Canonical measures and Kähler-Ricci flow

Song, Jian; Tian, Gang

doi:10.1090/S0894-0347-2011-00717-0

Canonical measures and Kähler-Ricci flow
HTML articles powered by AMS MathViewer

by Jian Song and Gang Tian;

J. Amer. Math. Soc. 25 (2012), 303-353

DOI: https://doi.org/10.1090/S0894-0347-2011-00717-0

Published electronically: October 12, 2011

PDF | Request permission

Abstract:

We show that the Kähler-Ricci flow on a projective manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on a projective manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.

References

Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95 (French, with English summary). MR 494932
Shigetoshi Bando and Toshiki Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 11–40. MR 946233, DOI 10.2969/aspm/01010011
Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359–372. MR 799272, DOI 10.1007/BF01389058
Bennett Chow, The Ricci flow on the $2$-sphere, J. Differential Geom. 33 (1991), no. 2, 325–334. MR 1094458
X. X. Chen and G. Tian, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002), no. 3, 487–544. MR 1893004, DOI 10.1007/s002220100181
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
Herbert Clemens, János Kollár, and Shigefumi Mori, Higher-dimensional complex geometry, Astérisque 166 (1988), 144 pp. (1989) (English, with French summary). MR 1004926
Jean-Pierre Demailly and Nefton Pali, Degenerate complex Monge-Ampère equations over compact Kähler manifolds, Internat. J. Math. 21 (2010), no. 3, 357–405. MR 2647006, DOI 10.1142/S0129167X10006070
Sławomir Dinew and Zhou Zhang, On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds, Adv. Math. 225 (2010), no. 1, 367–388. MR 2669357, DOI 10.1016/j.aim.2010.03.001
S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. MR 1916953
Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Singular Kähler-Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. MR 2505296, DOI 10.1090/S0894-0347-09-00629-8
Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, A priori $L^\infty$-estimates for degenerate complex Monge-Ampère equations, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 070, 8. MR 2439574, DOI 10.1093/imrn/rnn070
Hao Fang and Zhiqin Lu, Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli, J. Reine Angew. Math. 588 (2005), 49–69. MR 2196728, DOI 10.1515/crll.2005.2005.588.49
Joel Fine, Constant scalar curvature Kähler metrics on fibred complex surfaces, J. Differential Geom. 68 (2004), no. 3, 397–432. MR 2144537
Mark Gross and P. M. H. Wilson, Large complex structure limits of $K3$ surfaces, J. Differential Geom. 55 (2000), no. 3, 475–546. MR 1863732
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69–117. MR 1618325, DOI 10.1007/BF02392879
Sławomir Kołodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, 667–686. MR 1986892, DOI 10.1512/iumj.2003.52.2220
Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980, pp. 205–239. MR 573435
Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
John Lott, On the long-time behavior of type-III Ricci flow solutions, Math. Ann. 339 (2007), no. 3, 627–666. MR 2336062, DOI 10.1007/s00208-007-0127-x
Perelman, P., The entropy formula for the Ricci flow and its geometric applications, preprint math.DG/0211159.
Duong H. Phong and Jacob Sturm, On stability and the convergence of the Kähler-Ricci flow, J. Differential Geom. 72 (2006), no. 1, 149–168. MR 2215459
Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673. MR 1660941, DOI 10.1007/s002220050276
Yum-Tong Siu, Multiplier ideal sheaves in complex and algebraic geometry, Sci. China Ser. A 48 (2005), no. suppl., 1–31. MR 2156488, DOI 10.1007/bf02884693
Siu, Y-T., A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring, arXiv:math/0610740.
Jian Song and Gang Tian, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, 609–653. MR 2357504, DOI 10.1007/s00222-007-0076-8
Jian Song and Ben Weinkove, On the convergence and singularities of the $J$-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math. 61 (2008), no. 2, 210–229. MR 2368374, DOI 10.1002/cpa.20182
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
Gang Tian, On Kähler-Einstein metrics on certain Kähler manifolds with $C_1(M)>0$, Invent. Math. 89 (1987), no. 2, 225–246. MR 894378, DOI 10.1007/BF01389077
G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172. MR 1055713, DOI 10.1007/BF01231499
Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. MR 1064867
Gang Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629–646. MR 915841
Gang Tian and Zhou Zhang, On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. MR 2243679, DOI 10.1007/s11401-005-0533-x
Gang Tian and Xiaohua Zhu, Convergence of Kähler-Ricci flow, J. Amer. Math. Soc. 20 (2007), no. 3, 675–699. MR 2291916, DOI 10.1090/S0894-0347-06-00552-2
Valentino Tosatti, Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 4, 755–776. MR 2538503, DOI 10.4171/JEMS/165
Hajime Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), no. 1, 123–133. MR 944606, DOI 10.1007/BF01449219
Hajime Tsuji, Analytic Zariski decomposition, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 7, 161–163. MR 1193172
Tsuji, H., Generalized Bergmann Metrics and Invariance of Plurigenera, arXiv:math/960448.
Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 506253
Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 486659, DOI 10.2307/2373880
Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304
Zhou Zhang, On degenerate Monge-Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. , posted on (2006), Art. ID 63640, 18. MR 2233716, DOI 10.1155/IMRN/2006/63640