Coherent sheaves in logarithmic geometry

TL;DR

This paper develops a new framework for logarithmic coherent sheaves using the full logarithmic étale topology, unifying various logarithmic moduli spaces and providing tools for homological algebra calculations.

Contribution

It introduces an abelian category of logarithmic coherent sheaves and tools to simplify homological algebra evaluations in logarithmic geometry.

Findings

Defines an abelian category of logarithmic coherent sheaves.

Provides tools reducing homological algebra to calculable logarithmic alterations.

Connects logarithmic sheaves with moduli spaces like logarithmic Quot and Picard groups.

Abstract

This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent sheaves in the full logarithmic \'etale topology. We develop a suite of tools that reduces the evaluation of the basic functors of homological algebra to the conventional calculation on a computable logarithmic alteration. A second paper will establish good properties of the associated logarithmic derived category. We thus offer a unified perspective on logarithmic moduli spaces of coherent sheaves: The logarithmic Quot spaces motivated by Maulik and Ranganathan's logarithmic Donaldson--Thomas theory, the logarithmic Picard group constructed by Molcho and Wise, and moduli spaces of logarithmic parabolic sheaves as developed by Borne, Talpo, and Vistoli. In establishing the connection with logarithmic Picard groups, we offer a new interpretation of chip firing as the combinatorial shadow to a logarithmic version of S-equivalence.