Geometric polarizable Hodge variations on algebraic bases

Let $S$ be a connected smooth quasiprojective complex variety. A geometric polarizable rational Hodge variation on $S$ consists of a finite-rank rational local system $H$ (Local systems and pullback), a continuous pure Hodge decomposition of its fibers, or a finite direct sum of pure weights, in the sense of Rational mixed Hodge structures, and an algebraic realization on a nonempty Zariski open $U\subseteq S$: there are smooth projective morphisms $f_i:X_i\to U$, degrees $m_i$ and integers $r_i$ such that $H{\mathrm{\mid }}_U$ is a Hodge direct summand of ${⨁}_i{R}^{m_i}(f_i{)}_{\ast }\mathbb{Q}(r_i),$ where $\mathbb{Q}(r)$ is the Tate structure of Rational mixed Hodge structures. A Hodge direct summand means the image of a flat rational idempotent that is a Hodge morphism at every point.

A morphism of variations is a flat rational map that is a Hodge morphism at every point. A polarization of a pure object is a flat rational bilinear form with the Hodge positivity property on every fiber; it is not postulated for arbitrary local subsystems: a geometric realization carries one by relative ampleness, and Hodge direct summands inherit it.

The closures recorded here under finite sums, tensor products, Hodge direct summands and Tate twists, the exterior-power realizability, the duals via the polarization, and the full monodromy-stable integral lattices are proved by the family construction later on this page (Geometric Gauss-Manin Hodge variations and integral lattices ↗); that forward mention is explanatory and is not a prerequisite of the definition. The algebraic-open fundamental-group assertion and the Gauss-Manin construction remain separate proof suppliers, and no claim is made for arbitrary analytic variations on a punctured disk. No choice principle is needed: the definition quantifies over finite data.