×

zbMATH — the first resource for mathematics

Dynamical construction of Kähler-Einstein metrics. (English) Zbl 1205.32021
Author’s abstract: 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 \(f : X\rightarrow S\) 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 \(h_{E,m}\) on \(f_{*}{\mathcal O}_{X}(mK_{X/S})\) with semipositive curvature in the sense of Nakano.

MSC:
32Q20 Kähler-Einstein manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32G07 Deformations of special (e.g., CR) structures
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E11 Critical metrics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] T. Aubin, Equations du type Monge-Ampère sur les varietés kähleriennes compactes , C. R. Acad. Sci. Paris Sér. A-B 283 (1976), A119-A121. · Zbl 0333.53040
[2] B. Berndtsson, Curvature of vector bundles associated to holomorphic fibrations , · Zbl 1195.32012
[3] B. Berndtsson and M. Paun, Bergman kernels and the pseudoeffectivity of relative canonical bundles , · Zbl 1181.32025
[4] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type , · Zbl 1210.14019
[5] D. Catlin, “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. · Zbl 0941.32002
[6] J. P. Demailly, Complex analytic and algebraic geometry , (2009), 121.
[7] J. P. Demailly, T. Peternell, and M. Schneider, Pseudo-effective line bundles on compact Kähler manifolds , · Zbl 1111.32302
[8] S. K. Donaldson, Scalar curvature and projective embeddings I , J. Differential Geom. 59 (2001), 479-522. · Zbl 1052.32017
[9] S. Krantz, Function Theory of Several Complex Variables , New York, Wiley, 1982. · Zbl 0471.32008
[10] A. M. Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature , Ann. of Math. 132 (1990), 549-596. JSTOR: · Zbl 0731.53063
[11] J. Song and G. Tian, Canonical measures and Kähler-Ricci flow , · Zbl 1239.53086
[12] K. Sugiyama, “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. · Zbl 0770.53021
[13] H. Tsuji, Existence and degeneration of Kahler-Einstein metrics on minimal algebraic varieties of general type , Math. Ann. 281 (1988), 123-133. · Zbl 0631.53051
[14] H. Tsuji, Analytic Zariski decomposition , Proc. Japan Acad. Ser. A Math. Sci. 61 (1992), 161-163. · Zbl 0786.14005
[15] H. Tsuji, “Existence and applications of analytic Zariski decompositions” in Analysis and Geometry in Several Complex Variables , (Katata 1997), Trends Math., Birkhäuser, Boston, 1999, 253-272. · Zbl 0965.32022
[16] H. Tsuji, Deformation invariance of plurigenera , Nagoya Math. J. 166 (2002), 117-134. · Zbl 1064.14035
[17] H. Tsuji, Variation of Bergman kernels of adjoint line bundles , · Zbl 0072.11101
[18] H. Tsuji, Dynamical construction of Kähler-Einstein metrics , · Zbl 1205.32021
[19] H. Tsuji, Canonical measures and dynamical systems of Bergman kernels , · Zbl 0072.11101
[20] H. Tsuji, Ricci iterations and canonical Kähler-Einstein currents on log canonical pairs , · Zbl 0072.11101
[21] H. Tsuji, Global generation of the direct images of relative pluri log canonical systems , preprint, 2010.
[22] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation , Comm. Pure Appl. Math. 31 (1978), 339-441. · Zbl 0369.53059
[23] S. Zelditch, Szögo kernel and a theorem of Tian , Int. Res. Notice 6 (1998), 317-331. · Zbl 0922.58082
[24] S. Zhang, Heights and reductions of semistable varieties , Compos. Math. 104 (1996), 77-105. · Zbl 0924.11055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.