If two complex projective manifolds are homotopy equivalent are they homeomorphic?



Complex projective manifolds are homeomorphic if homotopy equivalent
For curves this follows from the classification of (2-dimensional topological) surfaces, and for simply-connected surfaces this follows from [Freedman's theorem.](https://mathworld.wolfram.com/FreedmanTheorem.html)


My former colleagues Anatoly Libgober and John Wood found examples of pairs of 3-folds which are complete intersections and are homotopy equivalent but not diffeomorphic, in fact have distinct Pontryagin classes. See [Example 9.2](https://doi.org/10.1016/0040-9383(82)90024-6). Since in this case $H^4(M;\mathbb{Z})\cong \mathbb{Z}$, this implies that the manifolds are not homeomorphic by the [topological invariance of rational Pontryagin classes](https://encyclopediaofmath.org/wiki/Pontryagin_class#Topological_invariance.) (see Ben Wieland's comment).


For the higher dimensional case see:


*Fang, Fuquan*, [**Topology of complete intersections**](http://dx.doi.org/10.1007/s000140050028), Comment. Math. Helv. 72, No. 3, 466-480 (1997). [ZBL0896.14028](https://zbmath.org/?q=an:0896.14028).


