The local fundamental group of a Kawamata log terminal singularity is finite
Let \(X\) be a (normal) complex algebraic variety and \(x \in X\) a (closed) point. The \textit{local fundamental group} \(\pi_1^{\mathrm{loc}}(X, x)\) is defined to be \(\pi_1(B \setminus \{x\})\), where \(B\) is the intersection of \(X\) with a sufficiently small euclidean ball around \(x\) (with respect to a local embedding \(X \hookrightarrow \mathbb C^n\)). Even if \(X\) has only log canonical singularities, it is known that the local fundamental group can be arbitrarily complicated. On the other hand, if \((X, \Delta)\) has only Kawamata log terminal (klt) singularities, then Kollár has conjectured that \(\pi_1^{\mathrm{loc}}(X, x)\) is finite. This epic paper proves (generalizations of) this conjecture. More precisely, in the above setting one can also define the \textit{regional fundamental group} \(\pi_1^{\mathrm{reg}}(X, x)\) as \(\pi_1(B \setminus X_{\mathrm{sg}})\) (local and regional fundamental group coincide for isolated singularities, but in general finiteness of the regional fundamental group is stronger). Then Theorem~1 states that the regional fundamental group of a klt singularity is finite. There is also a more general version concerning the \textit{orbifold} regional fundamental group. The proof of Theorem~1 relies on the following Theorem~2, which generalizes and proves another conjecture: let \((X, \Delta = \Delta' + \Delta'')\) be a klt weakly Fano pair (i.e.~\(-(K_X + \Delta)\) is big and nef). Then the orbifold fundamental group of the smooth locus is finite. The simultaneous proof of the two theorems is contained in Theorems~7 and~8: Theorem~1 in dimension \(n\) implies Theorem~2 in dimension \(n\), and Theorem~2 in dimension \(n\) implies Theorem~1 in dimension \(n + 1\). Previous results on the topic (on which this paper partly builds) include [\textit{C. Xu}, Compos. Math. 150, No. 3, 409--414 (2014; Zbl 1291.14057); \textit{Z. Tian} and \textit{C. Xu}, Compos. Math. 153, No. 2, 257--273 (2017; Zbl 1453.14014); \textit{D. Greb} et al., Duke Math. J. 165, No. 10, 1965--2004 (2016; Zbl 1360.14094)].
scientific article