Abstract
We prove several approximation theorems of the complex Monge–Ampère operator on a compact Kähler manifold. As an application we prove the Cegrell type theorem on a complete description of the range of the complex Monge–Ampère operator in the class of ω-plurisubharmonic functions with vanishing complex Monge–Ampère mass on all pluripolar sets. As a by-product we obtain a stability theorem of solutions of complex Monge–Ampère equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blocki Z.: On the definition of the complex Monge–Ampère operator in \(\mathbb{C^2}\) . Math. Ann. 328, 415–423 (2004)
Blocki Z.: The domain of definition of the complex Monge–Ampère operator. Am. J. Math. 128, 519–530 (2006)
Blocki Z.: Uniqueness and stability for the complex Monge–Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52, 1697–1701 (2003)
Blocki Z., Kolodziej S.: On regularization of plurisubharmonic functions on manifolds. Proc. Am. Math. Soc. (7) 135, 2089–2093 (2007)
Bedford E., Taylor B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)
Bedford E., Taylor B.A.: Uniqueness for the complex Monge–Ampèr equation for functions of logarithmic growth. Indiana Univ. Math. J. 38, 455–469 (1989)
Bedford E., Taylor B.A.: Fine topology, Šilov boundary and \({(dd^c)^n}\) . J. Funct. Anal. 72, 225–251 (1987)
Cegrell U.: Pluricomplex energy. Acta Math. 180(2), 187–217 (1998)
Cegrell U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier 54, 159–179 (2004)
Cegrell, U.: Convergence in capacity. Isaac Newton Institute for Math. Science Preprint Series NI01046-NPD, 2001, also available at arxiv.org: math. CV/0505218
Calabi E.: On Kähler manifolds with vanishing canonical class. Algebraic geometry and topology. Asymposium in honor of S.Lefschetz, pp. 78–89. Princeton University Press, Princeton (1957)
Cegrell U., Kolodziej S.: The equation of complex Monge–Ampère type and stability of solutions. Math. Ann. 334, 713–729 (2006)
Guedj, V., Zeriahi, A.: The weighted Monge–Ampère energy of quasiplurisubhar-monic functions. Preprint, arXiv math.CV/061230
Guedj V., Zeriahi A.: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15(4), 607–639 (2005)
Kiselman, C.O.: Sur la definition de l’opérateur de Monge–Ampère complexe. Analyse Complexe: Proceedings, Toulouse, pp. 139–150. LNM 1094, Springer, New York (1983)
Kolodziej, S.: The complex Monge–Ampère equation and pluripotential theory. Mem. Am. Math. Soc. 178(840) (2005)
Kolodziej S.: The complex Monge–Ampère equation. Acta Math. 180, 69–117 (1998)
Kolodziej S.: The Monge–Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52(3), 667–686 (2003)
Kolodziej S.: The set of measures given by bounded solutions of the complex Monge–Ampère equation on compact Kähler manifolds. J. Lond. Math. Soc. (2) 72, 225–238 (2005)
Xing Y.: Continuity of the complex Monge–Ampère operator. Proc. Am. Math. Soc. 124, 457–467 (1996)
Xing Y.: Convergence in capacity. Ann. Inst. Fourier 58(5), 1839–1861 (2008)
Xing, Y.: A strong comparison principle of plurisubharmonic functions with finite pluricomplex energy. Michigan J. Math (to appear)
Xing, Y.: The general definition of the complex Monge–Ampère operator on compact Kähler manifolds. Canad. J. Math. (to appear)
Yau S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. Comm. Pure Appl. Math. 31, 339–411 (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xing, Y. Continuity of the Complex Monge–Ampère operator on compact Kähler manifolds. Math. Z. 263, 331–344 (2009). https://doi.org/10.1007/s00209-008-0420-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0420-8