The fundamental group of the smooth part of a log Fano variety (Q1911583)

Let \(X\) be a normal projective variety over the complex number field \(\mathbb{C}\). We call \(X\) a Fano variety if \(X\) is \(\mathbb{Q}\)-Gorenstein and the anti-canonical divisor \(-K_X\) is ample. A Fano variety \(X\) is said to be a log Fano variety if \(X\) has only log terminal singularities. A Fano variety \(X\) is called a canonical Fano variety if \(X\) has only canonical singularities. The Cartier index \(c(X)\) is the smallest positive integer such that \(c(X)K_X\) is a Cartier divisor. The Fano index, denoted by \(r(X)\), is the largest positive rational number such that \(-K_X \sim_\mathbb{Q} r(X)H\) \((\mathbb{Q}\)-linear equivalence) for a Cartier divisor \(H\). This note consists of two sections. In \S 1, we shall consider canonical Fano 3-folds and prove the following: Theorem 1. Let \(X\) be a canonical Fano 3-fold. Let \(X^0: =X- \text{Sing} X\) be the smooth part of \(X\). Assume that \(X\) has only isolated singularities. Then we have: (1) Suppose the Fano index \(r(X)\) is 1. Then \(\pi_1(X^0) =\mathbb{Z}/c (X)\mathbb{Z}\). (2) Suppose that the canonical divisor \(K_X\) is a Cartier divisor. Then \(X^0\) is simply connected. In ``Log Fano threefolds of Fano index one and quotients of \(K3\)-surfaces'' [Proc. Pacific Rim Geom. Conf. (Singapore 1994)], we give a universal bound for \(c(X)\). Under much stronger condition \textit{T. Sano} proved that \(c(X)\leq 2\). -- In \S 2, we shall consider \(n\)-dimensional log Fano varieties and prove the following: Theorem 2. Let \(X\) be a log Fano variety of Fano index \(r(X)> \dim X-2\). Let \(X^0: =X- \text{Sing} X\) be the smooth part of \(X\). Then we have: (1) The fundamental group \(\pi_1(X^0)\) of \(X^0\) is a finite group. (2) Suppose that \(X\) has only canonical singularities. Then \(\pi_1(X^0)\) is an abelian group of order \(\leq 9\). (3) Suppose that \(r(X)\geq \dim X-1\). Then \(\pi_1(X^0)\) is a finite abelian group generated by two elements, and has order \(\leq 9\). (4) Suppose \(r(X)> \dim X-1\). Then the smooth part \(X^0\) of \(X\) is simply connected. We want to remove the condition about the Fano index \(r(X)\) in the above theorems and raise the following question: Let \(X\) be a Fano variety. Let \(X^0: =X- \text{Sing} X\) be the smooth part of \(X\). Suppose that \(X\) has only log-terminal (or canonical, or terminal) singularities. Is \(\pi_1(X^0)\) a finite group?