We discuss compact 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 ${\bf CP}^n$ is biholomorphic to ${\bf CP}^n$. 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 $Hsup*/sup(X,{\bf Z})$ and $Hsup*/sup({\bf CP}^n,{\bf Z})$ 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 $Hsup1/sup(X,{\bf Z})$ and asked whether this remains true under the weaker assumptions that the rational cohomology ring is an exterior algebra on $Hsup1/sup(X,{\bf Q})$ (we call the corresponding compact K\ahler 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.