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Liftings modulo p^2 and decomposition of the de Rham complex

SPEAKER: Nick Howell

TITLE: Liftings modulo p^2 and decomposition of the de Rham complex (After Deligne and Illusie.)

ABSTRACT: If X is a smooth projective variety over the complex numbers, the Fr┼ílicher (or Hodge to de Rham) spectral sequence, with E? terms given by the sheaf cohomology groups of (holomorphic) de Rham forms, degenerates at E?. This was proven by Fr┼ílicher in a 1955 paper using the Hodge decomposition. The Deligne-Illusie theorem gives an elegant algebraic proof of this degeneration using the reduction to characteristic p method. Their main result: if X is a smooth variety over a perfect field of characteristic p and the dimension of X is less than p, then a lifting of X modulo p^2 gives rise to a quasi-isomorphism between the Frobenius push-forward of the (algebraic) de Rham complex of X and the direct sum of its cohomology sheaves. In this talk, I will review the essential notions of differential geometry in positive characteristic, give the proof of the decomposition in high characteristic, and explain its relation to the above spectral sequence. If time permits, I will give an application: a positive characteristic analogue of Kodaira’s vanishing theorem.