Categorical Logarithmic Hodge Theory, I

TL;DR

This paper introduces a new logarithmic quasicoherent category associated with a smooth open algebraic variety and its toroidal compactification, linking it to logarithmic Hodge theory and Hochschild homology.

Contribution

It defines a novel categorical framework for logarithmic Hodge theory, establishes its Hochschild homology correspondence with log-forms, and proves derived invariance under toroidal modifications.

Findings

Hochschild homology matches log-forms on the compactification.

The noncommutative Hodge-to de Rham sequence recovers log Hodge structures.

Derived invariance under toroidal changes of compactification.

Abstract

We write down a new "logarithmic" quasicoherent category $\operatorname{Qcoh}_{log}(U, X, D)$ attached to a smooth open algebraic variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This is a (large) symmetric monoidal Abelian category, which we argue can be thought of as the categorical substrate for logarithmic Hodge theory of $U$. We show that its Hochschild homology theory coincides with the theory of log-forms on $X$ with logarithmic structure induced by $D$, and in particular, that the noncommutative Hodge-to de Rham sequence on $\operatorname{Qcoh}_{log}(U, X, D)$ recovers known log Hodge structure on the de Rham cohomology of the open variety $U$. As an application, we compute the Hochschild homology of the category of coherent sheaves on the infinite root stack of Talpo and Vistoli in the toroidal setting. We prove a derived invariance result for this theory: namely, that strictly toroidal changes of compactification do not change the derived category of $\operatorname{Qcoh}_{log}(U, X, D)$. The definition is motivated by the coherent object appearing in the author's microlocal mirror symmetry result [20]. In this paper, the first in a series, we work over an algebraically closed field of characteristic zero. The next installment will develop the characteristic p and mixed-characteristic theories.