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 ${\displaystyle E_{\bullet }}$ denote a complex of vector bundles on Y

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

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

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

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

Example: localized Euler class

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

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

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

Bloch's conductor formula

See also: Grothendieck–Ogg–Shafarevich formula

References

1. 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.