#### Volume 21, issue 2 (2017)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Rational cohomology tori

### Appendix: William F Sawin

Geometry & Topology 21 (2017) 1095–1130
##### Abstract

We study normal compact varieties in Fujiki’s class $\mathsc{C}$ 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.

##### Keywords
complex tori, compact Kähler manifolds, rational cohomology ring
##### Mathematical Subject Classification 2010
Primary: 32J27, 32Q15, 32Q55
Secondary: 14F45, 14E99
##### 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
##### 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 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 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 http://webusers.imj-prg.fr/~marti.lahoz/ William F Sawin Department of Mathematics Princeton University Fine Hall Washington Road Princeton, NJ 08544 United States http://web.math.princeton.edu/~wsawin/