Rel�vements modulop 2 et d�composition du complexe de de Rham

Lectures on Vanishing Theorems

“…This was the state of the art until P. Deligne and L. Illusie [12] proved the degeneration of the Hodge to de Rham spectral sequence for projective manifolds X defined over a field k of characteristic p > 0 and liftable to the second Witt vectors W 2 (k).…”

“…Short time before [12] was written the two authors studied in [20] the relations between logarithmic de Rham complexes and vanishing theorems on complex algebraic manifolds and showed that quite generally vanishing theorems follow from the degeneration of certain Hodge to de Rham type spectral sequences. The interplay between topological and algebraic vanishing theorems thereby obtained is also reflected in J. Kollár's work [41] and in the vanishing theorems M. Saito obtained as an application of his theory of mixed Hodge modules (see [54]).…”

“…It is obvious that the combination of [12] and [20] give another algebraic approach to vanishing theorems and it is one of the aims of these lecture notes to present it in all details. Of course, after the Deligne-Illusie-Raynaud proof of the original Kodaira and Akizuki-Nakano vanishing theorems, the main motivation to present the methods of [20] along with those of [12] is that they imply as well some of the known generalizations.…”

“…Of course, after the Deligne-Illusie-Raynaud proof of the original Kodaira and Akizuki-Nakano vanishing theorems, the main motivation to present the methods of [20] along with those of [12] is that they imply as well some of the known generalizations.…”

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…Here are few: P. Deligne-D. Mumford's proof [66] that the moduli space of curves of a given genus is irreducible in any characteristic, S. Mori's proof [147] of Hartshorne's conjecture, P. Deligne and L. Illusie's algebraic proof [64] of the Kodaira vanishing theorem and of the degeneration of the Hodge to de Rham spectral sequence (see the nice survey [104]). …”