Ramification And Nearby Cycles For1-adic Sheaves

In this thesis, we study the nearby cycle complex of anι-adic sheaf on a scheme over a trait, using ramification theory of Abbes and Saito.The first part is devoted to a new proof of a formula of Deligne and Kato that computes the dimension of the stalks of the nearby cycle complex of anι-adic sheaf on a smooth relative curve over a strictly local trait. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. Our approach is based on a local notion of characteristic cycle defined using the refined Swan conductor of Abbes and Saito.In the second part, we prove a formula that computes the Swan conductor of the cohomology of the nearby cycle complex of an ι-adic sheaf on a smooth variety over a trait of equal characteristic, satisfying a certain ramification condition. Tsushima introduced the refined characteristic class of the sheaf and he proved that it computes the Swan conductor of the cohomology of its nearby cycle complex by a Lefschetz-Verdier type formula. We compute the refined characteristic class as an intersection product on the logarithmic cotangent bundle of the variety, involving the characteristic cycle of the sheaf defined by Abbes and Saito and the zero section.