Consider a smooth surface of degree . We are interested in determining its cohomology.
1. A fibration argument
A key observation is that all such ‘s are diffeomorphic. (When is replaced by , 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 of all degree homogeneous equations, so that is the space of all smooth surfaces of degree . There is a universal hypersurface consisting of pairs where is a point lying on the hypersurface . This admits a map
which is (at least intuitively) a fiber bundle over the locus of smooth hypersurfaces. Consequently, if corresponds to smooth hypersurfaces, we get an honest fiber bundle
But is connected, since we have thrown away a complex codimension subset to get from ; 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.
2. Three-fourths of the cohomology
OK, so let’s go back to the original situation. We are interested in determining the cohomology of . The first thing to note is that, by the Lefschetz hyperplane theorem, is simply connected. We consequently have
By Poincaré duality, we’ll also have
The main step now is to determine the second Betti number. There is a nice argument for this in a paper by Milnor “On simply connected 4-manifolds” which I’d like to describe here.
The key point is that to determine the second Betti number, we may as well determine the Euler characteristic of ; we can do this by computing the top Chern class of and evaluating on a fundamental class.
Unfortunately, we don’t really have a good grasp of the fundamental class of . We do, however, have a completely good grasp of the cohomology of the ambient space : the cohomology here is generated by the hyperplane class . If is the inclusion, we then observe that
because the intersection of with itself in has points (by Bezout’s theorem). As a result, if we run into something which is a multiple of , we can easily evaluate it on the fundamental class.
3. Computing Chern classes
The Chern classes of are known: the total Chern class is . That is,
which means that if we pull back along , we have
To get the Chern classes of , we’ll use the fact that the normal bundle of is easily computed via the adjunction formula.
Proposition 1 (Adjunction formula) Let be an inclusion of a smooth divisor in a smooth complex manifold. Then the normal bundle of is just the associated line bundle restricted to .
Consequently, we have
by adjunction. If we take Chern classes, we get that
It follows that and
In other words,
4. Finishing the computation
Now that we have the Chern class , we can obtain the Euler characteristic easily by evaluating the top Chern class on .
If we evaluate on the fundamental class of , we get
which means that the second Betti number of is . In Milnor’s paper, the case is handled.
This method seems to work to handle smooth hypersurfaces in for any , but it only gives the cohomology mod torsion; also, it does not give the ring structure. Does anyone know how to obtain this extra information?