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, 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 for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a degenerating family of algebraic varieties.

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 denote a complex of vector bundles on Y
that is exact on. The localized Chern class of this complex is a class in the bivariant Chow group of defined as follows. Let denote the tautological bundle of the Grassmann bundle of rank subbundles of. Let. Then the i-th localized Chern class is defined by the formula:
where is the projection and is a cycle obtained from by the so-called graph construction.

Example: localized Euler class

Let be as in #Definitions. If S is smooth over a field, then the localized Chern class coincides with the class
where, roughly, is the section determined by the differential of f and is the class of the singular locus of f.

Bloch's conductor formula