Rational cohomology tori

TL;DR

This paper investigates special compact Kähler spaces called rational cohomology tori, classifies low-dimensional cases with rational singularities, and explores their geometric properties and examples.

Contribution

It classifies rational cohomology tori with rational singularities up to dimension three and provides constraints on their Albanese morphism and factors for those of general type.

Findings

Classification of rational cohomology tori up to dimension three.

Constraints on Albanese morphism degree and factors for general type tori.

Construction of new examples of rational cohomology tori.

Abstract

We study normal compact K\"ahler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Euler characteristic. We finish with the construction of series of examples.