Lectures on Vanishing Theorems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1992
|
| Schriftenreihe: | DMV Seminar
20 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 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 > 0, except for certain lower dimensional manifolds. Worse, counterexamples due to M. Raynaud [52J showed that in characteristic p > 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 > 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) |
| Beschreibung: | 1 Online-Ressource (VIII, 166 p) |
| ISBN: | 9783034886000 9783764328221 |
| DOI: | 10.1007/978-3-0348-8600-0 |
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| 500 | |a 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 > 0, except for certain lower dimensional manifolds. Worse, counterexamples due to M. Raynaud [52J showed that in characteristic p > 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 > 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) | ||
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Datensatz im Suchindex
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| author | Esnault, Hélène |
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| discipline | Allgemeine Naturwissenschaft Mathematik |
| doi_str_mv | 10.1007/978-3-0348-8600-0 |
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| spelling | Esnault, Hélène Verfasser aut Lectures on Vanishing Theorems by Hélène Esnault, Eckart Viehweg Basel Birkhäuser Basel 1992 1 Online-Ressource (VIII, 166 p) txt rdacontent c rdamedia cr rdacarrier DMV Seminar 20 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 > 0, except for certain lower dimensional manifolds. Worse, counterexamples due to M. Raynaud [52J showed that in characteristic p > 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 > 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) Science (General) Science, general Naturwissenschaft Verschwindungssatz (DE-588)4187983-1 gnd rswk-swf 1\p (DE-588)1071861417 Konferenzschrift gnd-content Verschwindungssatz (DE-588)4187983-1 s 2\p DE-604 Viehweg, Eckart Sonstige oth https://doi.org/10.1007/978-3-0348-8600-0 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Esnault, Hélène Lectures on Vanishing Theorems Science (General) Science, general Naturwissenschaft Verschwindungssatz (DE-588)4187983-1 gnd |
| subject_GND | (DE-588)4187983-1 (DE-588)1071861417 |
| title | Lectures on Vanishing Theorems |
| title_auth | Lectures on Vanishing Theorems |
| title_exact_search | Lectures on Vanishing Theorems |
| title_full | Lectures on Vanishing Theorems by Hélène Esnault, Eckart Viehweg |
| title_fullStr | Lectures on Vanishing Theorems by Hélène Esnault, Eckart Viehweg |
| title_full_unstemmed | Lectures on Vanishing Theorems by Hélène Esnault, Eckart Viehweg |
| title_short | Lectures on Vanishing Theorems |
| title_sort | lectures on vanishing theorems |
| topic | Science (General) Science, general Naturwissenschaft Verschwindungssatz (DE-588)4187983-1 gnd |
| topic_facet | Science (General) Science, general Naturwissenschaft Verschwindungssatz Konferenzschrift |
| url | https://doi.org/10.1007/978-3-0348-8600-0 |
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