Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Collapsing of products along the Kähler-Ricci flow


Author: Matthew Gill
Journal: Trans. Amer. Math. Soc. 366 (2014), 3907-3924
MSC (2010): Primary 53C44; Secondary 53C55
DOI: https://doi.org/10.1090/S0002-9947-2013-06073-4
Published electronically: November 14, 2013
MathSciNet review: 3192623
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X = M \times E$, where $ M$ is an $ m$-dimensional Kähler manifold with negative first Chern class and $ E$ is an $ n$-dimensional complex torus. We obtain $ C^\infty $ convergence of the normalized Kähler-Ricci flow on $ X$ to a Kähler-Einstein metric on $ M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture.


References [Enhancements On Off] (What's this?)

  • [Au] 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 (81d:53047)
  • [C] 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 (87d:58051), https://doi.org/10.1007/BF01389058
  • [Ch] Albert Chau, Convergence of the Kähler-Ricci flow on noncompact Kähler manifolds, J. Differential Geom. 66 (2004), no. 2, 211-232. MR 2106124 (2005g:53118)
  • [DH] Ph. Delanoë and A. Hirschowitz, About the proofs of Calabi's conjectures on compact Kähler manifolds, Enseign. Math. (2) 34 (1988), no. 1-2, 107-122. MR 960195 (90g:32033)
  • [Ev] Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333-363. MR 649348 (83g:35038), https://doi.org/10.1002/cpa.3160350303
  • [F1] F. Fong, Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds, to appear in Trans. Amer. Math. Soc., arXiv: 1104.3924.
  • [F2] F. Fong, On the collapsing rate of Kähler-Ricci flow with finite-time singularity, preprint, arXiv: 1112.5987.
  • [FZ] F. Fong, Z. Zhang, The collapsing rate of the Kähler-Ricci flow with regular infinite time singularity, preprint, arXiv: 1202.3199.
  • [GTZ] M. Gross, V. Tosatti, Y. Zhang, Collapsing of abelian fibred Calabi-Yau manifolds, preprint, arXiv: 1108.0967.
  • [Kr] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487-523, 670 (Russian). MR 661144 (84a:35091)
  • [LSY] Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Canonical metrics on the moduli space of Riemann surfaces. II, J. Differential Geom. 69 (2005), no. 1, 163-216. MR 2169586 (2007g:32010)
  • [PSS] D. H. Phong, Natasa Sesum, and Jacob Sturm, Multiplier ideal sheaves and the Kähler-Ricci flow, Comm. Anal. Geom. 15 (2007), no. 3, 613-632. MR 2379807 (2009a:32037)
  • [Sh] Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223-301. MR 1001277 (90i:58202)
  • [SSW] J. Song, G. Székelyhidi, B. Weinkove, The Kähler-Ricci flow on projective bundles, preprint, arXiv: 1107:2144.
  • [ST1] 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 (2008m:32044), https://doi.org/10.1007/s00222-007-0076-8
  • [ST2] Jian Song and Gang Tian, Canonical measures and Kähler-Ricci flow, J. Amer. Math. Soc. 25 (2012), no. 2, 303-353. MR 2869020, https://doi.org/10.1090/S0894-0347-2011-00717-0
  • [ST3] J. Song, G. Tian, The Kähler-Ricci flow through singularities, preprint, arXiv: 0803.1613.
  • [SW1] Jian Song and Ben Weinkove, The Kähler-Ricci flow on Hirzebruch surfaces, J. Reine Angew. Math. 659 (2011), 141-168. MR 2837013 (2012g:53142), https://doi.org/10.1515/CRELLE.2011.071
  • [SW2] J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow, preprint, arXiv: 1003.0718.
  • [SW3] J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow II, preprint, arXiv: 1102.1759.
  • [SW4] J. Song, B. Weinkove, Lecture notes on the Kähler-Ricci flow.
  • [ShW] Morgan Sherman and Ben Weinkove, Interior derivative estimates for the Kähler-Ricci flow, Pacific J. Math. 257 (2012), no. 2, 491-501. MR 2972475, https://doi.org/10.2140/pjm.2012.257.491
  • [SY] J. Song, Y. Yuan Metric flips with Calabi ansatz, preprint, arXiv: 1011.1608.
  • [T] Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99-130. MR 1064867 (91j:32031)
  • [TZ] 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 (2007c:32029), https://doi.org/10.1007/s11401-005-0533-x
  • [To] Valentino Tosatti, Adiabatic limits of Ricci-flat Kähler metrics, J. Differential Geom. 84 (2010), no. 2, 427-453. MR 2652468 (2011m:32039)
  • [Ts] 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 (89e:53075), https://doi.org/10.1007/BF01449219
  • [Yau] 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 (81d:53045), https://doi.org/10.1002/cpa.3160310304

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 53C55

Retrieve articles in all journals with MSC (2010): 53C44, 53C55


Additional Information

Matthew Gill
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093
Address at time of publication: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840

DOI: https://doi.org/10.1090/S0002-9947-2013-06073-4
Received by editor(s): June 14, 2012
Received by editor(s) in revised form: December 17, 2012
Published electronically: November 14, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society