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Guedj, Vincent; Zeriahi, Ahmed

Intrinsic capacities on compact Kähler manifolds. (English) Zbl 1087.32020

J. Geom. Anal. 15, No. 4, 607-639 (2005).
The authors develop a global pluripotential theory in the context of compact Kähler manifolds. From the maximum principle it follows that there are no plurisubharmonic (psh) functions on compact complex manifolds. If \(\omega\) is a real closed smooth form of bidegree \((1,1)\) on the complex manifold \(X\), one can obtain positive closed currents \(\omega^\prime\) of bidegree \((1,1)\) which are cohomologous to \(\omega\). If \(X\) is Kähler we can write \(\omega^\prime =\omega +dd^c\varphi\), where \(\varphi\) is a \(\omega\)-plurisubharmonic (\(\omega\) -psh)function (or a quasiplurisubharmonic (qpsh) function). The authors define \(\omega\)-psh functions and gather useful facts about them, define the complex Monge-Ampère operator, establish Chern-Levine-Nirenberg inequalities, and study the Monge-Ampère capacity. Next they study the Alexander capacity and show that locally pluripolar sets can be defined by \(\omega\) psh functions when \(\omega\) is Kähler.
Reviewer: Vasile Oproiu (Iaşi)

Cited in 1 Review
Cited in 123 Documents

MSC:

32U15 General pluripotential theory
32U05 Plurisubharmonic functions and generalizations
32Q15 Kähler manifolds
31C10 Pluriharmonic and plurisubharmonic functions
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets

Keywords:

intrinsic capacities; Kähler manifolds; pluripotential theory; plurisubharmonic; quasiplurisubharmonic functions
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\textit{V. Guedj} and \textit{A. Zeriahi}, J. Geom. Anal. 15, No. 4, 607--639 (2005; Zbl 1087.32020)
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References:

