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

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.