Polarized variation of Hodge structure and period map

Let $S$ be a connected complex manifold and let ${\mathbb{V}}_{\mathbb{Z}}$ be a finite-rank integral local system modulo torsion, with ${\mathbb{V}}_{\mathbb{Q}}={\mathbb{V}}_{\mathbb{Z}}\otimes \mathbb{Q}$. Its associated holomorphic bundle is $\mathcal{H}={\mathbb{V}}_{\mathbb{Q}}{\otimes }_{\mathbb{Q}}{\mathcal{O}}_S$ with the flat Gauss–Manin connection $\mathrm{\nabla }$. A pure polarized variation of Hodge structure of weight $k$ is this data together with a decreasing filtration by holomorphic subbundles $F^p\mathcal{H}$ of fixed ranks such that every fibre is a rational pure Hodge structure of weight $k$, polarized by one flat rational bilinear form $Q$, and $\mathrm{\nabla }(F^p\mathcal{H})\subseteq {\mathrm{\Omega }}_S^1\otimes F^{p-1}\mathcal{H}\mathrm{.}$ Here “pure of weight $k$” means $V_{s,\mathbb{C}}={⨁}_{p+q=k}{H}_s^{p,q}$ with $H_s^{p,q}=F_s^p\cap \overline{F_s^q}$ and $\overline{H_s^{p,q}}=H_s^{q,p}$. In the polarization convention of the rational mixed-Hodge-structure definition, $Q(u,v)=(-1{)}^{k}Q(v,u)$, its complexification pairs $H_s^{p,q}$ only with $H_s^{k-p,k-q}$, and $h_s(u,v)=Q(C_{s}u,\bar{v})$ is positive definite, where $C_s{\mathrm{\mid }}_{H_s^{p,q}}=i^{p-q}$. Flatness means that parallel transport preserves $Q$. A family of several weights means a finite direct sum of pure variations. Polarization requires either a projective family with a selected relatively ample class or a separately supplied flat polarization; compact Kähler fibres alone do not select one.

On a simply connected open chart $U\subseteq S$, choose a flat marking of ${\mathbb{V}}_{\mathbb{Q}}{\mathrm{\mid }}_U$ by a fixed rational vector space $V_{\mathbb{Q}}$. The period domain $D$ is the space of polarized Hodge filtrations on $V_{\mathbb{C}}$ with the fixed Hodge numbers and polarization. The period map is the holomorphic map $\mathrm{\Phi }:U\to D$, $s\mapsto F^{\bullet }{\mathcal{H}}_s$ in that marking. The marking identifies the flat polarization with a fixed form on $V_{\mathbb{Q}}$. Two such markings differ by a constant rational $Q$-isometry $g$, and their period maps satisfy ${\mathrm{\Phi }}_2=g{\mathrm{\Phi }}_1$. This $g$ need not belong to the monodromy group. After fixing a base-point marking and continuing it along paths, changes of path are governed by the monodromy representation; a global marking is not assumed.

Rational mixed Hodge structures

Geometric polarizable Hodge variations on algebraic bases

Geometric Gauss-Manin Hodge variations and integral lattices

The Axiom of Choice

Deligne, §4.2

Voisin, §3.3

The geometric supplier above applies to smooth projective families. The definition also admits an analytic polarized variation when the local system, filtration, connection and polarization are supplied independently; it does not infer projectivity or a polarization from compact Kähler fibres.