Deligne--Illusie Classes as Arithmetic Kodaira--Spencer Classes

TL;DR

This paper demonstrates that Deligne--Illusie classes fulfill a modified version of Faltings' non-existent arithmetic Kodaira--Spencer classes, and explores related Torelli problems and CM Jacobians.

Contribution

It introduces a modified compatibility condition showing Deligne--Illusie classes satisfy an arithmetic Kodaira--Spencer type property, extending prior non-existence results.

Findings

Deligne--Illusie classes meet a modified compatibility condition

Discussion of Wittfinitesimal Torelli problem and CM Jacobians

Extension of Faltings' non-existence results to new classes

Abstract

Faltings showed that "arithmetic Kodaira--Spencer classes" satisfying a certain compatibility axiom cannot exist. By modifying his definitions slightly, we show that the Deligne--Illusie classes satisfy what could be considered an "arithmetic Kodaira--Spencer" compatibility condition. Afterwards we discuss a "wittfinitesimal Torelli problem" and its relation to CM Jacobians.