Chern Classes via Derived Determinant

2019/09/16 by Terentiuk Gleb, Terentiuk, Gleb
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1909.07415

openalex publication_date 2019/09/16 · openalex created_date 2019/09/26 · openalex updated_date 2026/07/28

Abstract

Motivated by the Chern-Weil theory, we prove that for a given vector bundle E on a smooth scheme X over a field k of any characteristic, the Chern classes of E in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \citei, we present another proof by means of derived algebraic geometry. Also, for a scheme X over a field k of characteristic p with a vector bundle E we construct elements c^cris_n (E, α(E)) ∈ H_dR²ⁿ (X) using an obstruction α(E) to a lifting of F^* E to a crystal modulo p² and prove that c^cris_n (E, α(E)) = n! ⋅ c_n^dR (E), where c_n^dR (E) are the Chern classes of E in the de Rham cohomology and F is the Frobenius map.

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