# Birational complex smooth varieties and homotopy equivalence

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)

• 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 $H_2$. – Moishe Kohan Jan 1 at 15:26
• The fundamental group is a birational invariant. This thread on MO should be of interest: mathoverflow.net/questions/33021/… – dOuUuq3podCuoArv Jan 1 at 21:19