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.
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References
Aubin T.: Équations du type Monge–Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2) 102(1), 63–95 (1978)
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, Berlin
E. Calabi, Métriques Kählériennes et fibrés holomorphes, Annales scientifiques de l’fÉ.N.S. 4e série, 12:2 (1979), 269–294.
E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, NJ (1982), 259–290.
Cao H.D.: Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math 81(2), 359–372 (1985)
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.
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)
X.-X. Chen, B. Wang, Kähler–Ricci flow on Fano manifolds (I), preprint; arXiv:0909.2391
Debarre O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer-Verlag, New York (2001)
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
Donaldson S.K.: Scalar curvature and stability of toric varieties. J. Differential Geom 62, 289–349 (2002)
Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differential Geometry 65(2), 169–209 (2003)
Griffiths P., Harris J.: Principles of Algebraic Geometry, Pure and Applied Mathematics. Wiley-Interscience, New York (1978)
Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differential Geom 17(2), 255–306 (1982)
Hamilton R.S.: Four-manifolds with positive isotropic curvature. Comm. Anal. Geom 5(1), 1–92 (1997)
Kawamata Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math 79, 567–588 (1985)
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.
Kleiner B., Lott J.: Notes on Perelman’s papers, Geom. Topol 12(5), 2587–2855 (2008)
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.
Kollár S., Mori S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics 134. Cambridge University Press, Cambridge (1998)
Kolodziej S.: The complex Monge–Ampère equation. Acta Math 180(1), 69–117 (1998)
Kolodziej S.: The Complex Monge–Ampère Equation and Pluripotential Theory. Mem. Amer. Math. Soc 178, 840 (2005)
G. La Nave, G. Tian, Soliton-type metrics and Kähler–Ricci flow on symplectic quotients, preprint; arXiv: 0903.2413
C. Li, On rotationally symmetric Kähler–Ricci solitons, preprint; arXiv:1004.4049
J. Morgan, G. Tian, Completion of the Proof of the Geometrization Conjecture, preprint; arXiv: 0809.4040
O. Munteanu, G. Székelyhidi, On convergence of the Kähler–Ricci flow, preprint; arXiv: 0904.3505
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint; arXiv:math.DG/0211159
G. Perelman, unpublished work on the Kähler–Ricci flow
Phong D.H., Sesum N., Sturm J.: Multiplier ideal sheaves and the Kähler–Ricci flow Comm. Anal. Geom 15(3), 613–632 (2007)
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)
Phong D.H., Song J., Sturm J., Weinkove B.: The Kähler–Ricci flow with positive bisectional curvature. Invent. Math. 173(3), 651–665 (2008)
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)
Phong D.H., Sturm J.: On stability and the convergence of the Kähler–Ricci flow. J. Differential Geometry 72(1), 149–168 (2006)
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.
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)
J. Song, Finite time extinction of the Kähler–Ricci flow, preprint; arXiv: 0905.0939
Song J., Tian G.: The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)
J. Song, G. Tian, Canonical measures and Kähler–Ricci flow, preprint; arXiv: 0802.2570
J. Song, G. Tian, The Kähler–Ricciflow through singularities, preprint; arXiv: 0909.4898
J. Song, B. Weinkove, The Kähler–Ricci flow on Hirzebruch surfaces, preprint; arXiv: 0903.1900
J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow, preprint; arXiv:0909.4898
J. Song, B. Weinkove, Contracting divisors by the Kähler–Ricci flow on \({\mathbb {P}^1}\) -bundles and minimal surfaces of general type, preprint.
Székelyhidi G.: The Kähler–Ricci flow and K-polystability. Amer. J. Math. 132, 1077–1090 (2010)
G. Székelyhidi, V. Tosatti, Regularity of weak solutions of a complex Monge–Ampère equation, Anal. PDE (2010), to appear.
Tian G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math 130(1), 1–37 (1997)
Tian G.: New results and problems on Kähler–Ricci flow, Géométrie différentielle, physique mathématique. mathématiques et société. II. Astérisque 322, 71–92 (2008)
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)
Tian G., Zhu X.: Convergence of Kähler–Ricci flow. J. Amer. Math. Soc 20(3), 675–699 (2007)
Tosatti V.: Kähler–Ricci flow on stable Fano manifolds. J. Reine Angew. Math 640, 67–84 (2010)
Tsuji H.: Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann 281, 123–133 (1988)
Wang X.J., Zhu X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Advances Math 188, 87–103 (2004)
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)
Yau S.T.: Open problems in geometry. Proc. Symposia Pure Math. 54, 1–28 (1993)
Y. Yuan, On the convergence of a modified Kähler–Ricci flow, Math. Z., to appear.
Z. Zhang, On degenerate Monge–Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. 2006, Art. ID 63640, 18 pp.
X. Zhu, Kähler–Ricci flow on a toric manifold with positive first Chern class, preprint, arXiv: math.DG/0703486
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The first named author is supported in part by National Science Foundation grant DMS-0847524 and a Sloan Foundation Fellowship.
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Song, J., Yuan, Y. Metric Flips with Calabi Ansatz. Geom. Funct. Anal. 22, 240–265 (2012). https://doi.org/10.1007/s00039-012-0151-1
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DOI: https://doi.org/10.1007/s00039-012-0151-1