Localized Chern class

In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's intersection theory,[1] as an algebraic counterpart of the similar construction in algebraic topology. The notion is used in particular in the Riemann–Roch-type theorem.

S. Bloch later generalized the notion in the context of arithmetic schemes (schemes over a Dedekind domain) for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a degenerating family of algebraic varieties (in the mixed characteristic case).

Definitions

Let Y be a pure-dimensional regular scheme of finite type over a field or discrete valuation ring and X a closed subscheme. Let $E_{\bullet }$ denote a complex of vector bundles on Y

0=E_{n-1}\to E_{n}\to \dots \to E_{m}\to E_{m-1}=0

that is exact on $Y-X$ . The localized Chern class of this complex is a class in the bivariant Chow group of $X\subset Y$ defined as follows. Let $\xi _{i}$ denote the tautological bundle of the Grassmann bundle $G_{i}$ of rank $\operatorname {rk} E_{i}$ subbundles of $E_{i}\otimes E_{i-1}$ . Let $\xi =\prod (-1)^{i}\operatorname {pr} _{i}^{*}(\xi _{i})$ . Then the i-th localized Chern class $c_{i,X}^{Y}(E_{\bullet })$ is defined by the formula:

c_{i,X}^{Y}(E_{\bullet })\cap \alpha =\eta _{*}(c_{i}(\xi )\cap \gamma )

where $\eta :G_{n}\times _{Y}\dots \times _{Y}G_{m}\to X$ is the projection and $\gamma$ is a cycle obtained from $\alpha$ by the so-called graph construction.

Example: localized Euler class

Let $f:X\to S$ be as in #Definitions. If S is smooth over a field, then the localized Chern class coincides with the class

(-1)^{\dim X}\mathbf {Z} (s_{f})

where, roughly, $s_{f}$ is the section determined by the differential of f and (thus) $\mathbf {Z} (s_{f})$ is the class of the singular locus of f.

Bloch's conductor formula

References

Fulton 1998, Example 18.1.3.

S. Bloch, “Cycles on arithmetic schemes and Euler characteristics of curves,” Algebraic geometry, Bowdoin, 1985, 421–450, Proc. Symp. Pure Math. 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.
Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, section B.7
K. Kato and T. Saito, “On the conductor formula of Bloch,” Publ. Math. IHES 100 (2005), 5-151.

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[1] Fulton 1998, Example 18.1.3.