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On the conductor formula of Bloch. (English) Zbl 1099.14009
This $$151$$-page paper proves a vast generalisation of a formula for the Artin conductor proved by S. Bloch [in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, No. 2, 421–450 (1987; Zbl 0654.14004)] in the case of curves and conjectured by him to hold in general. Let $$\mathfrak o$$ be a discrete valuation ring, $$K$$ its field of fractions, and $$F$$ the residue field. Recall that for a regular proper flat $$\mathfrak o$$-scheme $$X$$, the Artin conductor of $$X$$ is defined as $\text{ Art}(X)= \chi(X_{\bar K})-\chi(X_{\bar F})+{\text{ Sw}}(X_K)$ where $$\chi$$ is the $$l$$-adic Euler characteristic and $$\text{Sw}$$ is the $$l$$-adic Swan conductor, $$l$$ being a prime different from the charateristic of $$F$$. In [loc. cit.], Bloch defined the “localised self-intersection class of the diagonal” $$(\Delta_X,\Delta_X)$$ as an element of the group $$A_0(X_F)$$ of $$0$$-cycles on the closed fibre $$X_F$$ modulo rational equivalence. The “conductor formula” he conjectured was the equality ${\text{ Art}}(X)=-\deg(\Delta_X,\Delta_X).$ It has recently been observed that the formula for the Milnor number of an isolated singularity follows from this; see F. Orgogozo [Ann. Inst. Fourier 53, No. 6, 1739–1754 (2003; Zbl 1065.14005)].
The authors prove the conductor formula under the hypothesis that the reduced closed fibre has normal crossings as a divisor on $$X$$ (Theorem 6.2.3). The general case would follow from this special case if embedded resolution of singularities holds in a strong sense for the reduced closed fibre. The best earlier result required the more restrictive hypothesis that the multiplicities of the irreducible components of $$X_F$$ be prime to the characteristic of $$F$$ [T. Chinburg, G. Pappas, M. J. Taylor, Math. Res. Lett. 7, No. 4, 433–446 (2000; Zbl 1097.14501)]. In fact, the authors’ main result concerns a more general version – involving correspondences – of the conductor formula. A correspondence $$\Gamma$$ on $$X_K$$ gives rise to an endomorphism $$\Gamma^*$$ on the $$l$$-adic étale cohomology, and hence to a Swan conductor $$\text{ Sw}(\Gamma,X_K)$$. They show (Theorem 6.3.1) that $${\text{ Sw}}(\Gamma,X_K)$$ is an integer independent of $$l$$ and, if the reduced closed fibre has simple normal crossings as a divisor on $$X$$, the equality ${\text{ Sw}}(\Gamma,X_K)=-\deg[[X,\Gamma]]$ holds, where $$[[X,\Gamma]]$$ is the “logarithmic localised intersection product” (Section 5.4) in a suitable graded piece of the Grothendieck group of $$X_F$$. The special case $$\Gamma=\Delta_X$$ (Theorem 6.2.5) is equivalent to the previous theorem.
The proof makes extensive use of logarithmic geometry. The authors define a “logarithmic self-intersection class” $$(\Delta_X,\Delta_X)_{\log}$$ in $$A_0(X_F)$$ by replacing $$\Omega^1_{X|{\mathfrak o}}$$ in the definition of $$(\Delta_X,\Delta_X)$$ by the sheaf of differential $$1$$-forms with logarithmic poles and prove that $$\text{Sw}(X_K)=-\deg(\Delta_X,\Delta_X)_{\log}$$ (Theorem 6.2.5). For the main theorem, de Jong’s alterations are needed. The authors also provide emendations of some results from a related paper by the second author [Duke Math. J. 57, No.2, 555–578 (1988; Zbl 0687.14004)].

##### MSC:
 14G20 Local ground fields in algebraic geometry 11G25 Varieties over finite and local fields 14C15 (Equivariant) Chow groups and rings; motives 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
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