Geometric and Functional Analysis

, Volume 22, Issue 1, pp 240–265 | Cite as

Metric Flips with Calabi Ansatz

Article

Abstract

We study the limiting behavior of the Kähler–Ricci flow on \({{\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus(m+1)})}}\) for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to \({{\mathbb{P}^n}}\) or contracts a subvariety of codimension m + 1 in the Gromov–Hausdorff sense. We also show that the Kähler–Ricci flow resolves a certain type of cone singularities in the Gromov–Hausdorff sense.

Keywords and phrases

Kähler-Ricci flow Gromov-Hausdorff convergence small contraction flip 

2010 Mathematics Subject Classification

53c55 53c44 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.
    Aubin T.: Équations du type Monge–Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2) 102(1), 63–95 (1978)MathSciNetMATHGoogle Scholar
  2. BHPV.
    W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven, Compact Complex Surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, BerlinGoogle Scholar
  3. C1.
    E. Calabi, Métriques Kählériennes et fibrés holomorphes, Annales scientifiques de l’fÉ.N.S. 4e série, 12:2 (1979), 269–294.Google Scholar
  4. C2.
    E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, NJ (1982), 259–290.Google Scholar
  5. Ca1.
    Cao H.D.: Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math 81(2), 359–372 (1985)MathSciNetMATHCrossRefGoogle Scholar
  6. Ca2.
    H.D. Cao, Existence of gradient Kähler–Ricci solitons, Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A.K. Peters, Wellesley, MA, (1996), 1–16.Google Scholar
  7. CaZ.
    Cao H.D., Zhu X.-P.: A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow. Asian J. Math 10(2), 165–492 (2006)MathSciNetMATHGoogle Scholar
  8. ChW.
    X.-X. Chen, B. Wang, Kähler–Ricci flow on Fano manifolds (I), preprint; arXiv:0909.2391 Google Scholar
  9. D.
    Debarre O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer-Verlag, New York (2001)MATHGoogle Scholar
  10. De.
    J.P. Demailly, Applications of the theory of L 2 estimates and positive currents in algebraic geometry, Lecture Notes, École d’été de Mathématiques de Grenoble ‘Géométrie des variétés projectives complexes : programme du modèle minimal’ (June-July 2007), arXiv: 9410022 Google Scholar
  11. Do.
    Donaldson S.K.: Scalar curvature and stability of toric varieties. J. Differential Geom 62, 289–349 (2002)MathSciNetMATHGoogle Scholar
  12. FIK.
    Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differential Geometry 65(2), 169–209 (2003)MathSciNetMATHGoogle Scholar
  13. GH.
    Griffiths P., Harris J.: Principles of Algebraic Geometry, Pure and Applied Mathematics. Wiley-Interscience, New York (1978)Google Scholar
  14. H1.
    Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differential Geom 17(2), 255–306 (1982)MathSciNetMATHGoogle Scholar
  15. H2.
    Hamilton R.S.: Four-manifolds with positive isotropic curvature. Comm. Anal. Geom 5(1), 1–92 (1997)MathSciNetMATHGoogle Scholar
  16. K.
    Kawamata Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math 79, 567–588 (1985)MathSciNetMATHCrossRefGoogle Scholar
  17. KMM.
    Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam (1987), 283–360.Google Scholar
  18. KlL.
    Kleiner B., Lott J.: Notes on Perelman’s papers, Geom. Topol 12(5), 2587–2855 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. Ko.
    N. Koiso, On rotationally symmetric Hamilton’s equation for Kähler–Einstein metrics, Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math. 18-I, Academic Press, Boston, MA (1990), 327–337.Google Scholar
  20. KolM.
    Kollár S., Mori S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics 134. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  21. Kolo1.
    Kolodziej S.: The complex Monge–Ampère equation. Acta Math 180(1), 69–117 (1998)MathSciNetMATHCrossRefGoogle Scholar
  22. Kolo2.
    Kolodziej S.: The Complex Monge–Ampère Equation and Pluripotential Theory. Mem. Amer. Math. Soc 178, 840 (2005)MathSciNetGoogle Scholar
  23. LT.
    G. La Nave, G. Tian, Soliton-type metrics and Kähler–Ricci flow on symplectic quotients, preprint; arXiv: 0903.2413 Google Scholar
  24. Li.
    C. Li, On rotationally symmetric Kähler–Ricci solitons, preprint; arXiv:1004.4049 Google Scholar
  25. MT.
    J. Morgan, G. Tian, Completion of the Proof of the Geometrization Conjecture, preprint; arXiv: 0809.4040 Google Scholar
  26. MuS.
