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 Pn is isomorphic to Pn. 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*(Pn,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 H1(X,Z) and asked whether this remains true under the weaker assumption that the rational cohomology ring is an exterior algebra on H1(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.