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 ‘’${\lim }_{\epsilon \to 0}\log \epsilon :=0$’’ 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.)