Derived intersections and the Hodge theorem

TL;DR

This paper offers a geometric interpretation of the algebraic Hodge theorem using derived algebraic geometry, providing a criterion for triviality of a line bundle on a derived scheme and recovering key results in the field.

Contribution

It introduces a geometric perspective on the algebraic Hodge theorem via derived schemes and establishes a criterion for line bundle triviality, unifying several important results.

Findings

Recovered Deligne-Illusie's original result using derived geometry techniques.

Provided a general criterion for the triviality of line bundles on derived schemes.

Extended formality results to twisted de Rham complexes.

Abstract

The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central algebraic object of their proof can be understood geometrically as a line bundle on a derived scheme. In this interpretation, the Deligne-Illusie result can be seen as a proof that this line bundle is trivial under certain assumptions. We give a criterion for the triviality of this line bundle in a more general context. The proof uses techniques from derived algebraic geometry, specifically arguments which show the formality of certain derived intersections. Applying our criterion we recover Deligne and Illusie's original result. We also apply these techniques to the result of Barannikov-Kontsevich, Sabbah, and Ogus-Vologodsky concerning the formality of the twisted de Rham complex.