Grauert–Riemenschneider vanishing theorem
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Short description: Mathematical theorem
In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to Grauert and Riemenschneider (1970).
Grauert–Riemenschneider conjecture
The Grauert–Riemenschneider conjecture is a conjecture related to the Grauert–Riemenschneider vanishing theorem:
This conjecture was proved by (Siu 1985) using the Riemann–Roch type theorem (Hirzebruch–Riemann–Roch theorem) and by (Demailly 1985) using Morse theory.
Note
- ↑ (Siu 1985)
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, 155, Berlin, New York: Springer-Verlag, pp. 97–109, doi:10.1007/BFb0060317, ISBN 978-3-540-05183-1
- Grauert, Hans; Riemenschneider, Oswald (1970b), "Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen", Inventiones Mathematicae 11: 263–292, doi:10.1007/BF01403182, ISSN 0020-9910, Bibcode: 1970InMat..11..263G
- 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. 1111. pp. 169–192. doi:10.1007/BFB0084590. ISBN 978-3-540-15195-1.
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