# Mixed Hodge structures on cohomology jump ideals

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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03268018
Contributor : Louis-Clément Lefèvre <>
Submitted on : Tuesday, June 22, 2021 - 9:18:11 PM
Last modification on : Thursday, June 24, 2021 - 9:40:55 AM
Long-term archiving on: : Thursday, September 23, 2021 - 7:20:35 PM

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### Identifiers

• HAL Id : hal-03268018, version 1
• ARXIV : 2106.12339

### Citation

Louis-Clément Lefèvre. Mixed Hodge structures on cohomology jump ideals. 2021. ⟨hal-03268018⟩

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