[1] Alexander, H., Projective capacity. Recent developments in several complex variables, (Proc. Conf., Princeton Univ., 1979), Ann. of Math. Stud., 3-27 (1981), Princeton, N.J.: Princeton University Press, Princeton, N.J. · Zbl 0494.32001
[2] Alexander, H.; Taylor, B. A., Comparison of two capacities inC^n, Math. Z., 186, 3, 407-417 (1984) · Zbl 0576.32029 · doi:10.1007/BF01174894
[3] Bedford, E., Survey of pluri-potential theory. Several complex variables, (Stockholm, 1987/1988), Math. Notes, 48-97 (1993), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0786.31001
[4] Bedford, E.; Taylor, B. A., The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37, 1, 1-44 (1976) · Zbl 0315.31007 · doi:10.1007/BF01418826
[5] Bedford, E.; Taylor, B. A., A new capacity for plurisubharmonic functions, Acta Math., 149, 1-2, 1-40 (1982) · Zbl 0547.32012 · doi:10.1007/BF02392348
[6] Cegrell, U., An estimate of the complex Monge-Ampère operator,Analytic Functions, Bazejewko, 1982, (Baze-jewko, 1982), Lecture Notes in Math., 84-87 (1983), Berlin: Springer, Berlin · Zbl 0529.35018
[7] Cegrell, U., Capacities in complex analysis, Aspects of Mathematics, E14. Friedr (1988), Braunschweig: Vieweg and Sohn, Braunschweig · Zbl 0655.32001
[8] Chern, S. S.; Levine, H. I.; Nirenberg, L., Intrinsic norms on a complex manifold, Global Analysis, (Papers in Honor of K. Kodaira), 119-139 (1969), Tokyo: Univ. Tokyo Press, Tokyo · Zbl 0202.11603
[9] Chinburg, T.; Lau, C. F.; Rumely, R., Capacity theory and arithmetic intersection theory, Duke Math. J., 117, 2, 229-285 (2003) · Zbl 1026.11056 · doi:10.1215/S0012-7094-03-11722-6
[10] Demailly, J.-P., EstimationsL^2 pour l’opérateur \(\bar \partial\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. E.N.S. (4), 15, 3, 457-511 (1982) · Zbl 0507.32021
[11] Demailly, J.-P., Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France (N.S.), 19, 124-124 (1985) · Zbl 0579.32012
[12] Demailly, J.-P., Regularization of closed positive currents and intersection theory, J. Algebraic Geom., 1, 3, 361-409 (1992) · Zbl 0777.32016
[13] Demailly, J.-P., MéthodesL^2 et résultats effectifs en géométrie algébrique, Séminaire Bourbaki, 1998/99, Astérisque, 266, 59-90 (2000) · Zbl 0962.14014
[14] Demailly, J.-P., On the Ohsawa-Takegoshi-Manivel L^2 extension theorem,Complex Analysis and Geometry, (Paris, 1997), Progr. Math., 47-82 (2000), Basel: Birkhäuser, Basel · Zbl 0959.32019
[15] Demailly, J.-P. Complex analytic and differential geometry, free accessible book (http://www-fourier.ujf-grenoble.fr/ demailly/books.html).
[16] Demailly, J.-P.; Peternell, T.; Schneider, M., Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math., 12, 6, 689-741 (2001) · Zbl 1111.32302 · doi:10.1142/S0129167X01000861
[17] Dinh, T.-C.; Sibony, N., Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), 82, 4, 367-423 (2003) · Zbl 1033.37023
[18] Dinh, T.-C. and Sibony, N. Value distribution of meromorphic transforms and applications, preprint (2003).
[19] El, H., Fonctions plurisousharmoniques et ensembles polaires, Séminaire Pierre Lelong-Henri Skoda (Analyse). Années 1978/79, Lecture Notes in Math., 61-76 (1980), Berlin: Springer, Berlin · Zbl 0445.32015
[20] Favre, C.; Jonsson, M., Brolin’s theorem for curves in two complex dimensions, Ann. Inst. Fourier (Grenoble), 53, 5, 1461-1501 (2003) · Zbl 1113.32005
[21] Guedj, V., Approximation of currents on complex manifolds, Math. Ann., 313, 3, 437-474 (1999) · Zbl 0924.32014 · doi:10.1007/s002080050269
[22] Guedj, V., Equidistribution towards the Green Current, Bulletin S.M.F., 131, 359-372 (2003) · Zbl 1070.37026
[23] Guedj, V., Ergodic properties of rational mappings with large topological degree, Anal. Math., 161, 3, 1589-1607 (2005) · Zbl 1088.37020 · doi:10.4007/annals.2005.161.1589
[24] Guedj, V. and Zeriahi, A. Monge-Ampère operators on compact Kähler surfaces, preprint arXiv/math.CV/0504234. · Zbl 1137.32015
[25] Huckleberry, A., Subvarieties of homogeneous and almost homogeneous manifolds, contributions to complex analysis and analytic geometry, Aspects Math., 189-232 (1994), Braunschweig: Vieweg, Braunschweig · Zbl 0823.32016
[26] Kiselman, C., Plurisubharmonic functions and potential theory in several complex variables, development of mathematics, 1950/2000, 655-714 (2000), Basel: Birkhäuser, Basel · Zbl 0962.31001
[27] Klimek, M., Pluripotential Theory, London Mathematical Society Monographs (1991), New York: Oxford University Press, New York · Zbl 0742.31001
[28] Kolodziej, S., The complex Monge-Ampère equation, Acta Math., 180, 1, 69-117 (1998) · Zbl 0913.35043 · doi:10.1007/BF02392879
[29] Kolodziej, S. Stability of solutions to the complex Monge-Ampère equation on compact Kähler manifolds, preprint (2001).
[30] Lau, C. F.; Rumely, R.; Varley, R., Existence of the sectional capacity, Mem. Amer. Math. Soc., 145, 690, viii-130 (2000) · Zbl 0987.14018
[31] Molzon, R. and Shiffman, B. Capacity, Tchebycheff constant, and transfinite hyperdiameter on complex projective space, Seminar Pierre Lelong-Henri Skoda, (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981, 337-357,Lecture Notes in Math. 919, Springer-Verlag, (1982). · Zbl 0489.31006
[32] Molzon, R.; Shiffman, B.; Sibony, N., Average growth estimates for hyperplane sections of entire analytic sets, Math. Ann., 257, 1, 43-59 (1981) · Zbl 0537.32009 · doi:10.1007/BF01450654
[33] Sibony, N. Dynamique des applications rationnelles deP^k, inDynamique et Géométrie Complexes, (Lyon, 1997),Panor. Synthèses Soc. Math. France, Paris, 97-185, (1999). · Zbl 1020.37026
[34] Sibony, N.; Wong, P. M., Some remarks on the Casorati-Weierstrass theorem, Ann. Polon. Math., 39, 165-174 (1981) · Zbl 0476.32005
[35] Siciak, J., On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc., 105, 322-357 (1962) · Zbl 0111.08102 · doi:10.2307/1993631
[36] Siciak, J., Extremal plurisubharmonic functions and capacities inC^n (1982), Tokyo: Sophia Univ., Tokyo · Zbl 0579.32025
[37] Tian, G., Canonical metrics in Kähler geometry, Lect. Mat. ETH Zürich, vi-101 (2000), Basel: Birkhäuser, Verlag, Basel · Zbl 0978.53002
[38] Zaharjuta, V. P., Extremal plurisubharmonic functions, orthogonal polynomials and Berstein-Walsh theorem for analytic functions of several variables, Ann. Polon. Math., 33, 137-148 (1976) · Zbl 0341.32011
[39] Zeriahi, A., A criterion of algebraicity for Lelong classes and analytic sets, Acta Math., 184, 1, 113-143 (2000) · Zbl 1016.32015 · doi:10.1007/BF02392783
[40] Zeriahi, A., Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J., 50, 1, 671-703 (2001) · Zbl 1138.31302 · doi:10.1512/iumj.2001.50.2062
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