${\pi }_1$-small divisors and fundamental groups of varieties

Abstract

Lasell and Ramachandran show that the existence of rational curves of positive self-intersection on a smooth projective surface $X$ implies that all the finite dimensional linear representations of the fundamental group ${\pi }_1(X)$ are finite. In this article, we generalize Lasell and Ramachandran's result to the case of ${\pi }_1$-small divisors on quasiprojective varieties. We also study ${\pi }_1$-small curves and hyperbolicity properties of smooth projective surfaces of general type with infinite fundamental groups.

26pages, comments are welcome