Le Potier's vanishing theorem

Short description: Generalizes the Kodaira vanishing theorem for ample vector bundle

In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles. 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 statements 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 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

References

• Demailly, Jean-Pierre (1988). "Vanishing theorems for tensor powers of an ample vector bundle". Inventiones Mathematicae 91: 203–220. doi:10.1007/BF01404918. Bibcode: 1988InMat..91..203D. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/vanishing.pdf. 
• Laytimi, F.; Nahm, W. (2004). "A generalization of le Potier's vanishing theorem". Manuscripta Mathematica 113 (2): 165–189. 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. https://books.google.com/books?id=rd4sIp0f79cC&pg=PA91. 
• Laytimi, F.; Nagaraj, D. S. (2018). "Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem". Indian Journal of Pure and Applied Mathematics 49 (2): 257–263. 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. https://books.google.com/books?id=Cx75zepMPewC&pg=PA248. 
• 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. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002311917. 
• 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. http://eudml.org/doc/162959. 
• 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. https://books.google.com/books?id=Bhr3BwAAQBAJ&pg=PA96. 
• Verdier, J. L. (1974). ""Le théorème de Le Potier." Différents aspects de la positivité". Soc. Math. France, Paris 17: 68–78. http://www.numdam.org/item/AST_1974__17__68_0.pdf. 
• Manivel, Laurent (1997). "Vanishing theorems for ample vector bundles". Inventiones Mathematicae 127 (2): 401–416. doi:10.1007/s002220050126. Bibcode: 1997InMat.127..401M. 
• Peternell, Th.; Le Potier, J.; Schneider, M. (1987). "Vanishing theorems, linear and quadratic normality". Inventiones Mathematicae 87 (3): 573–586. doi:10.1007/BF01389243. Bibcode: 1987InMat..87..573P. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002103761. 
• Sommese, Andrew John (1978). "Submanifolds of Abelian varieties to Rebecca". Mathematische Annalen 233 (3): 229–256. doi:10.1007/BF01405353. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002315556. 
• Schneider, Michael (1974). "Ein einfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel". Manuscripta Mathematica 11: 95–101. doi:10.1007/BF01189093. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002217643. 
• 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. http://www.numdam.org/item?id=ASNSP_1992_4_19_4_515_0. 
• GIRBAU, J. (1976). "Sur le theoreme de Le Potier d'annulation de la cohomologie". C. R. Acad. Sci. Paris Sér. A 283: 355–358. https://gallica.bnf.fr/ark:/12148/bpt6k62355157/f367.item. 
• 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. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002215225. 
• 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. doi:10.1007/BFb0094302. ISBN 978-3-540-62038-9. 
• Litt, Daniel (2018). "Non-Abelian Lefschetz hyperplane theorems". Journal of Algebraic Geometry 27 (4): 593–646. doi:10.1090/jag/704. 
• Debarre, Olivier (2005). "Varieties with ample cotangent bundle". Compositio Mathematica 141 (6): 1445–1459. doi:10.1112/S0010437X05001399. 

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. doi:10.1142/S0129167X09005182. 

External links

• Demailly, Jean-Pierre, Complex Analytic and Differential Geometry, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth_book.pdf  (OpenContent book)