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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.

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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), Invent. Math. 170, No. 3, 483-505 (2007). ZBL1126.14049.

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  • $\begingroup$ Thank you very much! After looking through this paper and ones that cite it, it seems to me like the following is true. Suppose $X_0$ has an isolated $T$-singularity with minimal resolution $Y \to X_0$ and $\mathbb{Q}$-Gorenstein smoothing $X_t$. Then $X_t$ is diffeomorphic to the rational blowdown of $Y$ along the exceptional locus of the resolution. Then one can apply Seifert-Van Kampen to compute the fundamental group of the rational blowdown pretty explicitly. $\endgroup$ Commented Dec 13, 2022 at 4:08

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