  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.
