Nadel vanishing theorem

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 $\mathbb {Q}$-divisor and L a line bundle on X, and ${\mathcal {J}}(D)$ is a multiplier ideal sheaves. Assume that $L-D$ is big and nef. Then

$H^{i}\left(X,{\mathcal {O}}_{X}(K_{X}+L)\otimes {\mathcal {J}}(D)\right)=0\;\;{\text{for}}\;\;i>0.$

Nadel vanishing theorem in the analytic setting: Let $(X,\omega )$ 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 $\varphi$. Assume that ${\sqrt {-1}}\cdot \theta (F)>\varepsilon \cdot \omega$ for some continuous positive function $\varepsilon$ on X. Then

$H^{i}\left(X,{\mathcal {O}}_{X}(K_{X}+F)\otimes {\mathcal {J}}(\varphi )\right)=0\;\;{\text{for}}\;\;i>0.$

Let arbitrary plurisubharmonic function $\phi$ on $\Omega \subset X$, then a multiplier ideal sheaf ${\mathcal {J}}(\phi )$ is a coherent on $\Omega$, and therefore its zero variety is an analytic set.

References

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.

Further reading

• Ohsawa, Takeo (2018). "Analyzing the Analyzing the $L^{2}{\bar {\partial }}$ - Cohomology". L² Approaches in Several Complex Variables. Springer Monographs in Mathematics. pp. 47–114. doi:10.1007/978-4-431-56852-0_2. ISBN 978-4-431-56851-3.
• Matsumura, Shin-Ichi (2015). "A Nadel vanishing theorem for metrics with minimal singularities on big line bundles". Advances in Mathematics. 280: 188–207. arXiv:1306.2497. doi:10.1016/j.aim.2015.03.019. S2CID 119297787.
• Matsumura, Shin-Ichi (2017). "An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities". Journal of Algebraic Geometry. 27 (2): 305–337. arXiv:1308.2033. doi:10.1090/jag/687. S2CID 119323658.
• Demailly, Jean-Pierre; Ein, Lawrence; Lazarsfeld, Robert (2000). "A subadditivity property of multiplier ideals". Michigan Mathematical Journal. 48. arXiv:math/0002035. doi:10.1307/mmj/1030132712. S2CID 11443349.
• Demailly, Jean-Pierre (1993). "A numerical criterion for very ample line bundles". Journal of Differential Geometry. 37 (2). doi:10.4310/jdg/1214453680. S2CID 18938872.
• Demailly, Jean-Pierre (1995). "L2-Methods and Effective Results in Algebraic Geometry". Proceedings of the International Congress of Mathematicians. pp. 817–827. doi:10.1007/978-3-0348-9078-6_75. ISBN 978-3-0348-9897-3.
• Demailly, Jean-Pierre (2000). "On the Ohsawa-Takegoshi-Manivel L 2 extension theorem". Complex Analysis and Geometry. Progress in Mathematics. Vol. 188. pp. 47–82. doi:10.1007/978-3-0348-8436-5_3. ISBN 978-3-0348-9566-8.
• Cao, Junyan (2014). "Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds". Compositio Mathematica. 150 (11): 1869–1902. arXiv:1210.5692. doi:10.1112/S0010437X14007398. S2CID 17960658.
• Matsumura, Shin-Ichi (2014). "A Nadel vanishing theorem via injectivity theorems". Mathematische Annalen. 359 (3–4): 785–802. doi:10.1007/s00208-014-1018-6. S2CID 253718483.