Variation of Hodge structure for generalized complex manifolds

TL;DR

This paper investigates how Hodge structures vary in families of generalized complex manifolds, establishing period maps, Griffiths transversality, and holomorphicity results, with applications to special cases like generalized Kähler manifolds.

Contribution

It introduces a Courant algebroid approach to study Hodge decomposition variation, defining period maps and proving Griffiths transversality and holomorphicity in this context.

Findings

Hodge decomposition exists in twisted cohomology for certain generalized complex manifolds.

Period maps are well-defined and satisfy Griffiths transversality in families.

Holomorphicity of period maps is established for holomorphic families.

Abstract

A generalized complex manifold which satisfies the $\partial \overline{\partial}$-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we define period maps, prove a Griffiths transversality theorem and show that for holomorphic families the period maps are holomorphic. Further results on the Hodge decomposition for various special cases including the generalized K\"ahler case are obtained.