Regularized integrals and manifolds with log corners

Clément Dupont; Erik Panzer; Brent Pym

doi:10.5802/jep.335

Regularized integrals and manifolds with log corners
[Intégrales régularisées et variétés différentielles à coins logarithmiques]

Clément Dupont ¹ ; Erik Panzer ² ; Brent Pym ³

¹Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, CNRS, Place Eugène Bataillon, 34090 Montpellier, France
²Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG Oxford, UK
³Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada

Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 687-757

Abstract (VO)
Résumé

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 “$\lim _{\varepsilon \rightarrow 0} \log \varepsilon := 0$” 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.)

Nous introduisons un cadre géométrique naturel pour l’étude des intégrales à divergences logarithmiques sur les variétés différentielles à coins et les variétés algébriques, en utilisant les techniques de la géométrie logarithmique. L’élément clé de cette construction est une nouvelle notion de morphisme en géométrie logarithmique, introduite par Howell, qui nous permet d’interpréter la prescription classique « $\lim _{\varepsilon \rightarrow 0} \log \varepsilon := 0$ » comme une restriction à une sous-variété. Grâce à une version du théorème de de Rham en présence de divergences logarithmiques, nous obtenons une caractérisation fonctorielle de la théorie classique de l’« intégration régularisée » : c’est l’unique manière d’étendre l’intégration ordinaire au cadre des divergences logarithmiques tout en respectant les lois fondamentales du calcul intégral (changement de variables, théorème de Fubini et formule de Stokes).

Reçu le : 2024-09-09
Accepté le : 2026-02-11
Publié le : 2026-03-13

DOI : 10.5802/jep.335

Classification : 58C35, 40A10, 14A21, 14F40
Keywords: Divergent integrals, logarithmic regularization, manifolds with corners, logarithmic geometry
Mots-clés : Intégrales divergentes, régularisation logarithmique, variétés différentielles à coins, géométrie logarithmique

Affiliations des auteurs :

Clément Dupont ¹ ; Erik Panzer ² ; Brent Pym ³

¹ Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, CNRS, Place Eugène Bataillon, 34090 Montpellier, France
² Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG Oxford, UK
³ Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Regularized integrals and manifolds~with~log corners},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Clément Dupont; Erik Panzer; Brent Pym. Regularized integrals and manifolds with log corners. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 687-757. doi: 10.5802/jep.335

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