# Olivier Debarre : Fake projective spaces and fake tori

We discuss compact complex manifolds which ``look like'' complex projective
spaces or complex tori.
Hirzebruch and Kodaira proved in 1957 that when *n* is odd, any compact
Kähler manifold *X* which is homeomorphic to **P**^{n} is isomorphic to
**P**^{n}. This holds for all *n* by Aubin and Yau's proofs of the Calabi
conjecture. One may conjecture that it should be sufficient to assume that
the integral cohomology rings *H*^{*}(*X*,**Z**) and *H*^{*}(**P**^{n},**Z**) are isomorphic.

Catanese observed that complex tori are characterized among compact Kähler
manifolds *X* by the fact that their integral cohomology rings are exterior
algebras on *H*^{1}(*X*,**Z**) and asked whether this remains true under the
weaker assumption that the rational cohomology ring is an exterior algebra
on *H*^{1}(*X*,**Q**). (We call the corresponding compact Kähler manifolds
``rational cohomology tori".)
We give a negative answer to Catanese's question by producing explicit
examples. We also prove some structure theorems for rational cohomology
tori. This is work in collaboration with Z. Jiang, M. Lahoz, and W. F.
Sawin.

**Category**: Algebraic Geometry**Duration**: 01:34:34**Date**: April 14, 2017 at 3:10 PM**Views**: 136-
**Tags:**seminar, Algebraic Geometry Seminar

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