Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$ such that $X(\mathbb{C})$ is homeomorphic to $Y(\mathbb{C})$. Is it true that $\dim_{\mathbb{C}} H^k(X,\mathcal{O}_X)=\dim_{\mathbb{C}} H^k(Y,\mathcal{O}_Y)$ for $k\ge 2$? (Note that for $k=1$ it follows by Hodge theory).
1 Answer
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The answer is no.
There are counterexamples already for surfaces, due to X. Gang and F. Campana (unpublished). The link to Campana's article is here, and the relevant result is
Proposition 0.1 Il existe des surfaces projectives complexes $S$, $S_0$ simplement connexes et de type général homéomorphes, mais ayant des nombres de Hodge différents.
You can look at this paper and at the references cited therein for further details.