On regularization of plurisubharmonic functions on manifolds

Blocki, Zbigniew; Kolodziej, Slawomir

doi:10.1090/S0002-9939-07-08858-2

On regularization of plurisubharmonic functions on manifolds
HTML articles powered by AMS MathViewer

by Zbigniew Blocki and Slawomir Kolodziej

Proc. Amer. Math. Soc. 135 (2007), 2089-2093

DOI: https://doi.org/10.1090/S0002-9939-07-08858-2

Published electronically: February 2, 2007

PDF | Request permission

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

Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
Jean-Pierre Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. MR 1158622
J.-P. Demailly, Complex Analytic and Differential Geometry, 1997, see http://www-fourier. ujf-grenoble.fr/$\widetilde {\phantom {a}}$demailly/books.html.
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, DOI 10.4007/annals.2004.159.1247
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, DOI 10.1142/S0129167X01000861
Vincent Guedj and Ahmed Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607–639. MR 2203165, DOI 10.1007/BF02922247
V. Guedj, A. Zeriahi, Monge-Ampère operators on compact Kähler manifolds, see http://arxiv.org/PS_ cache/math/pdf/0504/0504234.pdf.
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, DOI 10.4064/ap-60-2-173-197
Sławomir Kołodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, 667–686. MR 1986892, DOI 10.1512/iumj.2003.52.2220
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, DOI 10.1007/s00222-006-0512-1
Rolf Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257–286 (German). MR 222334, DOI 10.1007/BF02063212
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, DOI 10.1002/cpa.3160310304