π1—small Divisors and Fundamental Groups of Varieties

Abstract

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.