Chern slopes of simply connected complex surfaces of general type are dense in [2,3]

Abstract

We prove that for any number $r \in [2,3]$, there are spin (resp. nonspin and minimal) simply connected complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$. In particular, this shows the existence of simply connected surfaces of general type arbitrarily close to the Bogomolov-Miyaoka-Yau line. In addition, we prove that for any $r \in [1,3]$ and any integer $q\geq 0$, there are minimal complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$ and $\pi_1(X)$ isomorphic to the fundamental group of a compact Riemann surface of genus $q$. %A central ingredient is a new family of special arrangements of elliptic curves in the projective plane.

Authors

Xavier Roulleau

Unité de Formation Mathématiques, Université de Poitiers, Poitiers, France

Giancarlo Urzúa

Pontificia Universidad, Católica de Chile, Santiago, Chile