[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/localized-chern-class-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/localized-chern-class-wikipedia\/","headline":"Localized Chern class – Wikipedia","name":"Localized Chern class – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In algebraic geometry, a localized Chern class is a variant of a Chern","datePublished":"2018-01-22","dateModified":"2018-01-22","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/941a4fe23e8dc4f0218d49071448dd96c35dcb1c","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/941a4fe23e8dc4f0218d49071448dd96c35dcb1c","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/localized-chern-class-wikipedia\/","wordCount":3429,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In 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\u2013Roch-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).       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsDefinitions[edit]Example: localized Euler class[edit]Bloch’s conductor formula[edit]References[edit]Definitions[edit]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\u2219{displaystyle E_{bullet }} denote a complex of vector bundles on Y       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x40=En\u22121\u2192En\u2192\u22ef\u2192Em\u2192Em\u22121=0{displaystyle 0=E_{n-1}to E_{n}to dots to E_{m}to E_{m-1}=0}that is exact on Y\u2212X{displaystyle Y-X}. The localized Chern class of this complex is a class in the bivariant Chow group of X\u2282Y{displaystyle Xsubset Y} defined as follows. Let \u03bei{displaystyle xi _{i}} denote the tautological bundle of the Grassmann bundle Gi{displaystyle G_{i}} of rank rk\u2061Ei{displaystyle operatorname {rk} E_{i}} sub-bundles of Ei\u2297Ei\u22121{displaystyle E_{i}otimes E_{i-1}}. Let \u03be=\u220f(\u22121)ipri\u2217\u2061(\u03bei){displaystyle xi =prod (-1)^{i}operatorname {pr} _{i}^{*}(xi _{i})}. Then the i-th localized Chern class ci,XY(E\u2219){displaystyle c_{i,X}^{Y}(E_{bullet })} is defined by the formula:ci,XY(E\u2219)\u2229\u03b1=\u03b7\u2217(ci(\u03be)\u2229\u03b3){displaystyle c_{i,X}^{Y}(E_{bullet })cap alpha =eta _{*}(c_{i}(xi )cap gamma )}where \u03b7:Gn\u00d7Y\u22ef\u00d7YGm\u2192X{displaystyle eta :G_{n}times _{Y}dots times _{Y}G_{m}to X} is the projection and \u03b3{displaystyle gamma } is a cycle obtained from \u03b1{displaystyle alpha } by the so-called graph construction.Example: localized Euler class[edit]Let f:X\u2192S{displaystyle f:Xto S} be as in #Definitions. If S is smooth over a field, then the localized Chern class coincides with the class(\u22121)dim\u2061XZ(sf){displaystyle (-1)^{dim X}mathbf {Z} (s_{f})}where, roughly, sf{displaystyle s_{f}} is the section determined by the differential of f and (thus) Z(sf){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 PDE\u2019s 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 \u201cright\u201d fundamental cycle even if the section is no longer transversal.Bloch’s conductor formula[edit]This section needs expansion. You can help by adding to it.  (November 2019)This formula enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms.  S. Bloch conjectures a formula for the Artin conductor of the \u2113-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The deepest result about the Bloch conductor is its equality with the Artin conductor, defined in terms of the l-adic cohomology of X, in certain cases.References[edit]S. Bloch, \u201cCycles on arithmetic schemes and Euler characteristics of curves,\u201d Algebraic geometry, Bowdoin, 1985, 421\u2013450, 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], vol.\u00a02, Berlin, New York: Springer-Verlag, ISBN\u00a0978-3-540-62046-4, MR\u00a01644323, section B.7K. Kato and T. Saito, \u201cOn the conductor formula of Bloch,\u201d Publ. Math. IHES 100 (2005), 5-151.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/localized-chern-class-wikipedia\/#breadcrumbitem","name":"Localized Chern class – Wikipedia"}}]}]