Grauert–Riemenschneider vanishing 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:

Grauert & Riemenschneider (1970a); Let M be an n-dimensional compact complex manifold. M is Moishezon if and only if there exists a smooth Hermitian line bundle L over M whose curvature form which is semi-positive everywhere and positive on an open dense set.[1]

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

1. (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, 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.