Let $X\_0$ be a compact complex algebraic surface with an isolated singularity and let $X\_t$ be a smoothing of $X\_0$ over the disc. How can we compute the fundamental group of $X\_t$ say in terms of the topology of a minimal resolution and some local information of the singularity? If it helps we can assume the singularity is rational and that the smoothing is $\mathbb{Q}$-Gorenstein.



Fundamental group of a smoothing of a complex surface
Non surprisingly, this usually involves Seifert-Van Kampen theorem, but the actual computation can be a tricky one. However, since you have just one singularity, life will be probably easier.


For an example where the fundamental group turns out to be trivial, you can look at the celebrated paper by Lee and Park, in which they construct a simply-connected Campedelli surface via $\mathbb{Q}$-Gorenstein smoothing of a family of rational surfaces with isolated $T$-singularities:


*Lee, Yongnam; Park, Jongil*, [**A simply connected surface of general type with (p\_g=0) and (K^2=2)**](http://dx.doi.org/10.1007/s00222-007-0069-7), Invent. Math. 170, No. 3, 483-505 (2007). [ZBL1126.14049](https://zbmath.org/?q=an:1126.14049).


