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 (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 ∙ {\displaystyle E_{\bullet }} denote a complex of vector bundles on Y
0 = E n − 1 → E n → ⋯ → E m → E m − 1 = 0 {\displaystyle 0=E_{n-1}\to E_{n}\to \dots \to E_{m}\to E_{m-1}=0}
that is exact on Y − X {\displaystyle Y-X} . The localized Chern class of this complex is a class in the bivariant Chow group of X ⊂ Y {\displaystyle X\subset Y} defined as follows. Let ξ i {\displaystyle \xi _{i}} denote the tautological bundle of the Grassmann bundle G i {\displaystyle G_{i}} of rank rk E i {\displaystyle \operatorname {rk} E_{i}} sub-bundles of E i ⊗ E i − 1 {\displaystyle E_{i}\otimes E_{i-1}} . Let ξ = ∏ ( − 1 ) i pr i ∗ ( ξ i ) {\displaystyle \xi =\prod (-1)^{i}\operatorname {pr} _{i}^{*}(\xi _{i})} . Then the i-th localized Chern class c i , X Y ( E ∙ ) {\displaystyle c_{i,X}^{Y}(E_{\bullet })} is defined by the formula:
c i , X Y ( E ∙ ) ∩ α = η ∗ ( c i ( ξ ) ∩ γ ) {\displaystyle c_{i,X}^{Y}(E_{\bullet })\cap \alpha =\eta _{*}(c_{i}(\xi )\cap \gamma )}
where η : G n × Y ⋯ × Y G m → X {\displaystyle \eta :G_{n}\times _{Y}\dots \times _{Y}G_{m}\to X} is the projection and γ {\displaystyle \gamma } is a cycle obtained from α {\displaystyle \alpha } by the so-called graph construction.
Example: localized Euler class Let f : X → S {\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
( − 1 ) dim X Z ( s f ) {\displaystyle (-1)^{\dim X}\mathbf {Z} (s_{f})}
where, roughly, s f {\displaystyle s_{f}} is the section determined by the differential of f and (thus) Z ( s f ) {\displaystyle \mathbf {Z} (s_{f})} is the class of the singular locus of f. Consider an infinite dimensional bundle E over an infinite dimensional manifold M with a section s with Fredholm derivative. In practice this situation occurs whenever we have system of PDEs which are elliptic when considered modulo some gauge group action. The zero set Z(s) is then the moduli space of solutions modulo gauge, and the index of the derivative is the virtual dimension. The localized Euler class of the pair (E,s) is a homology class with closed support on the zero set of the section. Its dimension is the index of the derivative. When the section is transversal, the class is just the fundamental class of the zero set with the proper orientation. The class is well behaved in one parameter families and therefore defines the “right” fundamental cycle even if the section is no longer transversal.
Bloch's conductor formula
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