The Hodge decomposition, diamond, and Euler characteristics

 Seminar topic

Recall the sheaf of $r$-differential forms $\Omega^r_X$ on $X$ (with $\Omega^r_X(U) = \{fdx_{i_1}\wedge \cdots \wedge dx_{i_r}\ :\ f\ $is well-defined on $U\}$ and such sums) and the structure sheaf $\mathcal O_X$ on $X$ (with $\mathcal O_X(U) = \{f/g\ :\ f,g\in k[U],\ g\neq 0\ $on$\ U\}$). Then we may consider the sheaf cohomology of $X$, with values in $\Omega^r_X$ or $\mathcal O_X$.

Definition: Let $X$ be a smooth manifold of dimension $n$. The $(p,q)$th Hodge number is $h^{p,q}=\dim(H^{p,q})$, where $H^{p,q} = H^q(X,\Omega^p_X)$. These numbers are arranged in a Hodge diamond as below.

The Hodge diamond has a lot of repetition - by complex conjugation, we get that $h^{p,q}=h^{q,p}$, so it is symmetric about its vertical axis. By the Hard Lefschetz theorem (or the Hodge star operator, or Poincare duality), we get that $h^{p,q}=h^{n-q,n-p}$, so it is symmetric about its horizontal axis.

Proposition: Let $X$ be a Kähler manifold (note that all smooth projective varieties are Kähler) of dimension $n$. Then the cohomology groups of $X$ decompose as
$$H^k(X,\C) = \bigoplus_{p+q=k}H^{p,q}(X),$$
for all $0\leqslant k\leqslant 2n$. This is called the Hodge decomposition of $X$.

This decomposition immediately gives all the Hodge numbers for $\P^n$, knowing its cohomology. For a manifold of complex dimension $n$, there are several numbers and polynomials that may be defined. These are:
$$\begin{align*} \chi_{top}(X) & = \sum_{i=1}^{2n}(-1)^i \dim(H^i(X,\C)) & \text{the (topological) Euler characteristic} \\ \chi^p(X) & = \sum_{q=0}^{n-1}(-1)^qh^{p,q} & \text{the chi-$p$ characteristic} \\ \chi_y(X) & = \sum_{p=0}^{n-1}\chi^py^p & \text{the chi-$y$ characteristic} \end{align*}$$
Note the Euler characteristic is the alternating sum of the rows of the Hodge diamond, and the chi-$p$ characteristic is the alternating sum of the left-right diagonals of the diamond.

Example: In the case $X$ is a hypersurface in projective $n$-space $\P^n$ defined by a degree $d$ polynomial,
$$\chi_y = [z^n]\frac{1}{(1+zy)(1-z)^2}\cdot\frac{(1+zy)^d-(1-z)^d}{(1+zy)^d+y(1-z)^d}.$$
Since every row except the middle row of the Hodge diamond of a hypersurface is known (as it comes from the Hodge diamond of $\P^n$ by the Lefschetz hyperplane theorem), this expression gives all the unknown numbers. This particular formula is a simplification  of Theorem 22.1.1 in Hirzebruch, which itself comes from the Riemann--Roch theorem.

References: Huybrechts (Complex Geometry: An Introduction, Chapters 3.2, 3.3), Hirzebruch (Topological Methods in Algebraic Geometry, Appendix 1, Section 22)