{"product_id":"lectures-on-vanishing-theorems-esnault-9783764328221","title":"Lectures on Vanishing Theorems","description":"Introduction M. Kodaira's vanishing theorem, saying that the inverse of an ample invert- ible sheaf on a projective complex manifold X has no cohomology below the dimension of X and its generalization, due to Y. Akizuki and S. Nakano, have been proven originally by methods from differential geometry ( 39J and  1]). Even if, due to J.P. Serre's GAGA-theorems  56J and base change for field extensions the algebraic analogue was obtained for projective manifolds over a field k of characteristic p = 0, for a long time no algebraic proof was known and no generalization to p \u0026gt; 0, except for certain lower dimensional manifolds. Worse, counterexamples due to M. Raynaud  52J showed that in characteristic p \u0026gt; 0 some additional assumptions were needed. This was the state of the art until P. Deligne and 1. Illusie  12J proved the degeneration of the Hodge to de Rham spectral sequence for projective manifolds X defined over a field k of characteristic p \u0026gt; 0 and liftable to the second Witt vectors W2(k). Standard degeneration arguments allow to deduce the degeneration of the Hodge to de Rham spectral sequence in characteristic zero, as well, a re- sult which again could only be obtained by analytic and differential geometric methods beforehand. As a corollary of their methods M. Raynaud (loc. cit.) gave an easy proof of Kodaira vanishing in all characteristics, provided that X lifts to W2(k).\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Esnault, Vieweg\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 3764328223\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9783764328221\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Birkhauser\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 12\/01\/1992\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 166\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Paperback\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 0.64lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.61h x 6.69w x 0.38d","brand":"Esnault","offers":[{"title":"Paperback","offer_id":46747682242815,"sku":"9783764328221","price":99.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_a09eaf38-b6a8-4673-a598-c53fb129698a.jpg?v=1743584423","url":"https:\/\/www.whiterainbookhouse.com\/products\/lectures-on-vanishing-theorems-esnault-9783764328221","provider":"WR Book House","version":"1.0","type":"link"}