# On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties

### Mattia Cavicchi

LAGA, Institut Galilée, Université Paris 13, F-93430 Villetaneuse, France

## Abstract

We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mathrm{G}\mathrm{Sp}_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^\lambda\mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_\lambda$ of $\mathrm{G}\mathrm{Sp}_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $\lambda$, potentially generalisable to other families of Shimura varieties, which characterizes the absence of the *middle weights* 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over $\mathbb{Q}$, whose realizations equal interior (or intersection) cohomology of $S_K$ with $V_{\lambda}$-coefficients. We give applications to the construction of homological motives associated to automorphic representations.

## Cite this article

Mattia Cavicchi, On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties. Doc. Math. 24 (2019), pp. 1033–1098

DOI 10.4171/DM/699