Simplicial Chern-Weil theory for coherent analytic sheaves. I. (La théorie de Chern-Weil simpliciale pour les faisceaux analytiques cohérents. I.) (English. French summary) Zbl 1539.14005
Summary: In [Chern classes for coherent sheaves. Coventry: Univ. Warwick (PhD Thesis) (1980), http://webcat.warwick.ac.uk/record=b1751746~S1], H. I. Green constructed Chern classes in de Rham cohomology of coherent analytic sheaves. We construct here a formal \((\infty,1)\)-categorical framework into which we can place Green’s work and generalise it, also obtaining a better idea as to what exactly a simplicial connection should be. The result will be the ability to work with generalised invariant polynomials (which will be introduced in the sequel to this paper) evaluated at the curvature of so-called admissible simplicial connections to get explicit Čech representatives in de Rham cohomology of characteristic classes of coherent analytic sheaves.
MSC:
| 14A30 | Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.) |
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 18N60 | \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories |
| 32C35 | Analytic sheaves and cohomology groups |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
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