Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-07-08858-2
Published electronically: February 2, 2007
MathSciNet review: 2299485
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the question of when a $ \gamma $-plurisubharmonic function on a complex manifold, where $ \gamma $ is a fixed $ (1,1)$-form, can be approximated by a decreasing sequence of smooth $ \gamma $-plurisubharmonic functions. We show in particular that it is always possible in the compact Kähler case.


References [Enhancements On Off] (What's this?)

  • [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/ $ \widetilde {\phantom{a}}$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\lodziej, 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)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32U05, 32Q15, 32U25

Retrieve articles in all journals with MSC (2000): 32U05, 32Q15, 32U25


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: https://doi.org/10.1090/S0002-9939-07-08858-2
Keywords: Plurisubharmonic functions, K\"{a}hler manifolds, Lelong numbers
Received by editor(s): March 8, 2006
Published electronically: 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

American Mathematical Society