Hodge–Riemann bilinear relations - Imported Knowlpedia Content

Statement

Let $X$ be a compact Kähler manifold of complex dimension $n$ , with Kähler class $\omega$ . If $0\neq\alpha\in P^{p,q}(X)$ is a primitive cohomology class and $k=p+q\leq n$ , then the Hodge–Riemann bilinear relations assert

i^{\,p-q}(-1)^{k(k-1)/2} \int_X\alpha\wedge\overline{\alpha}\wedge\omega^{n-k}>0.

Moreover, the corresponding bilinear intersection form pairs $H^{p,q}$ orthogonally to $H^{r,s}$ unless $(r,s)=(q,p)$ . The displayed sign fixes the convention used here; changing the normalization of the intersection form moves, but does not remove, this sign.

Relation to hard Lefschetz

The hard Lefschetz theorem supplies the primitive decomposition, while the Hodge–Riemann relations determine the sign of the intersection form on every primitive summand. Together they imply that cohomology carries a polarized Hodge structure when the Kähler class is rational Voisin, §6.3, Theorem 6.33.

Consequences

For a compact Kähler surface, the degree-two relation implies the Hodge index theorem: the intersection form has one positive direction on the real $(1,1)$ -classes generated by a Kähler class and is negative definite on its primitive real $(1,1)$ -complement. More generally, the theorem gives nondegeneracy and controlled signatures on the primitive pieces rather than merely positivity of total cohomology.

Conventions and boundary cases

The power $\omega^{n-k}$ is essential: without it the integrand does not have top degree unless $k=n$ . The positivity statement is made on primitive classes; a nonprimitive class can have a different sign after Lefschetz decomposition. Authors who build $(-1)^{k(k-1)/2}$ into the definition of the bilinear form state the same theorem without displaying that factor.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Publisher record. Relevant: §6.3, especially Theorem 6.33, and §7.1.2 on polarizations.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: Chapter 3, the hard Lefschetz theorem and Hodge–Riemann bilinear relations.