Grothendieck topologies with logarithmic modifications
Xianyu Hu, Maximilian Schimpf
TL;DR
This paper develops a Grothendieck topology framework for logarithmic geometry that is invariant under log modifications. It defines the $m$-topologies ($m$-open, $m$-étale, $m$-smooth, $m$-fppf, $m$-fpqc) via universally surjective log étale monomorphisms and proves their basic properties, including a lifting criterion and strict étale locality. It also refines the notion of universal surjectivity to fix gaps in the traditional log étale topology, establishes an equivalence between sheaves on a valuative log space $X^{ ext{val}}$ and on the $m$-open site $X_{ ext{mop}}$, and develops a logarithmic dimension theory with $ ext{log-dim}(X) = ext{dim}(X^{ ext{val}})$ and cohomology vanishing for $p > ext{log-dim}(X)$. These results provide a robust, functorial framework for invariants in logarithmic geometry and have implications for log moduli and log Gromov–Witten theory.
Abstract
Many concepts in logarithmic geometry are invariant under log blowups. To formalize this invariance, we introduce the m-open, m-étale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes. These refine the standard topologies from scheme theory by treating log modifications as covers. In constructing them, we identify and correct errors in the definitions of log modifications and the full log étale topology. Our m-topologies are variants of those introduced by Niziol and Park; specifically, the m-étale topology is a subtopology of Kato's full log étale topology, characterized by a stronger lifting property than for log étale maps. This strengthening ensures the functoriality of the corresponding small site. We also characterize the sheaves for all these sites and connect the m-open site to Kato's valuative space.