Geometry & Topology

Rational cohomology tori

Olivier Debarre, Zhi Jiang, and Martí Lahoz

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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

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

Received: 14 September 2015
Revised: 11 April 2016
Accepted: 13 May 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32J27: Compact Kähler manifolds: generalizations, classification 32Q15: Kähler manifolds 32Q55: Topological aspects of complex manifolds
Secondary: 14F45: Topological properties 14E99: None of the above, but in this section

complex tori compact Kähler manifolds rational cohomology ring


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

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