Chern Classes via Derived Determinant

TL;DR

This paper connects Chern classes of vector bundles to the Atiyah class using derived algebraic geometry, providing new proofs and constructions in characteristic p and relating crystalline and de Rham cohomology.

Contribution

It offers a new proof of Chern class recovery from the Atiyah class via derived algebraic geometry and constructs crystalline Chern classes related to de Rham classes in characteristic p.

Findings

Chern classes can be recovered from the Atiyah class using derived algebraic geometry.

Constructs crystalline Chern classes in characteristic p that relate to de Rham Chern classes.

Proves equality of crystalline and de Rham Chern classes up to factorial factors.

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