thanks! but how can this happen if the quotient $SL(2,R)/B_+$ is compact? Also, I would be very grateful for any reference to anything related to the stuff

It all depends on your definitions, but if you want convergence in Barlet space and you consider all points in all cycles with multiplicities > 1 as "singular", then I suppose there should be such a result. I think that the Lelong number of a singular variety, considered as a current, is bigger in its singular point, and Lelong numbers are semi-continuous, so a limit of a singularity should be a singular point or a point with multiplicity > 1 (here I observe that convergence of currents is compatible with the Barlet space convergence). I never saw such a result, though.

This paper of LeBrun seems to be relevant: LeBrun, Claude Fano manifolds, contact structures, and quaternionic geometry. Internat. J. Math. 6 (1995), no. 3, 419–437, apparently, it answers positively to my question, but LeBrun assumes that X is smooth...

this is what I surmised when I read the paper

It was conjectured by several people (Bogomolov, for instance) that the fiber of the general point of the discriminant has no multiplicities. Somebody told me that this paper contains a counterexample arxiv.org/abs/1910.13127 but I don't see why.

The open ball in $C^n$ can be the universal cover of a Kahler manifold, and it does not have an algebraic structure