Derived Log Albanese Sheaves

TL;DR

This paper introduces higher pro-Albanese functors for effective log motives over a field of characteristic zero, extending classical Albanese varieties to a logarithmic setting and computing them explicitly for smooth log schemes.

Contribution

It defines and computes higher pro-Albanese functors for log motives, generalizing previous higher Albanese sheaves to the logarithmic context.

Findings

Involves an inverse system of coherent cohomology.

Defines a pro-group scheme extending Serre's semi-abelian Albanese.

Generalizes higher Albanese sheaves to log schemes.

Abstract

We define higher pro-Albanese functors for every effective log motive over a field $k$ of characteristic zero, and we compute them for every smooth log smooth scheme $X=(\underline{X}, \partial X)$. The result involves an inverse system of the coherent cohomology of the underlying scheme as well as a pro-group scheme $\mathrm{Alb}^{\log}(X)$ that extends Serre's semi-abelian Albanese variety of $\underline{X}-|\partial X|$. This generalizes the higher Albanese sheaves of Ayoub, Barbieri-Viale and Kahn.