UC Berkeley

Mikio Sato’s fundamental idea of viewing objects, a priori defined on a space, as living on the cotangent bundle of that space led to the birth of the subject of microlocal analysis and spread to other fields of mathematics. It has been applied to and greatly enriched the theories of D-modules and constructible sheaves in the real or complex analytic context, with important applications to geometric representation theory and much more. In this dissertation, we study étale sheaves in positive characteristic from the microlocal point of view. The main results are: i) generically on a smooth surface, the vanishing cycle form a local system with respect to the variation of transverse test functions in high enough order terms; ii) the vanishing cycle of a tame simple normal crossing sheaf has the same stability as in the complex constructible case; iii) for a monodromic sheaf on a finite dimensional vector space, its characteristic cycle is canonically identified with that of the Fourier transform of the sheaf. In the Introduction, we also discuss the implications of these results in a broader context and an application of iii) to the study of character sheaves in positive characteristic.