Deformations and Hodge Theory

Abstract

We have now come to the penultimate chapter. Over the course of this book, we have introduced a number of invariants, such as Betti numbers, Hodge numbers, and Picard numbers, that can be used to distinguish (complex smooth projective) varieties from one another. Varieties tend to occur in families. For example, we have encountered the Legendre family $y^2 = x(x-1)(x-\lambda)$ of elliptic curves. We have, at least implicitly, considered the family of all nonsingular hypersurfaces $\Sigma a_{d_0\cdots d_n}x^{d_n}_n \cdots x^{d_n}_n = 0$ of fixed degree. So the question is, what happens to these invariants as the coefficients vary? Or in more geometric language, what happens as the variety deforms? For Betti numbers of complex smooth projective varieties, we have already seen that as a consequence of Ehresmann–s theorem, they will not change in a family, because all the fibers are diffeomorphic. However, as algebraic varieties, or as complex manifolds, they can be very different. So it is perhaps surprising that the Hodge numbers will not change either. This is a theorem of Kodaira and Spencer, whose proof we outline. The proof will make use of pretty much everything we have done, plus one more thing. We will need a basic result due to Grothendieck in the algebraic setting, and Grauert in the analytic, that under the appropriate assumptions, the dimensions of coherent cohomology are upper semicontinuous. This means that the Hodge numbers could theoretically jump upward at special values of the parameters. On the other hand, by the Hodge decomposition, their sums, which are the Betti numbers, cannot. That is the basic idea. In the last section, we look at the behavior of the Picard number. Here the results are much less definitive. We end with the Noether–Lefschetz theorem, which explains what happens for general surfaces in $\mathbb{P}^3_{\mathbb C}$.