Kähler–Einstein metrics with edge singularities

Abstract

This article considers the existence and regularity of Kähler–Einstein metrics on a compact Kähler manifold $M$ with edge singularities with cone angle $2\pi \beta$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi \beta \leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi \beta < 2\pi$.

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      mrclass = {32L07 (14J45 53C25)},
      mrnumber = {1163733},
      mrreviewer = {Jaros{\l}aw A. Wi{\'s}niewski},
      doi = {10.1142/S0129167X92000175},
      zblnumber = {0779.53009},
      }
  • [T1994] G. Tian, "The $K$-energy on hypersurfaces and stability," Comm. Anal. Geom., vol. 2, iss. 2, pp. 239-265, 1994.
    @article{T1994, mrkey = {1312688},
      author = {Tian, Gang},
      title = {The {$K$}-energy on hypersurfaces and stability},
      journal = {Comm. Anal. Geom.},
      fjournal = {Communications in Analysis and Geometry},
      volume = {2},
      year = {1994},
      number = {2},
      pages = {239--265},
      issn = {1019-8385},
      mrclass = {32G13 (32L07 53C07 53C25 58G26)},
      mrnumber = {1312688},
      mrreviewer = {Kimio Miyajima},
      zblnumber = {0846.32019},
      }
  • [T] Go to document G. Tian, "Kähler-Einstein metrics on algebraic manifolds," in Transcendental Methods in Algebraic Geometry, New York: Springer-Verlag, 1996, vol. 1646, pp. 143-185.
    @incollection{T, mrkey = {1603624},
      author = {Tian, Gang},
      title = {Kähler-{E}instein metrics on algebraic manifolds},
      booktitle = {Transcendental Methods in Algebraic Geometry},
      venue = {{C}etraro, 1994},
      series = {Lecture Notes in Math.},
      volume = {1646},
      pages = {143--185},
      publisher = {Springer-Verlag},
      year = {1996},
      mrclass = {32L07 (53C25 53C55)},
      mrnumber = {1603624},
      mrreviewer = {Thalia D. Jeffres},
      doi = {10.1007/BFb0094304},
      address = {New York},
      zblnumber = {0896.32003},
      }
  • [T97] Go to document G. Tian, "Kähler-Einstein metrics with positive scalar curvature," Invent. Math., vol. 130, iss. 1, pp. 1-37, 1997.
    @article{T97, mrkey = {1471884},
      author = {Tian, Gang},
      title = {Kähler-{E}instein metrics with positive scalar curvature},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {130},
      year = {1997},
      number = {1},
      pages = {1--37},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {53C25 (32L07 53C55)},
      mrnumber = {1471884},
      mrreviewer = {Thalia D. Jeffres},
      doi = {10.1007/s002220050176},
      zblnumber = {0892.53027},
      }
  • [Tbook] Go to document G. Tian, Canonical Metrics in Kähler Geometry, Birkhäuser, Basel, 2000.
    @book{Tbook, mrkey = {1787650},
      author = {Tian, Gang},
      title = {Canonical Metrics in {K}ähler Geometry},
      series = {Lectures in Math. ETH Zürich},
      publisher = {Birkhäuser, Basel},
      year = {2000},
      pages = {vi+101},
      isbn = {3-7643-6194-8},
      mrclass = {32Q20 (53C25 58E11)},
      mrnumber = {1787650},
      mrreviewer = {Azniv Kasparian},
      doi = {10.1007/978-3-0348-8389-4},
      zblnumber = {0978.53002},
      }
  • [T12] G. Tian, "K-stability and Kähler-Einstein metrics," Comm. Pure Appl. Math., vol. 68, iss. 7, pp. 1085-1156, 2015.
    @article{T12, mrkey = {3352459},
      author = {Tian, Gang},
      title = {K-stability and {K}ähler-{E}instein metrics},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {68},
      year = {2015},
      number = {7},
      pages = {1085--1156},
      issn = {0010-3640},
      mrclass = {53C55 (53C25)},
      mrnumber = {3352459},
      }
  • [Tian-unpublished] G. Tian, A 3rd derivative estimate for conic Kähler metrics.
    @misc{Tian-unpublished,
      author = {Tian, Gang},
      title = {A 3rd derivative estimate for conic {K}ähler metrics},
      note = {unpublished notes},
      sortyear = {2020},
      }
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    @incollection{TY, mrkey = {0915840},
