Regularized integrals and manifolds with log corners
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917378704539648 |
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| author | Dupont, Clément Panzer, Erik Pym, Brent |
| author_facet | Dupont, Clément Panzer, Erik Pym, Brent |
| contents | 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 ''$\lim_{ε\to 0} \log ε:= 0$'' 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.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17720 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularized integrals and manifolds with log corners Dupont, Clément Panzer, Erik Pym, Brent Differential Geometry Mathematical Physics Algebraic Geometry Number Theory 58C35, 40A10, 14A21, 14F40 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 ''$\lim_{ε\to 0} \log ε:= 0$'' 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.) |
| title | Regularized integrals and manifolds with log corners |
| topic | Differential Geometry Mathematical Physics Algebraic Geometry Number Theory 58C35, 40A10, 14A21, 14F40 |
| url | https://arxiv.org/abs/2312.17720 |