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Pré-Publication, Document De Travail Année : 2023

Regularized integrals and manifolds with log corners

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, which allows us to interpret the ubiquitous rule of thumb "$\lim_{\epsilon\to 0} \log \epsilon := 0$" as the natural 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.)
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Dates et versions

hal-04407110 , version 1 (20-01-2024)

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Clément Dupont, Erik Panzer, Brent Pym. Regularized integrals and manifolds with log corners. 2023. ⟨hal-04407110⟩
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