The degeneration of the Hodge spectral sequence for a smooth proper scheme of characteristic 0 is a basic fact in algebraic geometry. Since the statement can be made purely algebraic (i.e. it can be stated without using any complex analysis), it is natural to expect a purely algebraic proof.
In this talk, I would like to present a purely algebraic proof of this fact, following Deligne-Illusie('87). This proof is quite surprising in two aspects. First, the proof is by reduction to the case where the base field is of positive characteristic. Note that there are many examples of smooth projective surfaces in positive characteristic for which the degeneration of the Hodge spectral sequence fails to hold. Deligne-Illusie proved the degeneration in positive characteristic under extra hypotheses (upper bound on dimension and liftability), from which they deduced the statement in characteristic 0. Second, the proof is short and elementary, contrary to our expections on Deligne, Illusie and characteristic p. The key technical inputs are 1) Cartier Isomorphism ('57 by Cartier, different proof by Grothendieck) and 2) basic deformation theory, and none of them are hard results.
From the characteristic p statement, one can deduce Kodaira vanishing theorem (for smooth projective schemes), both in characteristic p under extra hypotheses and in characteristic 0, by the argument due to Raynaud. It is worth noting that the proof is important not just because it gives a purely algebraic proof of the degeneration of Hodge spectral sequence, but because it provides a useful tool to study algebraic geometry in positive characteristic.
To minimize the background, I will state (and hopefully provide some intuition on) certain properties of smooth morphisms we need. Spectral sequences will not appear in the proof.