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…)

Today’s main goal is the Leray theorem (though at the end I have to ask a question):

Theorem 1 Let ${\mathcal{F}}$ be a sheaf on ${X}$, and ${\mathfrak{U} = \{ U_i, i \in I\}}$ an open cover of ${X}$. Suppose$\displaystyle H^n( U_{i_1} \cap \dots \cap U_{i_k}, \mathcal{F}|_{ U_{i_1} \cap \dots \cap U_{i_k}}) = 0$

for all ${k}$-tuples ${i_1, \dots , i_k \in I}$, and all ${n>0}$. Then the canonical morphism$\displaystyle H^n( \mathfrak{U}, \mathcal{F}) \rightarrow H^n( X, \mathcal{F})$

is an isomorphism for all ${n}$

This seems rather useless, because the theorem presupposes the vanishing of (regular) cohomology on the covering. However, in many cases it turns out to be helpful. If ${X}$ is a separated scheme, ${U_i}$ an open affine cover of ${X}$, and ${\mathcal{F}}$ quasi-coherent, it applies. The reason is that each of the intersections ${ U_{i_1} \cap \dots \cap U_{i_k}}$ are all affine by separatedness, so ${\mathcal{F}}$ has no cohomology on them by a basic property of quasi-coherent sheaves. This gives a practical way of computing sheaf cohomology in algebraic geometry. Hartshorne uses it to compute the cohomology of line bundles on projective space.

Another instance arises when ${\mathcal{O}}$ is the sheaf of holomorphic functions over some Riemann surface ${X}$. In this case ${\{U_i\}}$ is a covering of charts. It is a theorem (which I will eventually prove) that for any open subset of ${\mathbb{C}}$ (which any intersection of the ${U_i}$‘s is isomorphic to), the sheaf ${\mathcal{O}}$ has trivial cohomology. (more…)

To continue, I am now going to have to use the language of sheaves. For it, and for all details I will omit here, I refer the reader to Charles Siegel’s post at Rigorous Trivialties and Hartshorne’s Algebraic Geometry. When I talk about sheaf cohomology, it will always be the derived functor cohomology. I will briefly review some of these ideas.

Sheaf cohomology

The basic properties of this are as follows.

First, if ${X}$ is a topological space and ${i \in \mathbb{Z}_{\geq 0}}$, then ${H^i(X, \cdot)}$ is a covariant additive functor from sheaves on ${X}$ to the category of abelian groups. We have

$\displaystyle H^0(X,\mathcal{F}) = \Gamma(X,\mathcal{F}),$

that is to say, the global sections. Also, if

$\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0$

is a short exact sequence of sheaves, there is a long exact sequence

$\displaystyle H^i(X,\mathcal{F}) \rightarrow H^i(X, \mathcal{G})\rightarrow H^i(X, \mathcal{H}) \rightarrow H^{i+1}(X,\mathcal{F}) \rightarrow \dots .$

Finally, sheaf cohomology (except at 0) vanishes on injectives in the category of sheaves.

In other words, sheaf cohomology consists of the derived functors of the (left-exact) global section functor. (more…)