On the interior motive of certain Shimura varieties: the case of Picard surfaces

TL;DR

This paper constructs a Hecke-equivariant Chow motive for Picard surfaces, linking it to their intersection cohomology and enabling the definition of Grothendieck motives for Picard modular forms.

Contribution

It introduces a new motive construction for Picard surfaces that connects intersection cohomology with algebraic cycles, advancing the understanding of motives in this context.

Findings

Constructed a Hecke-equivariant Chow motive for Picard surfaces.

Linked the motive's realizations to intersection cohomology.

Enabled the definition of Grothendieck motives for Picard modular forms.

Abstract

The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Picard surfaces with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Picard modular forms.