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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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