On cotangent manifolds, complex structures and generalized geometry

TL;DR

This paper explores properties of symmetric generalized complex structures, their relation to cotangent manifolds, and extends classical results in complex geometry to a broader generalized geometric framework.

Contribution

It introduces a new construction of almost complex structures on cotangent bundles associated with generalized complex structures and studies their integrability.

Findings

Relation between Courant integrability and cotangent bundle complex structures

Construction of almost complex structures on cotangent manifolds

Extension of classical complex geometry results

Abstract

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given a symmetric or skew-symmetric generalized complex structure \mathcal J and a connection D on a manifold M, we construct an almost complex structure J^{\mathcal J,D} on the cotangent manifold T^{*}M and we study its integrability. For \mathcal J skew-symmetric, we relate the Courant integrability of \mathcal J with the integrability of J^{\mathcal J, D}. We consider in detail the case when M=G is a Lie group and \mathcal J , D are left-invariant. We also show that our approach generalizes various well-known results from special complex geometry.