Le Potier's vanishing theorem explained
In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles. The theorem states the following
In case of r = 1, and let E is an ample (or positive) line bundle on X, this theorem is equivalent to the Nakano vanishing theorem. Also, found another proof.
generalizes Le Potier's vanishing theorem to k-ample and the statement as follows:
gave a counterexample, which is as follows:
See also
References
- 10.1007/BF01404918 . Vanishing theorems for tensor powers of an ample vector bundle . 1988 . Demailly . Jean-Pierre . Inventiones Mathematicae . 91 . 203–220 . 1988InMat..91..203D . 18984867 .
- 10.1007/s00229-003-0432-y. A generalization of le Potier's vanishing theorem . 2004 . Laytimi . F. . Nahm . W. . Manuscripta Mathematica . 113 . 2 . 165–189 . 14203286. math/0210010 .
- Book: 10.1007/978-3-642-18810-7. [{{Google books|rd4sIp0f79cC|page=91|plainurl=yes}} Positivity in Algebraic Geometry II ]. 2004 . Lazarsfeld . Robert . 978-3-540-22531-7 .
- Laytimi . F. . Nagaraj . D. S. . Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem . Indian Journal of Pure and Applied Mathematics . 2018 . 49 . 2 . 257–263 . 10.1007/s13226-018-0267-6. 1702.04476 . 119147594 .
- Book: 10.1007/978-3-662-09873-8_6. Pseudoconvexity, the Levi Problem and Vanishing Theorems . [{{Google books|Cx75zepMPewC|page=248|plainurl=yes}} Several Complex Variables VII ]. Encyclopaedia of Mathematical Sciences . 1994 . Peternell . Th. . 74 . 221–257 . 978-3-642-08150-7 .
- 10.1007/BF01350066. Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque . 1975 . Le Potier . J. . Mathematische Annalen . 218 . 35–53 . 122814022.
- Cohomologie de la grassmannienne à valeurs dans les puissances extérieures et symétriques du fibré universel . Mathematische Annalen . 1977 . 226 . 257–270 . Le Potier . J. . 3 . 10.1007/BF01362429 . 117285630 .
- Book: 10.1007/978-1-4899-6680-3_5. Vector Bundles: Ampleness . Vanishing Theorems on Complex Manifolds . Progress in Mathematics . 1985 . Shiffman . Bernard . Sommese . Andrew John . 56 . 89–116 . 978-1-4899-6682-7. .
- Verdier . J. L. . "Le théorème de Le Potier." Différents aspects de la positivité . Soc. Math. France, Paris . 1974 . 17 . 68–78. 367312.
- 10.1007/s002220050126. Vanishing theorems for ample vector bundles . 1997 . Manivel . Laurent . Inventiones Mathematicae . 127 . 2 . 401–416 . alg-geom/9603012 . 1997InMat.127..401M . 14052238 .
- 10.1007/BF01389243. Vanishing theorems, linear and quadratic normality . 1987 . Peternell . Th. . Le Potier . J. . Schneider . M. . Inventiones Mathematicae . 87 . 3 . 573–586 . 1987InMat..87..573P . 120949227.
- 10.1007/BF01405353. Submanifolds of Abelian varieties to Rebecca . 1978 . Sommese . Andrew John . Mathematische Annalen . 233 . 3 . 229–256 . 120704169.
- 10.1007/BF01189093. Ein einfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel . 1974 . Schneider . Michael . Manuscripta Mathematica . 11 . 95–101 . 120722017.
- Théorèmes d'annulation pour les fibrés associés à un fibré ample . Annali della Scuola Normale Superiore di Pisa - Classe di Scienze . 1992 . 19 . 4 . 515–565 . Manivel . Laurent .
- GIRBAU . J. . Sur le theoreme de Le Potier d'annulation de la cohomologie . C. R. Acad. Sci. Paris Sér. A . 1976 . 283 . 355–358 .
- Broer. Abraham. 10.1515/crll.1997.493.153. A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles . Journal für die reine und angewandte Mathematik (Crelle's Journal) . 1997 . 1997 . 493 . 153–170 . 117547554.
- Book: 10.1007/BFb0094302. L2 vanishing theorems for positive line bundles and adjunction theory . Transcendental Methods in Algebraic Geometry . Lecture Notes in Mathematics . 1996 . Demailly . Jean-Pierre . 1646 . 1–97 . 978-3-540-62038-9 . 117583140. alg-geom/9410022 .
- 10.1090/jag/704. Non-Abelian Lefschetz hyperplane theorems . 2018 . Litt . Daniel . Journal of Algebraic Geometry . 27 . 4 . 593–646 . 16039153. 1601.07914.
- 10.1112/S0010437X05001399. Varieties with ample cotangent bundle . 2005 . Debarre . Olivier . Compositio Mathematica . 141 . 6 . 1445–1459 . 2644826 . math/0306066 .
Further reading
- 10.1007/BF03026551. The theorem of Barth-Lefschetz as a consequence of le Potier's vanishing theorem . 1993 . Schneider . Michael . Zintl . Jörg . Manuscripta Mathematica . 80 . 259–263 . 119887533 .
- 10.1093/imrn/rnac204. Vanishing Theorems for Sheaves of Logarithmic Differential Forms on Compact Kähler Manifolds . 2022 . Huang . Chunle . Liu . Kefeng . Wan . Xueyuan . Yang . Xiaokui . International Mathematics Research Notices .
- 10.1142/S0129167X09005182. A Barth–Lefschetz Theorem for Submanifolds of a Product of Projective Spaces . 2009 . Bădescu . Lucian . Repetto . Flavia . International Journal of Mathematics . 20 . 77–96 . math/0701376 . 10539504 .
External links
