Functoriality of logarithmic Hochschild homology of log smooth pairs

TL;DR

This paper develops a functorial framework for logarithmic Hochschild homology of log smooth pairs using Fourier--Mukai transforms, overcoming key technical challenges in logarithmic geometry.

Contribution

It introduces a new approach based on logarithmic Fourier--Mukai transforms and constructs a dg bicategory of logarithmic correspondences with categorical invariants.

Findings

Logarithmic Hochschild homology is functorial under the new transforms.

Constructed explicit unit- and counit-type morphisms for adjunctions.

Defined logarithmic Chern characters and Euler pairing compatible with the formalism.

Abstract

The construction of a satisfactory dg category of logarithmic coherent sheaves remains a central open problem in logarithmic geometry. In this paper, we propose an alternative correspondence-theoretic approach based on logarithmic Fourier--Mukai transforms. For smooth proper log pairs, we introduce strong log Fourier--Mukai kernels supported on canonical blow-up compactifications and prove that logarithmic Hochschild homology is functorial with respect to the induced transforms. Unlike the classical setting, logarithmic correspondences do not naturally live on ordinary products, and the standard adjunction formalism fails because of blow-up discrepancies. We overcome these difficulties by constructing explicit unit- and counit-type morphisms that provide the necessary adjunction data without requiring an ambient dg category of logarithmic sheaves. As applications, we construct a dg bicategory of logarithmic correspondences in which logarithmic Hochschild homology and cohomology become categorical invariants. We also define logarithmic Chern characters and a logarithmic Euler pairing compatible with the logarithmic Fourier--Mukai formalism.