Work

Regularized integrals and manifolds with log corners

Year: 2026

Type: article

Abstract: We introduce a natural geometric framework for the study of logarithmically divergent integrals on manifolds with corners and algebraic varieties, using the techniques of logarithmic geometry. Key to the construction is a new notion of morphism in logarithmic geometry itself, introduced by Howell, which allows us to interpret the ubiquitous rule of thumb “ lim ε → 0 log ε : = 0 ” as the restriction to a submanifold. Via a version of de Rham’s theorem with logarithmic divergences, we obtain a functorial characterization of the classical theory of “regularized integration”: it is the unique way to extend the ordinary integral to the logarithmically divergent context while respecting the basic laws of calculus (change of variables, Fubini’s theorem, and Stokes’ formula.)

Source: Journal de l’École polytechnique — Mathématiques

DOI: 10.5802/jep.335

Authors: Clément Dupont, Erik Panzer, Brent Pym

Institutions: Centre National de la Recherche Scientifique, Université de Montpellier, Institut Montpelliérain Alexander Grothendieck, University of Oxford, McGill University

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond