ar X iv : 0 70 5 . 27 99 v 1 [ m at h . A G ] 1 9 M ay 2 00 7 Wild ramification and the characteristic cycle of an l - adic sheaf

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an l-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number. Let X be a separated scheme of finite type over a perfect field k of characteristic p > 0. We consider a smooth l-adic étale sheaf F on a smooth dense open subscheme U ⊂ X for a prime l 6= p. The ramification of F along the boundary X \ U has been studied traditionally by using a finite étale covering of U trivializing F modulo l. In this paper, we propose a new geometric method, inspired by the definition of the ramification groups [1], [2] and [4]. The basic geometric construction used in this paper is the blowing-up at the ramification divisor embedded diagonally in the self log product. A precise definition will be given at the beginning of §2.3. We will consider two types of blow-up. The preliminary one, called the log blow-up, is the blow-up (X×X) → X×X at every Di×Di where Di denotes an irreducible component of a divisor D = X \U with simple normal crossings in a smooth scheme X over k. The second one is the blow-up (X ×X) → (X ×X) at R = ∑ i riDi, with some rational multiplicities ri ≥ 0, embedded in the log diagonal X → (X × X). This construction globalizes that used in the definition of the ramification groups in [1] and [2] recalled in §1. Inspired by [10], we consider the ramification along the boundary of the smooth sheaf H = Hom(pr2F , pr ∗ 1F) on the dense open subscheme U × U ⊂ (X ×X) . We introduce a measure of wild ramification by using the extension property of the identity regarded as a section of the restriction on the diagonal of the sheaf H, in Definition 2.3.1. Let j : U × U → (X × X) denote the open immersion. A key property of the sheaf H established in Propositions 2.3.7 and 2.3.8 is that the restriction of j ∗ H