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Geometric Constructions of Extremal Metrics on Complex Manifolds

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Trends in Contemporary Mathematics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 8))

Abstract

In this note we review recent progresses on the existence problem of Kähler constant scalar curvature metrics on complex manifolds. The content of this note is an expanded version of author’s talk “Constant curvature metrics on algebraic manifolds” at the Giornata Indam at L’Aquila on June 9th 2011.

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References

  1. C. Arezzo, A. Della Vedova, G. La Nave, Geometric flows and Kähler reduction. J. Symplectic Geom. (to appear)

    Google Scholar 

  2. C. Arezzo, F. Pacard, Blowing up and desingularizing Kähler orbifolds with constant scalar curvature. Acta Math. 196(2), 179–228 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Arezzo, F. Pacard, Blowing up Kähler manifolds with constant scalar curvature. II. Ann. Math. 170(2), 685–738 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Arezzo, A. Ghigi, G.P. Pirola, Symmetries, quotients and Kähler-Einstein metrics. Crelle’s J. 157(1), 1–51 (2006)

    Google Scholar 

  5. C. Arezzo, F. Pacard, M. Singer, Extremal metrics on blowups. Duke Math. J. 157(1), 1–51 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Bando, T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, in Algebraic Geometry, Sendai, 1985, ed. by T. Oda. Advanced Studies in Pure Mathematics, vol. 10, 1987

    Google Scholar 

  7. O. Biquard, Y. Rollin, Smoothing singular extremal Kähler surfaces and minimal Lagrangians (pre-print). arXiv:1211.6957

    Google Scholar 

  8. S. Bouksom, P. Eyssidieux, V. Guedj, A. Zerihai, Monge-Ampère equations in big cohomology classes. Acta Math. 205, 199–262 (2010)

    Article  MathSciNet  Google Scholar 

  9. D. Burns, P. de Bartolomeis, Stability of vector bundles and extremal metrics. Invent. Math. 92, 403–407 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Calabi, Métriques kählériennes et fibrés holomorphes. Ann. Sci. École Norm. Sup. 4 12(2), 269–294 (1979)

    Google Scholar 

  11. E. Calabi, Extremal Kähler metrics, in Seminar on Differential Geometry, ed. by S.-T. Yau. Annals of Mathematics Studies, vol. 102 (Princeton University Press, Princeton, 1982), pp. 259–290

    Google Scholar 

  12. E. Calabi, Extremal Kähler metrics II, in Differential Geometry and Its Complex Analysis, ed. by I. Chavel, H.M. Farkas (Springer, Berlin/Heidelberg, 1985)

    Google Scholar 

  13. D. Calderbank, M. Singer, Einstein metrics and complex singularities. Invent. Math. 156(2), 405–443 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. È. Cartan, Sur une classe remarquable d’espaces de Riemann, II. Bulletin de la Socit́ Ḿathḿatique de France 55, 114–134 (1927)

    Google Scholar 

  15. I. Cheltsov, K.A. Shramov, Log canonical thresholds of smooth Fano threefolds, with an appendix by J.P. Demailly. Russ. Math. Surv. 63(5), 859–958 (2008)

    Google Scholar 

  16. X.X. Chen, G. Tian, Geometry of Kähler metrics and holomorphic foliation by discs. Publ. Math. Inst. Hautes Études Sci. 107, 1–107 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. X.X. Chen, S.K. Donaldson, S. Sun, Kähler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities. http://arxiv.org/pdf/1211.4566.pdf

  18. X.X. Chen, S.K. Donaldson, S. Sun, Kähler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2π. http://arxiv.org/pdf/1212.4714.pdf

  19. X.X. Chen, S.K. Donaldson, S. Sun, Kähler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof. http://arxiv.org/pdf/1302.0282.pdf

  20. X.X. Chen, C. LeBrun, B. Weber, On conformally Kähler, Einstein manifolds. J. Am. Math. Soc. 21(4), 1137–1168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. A. DellaVedova, CM-stability of blow-ups and canonical metrics. arXiv:0810.5584

    Google Scholar 

  22. S.K. Donaldson, Scalar curvature and projective embeddings I. J. Differ. Geom. 59(3), 479–522 (2001)

    MATH  MathSciNet  Google Scholar 

  23. S.K. Donaldson, Lower bounds on the Calabi functional. J. Differ. Geom. 70, 453–472 (2005)

    MATH  MathSciNet  Google Scholar 

  24. S.K. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, in Handbook of Geometric Analysis, No. 1, ed. by L. Lin, P. Li, R.M. Schoen, L. Simon. Advanced Lectures in Mathematics (ALM), vol. 7 (International Press, Somerville, 2008), pp. 29–75

    Google Scholar 

  25. A. Futaki, Kähler-Einstein Metrics and Integral Invariants. Lecture Notes in Mathematics, vol. 1314 (Springer, Berlin/Heidelberg, 1988)

