Complex analysis
Weak solutions to complex Monge–Ampère equations on compact Kähler manifolds
[Solutions faibles des équations de Monge–Ampère complexes sur des variétés de Kähler compactes]
Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 589-592.

Voir la notice de l'article provenant de la source Numdam

We show a general existence theorem of solutions to the complex Monge–Ampère type equation on compact Kähler manifolds.

Nous montrons un théorème général d'existence et d'unicité de solution d'une équation de type Monge–Ampère complexe sur des variétés de Kähler compactes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.06.003

Benelkourchi, Slimane 1

1 Université de Montréal, Pavillon 3744, rue Jean-Brillant, Montréal QC H3C 3J7, Canada
@article{CRMATH_2014__352_7-8_589_0,
     author = {Benelkourchi, Slimane},
     title = {Weak solutions to complex {Monge{\textendash}Amp\`ere} equations on compact {K\"ahler} manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {589--592},
     publisher = {Elsevier},
     volume = {352},
     number = {7-8},
     year = {2014},
     doi = {10.1016/j.crma.2014.06.003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2014.06.003/}
}
TY  - JOUR
AU  - Benelkourchi, Slimane
TI  - Weak solutions to complex Monge–Ampère equations on compact Kähler manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 589
EP  - 592
VL  - 352
IS  - 7-8
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2014.06.003/
DO  - 10.1016/j.crma.2014.06.003
LA  - en
ID  - CRMATH_2014__352_7-8_589_0
ER  - 
%0 Journal Article
%A Benelkourchi, Slimane
%T Weak solutions to complex Monge–Ampère equations on compact Kähler manifolds
%J Comptes Rendus. Mathématique
%D 2014
%P 589-592
%V 352
%N 7-8
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2014.06.003/
%R 10.1016/j.crma.2014.06.003
%G en
%F CRMATH_2014__352_7-8_589_0
Benelkourchi, Slimane. Weak solutions to complex Monge–Ampère equations on compact Kähler manifolds. Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 589-592. doi : 10.1016/j.crma.2014.06.003. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2014.06.003/

[1] Aubin, T. Équations du type Monge–Ampère sur les variétés kählériennes compactes, C. R. Acad. Sci. Paris, Volume 283 (1976), pp. 119-121

[2] Aubin, T. Équations du type Monge–Ampère sur les variétës kählériennes compactes, Bull. Sci. Math., Volume 102 (1978), pp. 63-95

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

[4] Bedford, E.; Taylor, B.A. Fine topology, Šilov boundary, and (ddc)n, J. Funct. Anal., Volume 72 (1987) no. 2, pp. 225-251

[5] Benelkourchi, S.; Guedj, V.; Zeriahi, A. A priori estimates for weak solutions of complex Monge–Ampère equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume VII (2008), pp. 1-16

[6] Berman, R.J.; Boucksom, S.; Guedj, V.; Zeriahi, A. A variational approach to complex Monge–Ampère equations, Publ. Math. Inst. Hautes Études Sci., Volume 117 (2013), pp. 179-245

[7] Boucksom, S.; Eyssidieux, P.; Guedj, V.; Zeriahi, A. Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262

[8] Cegrell, U.; Kołodziej, S. The equation of complex Monge–Ampère type and stability of solutions, Math. Ann., Volume 334 (2006) no. 4, pp. 713-729

[9] Dinew, S. Uniqueness in E(X,ω), J. Funct. Anal., Volume 256 (2009) no. 7, pp. 2113-2122

[10] Edwards, R.E. Functional Analysis: Theory and Applications, Holt-Rinehart and Winston, 1965

[11] Guedj, V.; Zeriahi, A. Intrinsic capacities on compact Kahler manifolds, J. Geom. Anal., Volume 15 (2005) no. 4, pp. 607-639

[12] Guedj, V.; Zeriahi, A. The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 442-482

[13] Kołodziej, Slawomir Weak solutions of equations of complex Monge–Ampère type, Ann. Pol. Math., Volume 73 (2000) no. 1, pp. 59-67

[14] Lu, Hoang Chinh Solutions to degenerate complex Hessian equations, J. Math. Pures Appl., Volume 100 (2013), pp. 785-805

[15] Kołodziej, S. The complex Monge–Ampère equation and pluripotential theory, Mem. Amer. Math. Soc., Volume 178 (2005) no. 840 (x+64 pp.)

[16] Yau, S.T. On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Commun. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411

Cité par Sources :