In mathematics, the Nadel vanishing theorem is a global vanishing theorem for multiplier ideals, introduced by A. M. Nadel in 1989. 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 The theorem can be stated as follows. Let X be a smooth complex projective variety, D an effective Q {\displaystyle \mathbb {Q} } -divisor and L a line bundle on X, and J ( D ) {\displaystyle {\mathcal {J}}(D)} is a multiplier ideal sheaves. Assume that L − D {\displaystyle L-D} is big and nef. Then
H i ( X , O X ( K X + L ) ⊗ J ( D ) ) = 0 for i > 0. {\displaystyle H^{i}\left(X,{\mathcal {O}}_{X}(K_{X}+L)\otimes {\mathcal {J}}(D)\right)=0\;\;{\text{for}}\;\;i>0.}
Nadel vanishing theorem in the analytic setting: Let ( X , ω ) {\displaystyle (X,\omega )} 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 φ {\displaystyle \varphi } . Assume that − 1 ⋅ θ ( F ) > ε ⋅ ω {\displaystyle {\sqrt {-1}}\cdot \theta (F)>\varepsilon \cdot \omega } for some continuous positive function ε {\displaystyle \varepsilon } on X. Then
H i ( X , O X ( K X + F ) ⊗ J ( φ ) ) = 0 for i > 0. {\displaystyle H^{i}\left(X,{\mathcal {O}}_{X}(K_{X}+F)\otimes {\mathcal {J}}(\varphi )\right)=0\;\;{\text{for}}\;\;i>0.}
Let arbitrary plurisubharmonic function ϕ {\displaystyle \phi } on Ω ⊂ X {\displaystyle \Omega \subset X} , then a multiplier ideal sheaf J ( ϕ ) {\displaystyle {\mathcal {J}}(\phi )} is a coherent on Ω {\displaystyle \Omega } , and therefore its zero variety is an analytic set.
References
Citations
Bibliography 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.
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