Intersection complex of any threefold as a Chow motive

TL;DR

This paper constructs a Chow motive that lifts the motivic intersection complex for any threefold, extending previous characterizations and demonstrating functoriality within motivic sheaves.

Contribution

It introduces a functorial Chow motive for threefolds that generalizes the motivic intersection complex using weight truncations.

Findings

The constructed motive satisfies Wildeshaus's characterization.

The motive lifts the motivic intersection complex for arbitrary threefolds.

The construction is functorial.

Abstract

Motivated by the characterization of the intersection complex in terms of S$.$Morel's weight truncations, we introduced an object $EM^{F}_{X}$ in the setting of motivic sheaves for certain schemes $X$ and weight profiles $F$. In this article, we show that when $X$ is any threefold, this object satisfies Wildeshaus's characterization of a motivic intersection complex. In particular, we demonstrate that the construction is a suitably functorial Chow motive lifting the motivic intersection complex for an arbitrary threefold.