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

top
Let X be a compact Kähler manifold and Δ be a -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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.