Regularized integrals and manifolds with log corners

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 “ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo form="prefix" movablelimits="true">lim</mml:mo> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo form="prefix">log</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ” 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.)