Two questions about complex projective varieties

Question

Here are two questions about complex proj. varieties:

1. For two nonsingular complex projective varieties, can birational equivalence imply homeomorphism in the sense of complex topology?

2. For a n-dimensional complex proj. variety $X$, does $H_{2n}(X,\mathbb{Z})\neq0$ always hold?

Anyone can help or give some helpful guide will be very appreciated!

Answer 1

For the first question, the answer is no. Indeed, if you blow up $\mathbb P^2$ at one point, you get a birational variety with $b_2=2$. For the second question, your homology space is $\mathbb Z$, generated by the fundamental class of the variety, so this is true. (this is true more generally for a connected oriented manifold of (real) dimension $2n$).