Jan Wiegerinck
Plurifinely plurisubharmonic and holomorphic functions
As is well known, the fine toplogy is the weakest topology on domains in $\mathbb R^n$ that makes all subharmonic functions continuous. It allows naturally for finely subharmonic - and in the 2-dimensional case also finely holomorphic - functions.
In $\mathbb C^n$ the plurifine topology, which makes all plurisubharmonic functions continuous, is challenging. In this setting we introduce a weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions. Strong will imply weak, but it is unknown whether the two concepts are the same.
In this lecture we will discuss the plurifine topology and present our results on plurifinely plurisubharmonic and holomorphic functions.
All this is joint work, partly with Said El Marzguioui, and partly with Mohamed El Kadiri and Bent Fuglede, and it includes
$\bullet$ Every bounded finely plurisubharmonic function can be locally (in the plurifine topology) written as the difference of two usual plurisubharmonic functions. As a consequence finely plurisubharmonic functions are continuous with respect to the plurifine topology.
$\bullet$ The $-\infty$ sets of finely plurisubharmonic functions are pluripolar, hence graphs of finely holomorphic functions are pluripolar.
$\bullet$ A function $f$ is weakly plurifinely plurisubharmonic if and only if it is locally bounded from above in the plurifine topology and $f\circ h$ is finely subharmonic for all complex affine-linear maps $h$.
$\bullet$ Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.