Mixed Hodge structures on cohomology jump ideals

TL;DR

This paper establishes that cohomology jump loci in the moduli space of representations of fundamental groups carry natural mixed Hodge structures, revealing their motivic nature and geometric structure.

Contribution

It proves that jump ideals are sub-mixed Hodge structures and identifies cases where these loci are translated sub-tori, combining Hodge theory with homotopy and algebraic methods.

Findings

Jump ideals are sub-mixed Hodge structures.

Cohomology jump loci are translated sub-tori in rank one cases.

Methods combine transcendental Hodge theory with algebraic and homotopy techniques.

Abstract

In previous work, we constructed for a smooth complex variety $X$ and for a linear algebraic group $G$ a mixed Hodge structure on the complete local ring $\widehat{\mathcal{O}}_\rho$ to the moduli space of representations of the fundamental group $\pi_1(X,x)$ into $G$ at a representation $\rho$ underlying a variation of mixed Hodge structure. We now show that the jump ideals $J_k^i \subset \widehat{\mathcal{O}}_\rho$, defining the locus of representations such the the dimension of the cohomology of $X$ in degree $i$ of the associated local system is greater than $k$, are sub-mixed Hodge structures; this is in accordance with various known motivicity results for these loci. In rank one we also recover, and find new cases, where these loci are translated sub-tori of the moduli of representations. Our methods are first transcendental, relying on Hodge theory, and then combined with tools of homotopy and algebra.