Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts

Abstract

Abstract. Let (S, ∆) be a log surface with at worst log canonical singularities and reduced boundary ∆ such that −(K S + ∆) is nef and big. We shall prove that S o = S − SingS −∆ either has finite fundamental group or is affine-ruled. Moreover, π 1 (S o ) and the structure of S are determined in some sense when ∆ = 0. IntroductionLet S be a normal projective algebraic surface over the complex number field C with nef and big anti-canonical divisor −K S . Recently, in [11] and [15] (see also [12,13]) we have proved that the fundamental group of the smooth part S o = S − SingS is finite provided that S has at worst log-terminal singularities (see [5] or [6] for its definition).In this short note, we shall consider the case when S has at worst log canonical singularities (see (4) of Remark (2.4) below for background and see [4] or [6] for the complete list of log-canonical singularities). Then π 1 (S o ) is no longer finite. However, we shall describe precisely, in the case when π 1 (S o ) is infinite, this fundamental group and the structure of S.The following Lemma (2.1) is rather easy. Lemma (2.1). Let S be a normal projective algebraic surface with at worst log canonical singularities and with nef and big anti-canonical divisor −K S . Suppose that S has at least one non-rational singularity. Then we have:(1) S is obtained from a nonsingular elliptic ruled surface S by contracting a cross-section of the ruling to a simple elliptic singularity and several P 1 's in fibers into rational singularities.(2) We have the following surjective homomorphism, where S o = S − SingS is the smooth part, and hence the infiniteness ofThe following is one of our main theorems. It says that when π 1 (S o ) is infinite S has a finite Galois covering unramified over S o and with the covering surface described in Lemma (2.1). In Theorems (2.3) and (1.1) below, we refer to [5] or [6]

Characterization of log del Pezzo pairs via anticanonical models

“…There exists a smooth projective surface whose anticanonical model is a non-klt lc del Pezzo surface (see [Zh,Section 3] for the rational case). In contrast to the klt case, there are non-rational lc del Pezzo surfaces, but it turns out that they are very special.…”

Quotients of K3 surfaces modulo involutions

“…Remark 1.10. (1) In view of [S], log del Pezzo surfaces are rational (see also [Z4,Lemma 1.1] or [Z5,Lemma 1.3]).…”