The fundamental group of the smooth part of a log Fano variety

Abstract

Let X be a normal projective variety over the complex number field C. We call X a Fano variety if X is Q-Gorenstein and the anti-canonical divisor — Kx is ample. A Fano variety X is said to be a log Fano variety if X has only log terminal singularities (cf. [6]). A Fano variety X is called a canonical Fano variety if X has only canonical singularities (cf. [6]). The Cartier index c{X) is the smallest positive integer such that c{X)Kx is a Cartier divisor. The Fano index, denoted by r(X), is the largest positive rational number such that —Kχ~~Q r{X)H (Q-linear equivalence) for a Cartier divisor H. This note consists of two sections. In §1, we shall consider canonical Fano 3-folds and prove the following: