Variation of Hodge structure - Imported Knowlpedia Content

Definition

A real variation of Hodge structure of weight $n$ on a complex manifold $S$ consists of a finite-rank real local system $\mathbb V_{\mathbb R}$ , its holomorphic bundle $\mathcal V=\mathbb V_{\mathbb R}\otimes_{\mathbb R}\mathcal O_S$ with flat connection $\nabla$ , and holomorphic subbundles forming a decreasing filtration $F^\bullet\mathcal V$ . At every $s\in S$ , the filtration defines a pure real Hodge structure of weight $n$ , and it satisfies Griffiths transversality

\nabla(F^p\mathcal V)\subseteq \Omega_S^1\otimes F^{p-1}\mathcal V.

A polarized variation additionally has a flat bilinear form that polarizes every fiber.

Period-map interpretation

After locally trivializing the flat bundle, $F^\bullet$ determines a holomorphic map from $S$ to a flag variety. Fiberwise Hodge opposedness restricts its image to a period domain, and Griffiths transversality says that its differential lies in the horizontal subbundle. This is the infinitesimal period relation established in Griffiths, §1, Theorems 1.27 and 1.34.

Geometric construction

For a smooth proper holomorphic family $f:\mathcal X\to S$ of compact Kähler manifolds, the local system $R^nf_*\mathbb R$ , its Gauss–Manin connection, and the fiberwise Hodge filtrations form a variation of weight $n$ . The locally constant lattice $R^nf_*\mathbb Z$ gives an integral structure when torsion is removed. Voisin develops this construction in Chapter 9 and §10.2.

Conventions and near-misses

A holomorphic family of filtrations with pure Hodge fibers is not a variation unless it is tied to a flat local system and obeys transversality. “VHS” may mean real, rational, integral, or complex variation, and polarization is sometimes included by default; both choices must be stated. Variations are usually assumed to have locally constant Hodge numbers so that the $F^p$ are subbundles.

References

Phillip A. Griffiths, “Periods of Integrals on Algebraic Manifolds, II: Local Study of the Period Mapping,” American Journal of Mathematics 90 (1968), 805–865. DOI record. Relevant: §1, especially Theorems 1.27 and 1.34, for the period map and its infinitesimal relation.
Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Publisher record. Relevant: Chapter 9, especially §9.2, and Chapter 10, especially §10.2.