Le Potier's vanishing theorem

In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundle. The theorem states the following

Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here

$H^{p,q}(X,E)$

is Dolbeault cohomology group, where

${\mathrm{\Omega }}_X^p$

denotes the sheaf of holomorphic p-forms on X. If E is an ample, then

$H^{p,q}(X,E)=0$ for $p+q\ge n+r$ .

from Dolbeault theorem,

$H^q(X,{\mathrm{\Omega }}_X^p\otimes E)=0$ for $p+q\ge n+r$ .

By Serre duality, the staements are equivalent to the assertions:

$H^i(X,{\mathrm{\Omega }}_X^j\otimes E^{*})=0$ for $j+i\le n-r$ .

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, Schneider (1974) found the another proof

Sommese (1978) generalizes Le Potier's vanishing theorem to k-ample and the statement as follows:

Le Potier–Sommese vanishing theorem: Let X be a n-dimensional algebraic manifold and E is a k-ample holomorphic vector bundle of rank r over X, then

$H^{p,q}(X,E)=0$ for $p+q\ge n+r+k$ .

Demailly (1988) gave a counterexample, which is as follows:

Conjecture of Sommese (1978): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X. If E is an ample, then

$H^{p,q}(X,{\mathrm{\Lambda }}^{a}E)=0$ for $p+q\ge n+r-a+1$ is false for $n=2r\ge 6.$

See also

• vanishing theorem
• Barth–Lefschetz theorem

Reference

• Demailly, Jean-Pierre (1988). "Vanishing theorems for tensor powers of an ample vector bundle" (PDF). Inventiones Mathematicae. 91: 203–220. Bibcode:1988InMat..91..203D. doi:10.1007/BF01404918.
• Laytimi, F.; Nahm, W. (2004). "A generalization of le Potier's vanishing theorem". Manuscripta Mathematica. 113 (2): 165–189. arXiv:math/0210010. doi:10.1007/s00229-003-0432-y.
• Lazarsfeld, Robert (2004). Positivity in Algebraic Geometry II. doi:10.1007/978-3-642-18810-7. ISBN 978-3-540-22531-7. Search this book on
• Laytimi, F.; Nagaraj, D. S. (2018). "Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem". Indian Journal of Pure and Applied Mathematics. 49 (2): 257–263. arXiv:1702.04476. doi:10.1007/s13226-018-0267-6.
• Peternell, Th. (1994). "Pseudoconvexity, the Levi Problem and Vanishing Theorems". Several Complex Variables VII. Encyclopaedia of Mathematical Sciences. 74. pp. 221–257. doi:10.1007/978-3-662-09873-8_6. ISBN 978-3-642-08150-7. Search this book on
• Le Potier, J. (1975). "Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque". Mathematische Annalen. 218: 35–53. doi:10.1007/BF01350066.
• Le Potier, J. (1977). "Cohomologie de la grassmannienne à valeurs dans les puissances extérieures et symétriques du fibré universel". Mathematische Annalen. 226 (3): 257–270. doi:10.1007/BF01362429.
• Shiffman, Bernard; Sommese, Andrew John (1985). "Vector Bundles: Ampleness". Vanishing Theorems on Complex Manifolds. Progress in Mathematics. 56. pp. 89–116. doi:10.1007/978-1-4899-6680-3_5. ISBN 978-1-4899-6682-7. Search this book on
• Verdier, J. L. (1974). ""Le théorème de Le Potier." Différents aspects de la positivité" (PDF). Soc. Math. France, Paris. 17: 68–78. MR 0367312.
• Manivel, Laurent (1997). "Vanishing theorems for ample vector bundles". Inventiones Mathematicae. 127 (2): 401–416. arXiv:alg-geom/9603012. Bibcode:1997InMat.127..401M. doi:10.1007/s002220050126.
• Peternell, Th.; Le Potier, J.; Schneider, M. (1987). "Vanishing theorems, linear and quadratic normality". Inventiones Mathematicae. 87 (3): 573–586. Bibcode:1987InMat..87..573P. doi:10.1007/BF01389243.
• Sommese, Andrew John (1978). "Submanifolds of Abelian varieties to Rebecca". Mathematische Annalen. 233 (3): 229–256. doi:10.1007/BF01405353.
• Schneider, Michael (1974). "Ein einfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel". Manuscripta Mathematica. 11: 95–101. doi:10.1007/BF01189093.
• Manivel, Laurent (1992). "Théorèmes d'annulation pour les fibrés associés à un fibré ample". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 19 (4): 515–565.
• GIRBAU, J. (1976). "Sur le theoreme de Le Potier d'annulation de la cohomologie". C. R. A. S., Paris Serie A. 283: 355–358.
• Broer, Abraham (1997). "A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles". Journal für die reine und angewandte Mathematik (Crelle's Journal). 1997 (493): 153–170. doi:10.1515/crll.1997.493.153.
• Demailly, Jean-Pierre (1996). "L2 vanishing theorems for positive line bundles and adjunction theory". Transcendental Methods in Algebraic Geometry. Lecture Notes in Mathematics. 1646. pp. 1–97. arXiv:alg-geom/9410022. doi:10.1007/BFb0094302. ISBN 978-3-540-62038-9. Search this book on

Further reading

• Schneider, Michael; Zintl, Jörg (1993). "The theorem of Barth-Lefschetz as a consequence of le Potier's vanishing theorem". Manuscripta Mathematica. 80: 259–263. doi:10.1007/BF03026551.
• Huang, Chunle; Liu, Kefeng; Wan, Xueyuan; Yang, Xiaokui (2022). "Vanishing Theorems for Sheaves of Logarithmic Differential Forms on Compact Kähler Manifolds". International Mathematics Research Notices. doi:10.1093/imrn/rnac204.
• Bădescu, Lucian; Repetto, Flavia (2009). "A Barth–Lefschetz Theorem for Submanifolds of a Product of Projective Spaces". International Journal of Mathematics. 20: 77–96. arXiv:math/0701376. doi:10.1142/S0129167X09005182.

External links

• Demailly, Jean-Pierre, Complex Analytic and Differential Geometry (PDF) (OpenContent book)

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