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 $H^{p,q}(X,E)$ is Dolbeault cohomology group, where $\Omega _{X}^{p}$ denotes the sheaf of holomorphic p-forms on X. If E is an ample, then
$H^{p,q}(X,E)=0$ for $p+q\geq n+r$ .

from Dolbeault theorem,

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

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

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

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

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

Note

^ a b (Lazarsfeld 2004)
^ a b (Shiffman & Sommese 1985)
^ (Demailly 1988)
^ (Peternell 1994)
^ (Laytimi & Nahm 2004)
^ (Verdier 1974)
^ (Schneider 1974)
^ (Broer 1997)
^ (Demailly 1996, p.30)
^ (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.

External links

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

[Lazarsfeld2004-1] (Lazarsfeld 2004)

[ShiffmanSommese1985-2] (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)

Le Potier's vanishing theorem

See also

Note

References

Further reading

External links