Le Potier's vanishing theorem

In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles. The theorem states the following[1][2][3][4][5][6][7][8][9]

Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here ${\displaystyle H^{p,q}(X,E)}$ is Dolbeault cohomology group, where ${\displaystyle \Omega _{X}^{p}}$ denotes the sheaf of holomorphic p-forms on X. If E is an ample, then

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

from Dolbeault theorem,

${\displaystyle H^{q}(X,\Omega _{X}^{p}\otimes E)=0}$ for ${\displaystyle p+q\geq n+r}$ .

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

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

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:[2]

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

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

Demailly (1988) gave a counterexample, which is as follows:[1][10]

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

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

See also

• vanishing theorem
• Barth–Lefschetz theorem

Note

1. (Lazarsfeld 2004)
2. (Shiffman & Sommese 1985)
3. (Demailly 1988)
4. (Peternell 1994)
5. (Laytimi & Nahm 2004)
6. (Verdier 1974)
7. (Schneider 1974)
8. (Broer 1997)
9. (Demailly 1996, p.30)
10. (Manivel 1997)

References

• 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. S2CID 18984867.
• 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. S2CID 14203286.
• Lazarsfeld, Robert (2004). Positivity in Algebraic Geometry II. doi:10.1007/978-3-642-18810-7. ISBN 978-3-540-22531-7.
• 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. S2CID 119147594.
• Peternell, Th. (1994). "Pseudoconvexity, the Levi Problem and Vanishing Theorems". Several Complex Variables VII. Encyclopaedia of Mathematical Sciences. Vol. 74. pp. 221–257. doi:10.1007/978-3-662-09873-8_6. ISBN 978-3-642-08150-7.
• 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. S2CID 122814022.
• 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. S2CID 117285630.
• Shiffman, Bernard; Sommese, Andrew John (1985). "Vector Bundles: Ampleness". Vanishing Theorems on Complex Manifolds. Progress in Mathematics. Vol. 56. pp. 89–116. doi:10.1007/978-1-4899-6680-3_5. ISBN 978-1-4899-6682-7.
• 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. S2CID 14052238.
• 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. S2CID 120949227.
• Sommese, Andrew John (1978). "Submanifolds of Abelian varieties to Rebecca". Mathematische Annalen. 233 (3): 229–256. doi:10.1007/BF01405353. S2CID 120704169.
• Schneider, Michael (1974). "Ein einfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel". Manuscripta Mathematica. 11: 95–101. doi:10.1007/BF01189093. S2CID 120722017.
• 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. Acad. Sci. Paris Sér. 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. S2CID 117547554.
• Demailly, Jean-Pierre (1996). "L2 vanishing theorems for positive line bundles and adjunction theory". Transcendental Methods in Algebraic Geometry. Lecture Notes in Mathematics. Vol. 1646. pp. 1–97. arXiv:alg-geom/9410022. doi:10.1007/BFb0094302. ISBN 978-3-540-62038-9. S2CID 117583140.
• Litt, Daniel (2018). "Non-Abelian Lefschetz hyperplane theorems". Journal of Algebraic Geometry. 27 (4): 593–646. arXiv:1601.07914. doi:10.1090/jag/704. S2CID 16039153.
• Debarre, Olivier (2005). "Varieties with ample cotangent bundle". Compositio Mathematica. 141 (6): 1445–1459. arXiv:math/0306066. doi:10.1112/S0010437X05001399. S2CID 2644826.

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. S2CID 119887533.
• 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. S2CID 10539504.

External links

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