The Poletsky-Rosay theorem on singular complex spaces (Q2851020)

It is shown that the Poisson envelope of an upper semicontinuous function on a complex space is plurisubharmonic. More precisely, the authors prove the following theorem: if \(X\) is an irreducible, locally irreducible, reduced, paracompact complex space and \(u:X\to[-\infty,+\infty)\) is an upper semicontinuous function, then the function NEWLINE\[NEWLINEPu(x)=\inf_f\frac{1}{2\pi}\,\int_0^{2\pi}u(f(e^{it}))\,dt\,,\;\;x\in X\,,NEWLINE\]NEWLINE is plurisubharmonic or identically \(-\infty\) on \(X\), and \(Pu\) is the largest such function which is less than or equal to \(u\) on \(X\). Here the infimum is taken over the set of all analytic discs \(f\) in \(X\), i.e. continuous maps \(f:\overline{\mathbb D}\to X\) which are holomorphic on \(\mathbb D\), where \(\mathbb D\) is the unit disc in \(\mathbb C\) , such that \(f(0)=x\). The authors also give some nice applications of this theorem, for example, a characterization of the plurisubharmonic hull of a compact set in \(X\) using analytic discs.NEWLINENEWLINEThis theorem was first proved in the case of domains in \(\mathbb C^n\) by \textit{E. A. Poletsky}, as an application of his theory of holomorphic currents [Indiana Univ. Math. J. 42, No. 1, 85--144 (1993; Zbl 0811.32010)]. Later it was extended to certain classes of complex manifolds by \textit{F. Lárusson} and \textit{R. Sigurdsson} [J. Reine Angew. Math. 501, 1--39 (1998; Zbl 0901.31004)] (see also [J. Reine Angew. Math. 555, 27--38 (2003; Zbl 1023.32019)]). It was shown to hold for all complex manifolds \(X\) by \textit{J.-P. Rosay} [Indiana Univ. Math. J. 52, No. 1, 157--169 (2003; Zbl 1033.31006)] (see also [J. Korean Math. Soc. 40, No. 3, 423--434 (2003; Zbl 1040.32015)]).NEWLINENEWLINEThe main step of the proof is to show that the function \(Pu\) satisfies the sub-mean value property along analytic disks in \(X\). For this, the authors use holomorphic sprays and some of their properties which they developed in earlier work [Duke Math. J. 139, No. 2, 203--253 (2007; Zbl 1133.32002)].