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.

This conjecture was proved by Siu (1985) using the Riemann–Roch type theorem (Hirzebruch–Riemann–Roch theorem) and by Demailly (1985) using Morse theory.

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.