Hodge index theorem

Remarks

Equivalently, the intersection form on the real vector space $\operatorname{NS}(S) \otimes \R$, where $\operatorname{NS}(S)$ is the Néron–Severi group, has signature $(1, \rho - 1)$, where $\rho$ is its dimension. The proof uses Riemann–Roch on the surface, which shows that if $D^2 > 0$, then $nD$ or $-nD$ is linearly equivalent to an effective divisor for some $n \ge 1$. A nonzero effective divisor meets the ample $H$ positively, which contradicts $D \cdot H = 0$. A useful consequence is the inequality $(D^2)(H^2) \le (D \cdot H)^2$ for $H$ ample and $D$ arbitrary, obtained by applying the theorem to $(H^2) D - (D \cdot H) H$. Example: on $S = \mathbb{P}^1 \times \mathbb{P}^1$ with fibre classes $F_1, F_2$, one has $F_1^2 = F_2^2 = 0$ and $F_1 \cdot F_2 = 1$; the ample $H = F_1 + F_2$ is orthogonal to $D = F_1 - F_2$, and indeed $D^2 = -2 < 0$. Over $\C$, Hodge theory gives the more general fact that the cup product on $H^2(S, \R)$ has signature $(2 p_g + 1, h^{1,1} - 1)$, where $p_g = h^{2,0}$.

Going deeper

For a student

The Hodge index theorem is the main structural fact about intersection theory on surfaces. It is used in a proof of the Riemann hypothesis for curves over finite fields, applied to the graph of Frobenius on a product of curves. It is also used in the study of the Néron–Severi lattice of K3 surfaces and in many inequalities between Chern numbers. Its higher-dimensional analogues are the Hodge–Riemann bilinear relations and the hard Lefschetz theorem.