On deformations of generalized Calabi--Yau and generalized $\mathrm{SU}(n)$-structures

Abstract

In this paper we will introduce the new notion of generalized geometric structures defined by systems of closed differential forms. From a cohomological point of view, we develop a unified approach to deformation problems and establish a criterion for unobstructed deformations of the generalized geometric structures. We construct the moduli spaces of the structures with the action of $d$-closed $b$-fields and show that the period map of the moduli space is locally injective under the certain cohomological condition (the local Torelli type theorem). We apply our approach to generalized Calabi--Yau structures and generalized $\mathrm{SU}(n)$-structures and obtain unobstructed deformations of generalized Calabi--Yau structures if the $dd^{\mathcal{J}}$-property is satisfied. We also have unobstructed deformations of generalized $\mathrm{SU}(n)$-structures and show that the period map of the moduli space of generalized $\mathrm{SU}(n)$-structures is locally injective.