Let $M$ be a Brody hyperbolic complex projective manifold with $\pi_1(M)=\{0\}$. Can $M$ be homeomorphic to $P\times S^2$ where $P$ is a manifold?
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$\begingroup$ There is a conjecture that hyperbolic projective manifolds have $K_{X}$ ample. Such manifolds do satisfy topological restrictions; from the Bogomolov-Miyaoka-Yau and Miyaoka-Yau inequalities in dimension 2 and 3 respectively. For example they can be used to give restrictions on the Betti numbers and allow one to rule out being diffeomorphic more than half of the Fano $3$-folds mathoverflow.net/questions/318900/…. On the other hand, the conjecture suggests that there is not an example of what you ask for in the current literature. $\endgroup$– Nick LCommented Sep 29, 2020 at 12:23
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