Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Henri Guenancia[1]

  • [1] Université Pierre et Marie Curie Institut de Mathématiques de Jussieu, Paris & École Normale Supérieure Département de Mathématiques et Applications Paris (France)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 3, page 1291-1330
  • ISSN: 0373-0956

Abstract

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Let X be a compact Kähler manifold and Δ be a ℝ -divisor with simple normal crossing support and coefficients between 1 / 2 and 1 . Assuming that K X + Δ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X ∖ Supp ( Δ ) having mixed Poincaré and cone singularities according to the coefficients of Δ . As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair ( X , Δ ) .

How to cite

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Guenancia, Henri. "Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor." Annales de l’institut Fourier 64.3 (2014): 1291-1330. <http://eudml.org/doc/275540>.

@article{Guenancia2014,
abstract = {Let $X$ be a compact Kähler manifold and $\Delta $ be a $\mathbb\{R\}$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta $ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \textrm\{Supp\}(\Delta )$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta $. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.},
affiliation = {Université Pierre et Marie Curie Institut de Mathématiques de Jussieu, Paris & École Normale Supérieure Département de Mathématiques et Applications Paris (France)},
author = {Guenancia, Henri},
journal = {Annales de l’institut Fourier},
keywords = {Kähler-Einstein metrics; cone singularities; Poincaré singularities; cusps; orbifold tensors; complex Monge-Ampère equation; cone singularities, Poincaré metric, complex Monge-Ampère equation},
language = {eng},
number = {3},
pages = {1291-1330},
publisher = {Association des Annales de l’institut Fourier},
title = {Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor},
url = {http://eudml.org/doc/275540},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Guenancia, Henri
TI - Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 1291
EP - 1330
AB - Let $X$ be a compact Kähler manifold and $\Delta $ be a $\mathbb{R}$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta $ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \textrm{Supp}(\Delta )$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta $. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.
LA - eng
KW - Kähler-Einstein metrics; cone singularities; Poincaré singularities; cusps; orbifold tensors; complex Monge-Ampère equation; cone singularities, Poincaré metric, complex Monge-Ampère equation
UR - http://eudml.org/doc/275540
ER -

References

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  1. Hugues Auvray, The space of Poincaré type Kähler metrics on the complement of a divisor, (2011) Zbl1268.53077
  2. E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40 Zbl0547.32012MR674165
  3. Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi, A priori estimates for weak solutions of complex Monge-Ampère equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 7 (2008), 81-96 Zbl1150.32011MR2413673
  4. R. Berman, S. Boucksom, Ph. Eyssidieux, V. Guedj, A. Zeriahi, Kähler-Einstein metrics and the Kähler-Ricci flow on log-Fano varieties, (2011) 
  5. Robert J. Berman, A thermodynamical formalism for Monge-Ampèe equations, Moser-Trudinger inequalities and Kähler-Einstein metrics, Adv. Math. 248 (2013), 1254-1297 Zbl1286.58010MR3107540
  6. Zbigniew Błocki, The Calabi-Yau theorem, Complex Monge-Ampère equations and geodesics in the space of Kähler metrics 2038 (2012), 201-227, Springer, Heidelberg Zbl1231.32017MR2932444
  7. Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi, Monge-Ampère equations in big cohomology classes, Acta Math. 205 (2010), 199-262 Zbl1213.32025MR2746347
  8. Simon Brendle, Ricci flat Kähler metrics with edge singularities, Int. Math. Res. Not. IMRN (2013), 5727-5766 Zbl1293.32029MR3144178
  9. F. Campana, Orbifoldes spéciales et classification biméromorphe des variétés kähleriennes compactes, (2009) Zbl1236.14039
  10. F. Campana, Special orbifolds and birational classification: a survey , (2010) Zbl1229.14011MR2779470
  11. Frédéric Campana, Henri Guenancia, Mihai Păun, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 879-916 Zbl1310.32029MR3134683
  12. James Carlson, Phillip Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557-584 Zbl0248.32018MR311935
  13. Shiu-Yuen Cheng, Shing-Tung Yau, On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation, Commun. Pure Appl. Math. 33 (1980), 507-544 Zbl0506.53031MR575736
  14. Benoît Claudon, Γ -reduction for smooth orbifolds, Manuscripta Math. 127 (2008), 521-532 Zbl1163.14020MR2457193
  15. Jean-Pierre Demailly, Potential theory in several complex variables Zbl1296.01027
  16. Jean-Pierre Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry—Santa Cruz 1995 62 (1997), 285-360, Amer. Math. Soc., Providence, RI Zbl0919.32014MR1492539
  17. David Gilbarg, Neil S. Trudinger, Elliptic partial differential equations of second order, (1977), Springer-Verlag, Berlin-New York Zbl0562.35001MR473443
  18. Phillip A. Griffiths, Entire holomorphic mappings in one and several complex variables, (1976), Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo Zbl0317.32023MR447638
  19. V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasi plurisubharmonic functions, J. Funct. An. 250 (2007), 442-482 Zbl1143.32022MR2352488
  20. Vincent Guedj, Ahmed Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), 607-639 Zbl1087.32020MR2203165
  21. T. Jeffres, Uniqueness of Kähler-Einstein cone metrics, Publ. Mat. 44 44 (2000), 437-448 Zbl0981.32015MR1800816
  22. Thalia Jeffres, Rafe Mazzeo, Yanir Rubinstein, Kähler-Einstein metrics with edge singularities, (2011) Zbl1337.32037
  23. R. Kobayashi, Kähler-Einstein metric on an open algebraic manifolds, Osaka 1. Math. 21 (1984), 399-418 Zbl0582.32011MR752470
  24. S. Kołodziej, The complex Monge-Ampère operator, Acta Math. 180 (1998), 69-117 Zbl0913.35043
  25. S. Kołodziej, Stability of solutions to the complex Monge-Ampère equations on compact Kähler manifolds, (2001) 
  26. R. Mazzeo, Kähler-Einstein metrics singular along a smooth divisor, Journées "Équations aux dérivées partielles" (Saint Jean-de-Monts, 1999) (1999) Zbl1009.32013MR1718970
  27. Yum Tong Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, 8 (1987), Birkhäuser Verlag, Basel Zbl0631.53004MR904673
  28. G. Tian, S.-T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986) 1 (1987), 574-628, World Sci. Publishing, Singapore Zbl0682.53064MR915812
  29. Shing-Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), 197-203 Zbl0424.53040MR486659

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