Hodge filtration - Imported Knowlpedia Content

Definition

Let $V$ carry a pure Hodge structure of weight $n$ , with $V_{\mathbb C}=\bigoplus_{p+q=n}V^{p,q}$ . Its Hodge filtration is the decreasing filtration

F^pV_{\mathbb C}=\bigoplus_{r\geq p}V^{r,n-r}.

Thus $F^{p+1}\subseteq F^p$ , the filtration is exhaustive and separated, and

V^{p,q}=F^pV_{\mathbb C}\cap\overline{F^qV_{\mathbb C}} \qquad (p+q=n).

For a compact complex manifold, the Hodge filtration on degree- $n$ de Rham cohomology is induced by differential forms whose holomorphic degree is at least $p$ ; in the Kähler case it agrees with the displayed direct-sum formula.

Recovering the decomposition

A decreasing filtration on the complexification of a real vector space comes from a pure Hodge structure of weight $n$ precisely when

V_{\mathbb C}=F^p\oplus\overline{F^{\,n-p+1}}

for every $p$ . This opposedness condition recovers the Hodge components and the conjugation symmetry, so the filtration and decomposition formulations contain the same information Voisin, §7.1.1.

Geometric role

For a holomorphic family of compact Kähler manifolds, the subspaces $F^p$ vary holomorphically even though the individual subspaces $H^{p,q}$ generally do not. This makes the filtration the natural formulation for period maps and variations of Hodge structure. Griffiths transversality measures its failure to be preserved by the flat connection.

Conventions and near-misses

The Hodge filtration is decreasing and indexed by the first bidegree. The weight filtration of a mixed Hodge structure is instead increasing and records weights; the two filtrations are not interchangeable. An arbitrary filtration with the correct dimensions need not be a Hodge filtration because it may fail the conjugate-opposedness condition.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Chapter record. Relevant: §7.1.1, equivalence of decomposition and filtration descriptions.
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 on the differential of the period filtration.