characterize generalized K"ahler structures using complex Dirac structures, proves an unobstructedness result for their deformations, and links these deformations to Hamiltonian Poisson structures.

Contribution

It provides a novel characterization of generalized K"ahler structures and demonstrates their deformation properties, extending Hitchin's unobstructedness results to a broader setting.

Findings

Generalized K"ahler structures can be characterized via complex Dirac structures.

Any generalized K"ahler structure can be deformed while fixing one holomorphic Poisson component.

Deformations relate closely to Hamiltonian families of Poisson structures.

Abstract

We give a new characterization of generalized K\"ahler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin's partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for arbitrary generalized K\"ahler structures. That is, we show that any generalized K\"ahler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin's deformation. Finally, we indicate a close relationship between this deformation and the notion of a Hamiltonian family of Poisson structures.