      author = {Tian, Gang and Yau, S.-T.},
      title = {Existence of {K}ähler-{E}instein metrics on complete {K}ähler manifolds and their applications to algebraic geometry},
      booktitle = {Mathematical Aspects of String Theory},
      venue = {{S}an {D}iego, {C}alif., 1986},
      series = {Adv. Ser. Math. Phys.},
      volume = {1},
      pages = {574--628},
      publisher = {World Sci. Publishing},
      address = {Singapore},
      year = {1987},
      mrclass = {32F99 (14J99 53C25 53C55 58E15 58G99)},
      mrnumber = {0915840},
      }
  • [TY1990] Go to document G. Tian and S. Yau, "Complete Kähler manifolds with zero Ricci curvature. I," J. Amer. Math. Soc., vol. 3, iss. 3, pp. 579-609, 1990.
    @article{TY1990, mrkey = {1040196},
      author = {Tian, Gang and Yau, Shing-Tung},
      title = {Complete {K}ähler manifolds with zero {R}icci curvature. {I}},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {3},
      year = {1990},
      number = {3},
      pages = {579--609},
      issn = {0894-0347},
      mrclass = {53C55 (32C10 53C25)},
      mrnumber = {1040196},
      mrreviewer = {Alan Michael Nadel},
      doi = {10.2307/1990928},
      zblnumber = {0719.53041},
      }
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    @article{Tr, mrkey = {1005085},
      author = {Troyanov, Marc},
      title = {Prescribing curvature on compact surfaces with conical singularities},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {324},
      year = {1991},
      number = {2},
      pages = {793--821},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {53C45 (30F10 58G30)},
      mrnumber = {1005085},
      mrreviewer = {Robert McOwen},
      doi = {10.2307/2001742},
      zblnumber = {0724.53023},
      }
  • [Wu] Go to document D. Wu, "Kähler-Einstein metrics of negative Ricci curvature on general quasi-projective manifolds," Comm. Anal. Geom., vol. 16, iss. 2, pp. 395-435, 2008.
    @article{Wu, mrkey = {2425471},
      author = {Wu, Damin},
      title = {Kähler-{E}instein metrics of negative {R}icci curvature on general quasi-projective manifolds},
      journal = {Comm. Anal. Geom.},
      fjournal = {Communications in Analysis and Geometry},
      volume = {16},
      year = {2008},
      number = {2},
      pages = {395--435},
      issn = {1019-8385},
      mrclass = {32Q20 (32W20 53C25)},
      mrnumber = {2425471},
      mrreviewer = {Julien Keller},
      doi = {10.4310/CAG.2008.v16.n2.a4},
      zblnumber = {1151.32009},
      }
  • [Y1] Go to document S. T. Yau, "A general Schwarz lemma for Kähler manifolds," Amer. J. Math., vol. 100, iss. 1, pp. 197-203, 1978.
    @article{Y1, mrkey = {0486659},
      author = {Yau, Shing Tung},
      title = {A general {S}chwarz lemma for {K}ähler manifolds},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {100},
      year = {1978},
      number = {1},
      pages = {197--203},
      issn = {0002-9327},
      mrclass = {32H25 (53C55)},
      mrnumber = {0486659},
      mrreviewer = {Samuel I. Goldberg},
      doi = {10.2307/2373880},
      zblnumber = {0424.53040},
      }
  • [Y2] Go to document S. T. Yau, "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I," Comm. Pure Appl. Math., vol. 31, iss. 3, pp. 339-411, 1978.
    @article{Y2, mrkey = {0480350},
      author = {Yau, Shing Tung},
      title = {On the {R}icci curvature of a compact {K}ähler manifold and the complex {M}onge-{A}mpère equation. {I}},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {31},
      year = {1978},
      number = {3},
      pages = {339--411},
      issn = {0010-3640},
      coden = {CPAMAT},
      mrclass = {53C55 (32C10 35J60)},
      mrnumber = {0480350},
      mrreviewer = {Robert E. Greene},
      doi = {10.1002/cpa.3160310304},
      zblnumber = {0369.53059},
      }

Authors

Thalia Jeffres

Wichita State University, Wichita, KS

Rafe Mazzeo

Stanford University, Stanford, CA

Yanir A. Rubinstein

University of Maryland, College Park, MD