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Let $X$, $Y$ be two birational complex smooth projective varieties. Note they are also compact complex manifolds. I wonder if they are homotopy equivalent? If not, what topological properties do they share? (I know they have the same geometric genus)

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    $\begingroup$ No, a blow-up of a point in, say, a surface $X$ is a connected sum with the complex-projective plane $CP^2$. (More precisely, the projective plane with the opposite orientation.) This changes $H_2$. $\endgroup$ – Moishe Kohan Jan 1 at 15:26
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    $\begingroup$ The fundamental group is a birational invariant. This thread on MO should be of interest: mathoverflow.net/questions/33021/… $\endgroup$ – dOuUuq3podCuoArv Jan 1 at 21:19

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