Nadel vanishing theorem

In mathematics, the Nadel vanishing theorem is a global vanishing theorem for multiplier ideals, introduced by A. M. Nadel in 1989.[1] It generalizes the Kodaira vanishing theorem using singular metrics with (strictly) positive curvature, and also it can be seen as an analytical analogue of the Kawamata–Viehweg vanishing theorem.

Statement

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The theorem can be stated as follows.[2][3][4] Let X be a smooth complex projective variety, D an effective  -divisor and L a line bundle on X, and   is a multiplier ideal sheaves. Assume that   is big and nef. Then

 

Nadel vanishing theorem in the analytic setting:[5][6] Let   be a Kähler manifold (X be a reduced complex space (complex analytic variety) with a Kähler metric) such that weakly pseudoconvex, and let F be a holomorphic line bundle over X equipped with a singular hermitian metric of weight  . Assume that   for some continuous positive function   on X. Then

 

Let arbitrary plurisubharmonic function   on  , then a multiplier ideal sheaf   is a coherent on  , and therefore its zero variety is an analytic set.

References

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Citations

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Bibliography

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  • Nadel, Alan Michael (1989). "Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature". Proceedings of the National Academy of Sciences of the United States of America. 86 (19): 7299–7300. Bibcode:1989PNAS...86.7299N. doi:10.1073/pnas.86.19.7299. JSTOR 34630. MR 1015491. PMC 298048. PMID 16594070.
  • Nadel, Alan Michael (1990). "Multiplier Ideal Sheaves and Kahler-Einstein Metrics of Positive Scalar Curvature". Annals of Mathematics. 132 (3): 549–596. doi:10.2307/1971429. JSTOR 1971429.
  • Lazarsfeld, Robert (2004). "Multiplier Ideal Sheaves". Positivity in Algebraic Geometry II. pp. 139–231. doi:10.1007/978-3-642-18810-7_5. ISBN 978-3-540-22531-7.
  • Fujino, Osamu (2011). "Fundamental Theorems for the Log Minimal Model Program". Publications of the Research Institute for Mathematical Sciences. 47 (3): 727–789. arXiv:0909.4445. doi:10.2977/PRIMS/50. S2CID 50561502.
  • Demailly, Jean-Pierre (1998–1999). "Méthodes L2 et résultats effectifs en géométrie algébrique". Séminaire Bourbaki. 41: 59–90.

Further reading

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