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Rational cohomology
tori
Olivier Debarre, Zhi Jiang and Martí LahozAppendix: William F Sawin |
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Geometry & Topology 21 (2017) 1095–1130
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Abstract |
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We study normal compact varieties in Fujiki’s class 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 –forms on smooth varieties of general type. |
Keywords
complex tori, compact Kähler manifolds, rational cohomology
ring
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Mathematical Subject Classification 2010
Primary: 32J27, 32Q15, 32Q55
Secondary: 14F45, 14E99
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References |
Publication
Received: 14 September 2015
Revised: 11 April 2016
Accepted: 13 May 2016
Published: 17 March 2017
Proposed: Richard Thomas
Seconded: Dan Abramovich, Frances Kirwan
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Authors |
| Olivier Debarre | |
| Département Mathématiques et
Applications, UMR CNRS 8553 PSL Research University École normale supérieure 45 rue d’Ulm 75230 Paris Cedex 05 France |
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| http://www.math.ens.fr/~debarre/ | |
| Zhi Jiang | |
| Département de Mathématiques
d’Orsay, UMR CNRS 8628 Université Paris-Sud Bâtiment 425 91405 Orsay Cedex France |
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| http://www.math.u-psud.fr/~jiang/ | |
| Martí Lahoz | |
| Institut de Mathématiques
Jussieu Université Paris Diderot Paris Rive Gauche - Paris 7 Bâtiment Sophie Germain, Case 7012 75205 Paris Cedex 13 France |
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| http://webusers.imj-prg.fr/~marti.lahoz/ | |
| William F Sawin | |
| Department of Mathematics Princeton University Fine Hall Washington Road Princeton, NJ 08544 United States |
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| http://web.math.princeton.edu/~wsawin/ | |
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