Slopes of fibrations with trivial vertical fundamental groups

TL;DR

This paper establishes that the maximum slope for fibrations with trivial vertical fundamental groups is 12, and demonstrates the existence of such fibrations with slopes approaching this bound for genus g ≥ 3.

Contribution

It proves that 12 is the sharp upper bound for slopes of fibrations with trivial vertical fundamental groups and constructs examples approaching this bound for all genera g ≥ 3.

Findings

12 is the maximum slope for fibrations with trivial vertical fundamental groups.

Existence of fibrations with trivial vertical fundamental groups approaching slope 12 for all g ≥ 3.

Provides a relative analogue to known results on surfaces of general type with trivial fundamental groups.

Abstract

Kodaira fibrations have non-trivial vertical fundamental groups and their slopes are all $12$. In this paper, we show that $12$ is indeed the sharp upper bound for the slopes of fibrations with trivial vertical fundamental groups. Precisely, for each $g\geq3$ we prove the existence of fibrations of genus $g$ with trivial vertical fundamental groups whose slopes can be arbitrarily close to $12$. This gives a relative analogy of Roulleau-Urz\'ua's work on the slopes of surfaces of general type with trivial fundamental groups.