Mixed Hodge structures on cohomology jump ideals

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.