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On regularization of plurisubharmonic functions on manifolds
Authors:
Zbigniew Blocki and Slawomir Kolodziej
Journal:
Proc. Amer. Math. Soc. 135 (2007), 2089-2093
MSC (2000):
Primary 32U05, 32Q15, 32U25
Posted:
February 2, 2007
MathSciNet review:
2299485
Full-text PDF Free Access
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Additional Information
Abstract: We study the question of when a  -plurisubharmonic function on a complex manifold, where  is a fixed  -form, can be approximated by a decreasing sequence of smooth  -plurisubharmonic functions. We show in particular that it is always possible in the compact Kähler case.
- [1]
Eric
Bedford and B.
A. Taylor, A new capacity for plurisubharmonic functions, Acta
Math. 149 (1982), no. 1-2, 1–40. MR 674165
(84d:32024), http://dx.doi.org/10.1007/BF02392348
- [2]
Jean-Pierre
Demailly, Regularization of closed positive currents and
intersection theory, J. Algebraic Geom. 1 (1992),
no. 3, 361–409. MR 1158622
(93e:32015)
- [3]
J.-P.Demailly, Complex Analytic and Differential Geometry, 1997, see http://www-fourier. ujf-grenoble.fr/
 demailly/books.html.
- [4]
Jean-Pierre
Demailly and Mihai
Paun, Numerical characterization of the Kähler cone of a
compact Kähler manifold, Ann. of Math. (2) 159
(2004), no. 3, 1247–1274. MR 2113021
(2005i:32020), http://dx.doi.org/10.4007/annals.2004.159.1247
- [5]
Jean-Pierre
Demailly, Thomas
Peternell, and Michael
Schneider, Pseudo-effective line bundles on compact Kähler
manifolds, Internat. J. Math. 12 (2001), no. 6,
689–741. MR 1875649
(2003a:32032), http://dx.doi.org/10.1142/S0129167X01000861
- [6]
Vincent
Guedj and Ahmed
Zeriahi, Intrinsic capacities on compact Kähler
manifolds, J. Geom. Anal. 15 (2005), no. 4,
607–639. MR 2203165
(2006j:32041), http://dx.doi.org/10.1007/BF02922247
- [7]
V.Guedj, A.Zeriahi, Monge-Ampère operators on compact Kähler manifolds, see http://arxiv.org/PS_ cache/math/pdf/0504/0504234.pdf.
- [8]
Christer
O. Kiselman, Attenuating the singularities of plurisubharmonic
functions, Ann. Polon. Math. 60 (1994), no. 2,
173–197 (English, with English and Esperanto summaries). MR 1301603
(95i:32024)
- [9]
Sławomir
Kołodziej, The Monge-Ampère equation on compact
Kähler manifolds, Indiana Univ. Math. J. 52
(2003), no. 3, 667–686. MR 1986892
(2004i:32062), http://dx.doi.org/10.1512/iumj.2003.52.2220
- [10]
D.
H. Phong and Jacob
Sturm, The Monge-Ampère operator and geodesics in the space
of Kähler potentials, Invent. Math. 166 (2006),
no. 1, 125–149. MR 2242635
(2007h:32036), http://dx.doi.org/10.1007/s00222-006-0512-1
- [11]
Rolf
Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann.
175 (1968), 257–286 (German). MR 0222334
(36 #5386)
- [12]
Shing
Tung Yau, On the Ricci curvature of a compact Kähler manifold
and the complex Monge-Ampère equation. I, Comm. Pure Appl.
Math. 31 (1978), no. 3, 339–411. MR 480350
(81d:53045), http://dx.doi.org/10.1002/cpa.3160310304
- [1]
- E.Bedford, B.A.Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-41. MR 0674165 (84d:32024)
- [2]
- J.-P.Demailly, Regularization of closed positive currents and intersection theory, J.Alg. Geom. 1 (1992), 361-409. MR 1158622 (93e:32015)
- [3]
- J.-P.Demailly, Complex Analytic and Differential Geometry, 1997, see http://www-fourier. ujf-grenoble.fr/ demailly/books.html.
- [4]
- J.-P.Demailly, M.Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann.of Math. 159 (2004), 1247-1274. MR 2113021 (2005i:32020)
- [5]
- J.-P.Demailly, T,Peternell, M.Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat.J.Math. 12 (2001), 689-741. MR 1875649 (2003a:32032)
- [6]
- V.Guedj, A.Zeriahi, Intrinsic capacities on compact Kähler manifolds, J.Geom.Anal. 15 (2005), 607-639. MR 2203165 (2006j:32041)
- [7]
- V.Guedj, A.Zeriahi, Monge-Ampère operators on compact Kähler manifolds, see http://arxiv.org/PS_ cache/math/pdf/0504/0504234.pdf.
- [8]
- C.O.Kiselman, Attenuating the singularities of plurisubharmonic functions, Ann.Pol.Math. 60 (1994), 173-197. MR 1301603 (95i:32024)
- [9]
- S.Ko
 odziej, Stability of solutions to the complex Monge-Ampère equation on compact Kähler manifolds, Indiana Univ.Math. J 52 (2003), 667-686. MR 1986892 (2004i:32062)
- [10]
- D.H.Phong, J.Sturm, The Monge-Ampère operator and geodesics in the space of Kähler potentials, Invent.Math. 166 (2006), 125-149. MR 2242635
- [11]
- R.Richberg, Stetige streng pseudokonvexe Funktionen, Math.Ann. 175 (1968), 257-286. MR 0222334 (36:5386)
- [12]
- S.-T.Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm.Pure Appl.Math. 31 (1978), 339-411. MR 0480350 (81d:53045)
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Additional Information
Zbigniew Blocki
Affiliation:
Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Email:
Zbigniew.Blocki@im.uj.edu.pl
Slawomir Kolodziej
Affiliation:
Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Email:
Slawomir.Kolodziej@im.uj.edu.pl
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08858-2
PII:
S 0002-9939(07)08858-2
Keywords:
Plurisubharmonic functions,
K\"{a}hler manifolds,
Lelong numbers
Received by editor(s):
March 8, 2006
Posted:
February 2, 2007
Additional Notes:
Both authors were partially supported by KBN Grant #2 P03A 03726. The second author was also supported by the Rector of the Jagiellonian University Fund
Communicated by:
Mei-Chi Shaw
Article copyright:
© Copyright 2007 American Mathematical Society
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