On the convergence in capacity on compact Kahler manifolds and its applications

Author:
Pham Hoang Hiep

Journal:
Proc. Amer. Math. Soc. **136** (2008), 2007-2018

MSC (2000):
Primary 32W20; Secondary 32Q15

Published electronically:
February 12, 2008

MathSciNet review:
2383507

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Abstract | References | Similar Articles | Additional Information

Abstract: The main aim of the present note is to study the convergence in on a compact Kahler mainfold . The obtained results are used to study global extremal functions and describe the -pluripolar hull of an -pluripolar subset in .

**[Bl]**Zbigniew Błocki,*Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds*, Indiana Univ. Math. J.**52**(2003), no. 6, 1697–1701. MR**2021054**, 10.1512/iumj.2003.52.2346**[BT1]**Eric Bedford and B. A. Taylor,*The Dirichlet problem for a complex Monge-Ampère equation*, Invent. Math.**37**(1976), no. 1, 1–44. MR**0445006****[BT2]**Eric Bedford and B. A. Taylor,*A new capacity for plurisubharmonic functions*, Acta Math.**149**(1982), no. 1-2, 1–40. MR**674165**, 10.1007/BF02392348**[BT3]**E. Bedford and B. A. Taylor,*Plurisubharmonic functions with logarithmic singularities*, Ann. Inst. Fourier (Grenoble)**38**(1988), no. 4, 133–171 (English, with French summary). MR**978244****[BT4]**Eric Bedford and B. A. Taylor,*Uniqueness for the complex Monge-Ampère equation for functions of logarithmic growth*, Indiana Univ. Math. J.**38**(1989), no. 2, 455–469. MR**997391**, 10.1512/iumj.1989.38.38021**[Ce1]**Urban Cegrell,*Pluricomplex energy*, Acta Math.**180**(1998), no. 2, 187–217. MR**1638768**, 10.1007/BF02392899**[Ce2]**Urban Cegrell,*The general definition of the complex Monge-Ampère operator*, Ann. Inst. Fourier (Grenoble)**54**(2004), no. 1, 159–179 (English, with English and French summaries). MR**2069125****[Ce3]**U. Cegrell,*Convergence in capacity*, Technical report, Issac Newton Institute for Mathematical Sciences, 2001.**[GZ]**Vincent Guedj and Ahmed Zeriahi,*Intrinsic capacities on compact Kähler manifolds*, J. Geom. Anal.**15**(2005), no. 4, 607–639. MR**2203165**, 10.1007/BF02922247**[Ho]**Lars Hörmander,*Notions of convexity*, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR**1301332****[Ko1]**S. Kołodziej,*Capacities associated to the Siciak extremal function*, Ann. Polon. Math.**49**(1989), no. 3, 279–290. MR**997520****[Ko2]**Sławomir Kołodziej,*The Monge-Ampère equation on compact Kähler manifolds*, Indiana Univ. Math. J.**52**(2003), no. 3, 667–686. MR**1986892**, 10.1512/iumj.2003.52.2220**[Si]**Józef Siciak,*Extremal plurisubharmonic functions in 𝐶ⁿ*, Ann. Polon. Math.**39**(1981), 175–211. MR**617459****[Xi1]**Yang Xing,*Continuity of the complex Monge-Ampère operator*, Proc. Amer. Math. Soc.**124**(1996), no. 2, 457–467. MR**1322940**, 10.1090/S0002-9939-96-03316-3**[Xi2]**Yang Xing,*Complex Monge-Ampère measures of plurisubharmonic functions with bounded values near the boundary*, Canad. J. Math.**52**(2000), no. 5, 1085–1100. MR**1782339**, 10.4153/CJM-2000-045-x

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Additional Information

**Pham Hoang Hiep**

Affiliation:
Department of Mathematics, University of Education (Dai hoc Su Pham Ha Noi), CauGiay, Hanoi, Vietnam

Email:
phhiep_vn@yahoo.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09043-6

Keywords:
Complex Monge-Amp\`{e}re operator,
$\omega$-plurisubharmonic functions,
compact Kahler manifold

Received by editor(s):
September 30, 2006

Received by editor(s) in revised form:
December 11, 2006

Published electronically:
February 12, 2008

Additional Notes:
This work is supported by the National Research Program for Natural Sciences, Vietnam.

Communicated by:
Mei-Chi Shaw

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.