On the Swan conductor in positive characteristic

Abstract. Let X be a proper smooth surface over an algebraically closed field of positive character-istic and U be a complement of a simple normal crossing divisor. For a smooth l-adic sheaf F on U, Deligne proved a formula calculating the Euler characteristic of F by local invariants. Kato gave another formula for the Euler characteristic in case where F is of rank 1, using class field theory. In this paper, we show that local terms of these formulas coincide, admitting certain conjectures for vanishing cycles. 1. Introduction. 1.1. Let K be a complete discrete valuation field with residue field F and let L be a finite Galois extension of K. Classically the Swan conductor of a character of Gal(L=K) is defined in the case where the residue field of L is separable over F. In 1989, Kato gave a natural definition of the Swan conductor for a character o