Advent M. Kodaira's vanishing theorem, asserting that the inverse of an plentiful invert ible sheaf on a projective complicated manifold X has no cohomology less than the size of X and its generalization, as a result of Y. Akizuki and S. Nakano, were confirmed initially via tools from differential geometry ([39J and ). whether, because of J.P. Serre's GAGA-theorems [56J and base swap for box extensions the algebraic analogue used to be bought for projective manifolds over a box ok of attribute p = zero, for a very long time no algebraic facts used to be identified and no generalization to p > zero, aside from convinced reduce dimensional manifolds. Worse, counterexamples as a result of M. Raynaud [52J confirmed that during attribute p > zero a few extra assumptions have been wanted. This was once the cutting-edge till P. Deligne and 1. Illusie [12J proved the degeneration of the Hodge to de Rham spectral series for projective manifolds X outlined over a box ok of attribute p > zero and liftable to the second one Witt vectors W2(k). average degeneration arguments enable to infer the degeneration of the Hodge to de Rham spectral series in attribute 0, to boot, a re sult which back might basically be bought by means of analytic and differential geometric tools previously. As a corollary in their tools M. Raynaud (loc. cit.) gave a simple evidence of Kodaira vanishing in all features, only if X lifts to W2(k).
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