Grauert–Riemenschneider vanishing theorem
{{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 {{harvs|txt|last=Grauert|last2=Riemenschneider|year=1970}}.
Grauert–Riemenschneider conjecture
The Grauert–Riemenschneider conjecture is a conjecture related to the Grauert–Riemenschneider vanishing theorem:
{{blockquote|{{harvtxt|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.{{harv|Siu|1985}}}}
This conjecture was proved by {{harvtxt|Siu|1985}} using the Riemann–Roch type theorem (Hirzebruch–Riemann–Roch theorem) and by {{harvtxt|Demailly|1985}} using Morse theory.
Note
References
- {{Citation | last1=Grauert | first1=Hans | last2=Riemenschneider | first2=Oswald | authorlink1 = Hans Grauert| authorlink2= Oswald Riemenschneider | title=Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970) | publisher=Springer-Verlag | location=Berlin, New York | doi=10.1007/BFb0060317 |mr=0273066 | year=1970a | chapter=Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen | series=Lecture Notes in Mathematics | pages=97–109 | volume=155| isbn=978-3-540-05183-1 }}
- {{Citation | last1=Grauert | first1=Hans | last2=Riemenschneider | first2=Oswald | title=Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen | doi=10.1007/BF01403182 |mr=0302938 | year=1970b | journal=Inventiones Mathematicae | issn=0020-9910 | volume=11 | issue=4 | pages=263–292| bibcode=1970InMat..11..263G | s2cid=115238607 }}
- {{cite journal |doi=10.5802/aif.1034|title=Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie |year=1985 |last1=Demailly |first1=Jean-Pierre |journal=Annales de l'Institut Fourier |volume=35 |issue=4 |pages=189–229 |doi-access=free }}
- {{cite journal |doi=10.4310/JDG/1214438686|title=A vanishing theorem for semipositive line bundles over non-Kähler manifolds |year=1984 |last1=Siu |first1=Yam Tong |journal=Journal of Differential Geometry |volume=19 |issue=2 |doi-access=free }}
- {{cite book |doi=10.1007/BFB0084590|chapter=Some recent results in complex manifold theory related to vanishing theorems for the semipositive case |title=Arbeitstagung Bonn 1984 |series=Lecture Notes in Mathematics |year=1985 |last1=Siu |first1=Yum-Tong |volume=1111 |pages=169–192 |isbn=978-3-540-15195-1 }}
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Category:Theorems in algebraic geometry
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