Pure motives, mixed motives and extensions of motives associated to singular surfaces

TL;DR

This paper explores the construction and properties of motives associated with singular surfaces, using Voevodsky's motives, and provides a motivic interpretation of recent results on Hilbert--Blumenthal surfaces.

Contribution

It introduces a new approach to studying motives of singular surfaces and their exceptional divisors, extending Voevodsky's formalism to construct specific motive extensions.

Findings

Construction of Chow motives modeling intersection cohomology

Motivic interpretation of recent results on Hilbert--Blumenthal surfaces

Extension of Voevodsky's motives to singular surface contexts

Abstract

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert--Blumenthal surfaces, we recover a motivic interpretation of a recent construction of A. Caspar.