Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1291-1330.

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 XSupp(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).

Soit X une variété compacte kählerienne et Δ un -diviseur dont le support est à croisements normaux simples et à coefficients entre 1/2 et 1. En supposant K X +Δ ample, on prouve l’existence et l’unicité d’une métrique de Kähler-Einstein à courbure négative sur XSupp(Δ) ayant des singularités mixtes Poincaré et coniques suivant les coefficients de Δ. Nous appliquons ensuite ce résultat pour prouver un théorème d’annulation concernant certains champs de tenseurs holomorphes naturellement attachés à la paire (X,Δ).

DOI: 10.5802/aif.2881
Classification: 32Q05, 32Q10, 32Q15, 32Q20, 32U05, 32U15
Keywords: Kähler-Einstein metrics, cone singularities, Poincaré singularities, cusps, orbifold tensors, complex Monge-Ampère equation
Mot clés : métriques de Kähler-Einstein, singularités coniques, singularités Poincaré, cusps, tenseurs orbifoldes, équation de Monge-Ampère complexe
Guenancia, Henri 1

1 Université Pierre et Marie Curie Institut de Mathématiques de Jussieu, Paris & École Normale Supérieure Département de Mathématiques et Applications Paris (France)
@article{AIF_2014__64_3_1291_0,
     author = {Guenancia, Henri},
     title = {K\"ahler-Einstein metrics with mixed {Poincar\'e} and cone singularities along a normal crossing divisor},
     journal = {Annales de l'Institut Fourier},
     pages = {1291--1330},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {3},
     year = {2014},
     doi = {10.5802/aif.2881},
     zbl = {06387308},
     mrnumber = {3330171},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2881/}
}
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
SP  - 1291
EP  - 1330
VL  - 64
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2881/
DO  - 10.5802/aif.2881
LA  - en
ID  - AIF_2014__64_3_1291_0
ER  - 
%0 Journal Article
%A Guenancia, Henri
%T Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor
%J Annales de l'Institut Fourier
%D 2014
%P 1291-1330
%V 64
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2881/
%R 10.5802/aif.2881
%G en
%F AIF_2014__64_3_1291_0
Guenancia, Henri. Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1291-1330. doi : 10.5802/aif.2881. http://www.numdam.org/articles/10.5802/aif.2881/

[1] Auvray, Hugues The space of Poincaré type Kähler metrics on the complement of a divisor (2011) (arXiv:1109.3159)

[2] Bedford, E.; Taylor, B.A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982) no. 1-2, pp. 1-40 | MR | Zbl

[3] Benelkourchi, Slimane; Guedj, Vincent; Zeriahi, Ahmed A priori estimates for weak solutions of complex Monge-Ampère equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 7 (2008) no. 1, pp. 81-96 | Numdam | MR | Zbl

[4] Berman, R.; Boucksom, S.; Eyssidieux, Ph.; Guedj, V.; Zeriahi, A. Kähler-Einstein metrics and the Kähler-Ricci flow on log-Fano varieties (2011) (arXiv:1111.7158v2)

[5] Berman, Robert J. A thermodynamical formalism for Monge-Ampèe equations, Moser-Trudinger inequalities and Kähler-Einstein metrics, Adv. Math., Volume 248 (2013), pp. 1254-1297 | MR | Zbl

[6] Błocki, Zbigniew The Calabi-Yau theorem, Complex Monge-Ampère equations and geodesics in the space of Kähler metrics (Lecture Notes in Math.), Volume 2038, Springer, Heidelberg, 2012, pp. 201-227 | MR | Zbl

[7] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge-Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | MR | Zbl

[8] Brendle, Simon Ricci flat Kähler metrics with edge singularities, Int. Math. Res. Not. IMRN (2013) no. 24, pp. 5727-5766 | MR | Zbl

[9] Campana, F. Orbifoldes spéciales et classification biméromorphe des variétés kähleriennes compactes (2009) (arXiv:0705.0737)

[10] Campana, F. Special orbifolds and birational classification: a survey (2010) (arXiv:1001.3763) | MR | Zbl

[11] Campana, Frédéric; Guenancia, Henri; Păun, Mihai Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér. (4), Volume 46 (2013) no. 6, pp. 879-916 | EuDML | MR | Zbl

[12] Carlson, James; Griffiths, Phillip A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math., Volume 95 (1972), pp. 557-584 | MR | Zbl

[13] Cheng, Shiu-Yuen; Yau, Shing-Tung On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation, Commun. Pure Appl. Math., Volume 33 (1980), pp. 507-544 | MR | Zbl

[14] Claudon, Benoît Γ-reduction for smooth orbifolds, Manuscripta Math., Volume 127 (2008) no. 4, pp. 521-532 | MR | Zbl

[15] Demailly, Jean-Pierre Potential theory in several complex variables (Lecture given at the CIMPA in 1989, completed by a conference given in Trento, 1992; avalaible at the author’s webpage: http://www-fourier.ujf-grenoble.fr/~demailly/books.html)

[16] Demailly, Jean-Pierre Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 285-360 | MR | Zbl

[17] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977, pp. x+401 (Grundlehren der Mathematischen Wissenschaften, Vol. 224) | MR | Zbl

[18] Griffiths, Phillip A. Entire holomorphic mappings in one and several complex variables, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976, pp. x+99 (The fifth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N. J., October and November 1974, Annals of Mathematics Studies, No. 85) | MR | Zbl

[19] Guedj, V.; Zeriahi, A. The weighted Monge-Ampère energy of quasi plurisubharmonic functions, J. Funct. An., Volume 250 (2007), pp. 442-482 | MR | Zbl

[20] Guedj, Vincent; Zeriahi, Ahmed Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal., Volume 15 (2005) no. 4, pp. 607-639 | MR | Zbl

[21] Jeffres, T. Uniqueness of Kähler-Einstein cone metrics, Publ. Mat. 44, Volume 44 (2000) no. 2, pp. 437-448 | EuDML | MR | Zbl

[22] Jeffres, Thalia; Mazzeo, Rafe; Rubinstein, Yanir Kähler-Einstein metrics with edge singularities (2011) (arXiv:1105.5216, with an appendix by C. Li and Y. Rubinstein)

[23] Kobayashi, R. Kähler-Einstein metric on an open algebraic manifolds, Osaka 1. Math., Volume 21 (1984), pp. 399-418 | MR | Zbl

[24] Kołodziej, S. The complex Monge-Ampère operator, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | MR | Zbl

[25] Kołodziej, S. Stability of solutions to the complex Monge-Ampère equations on compact Kähler manifolds (2001) (Preprint)

[26] Mazzeo, R. Kähler-Einstein metrics singular along a smooth divisor, Journées "Équations aux dérivées partielles" (Saint Jean-de-Monts, 1999) (1999), pp. Exp. VI, 10 | EuDML | MR | Zbl

[27] Siu, Yum Tong Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar, 8, Birkhäuser Verlag, Basel, 1987, pp. 171 | MR | Zbl

[28] Tian, G.; Yau, S.-T. Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986) (Adv. Ser. Math. Phys.), Volume 1, World Sci. Publishing, Singapore, 1987, pp. 574-628 | MR | Zbl

[29] Yau, Shing-Tung A general Schwarz lemma for Kähler manifolds, Amer. J. Math., Volume 100 (1978), pp. 197-203 | MR | Zbl

Cited by Sources: