Abstract
The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.
References
[1] Michael T. Anderson. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc., 2(3):455-490, 1989.10.1090/S0894-0347-1989-0999661-1Search in Google Scholar
[2] Michael Atiyah and Claude Lebrun. Curvature, cones and characteristic numbers. Math. Proc. Cambridge Philos. Soc., 155(1):13-37, 2013.10.1017/S0305004113000169Search in Google Scholar
[3] Simon Brendle. Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. IMRN, (24):5727-5766, 2013.10.1093/imrn/rns228Search in Google Scholar
[4] Egbert Brieskorn and Horst Knörrer. Plane algebraic curves. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1986. Translated from the German original by John Stillwell, [2012] reprint of the 1986 edition.10.1007/978-3-0348-5097-1Search in Google Scholar
[5] Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.10.1090/gsm/033Search in Google Scholar
[6] Jeff Cheeger. Degeneration of Einstein metrics and metrics with special holonomy. In Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), Surv. Differ. Geom., VIII, pages 29-73. Int. Press, Somerville, MA, 2003.10.4310/SDG.2003.v8.n1.a2Search in Google Scholar
[7] Xiuxiong Chen, Simon Donaldson, and Song Sun. Kähler-Einstein metrics and stability. Int. Math. Res. Not. IMRN, (8):2119-2125, 2014.10.1093/imrn/rns279Search in Google Scholar
[8] Xiuxiong Chen, Simon Donaldson, and Song Sun. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc., 28(1):183-197, 2015.Search in Google Scholar
[9] Xiuxiong Chen, Simon Donaldson, and Song Sun. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. J. Amer. Math. Soc., 28(1):199-234, 2015.Search in Google Scholar
[10] Xiuxiong Chen and Yuanqi Wang. On the regularity problem of complex Monge-Ampere equations with conical singularities. arXiv preprint arXiv:1405.1021, 2014.Search in Google Scholar
[11] Ronan J Conlon, Hans-Joachim Hein, et al. Asymptotically conical Calabi-Yau manifolds, i. Duke Mathematical Journal, 162(15):2855-2902, 2013.10.1215/00127094-2382452Search in Google Scholar
[12] Martin de Borbon. Asymptotically conical Ricci-flat Kähler metrics with cone singularities. PhD thesis, Imperial College, London, UK, 2015.Search in Google Scholar
[13] S. K. Donaldson. Kähler metrics with cone singularities along a divisor. In Essays in mathematics and its applications, pages 49-79. Springer, Heidelberg, 2012.10.1007/978-3-642-28821-0_4Search in Google Scholar
[14] Simon Donaldson and Song Sun. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, ii. to appear in J. Differential Geom.Search in Google Scholar
[15] Simon Donaldson and Song Sun. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. Acta Math., 213(1):63-106, 2014.10.1007/s11511-014-0116-3Search in Google Scholar
[16] Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi. Singular Kähler-Einstein metrics. J. Amer. Math. Soc., 22(3):607-639, 2009.10.1090/S0894-0347-09-00629-8Search in Google Scholar
[17] Patricio Gallardo and Jesus Martinez-Garcia. Moduli of cubic surfaces and their anticanonical divisors. arXiv preprint arXiv:1607.03697, 2016.Search in Google Scholar
[18] Henri Guenancia and Mihai P˘aun. Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors. J. Differential Geom., 103(1):15-57, 2016.Search in Google Scholar
[19] Richard S. Hamilton. Three-manifolds with positive Ricci curvature. J. Differential Geom., 17(2):255-306, 1982.10.4310/jdg/1214436922Search in Google Scholar
[20] Hans-Joachim Hein and Song Sun. Calabi-Yau manifolds with isolated conical singularities. arXiv preprint arXiv:1607.02940, 2016.Search in Google Scholar
[21] Friedrich Hirzebruch. Algebraic surfaces with extreme Chern numbers (report on the thesis of Th. Höfer, Bonn 1984). Russian Mathematical Surveys, 40(4):135-145, 1985.10.1070/RM1985v040n04ABEH003620Search in Google Scholar
[22] Thalia Jeffres, Rafe Mazzeo, and Yanir A. Rubinstein. Kähler-Einstein metrics with edge singularities. Ann. of Math. (2), 183(1):95-176, 2016.10.4007/annals.2016.183.1.3Search in Google Scholar
[23] Dominic D Joyce. Compact manifolds with special holonomy. Oxford University Press, 2000.Search in Google Scholar
[24] János Kollár. Singularities of pairs. In Algebraic geometry-Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., pages 221-287. Amer. Math. Soc., Providence, RI, 1997.10.1090/pspum/062.1/1492525Search in Google Scholar
[25] P. B. Kronheimer and T. S. Mrowka. Gauge theory for embedded surfaces. I. Topology, 32(4):773-826, 1993.10.1016/0040-9383(93)90051-VSearch in Google Scholar
[26] Gustav I. Lehrer and Donald E. Taylor. Unitary reflection groups, volume 20 of Australian Mathematical Society Lecture Series. Cambridge University Press, Cambridge, 2009.Search in Google Scholar
[27] Chi Li. Remarks on logarithmic K-stability. Commun. Contemp. Math., 17(2):1450020, 17, 2015.10.1142/S0219199714500205Search in Google Scholar
[28] Chi Li and Song Sun. Conical Kähler-Einstein metrics revisited. Comm. Math. Phys., 331(3):927-973, 2014.10.1007/s00220-014-2123-9Search in Google Scholar
[29] Zhong-Dong Liu and Zhongmin Shen. Riemannian geometry of conical singular sets. Ann. Global Anal. Geom., 16(1):29-62, 1998.10.1023/A:1006597812394Search in Google Scholar
[30] Feng Luo and Gang Tian. Liouville equation and spherical convex polytopes. Proc. Amer. Math. Soc., 116(4):1119-1129, 1992.10.1090/S0002-9939-1992-1137227-5Search in Google Scholar
[31] John Milnor. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968.Search in Google Scholar
[32] Gabriele Mondello and Dmitri Panov. Spherical Metrics with Conical Singularities on a 2-Sphere: Angle Constraints. Int. Math. Res. Not. IMRN, (16):4937-4995, 2016. Search in Google Scholar
[33] Yuji Odaka, Cristiano Spotti, and Song Sun. Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics. J. Differential Geom., 102(1):127-172, 2016.10.4310/jdg/1452002879Search in Google Scholar
[34] Peter Orlik. Seifert manifolds. Lecture Notes in Mathematics, Vol. 291. Springer-Verlag, Berlin-New York, 1972.10.1007/BFb0060329Search in Google Scholar
[35] Dmitri Panov. Polyhedral Kähler manifolds. Geom. Topol., 13(4):2205-2252, 2009.10.2140/gt.2009.13.2205Search in Google Scholar
[36] Dmitri Panov. Real line arrangements with Hirzebruch property. arXiv preprint arXiv:1607.07709, 2016.Search in Google Scholar
[37] Jian Song and Xiaowei Wang. The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality. Geom. Topol., 20(1):49-102, 2016.10.2140/gt.2016.20.49Search in Google Scholar
[38] James Sparks. Sasaki-Einstein manifolds. In Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, volume 16 of Surv. Differ. Geom., pages 265-324. Int. Press, Somerville, MA, 2011.10.4310/SDG.2011.v16.n1.a6Search in Google Scholar
[39] Gábor Székelyhidi. A remark on conical Kähler-Einstein metrics. Math. Res. Lett., 20(3):581-590, 2013.10.4310/MRL.2013.v20.n3.a14Search in Google Scholar
[40] Gang Tian. Kähler-Einstein metrics on algebraic manifolds. In Transcendental methods in algebraic geometry (Cetraro, 1994), volume 1646 of Lecture Notes in Math., pages 143-185. Springer, Berlin, 1996.10.1007/BFb0094304Search in Google Scholar
[41] Marc Troyanov. Metrics of constant curvature on a sphere with two conical singularities. In Differential geometry (Peñíscola, 1988), volume 1410 of Lecture Notes in Math., pages 296-306. Springer, Berlin, 1989.10.1007/BFb0086431Search in Google Scholar
[42] Marc Troyanov. Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc., 324(2):793-821, 1991.10.1090/S0002-9947-1991-1005085-9Search in Google Scholar
[43] Hao Yin and Kai Zheng. Expansion formula for complex Monge-Ampere equation along cone singularities. arXiv preprint arXiv:1609.03111, 2016.Search in Google Scholar
© 2017
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
