I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.


Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $F$ be a family of plurisubharmonic functions on $\Omega.$ We may assume that these functions are continuous or smooth if we wish; also, I'm particularly interested in $n=2.$ 


**Q1.** In what convergence can we get a converging subsequence to a plurisubharmonic function? 


**Q2.** Are there further special conditions on the family $F$ that ensure compactness in stronger sense? 



Compactness properties of plurisubharmonic functions
First of all, I would say that there exists $\textit{one}$ good topology for psh functions, that is the $L^1\_{loc}$ topology. One of the main result is the following one : 


Let $(u\_n)$ be a sequence of psh functions on a connected open subset $\Omega \subset \mathbb{C} ^n$ with $u\_n \not \equiv -\infty $. We suppose that $(u\_n)$ converges to $u$ psh, in the weak topology of distributions. Then $(u\_n)$ is locally upper bounded and $u\_n \to u$ in $L^p\_{loc}(\Omega)$ for every $p\in[1,+\infty[$.


Other similar results are : 


-every bounded subset of $Psh(\Omega) \cap L^1\_{loc}(\Omega)$ is relatively compact;


-if $(u\_n)$ is locally upper bounded on $\Omega$, then either $(u\_n)$ converges locally uniformly to $-\infty$ on $\Omega$ (for the $L^1\_{loc}$ topology), either there exists some subsequence converging to a psh function on $\Omega$ (in $ L^1\_{loc}$ - or $L^p\_{loc}, p\geq 1$, this is the same-).


As for the references, the one I prefer is "Notions of convexity" by Hörmander, 94, around section 3.2.
The online book of Demailly is good too, but far less detailed about this topic.


