Polarized Hodge structure - Imported Knowlpedia Content

Definition

A polarization of a real pure Hodge structure $V$ of weight $n$ is a nondegenerate real bilinear form $Q$ such that

Q(v,w)=(-1)^nQ(w,v), \qquad Q(V^{p,q},V^{r,s})=0\ \text{unless }(r,s)=(q,p),

and, for every nonzero $v\in V^{p,q}$ ,

i^{\,p-q}Q(v,\overline v)>0.

A polarized Hodge structure is the pair $(V,Q)$ . For a rational or integral Hodge structure, $Q$ is required to be defined over the corresponding coefficient ring, with nondegeneracy interpreted after extending scalars. Thus $Q$ is symmetric in even weight and alternating in odd weight.

Weil-operator formulation

The Weil operator $C$ acts on $V^{p,q}$ by $i^{p-q}$ . The positivity condition says that

h(v,w)=Q(Cv,\overline w)

is a positive-definite Hermitian form. The orthogonality and positivity conditions are the abstract form of the Hodge–Riemann bilinear relations Voisin, §7.1.2.

Cohomological construction

For a compact Kähler manifold with a rational Kähler class, each primitive cohomology group becomes polarized after the signed intersection pairing is normalized by the factor $(-1)^{n(n-1)/2}$ appropriate to its weight. Lefschetz decomposition then expresses the full cohomology as a sum of polarized primitive pieces.

Conventions and near-misses

Some authors place a weight-dependent sign in $Q$ , in the Weil operator, or in the positivity inequality. These conventions describe equivalent structures only when all three formulas are adjusted together. A nondegenerate form satisfying the parity rule is not automatically a polarization: it must also satisfy Hodge orthogonality and positivity.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Chapter record. Relevant: §7.1.2, polarizations, and the primitive cohomology examples.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: Chapter 3, Hodge–Riemann bilinear relations and polarized cohomology.