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Working paper

Chern Classes via Derived Determinant

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 \cite{i}, 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^2n_dR(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^dR_n(E), where cdRn(E)are the Chern classes of E in the de Rham cohomology and F is the Frobenius map.