$\pi_1$-small divisors and fundamental groups of varieties

TL;DR

This paper generalizes a result linking rational curves and finite representations of fundamental groups from surfaces to quasiprojective varieties, also exploring hyperbolicity and fundamental group properties of surfaces of general type.

Contribution

It extends the understanding of how $ ext{pi}_1$-small divisors influence the fundamental group representations and hyperbolicity in higher-dimensional and quasiprojective varieties.

Findings

Finite-dimensional linear representations of $ ext{pi}_1$ are finite under certain conditions.

$ ext{pi}_1$-small divisors imply restrictions on fundamental group representations.

Hyperbolicity properties are studied for surfaces of general type with infinite fundamental groups.

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.