    O. Munteanu, G. Székelyhidi, On convergence of the Kähler–Ricci flow, preprint; arXiv: 0904.3505 Google Scholar
  27. P1.
    G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint; arXiv:math.DG/0211159 Google Scholar
  28. P2.
    G. Perelman, unpublished work on the Kähler–Ricci flowGoogle Scholar
  29. PhSS.
    Phong D.H., Sesum N., Sturm J.: Multiplier ideal sheaves and the Kähler–Ricci flow Comm. Anal. Geom 15(3), 613–632 (2007)MathSciNetMATHGoogle Scholar
  30. PhSSW1.
    Phong D.H., Song J., Sturm J., Weinkove B.: The Kähler–Ricci flow and the \({{\bar{\partial}}}\) operator on vector fields. J. Differential Geometry 81(3), 631–647 (2009)MathSciNetMATHGoogle Scholar
  31. PhSSW2.
    Phong D.H., Song J., Sturm J., Weinkove B.: The Kähler–Ricci flow with positive bisectional curvature. Invent. Math. 173(3), 651–665 (2008)MathSciNetMATHCrossRefGoogle Scholar
  32. PhSSW3.
    Phong D.H., Song J., Sturm J., Weinkove B.: On the convergence of the modified Kähler–Ricci flow and solitons. Comment. Math. Helv 86(1), 91–112 (2011)MathSciNetMATHCrossRefGoogle Scholar
  33. PhS.
    Phong D.H., Sturm J.: On stability and the convergence of the Kähler–Ricci flow. J. Differential Geometry 72(1), 149–168 (2006)MathSciNetMATHGoogle Scholar
  34. R.
    Rubinstein Y.: On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow, Trans. Amer. Math. Soc. 361:11 (2009), 5839–5850.Google Scholar
  35. ST.
    Sesum N., Tian G.: Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman). J. Inst. Math. Jussieu 7(3), 575–587 (2008)MathSciNetMATHCrossRefGoogle Scholar
  36. So.
    J. Song, Finite time extinction of the Kähler–Ricci flow, preprint; arXiv: 0905.0939 Google Scholar
  37. SoT1.
    Song J., Tian G.: The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)MathSciNetMATHCrossRefGoogle Scholar
  38. SoT2.
    J. Song, G. Tian, Canonical measures and Kähler–Ricci flow, preprint; arXiv: 0802.2570 Google Scholar
  39. SoT3.
    J. Song, G. Tian, The Kähler–Ricciflow through singularities, preprint; arXiv: 0909.4898 Google Scholar
  40. SoW1.
    J. Song, B. Weinkove, The Kähler–Ricci flow on Hirzebruch surfaces, preprint; arXiv: 0903.1900 Google Scholar
  41. SoW2.
    J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow, preprint; arXiv:0909.4898 Google Scholar
  42. SoW3.
    J. Song, B. Weinkove, Contracting divisors by the Kähler–Ricci flow on \({\mathbb {P}^1}\) -bundles and minimal surfaces of general type, preprint.Google Scholar
  43. Sz.
    Székelyhidi G.: The Kähler–Ricci flow and K-polystability. Amer. J. Math. 132, 1077–1090 (2010)MathSciNetMATHCrossRefGoogle Scholar
  44. SzT.
    G. Székelyhidi, V. Tosatti, Regularity of weak solutions of a complex Monge–Ampère equation, Anal. PDE (2010), to appear.Google Scholar
  45. T1.
    Tian G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math 130(1), 1–37 (1997)MathSciNetMATHCrossRefGoogle Scholar
  46. T2.
    Tian G.: New results and problems on Kähler–Ricci flow, Géométrie différentielle, physique mathématique. mathématiques et société. II. Astérisque 322, 71–92 (2008)Google Scholar
  47. TZ.
    Tian G., Zhang Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chinese Ann. Math. Ser. B 27(2), 179–192 (2006)MathSciNetMATHCrossRefGoogle Scholar
  48. TZh.
    Tian G., Zhu X.: Convergence of Kähler–Ricci flow. J. Amer. Math. Soc 20(3), 675–699 (2007)MathSciNetMATHCrossRefGoogle Scholar
  49. To.
    Tosatti V.: Kähler–Ricci flow on stable Fano manifolds. J. Reine Angew. Math 640, 67–84 (2010)MathSciNetMATHCrossRefGoogle Scholar
  50. Ts.
    Tsuji H.: Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann 281, 123–133 (1988)MathSciNetMATHCrossRefGoogle Scholar
  51. WZ.
    Wang X.J., Zhu X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Advances Math 188, 87–103 (2004)MATHCrossRefGoogle Scholar
  52. Y1.
    Yau S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I. Comm. Pure Appl. Math 31, 339–411 (1978)MathSciNetMATHCrossRefGoogle Scholar
  53. Y2.
    Yau S.T.: Open problems in geometry. Proc. Symposia Pure Math. 54, 1–28 (1993)Google Scholar
  54. Yu.
    Y. Yuan, On the convergence of a modified Kähler–Ricci flow, Math. Z., to appear.Google Scholar
  55. Z.
    Z. Zhang, On degenerate Monge–Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. 2006, Art. ID 63640, 18 pp.Google Scholar
  56. Zh.
    X. Zhu, Kähler–Ricci flow on a toric manifold with positive first Chern class, preprint, arXiv: math.DG/0703486 Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations