Generalized geometry and the Hodge decomposition

TL;DR

This paper reviews generalized geometry concepts and proves a Hodge decomposition for twisted cohomology on compact generalized Kähler manifolds, extending classical Kähler results.

Contribution

It introduces a Hodge decomposition for twisted cohomology and generalizes the dd^c-lemma within the framework of generalized geometry.

Findings

Hodge decomposition for twisted cohomology

Generalization of the dd^c-lemma

Extension of classical Kähler geometry results

Abstract

In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized K\"ahler manifold, as well as a generalization of the $dd^c$-lemma of K\"ahler geometry.