Abstract
The purpose of this paper is to establish Nadel type vanishing theorems with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu’s metrics). For this purpose, we generalize Kollár’s injectivity theorem to an injectivity theorem for line bundles equipped with singular metrics, by making use of the theory of harmonic integrals. Moreover we give asymptotic cohomology vanishing theorems for high tensor powers of line bundles.
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Andreotti, A., Grauert, H.: Théorème de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962)
Berndtsson, B.: An eigenvalue estimate for the \(\bar{\partial }\)-Laplacian. J. Differ. Geom. 60(2), 295–313 (2002)
Berndtsson, B.: The openness conjecture for plurisubharmonic functions. Preprint, arxiv:1305.5781v1
Cao, J.: Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact Kähler manifolds. Preprint, arxiv:1210.5692v1
Demailly, J.-P., Ein, L., Lazarsfeld, R.: A subadditivity property of multiplier ideals. Mich. Math. J. 48, 137–156 (2000)
Demailly, J.-P.: Analytic methods in algebraic geometry. Lecture Notes on the web page of the author
Demailly, J.-P.: Complex analytic and differential geometry. Lecture Notes on the web page of the author
Demaily, J.P.: Estimations \(L^{2}\) pour l’opérateur \(\overline{\partial }\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. 15, 457–511 (1982)
Demailly, J.P., J. Kollár, J.: Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. 34, 525–556 (2001)
Demailly, J.-P., Peternell, T., Schneider, M.: Holomorphic line bundles with partially vanishing cohomology. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), pp. 165–198, Bar-Ilan University (1996)
Enoki, I.: Kawamata–Viehweg vanishing theorem for compact Kähler manifolds. Einstein metrics and Yang-Mills connections, pp. 59–68 (1990)
de Fernex, T., Küronya, A., Lazarsfel, R.: Higher cohomology of divisors on a projective variety. Math. Ann. 337(2), 443–455 (2007)
Fujino, O.: A transcendental approach to Kollár’s injectivity theorem. Osaka J. Math. 49(3), 833–852 (2012)
Kawamata, Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann. 261(1), 43–46 (1982)
Kim, D.: The exactness of a general Skoda complex. Preprint, arxiv:1007.0551v1
Kim, D.: A remark on Siu type metric. Private Note of the author
Kollár, J.: Higher direct images of dualizing sheaves. I. Ann. Math. (2) 123(1), 11–42 (1986)
Lazarsfeld, R.: Positivity in algebraic geometry I–II. Springer Verlag, Berlin (2004)
Lesieutre, J.: The diminished base locus is not always closed. Preprint, arxiv:1212.3738v1
Matsumura, S.: On the curvature of holomorphic line bundles with partially vanishing cohomology. RIMS Kôkyûroku 1783, 155–168 (2012)
Matsumura, S.: Asymptotic cohomology vanishing and a converse to the Andreotti–Grauert theorem on surfaces. to appear in, Ann. Inst. Fourier 63 (2013)
Matsumura, S.: A Nadel vanishing theorem for metrics with minimal singularities on big line bundles. Preprint, arxiv:1306.2497v1
Matsumura, S.: An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. Preprint, arxiv:1308.2033v1
Nadel, A.M.: Multiplier ideal sheaves and existence of Kähler–Einstein metrics of positive scalar curvature. Proc. Nat. Acad. Sci. USA 86(19), 7299–7300 (1989)
Nadel, A.M.: Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature. Ann. Math. (2) 132(3), 549–596 (1990)
Nakayama, N.: Zariski-decomposition and abundance. MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, (2004)
Pǎun, M.: Relative critical exponents, non-vanishing and metrics with minimal singularities. Invent. Math. 187, 195–258 (2012)
Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134(3), 661–673 (1998)
Viehweg, E.: Vanishing theorems. J. Reine Angew. Math. 335, 1–8 (1982)
Acknowledgments
The author wishes to express his gratitude Professor Dano Kim for kindly giving some remarks on Siu’s metrics to him. He would like to thank the referee for carefully reading the manuscript. He is supported by the Grant-in-Aid for Young Scientists (B) \(\sharp \)25800051 from JSPS.
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Matsumura, Si. A Nadel vanishing theorem via injectivity theorems. Math. Ann. 359, 785–802 (2014). https://doi.org/10.1007/s00208-014-1018-6
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DOI: https://doi.org/10.1007/s00208-014-1018-6