Categories of complex variations of Hodge structure over compact K"ahler manifolds

TL;DR

This paper introduces a comprehensive framework for complex polarized variations of Hodge structure on compact Kähler manifolds, enabling control over all finite-dimensional variations and their tensor relations, with implications for cohomology algebras.

Contribution

It provides a universal variation of Hodge structure that governs all finite-dimensional cases and their tensor relations on compact Kähler manifolds.

Findings

Established a universal complex polarized variation of Hodge structure.

Derived the structure of cohomology algebras with local system values.

Connected variations of Hodge structure with multiplicative properties.

Abstract

We give a complex polarized variation of Hodge structure over a compact K"ahler manifold $M$ which controls all finite-dimensional complex polarized variations of Hodge structure over $M$ and their tensor relations. As a corollary, we obtain the cohomology algebra with values in a local system admitting multiplicative Hodge structures.