Nadel vanishing theorem explained

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 -divisor and L a line bundle on X, and l{J}(D) is a multiplier ideal sheaves. Assume that L - D is big and nef. ThenH ^{i} \left(X, l{O} _{X} (K _{X} + L) ⊗ l{J}(D) \right) = 0 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 } ⋅ \theta(F) > \varepsilon ⋅ \omega for some continuous positive function \varepsilon on X. ThenH ^{i} \left(X, l{O} _{X} (K _{X} + F) ⊗ l{J}(\varphi) \right) = 0 for i > 0 . \phi on \Omega \subset X , then a multiplier ideal sheaf l{J}(\phi) is a coherent on \Omega , and therefore its zero variety is an analytic set.

References

Bibliography

• Nadel . Alan Michael . Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature . 34630 . 1015491 . 1989 . . 86 . 19 . 7299–7300 . 10.1073/pnas.86.19.7299. 298048 . 1989PNAS...86.7299N . 16594070 . free .
• 10.2307/1971429. 1971429 . Nadel . Alan Michael . Multiplier Ideal Sheaves and Kahler-Einstein Metrics of Positive Scalar Curvature . Annals of Mathematics . 1990 . 132 . 3 . 549–596 .
• Book: 10.1007/978-3-642-18810-7_5 . Multiplier Ideal Sheaves . [{{Google books|PfJdDwAAQBAJ|page=188|plainurl=yes}} Positivity in Algebraic Geometry II ]. 2004 . Lazarsfeld . Robert . 139–231 . 978-3-540-22531-7 .
• 10.2977/PRIMS/50 . Fundamental Theorems for the Log Minimal Model Program . 2011 . Fujino . Osamu . Publications of the Research Institute for Mathematical Sciences . 47 . 3 . 727–789 . 50561502 . 0909.4445 .
• Méthodes L² et résultats effectifs en géométrie algébrique . Séminaire Bourbaki . 41 . 59–90 . 1998–1999. Demailly . Jean-Pierre .

Further reading

• Book: 10.1007/978-4-431-56852-0_2 . Analyzing the Analyzing the L ^{2} \bar\partial - Cohomology . [{{Google books|DIJ8DwAAQBAJ|page=68|plainurl=yes}} L² Approaches in Several Complex Variables ]. Springer Monographs in Mathematics . 2018 . Ohsawa . Takeo . 47–114 . 978-4-431-56851-3 .
• 10.1016/j.aim.2015.03.019 . free . A Nadel vanishing theorem for metrics with minimal singularities on big line bundles . 2015 . Matsumura . Shin-Ichi . . 280 . 188–207 . 119297787 . 1306.2497 .
• 10.1090/jag/687 . An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities . 2017 . Matsumura . Shin-Ichi . Journal of Algebraic Geometry . 27 . 2 . 305–337 . 119323658. 1308.2033.
• 10.1307/mmj/1030132712 . A subadditivity property of multiplier ideals . 2000 . Demailly . Jean-Pierre . Ein . Lawrence . Lazarsfeld . Robert . Michigan Mathematical Journal . 48 . 11443349 . math/0002035 .
• 10.4310/jdg/1214453680 . A numerical criterion for very ample line bundles . 1993 . Demailly . Jean-Pierre . Journal of Differential Geometry . 37 . 2 . 18938872 .
• Book: 10.1007/978-3-0348-9078-6_75 . L2-Methods and Effective Results in Algebraic Geometry . Proceedings of the International Congress of Mathematicians . 1995 . Demailly . Jean-Pierre . 817–827 . 978-3-0348-9897-3 .
• Book: 10.1007/978-3-0348-8436-5_3 . On the Ohsawa-Takegoshi-Manivel L 2 extension theorem . [{{Google books|hG4FCAAAQBAJ|page=73|plainurl=yes}} Complex Analysis and Geometry ]. Progress in Mathematics . 2000 . Demailly . Jean-Pierre . 188 . 47–82 . 978-3-0348-9566-8.
• 10.1112/S0010437X14007398 . Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds . 2014 . Cao . Junyan . Compositio Mathematica . 150 . 11 . 1869–1902 . 17960658 . 1210.5692 .
• 10.1007/s00208-014-1018-6 . A Nadel vanishing theorem via injectivity theorems . 2014 . Matsumura . Shin-Ichi . Mathematische Annalen . 359 . 3–4 . 785–802 . 253718483 .