Conductor formula of Bloch

Conductor is a numerical invariant of a variety over a local field measuring the wild ramification of the inertia action on the l-adic etale cohomology. In [3], S.Bloch proposes a conjectural formula, Conjecture 1.9, which we call the conductor formula of Bloch. To formulate it, he defines another numerical invariant as the degree of the self-intersection class, which is defined using the localized chern class of the sheaf of differential 1-forms of a regular proper model of the variety. The conductor formula asserts that the conductor of a regular proper model of a variety over a local field is equal to the minus of the degree of the self-intersection class. It enables us to compute a Galois theoretic invariant, the conductor, of ramification in terms of a de Rham theoretic invariant, the self-intersection class, of degeneration. In the same paper, he proves the formula for curves over a local field. For a finite extension of a local field, the conductor formula is nothing but the classical conductor-discriminant formula in algebraic number theory. For an elliptic curve, the formula is known in [31] Corollary 2 of Theorem 1 to be equivalent to the Tate-Ogg formula [28] for the relation between the conductor and the discriminant. In this paper, we prove the conductor formula in arbitrary dimension under a rather mild hypothesis. To describe the main results, we introduce some notations. Detailed explanation is given in the text. Let K be a discrete valuation field with perfect residue field F and let XK be a proper smooth scheme over K of dimension d. The Swan conductor Sw(XK/K) of XK is defined to be the alternating sum Sw(XK/K) = ∑2d q=0(−1) SwH(XK̄ ,Ql) of the Swan conductor of the l-adic etale cohomology for a prime l different from the characteristic p of F . The Swan conductor of an l-adic representation V is defined to be the intertwining number