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

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.

Intermediate extension of Chow motives of Abelian type

“…8.6] is induced by an irreducible representation of G, whose highest weight is regular. Then according to [W5,Cor. 3.9], the interior analogous to Lemma 8.4, one would need to analyze the restriction of the morphism of models of toroidal compactifications from [La2,Thm.…”

𝔸¹-equivalence of zero cycles on surfaces

“…Therefore, the conservativity conjecture implies the log Bloch conjecture for U . Part of our main theorem then follows from a special case of the conservativity conjecture proven by Wildeshaus [Wil15].…”

On weight complexes, pure functors, and detecting weights

“…3. In the current paper we use the "homological convention" for weight structures; it was previously used in [Wil15], [Wil18], [Heb11], [Bon14], [BoI15], [BoS18b], [Bon16], [BoK18], [Bon18a], and in [Bon19], whereas in [Bon10a], [Bon10b], and [BoT17] the "cohomological convention" was used. In the latter convention the roles of C w≤0 and C w≥0 are interchanged, i.e., one considers C w≤0 = C w≥0 and C w≥0 = C w≤0 .…”