EVENT DETAILS AND ABSTRACT

Title: Relative rounding: the geometry underlying the smoothness of logarithmic cohomology
Speaker: Arthur Ogus
Speaker Info: UC-Berkeley
Brief Description:
Special Note:
Abstract:

Logarithmic geometry was invented by Deligne, Faltings, Illusie, Fontaine, and Kato with the purpose of understanding bad reduction of varieties over number fields. It has had spectacular success: logarithmic crystalline coho- mology plays a key role in the study of p-adic Hodge theory and the for- mulation and proof of the B_{st}-conjecture. Since then Kato and Nakayama have developed other versions of logarithmic cohomology, including logarith- mic etale cohomology and logarithmic Hodge theory. Logarithmic structures seem to serve as kind of "magic powder" which makes singularities behave as if they are smooth. In this talk I will describe a geometric picture which helps to explain how this magic works. In this picture, logarithmic schemes can be viewed as "algebraic varieties with boundary," a notion which formerly seemed to be completely absent from algebraic and even analytic geometry. \Relative rounding," an analog of the implicit function theorem for exact log smooth morphisms, shows that this construction behaves well in families. The proof, joint work with C. Nakayama, depends in a key way on the moment mapping and Kato's notion of exactness.