Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T16:34:32.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical construction of Kähler-Einstein metrics

Published online by Cambridge University Press:  11 January 2016

Hajime Tsuji
Hajime Tsuji*
Affiliation:
Department of Mathematics, Sophia University, 7-1 Kioicho, Chiyoda-ku 102-8554, Japan, h-tsuji@h03.itscom.net
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphism f: X → S with connected fibers such that a general fiber has an ample canonical bundle, and for a positive integer m, we construct a canonical singular Hermitian metric hE,m on with semipositive curvature in the sense of Nakano.

Keywords

53C2532G0753C5558E11
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 199 , September 2010 , pp. 107 - 122
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[A] Aubin, T., Equations du type Monge-Ampère sur les varietés kähleriennes compactes, C. R. Acad. Sci. Paris Sér. A–B 283 (1976), A119–A121.Google Scholar
[B] Berndtsson, B., Curvature of vector bundles associated to holomorphic fibrations, arXiv:math/0511225 [math.CV]CrossRefGoogle Scholar
[BP] Berndtsson, B. and Paun, M., Bergman kernels and the pseudoeffectivity of relative canonical bundles, arXiv:math/0703344 [math.AG]Google Scholar
[BCHM] Birkar, C., Cascini, P., Hacon, C., and McKernan, J., Existence of minimal models for varieties of log general type, arXiv:math/0610203CrossRefGoogle Scholar
[C] Catlin, D., “The Bergman kernel and a theorem of Tian” in Analysis and Geometry in Several Complex Variables, (Katata 1997), Trends Math., Birkhäuser, Boston, 1999, 1–23.Google Scholar
[D] Demailly, J. P., Complex analytic and algebraic geometry, http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf (2009), 121.Google Scholar
[DPS] Demailly, J. P., Peternell, T., and Schneider, M., Pseudo-effective line bundles on compact Kähler manifolds, arXiv:math/0006025 [math.AG]Google Scholar
[Do] Donaldson, S. K., Scalar curvature and projective embeddings I, J. Differential Geom. 59 (2001), 479–522.Google Scholar
[Kr] Krantz, S., Function Theory of Several Complex Variables, New York, Wiley, 1982.Google Scholar
[N] Nadel, A. M., Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. 132 (1990), 549–596.Google Scholar
[ST] Song, J. and Tian, G., Canonical measures and Kähler-Ricci flow, arXiv:math/0802.2570Google Scholar
[S] Sugiyama, K., “Einstein-Kahler metrics on minimal varieties of general type and an inequality between Chern numbers” in Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math. 18-I, Academic Press, Boston, 1990, 417–433.Google Scholar
[T0] Tsuji, H., Existence and degeneration of Kahler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), 123–133.CrossRefGoogle Scholar
[T1] Tsuji, H., Analytic Zariski decomposition, Proc. Japan Acad. Ser. A Math. Sci. 61 (1992), 161–163.Google Scholar
[T2] Tsuji, H., “Existence and applications of analytic Zariski decompositions” in Analysis and Geometry in Several Complex Variables, (Katata 1997), Trends Math., Birkhäuser, Boston, 1999, 253–272.CrossRefGoogle Scholar
[T3] Tsuji, H., Deformation invariance of plurigenera, Nagoya Math. J. 166 (2002), 117–134.Google Scholar
[T5] Tsuji, H., Variation of Bergman kernels of adjoint line bundles, arXiv:math/0511342 [math.CV]Google Scholar
[T6] Tsuji, H., Dynamical construction of Kähler-Einstein metrics, arXiv:math/0606023 [math.AG]Google Scholar
[T7] Tsuji, H., Canonical measures and dynamical systems of Bergman kernels, arXiv:math/0805.1829Google Scholar
[T8] Tsuji, H., Ricci iterations and canonical Kähler-Einstein currents on log canonical pairs, arXiv:math/0903.5445Google Scholar
[T9] Tsuji, H., Global generation of the direct images of relative pluri log canonical systems, preprint, 2010.Google Scholar
[Y1] Yau, S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, Comm. Pure Appl. Math. 31 (1978), 339–441.Google Scholar
[Ze] Zelditch, S., Szögo kernel and a theorem of Tian, Int. Res. Notice 6 (1998), 317–331.Google Scholar
[Z] Zhang, S., Heights and reductions of semistable varieties, Compos. Math. 104 (1996), 77–105.Google Scholar