    Google Scholar 

  26. D. Joyce, Compact Manifolds with Special Holonomy (Oxford University Press, Oxford/ New York, 2000)

    MATH  Google Scholar 

  27. J. Kim, M. Pontecorvo, A new method of constructing scalar-flat Kähler surfaces. J. Differ. Geom. 41(2), 449–477 (1995)

    MATH  MathSciNet  Google Scholar 

  28. P. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29(3), 665–683 (1989)

    MATH  MathSciNet  Google Scholar 

  29. C. LeBrun, Scalar-flat Kähler metrics on blown-up ruled surfaces. J. Reine Angew. Math. 420, 161–177 (1991)

    MATH  MathSciNet  Google Scholar 

  30. C. LeBrun, On the scalar curvature of complex surfaces. Geom. Funct. Anal. 5, 619–628 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. C. LeBrun, Einstein metrics, four-manifolds, and differential topology, in Surveys in Differential Geometry, Boston, 2002. Surveys in Differential Geometry, vol. VIII (International Press, Somerville, 2003), pp. 235–255

    Google Scholar 

  32. C. LeBrun, S. Simanca, Extremal Kähler metrics and complex deformation theory. Geom. Funct. Anal. 4, 298–336 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. C. LeBrun, M. Singer, Existence and deformation theory for scalar flat Kähler metrics on compact complex surfaces. Invent. Math. 112, 273–313 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  34. R. Lena, On the desingularization of Kähler orbifolds with constant scalar curvature. PhD thesis, SISSA, 2013

    Google Scholar 

  35. M. Levine, A remark on extremal Kähler metrics. J. Differ. Geom. 21(1), 73–77 (1985)

    MATH  Google Scholar 

  36. A. Lichnerowicz, Sur les transformations analytiques des variétés kählériennes compactes. C. R. Acad. Sci. Paris 244, 3011–3013 (1957)

    MATH  MathSciNet  Google Scholar 

  37. T. Mabuchi, Einstein-Kähler forms, Futaki invariants and convex geometry on toric Fano varieties. Osaka J. Math. 24(4), 705–737 (1987)

    MATH  MathSciNet  Google Scholar 

  38. T. Mabuchi, An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I. Invent. Math. 159, 225–243 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. Y. Matsushima, Sur la structure du groupe d’homomorphismes analytiques d’une certaine varité kählériennee. Nagoya Math. J. 11, 145–150 (1957)

    MATH  MathSciNet  Google Scholar 

  40. S.S. Roan, Minimal resolution of Gorenstein orbifolds. Topology 35, 489–508 (1971)

    Article  MathSciNet  Google Scholar 

  41. Y. Rollin, M. Singer, Construction of Kaehler surfaces with constant scalar curvature. J. Eur. Math. Soc. 11(5), 979–997 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984)

    MATH  MathSciNet  Google Scholar 

  43. S. Simanca, Kähler metrics of constant scalar curvature on bundles over CP n−1. Math. Ann. 291(2), 239–246 (1991)

    Article  MathSciNet  Google Scholar 

  44. Y.T. Siu, The existence of Kähler-Einstein metrics on manifolds with positive anticanonnical line bundle and a suitably symmetry group. Ann. Math. 127, 585–627 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  45. C. Spotti, Deformations of Nodal Kähler-Einstein Del Pezzo surfaces with finite automorphism groups. J. Lond. Math. Soc. (2014). doi:10.1112/jlms/jdt076

    Google Scholar 

  46. J. Stoppa, K-stability of constant scalar curvature Kähler manifolds. Adv. Math. 221(4), 1397–1408 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  47. J. Stoppa, Unstable blowups. J. ALgebr. Geom. 19, 1–17 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  48. J. Stoppa, G. Szèkelyhidi, Relative K-stability of extremal metrics. J. Eur. Math. Soc. 13(4), 899–909 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  49. G. Szèkelyhidi, Extremal metrics and K-stability. Bull. Lond. Math. Soc. 39(1), 76–84 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  50. G. Szèkelyhidi, The Kähler-Ricci flow and K-polystability. Am. J. Math. 132(4), 1077–1090 (2010)

    Article  MATH  Google Scholar 

  51. G. Szèkelyhidi, On blowing up extremal Kähler manifolds. Duke Math. J. 161(8), 1411–1453 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  52. G. Tian, On Calabi’s conjectures for complex surfaces with positive first Chern class. Invent. Math. 101(1), 101–172 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  53. G. Tian, On Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  54. G. Tian, Canonical Metrics on Kähler Manifolds (Birkhauser, Boston, 1999)

    Google Scholar 

  55. G. Tian, Extremal metrics and geometric stability. Houst. J. Math. 28(2), 411–431 (2002)

    MATH  Google Scholar 

  56. G. Tian, K-stability and Kähler-Einstein metrics. http://arxiv.org/pdf/1211.4669v2.pdf

  57. X.-J. Wang, X.-H. Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Claudio Arezzo .

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Arezzo, C. (2014). Geometric Constructions of Extremal Metrics on Complex Manifolds. In: Ancona, V., Strickland, E. (eds) Trends in Contemporary Mathematics. Springer INdAM Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-05254-0_17

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