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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 is Dolbeault cohomology group, where denotes the sheaf of holomorphic p-forms on X. If E is an ample, then
from Dolbeault theorem,
By Serre duality, the statements are equivalent to the assertions:
References
- Grauert, Hans; Riemenschneider, Oswald (1970a), "Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen", Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970), Lecture Notes in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, pp. 97–109, doi:10.1007/BFb0060317, ISBN 978-3-540-05183-1, MR 0273066
- Grauert, Hans; Riemenschneider, Oswald (1970b), "Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen", Inventiones Mathematicae , 11: 263–292, Bibcode:1970InMat..11..263G, doi:10.1007/BF01403182, ISSN 0020-9910, MR 0302938, S2CID 115238607
- Demailly, Jean-Pierre (1985). "Champs magnétiques et inégalités de Morse pour la $d$-cohomologie". Annales de l'Institut Fourier. 35 (4): 189–229. doi:10.5802/aif.1034.
- Siu, Yam Tong (1984). "A vanishing theorem for semipositive line bundles over non-Kähler manifolds". Journal of Differential Geometry. 19 (2). doi:10.4310/JDG/1214438686.
- Siu, Yum-Tong (1985). "Some recent results in complex manifold theory related to vanishing theorems for the semipositive case". Arbeitstagung Bonn 1984. Lecture Notes in Mathematics. Vol. 1111. pp. 169–192. doi:10.1007/BFB0084590. ISBN 978-3-540-15195-1.
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