Type one generalized Calabi--Yaus

TL;DR

This paper investigates the properties and constraints of type one generalized complex and Calabi--Yau manifolds, introducing a cohomology class that obstructs certain structures and exploring their topological and geometric characteristics.

Contribution

It introduces the twisting class as an obstruction, characterizes the topology of compact type one manifolds, and establishes conditions for the existence and deformation of generalized Calabi--Yau structures.

Findings

Euler characteristic of compact 4n-dimensional manifolds is even and equals the signature mod four.

Type one generalized Calabi--Yau fibers over the 2-torus, with structures that can be deformed to have a compact leaf.

Every symplectic fibration over the 2-torus admits a type one generalized Calabi--Yau structure.

Abstract

We study type one generalized complex and generalized Calabi--Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type one, 4n-dimensional generalized complex manifold the Euler characteristic must be even and equal to the signature modulo four. The generalized Calabi--Yau condition places much stronger constrains: a compact type one generalized Calabi--Yau fibers over the 2-torus and if the structure has one compact leaf, then this fibration can be chosen to be the fibration by the symplectic leaves of the generalized complex structure. If the twisting class vanishes, one can always deform the structure so that it has a compact leaf. Finally we prove that every symplectic fibration over the 2-torus admits a type one generalized Calabi--Yau structure.