Chern Classes via Derived Determinant

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 \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 $c_n^{cris}(E,\alpha (E))\in H_{dR}^{2n}(X)$ using an obstruction $\alpha (E)$ to a lifting of $F^{\ast }E$ to a crystal modulo $p^2$ and prove that $c_n^{cris}(E,\alpha (E))=n!\cdot c_n^{dR}(E)$, where $c_n^{dR}(E)$ are the Chern classes of $E$ in the de Rham cohomology and $F$ is the Frobenius map.