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…)

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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…)