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

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.