Consider a smooth surface ${M \subset \mathop{\mathbb P}^3(\mathbb{C})}$ of degree ${d}$. We are interested in determining its cohomology.

1. A fibration argument

A key observation is that all such ${M}$‘s are diffeomorphic. (When ${\mathop{\mathbb P}^3}$ is replaced by ${\mathop{\mathbb P}^2}$, then this is just the observation that the genus is determined by the degree, in the case of a plane curve.) In fact, consider the space ${V}$ of all degree ${d}$ homogeneous equations, so that ${\mathop{\mathbb P}(V)}$ is the space of all smooth surfaces of degree ${d}$. There is a universal hypersurface ${H \subset \mathop{\mathbb P}^3 \times \mathop{\mathbb P}(V)}$ consisting of pairs ${(p, M)}$ where ${p}$ is a point lying on the hypersurface ${M}$. This admits a map

$\displaystyle \pi: H \rightarrow \mathop{\mathbb P}(V)$

which is (at least intuitively) a fiber bundle over the locus of smooth hypersurfaces. Consequently, if ${U \subset \mathop{\mathbb P}(V)}$ corresponds to smooth hypersurfaces, we get an honest fiber bundle

$\displaystyle \pi^{-1}(U) \rightarrow U .$

But ${U}$ is connected, since we have thrown away a complex codimension ${\geq 1}$ subset to get ${U}$ from ${\mathop{\mathbb P}(V)}$; this means that the fibers are all diffeomorphic.

This argument fails when one considers only the real points of a variety, because a codimension one subset of a real variety may disconnect the variety. (more…)