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Rational cohomology tori

Olivier Debarre, Zhi Jiang and Martí Lahoz

Appendix: William F Sawin

Geometry & Topology 21 (2017) 1095–1130

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.

complex tori, compact Kähler manifolds, rational cohomology ring
Mathematical Subject Classification 2010
Primary: 32J27, 32Q15, 32Q55
Secondary: 14F45, 14E99
Received: 14 September 2015
Revised: 11 April 2016
Accepted: 13 May 2016
Published: 17 March 2017
Proposed: Richard Thomas
Seconded: Dan Abramovich, Frances Kirwan
Olivier Debarre
Département Mathématiques et Applications, UMR CNRS 8553
PSL Research University
École normale supérieure
45 rue d’Ulm
75230 Paris Cedex 05
Zhi Jiang
Département de Mathématiques d’Orsay, UMR CNRS 8628
Université Paris-Sud
Bâtiment 425
91405 Orsay Cedex
Martí Lahoz
Institut de Mathématiques Jussieu
Université Paris Diderot
Paris Rive Gauche - Paris 7
Bâtiment Sophie Germain, Case 7012
75205 Paris Cedex 13
William F Sawin
Department of Mathematics
Princeton University
Fine Hall
Washington Road
Princeton, NJ 08544
United States