Geometry & Topology

Rational cohomology tori

Abstract

We study normal compact varieties in Fujiki’s class $C$ 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.

In an appendix, we show that there are no smooth rational cohomology tori of general type. The key technical ingredient is a result of Popa and Schnell on $1$–forms on smooth varieties of general type.

Article information

Source
Geom. Topol., Volume 21, Number 2 (2017), 1095-1130.

Dates
Revised: 11 April 2016
Accepted: 13 May 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859175

Digital Object Identifier
doi:10.2140/gt.2017.21.1095

Mathematical Reviews number (MathSciNet)
MR3626598

Zentralblatt MATH identifier
1378.32013

Citation

Debarre, Olivier; Jiang, Zhi; Lahoz, Martí. Rational cohomology tori. Geom. Topol. 21 (2017), no. 2, 1095--1130. doi:10.2140/gt.2017.21.1095. https://projecteuclid.org/euclid.gt/1510859175

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