Differential Gerstenhaber Algebras of Generalized Complex Structures

TL;DR

This paper investigates how differential Gerstenhaber algebras associated with generalized complex structures deform, identifying conditions for their invariance and providing solutions in specific geometric contexts.

Contribution

It establishes the integrability of infinitesimal invariance conditions and offers a general construction for solutions on holomorphic Poisson nilmanifolds and invariant geometries.

Findings

Infinitesimal invariance conditions are always integrable.

Solutions exist on holomorphic Poisson nilmanifolds under certain conditions.

Examples and counterexamples illustrate the existence or non-existence of solutions.

Abstract

Associated to every generalized complex structure is a differential Gerstenhaber algebra (DGA). When the generalized complex structure deforms, so does the associated DGA. In this paper, we identify the infinitesimal conditions when the DGA is invariant as the generalized complex structure deforms. We prove that the infinitesimal condition is always integrable. When the underlying manifold is a holomorphic Poisson nilmanifolds, or simply a group in the general, and the geometry is invariant, we find a general construction to solve the infinitesimal conditions under some geometric conditions. Examples and counterexamples of existence of solutions to the infinitesimal conditions are given.