ON THE SWAN CONDUCTOR IN POSITIVE CHARACTERISTIC

Let X be a proper smooth surface over an algebraically closed field of positive characteristic and U be a complement of a simple normal cros sing divisor. For a smooth l-adic sheaf F on U, Deligne proved a formu la 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 v anishing cycles.