Hodge theorem for the logarithmic de Rham complex via derived intersections

TL;DR

This paper extends the Hodge theorem for the de Rham complex to logarithmic schemes by employing derived intersection theory, building on previous geometric interpretations of the Deligne-Illusie degeneration result.

Contribution

It generalizes the geometric interpretation of the Deligne-Illusie theorem to the setting of logarithmic schemes using twisted derived intersections.

Findings

Proves the Hodge theorem for the logarithmic de Rham complex.

Establishes a link between derived intersections and Hodge theory in logarithmic geometry.

Provides a new geometric framework for understanding the degeneration of spectral sequences.

Abstract

In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, C\u{a}ld\u{a}raru and the author of this paper gave a geometric interpretation of the problem of Deligne-Illusie showing that the triviality of a certain line bundle on a derived scheme implies the the Deligne-Illusie result. In the present paper we generalize these ideas to logarithmic schemes and using the theory of twisted derived intersection of logarithmic schemes we obtain the Hodge theorem for the logarithmic de Rham complex.