Plurisubharmonically separable complex manifolds

Poletsky, Evgeny; Shcherbina, Nikolay

doi:10.1090/proc/14222

Plurisubharmonically separable complex manifolds
HTML articles powered by AMS MathViewer

by Evgeny A. Poletsky and Nikolay Shcherbina PDF

Proc. Amer. Math. Soc. 147 (2019), 2413-2424 Request permission

Abstract:

Let $M$ be a complex manifold and let $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when the functions from $PSH^{cb}(M)$ separate points. Our main results show that this property is equivalent to each of the following properties of $M$:

the core of $M$ is empty;
for every $w_0\in M$ there is a continuous plurisubharmonic function $u$ with the logarithmic singularity at $w_0$.

Moreover, the core of $M$ is the disjoint union of the sets $E_j$ that are 1-pseudoconcave in the sense of Rothstein and have the following Liouville property: every function from $PSH^{cb}(M)$ is constant on each $E_j$.

References

A. Aytuna and A. Sadullaev, Parabolic Stein manifolds, Math. Scand. 114 (2014), no. 1, 86–109. MR 3178107, DOI 10.7146/math.scand.a-16640
Tobias Harz, Nikolay Shcherbina, and Giuseppe Tomassini, On defining functions and cores for unbounded domains I, Math. Z. 286 (2017), no. 3-4, 987–1002. MR 3671568, DOI 10.1007/s00209-016-1792-9
Tobias Harz, Nikolay Shcherbina, and Giuseppe Tomassini, On defining functions and cores for unbounded domains II, J. Geom. Anal. (2017).
Tobias Harz, Nikolay Shcherbina, and Giuseppe Tomassini, On defining functions and cores for unbounded domains III. Submitted for publication.
Tobias Harz, Nikolay Shcherbina, and Giuseppe Tomassini, Wermer type sets and extension of CR functions, Indiana Univ. Math. J. 61 (2012), no. 1, 431–459. MR 3029404, DOI 10.1512/iumj.2012.61.4507
Raghavan Narasimhan and Yves Nievergelt, Complex analysis in one variable, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1803086, DOI 10.1007/978-1-4612-0175-5
Evgeny A. Poletsky, Plurisubharmonic functions as solutions of variational problems, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 163–171. MR 1128523
Jean-Pierre Rosay, Poletsky theory of disks on holomorphic manifolds, Indiana Univ. Math. J. 52 (2003), no. 1, 157–169. MR 1970025, DOI 10.1512/iumj.2003.52.2170
Jean-Pierre Rosay, Discs in complex manifolds with no bounded plurisubharmonic functions, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2315–2319. MR 2052407, DOI 10.1090/S0002-9939-04-07460-X
Zbigniew Slodkowski, Local maximum property and $q$-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl. 115 (1986), no. 1, 105–130. MR 835588, DOI 10.1016/0022-247X(86)90027-2
Z. Slodkowski, Preliminary notes, Spring 2017.
Zibgniew Slodkowski and Giuseppe Tomassini, Minimal kernels of weakly complete spaces, J. Funct. Anal. 210 (2004), no. 1, 125–147. MR 2052116, DOI 10.1016/S0022-1236(03)00182-4