On fibrations approaching the Arakelov equality

TL;DR

This paper investigates whether sequences of semi-stable fibrations can asymptotically reach the Arakelov bound, concluding they cannot in certain cases and providing examples related to Teichmüller curves.

Contribution

It proves that sequences of fibrations do not approach the Arakelov bound under specific conditions and constructs examples related to Teichmüller curves.

Findings

Sequences of fibrations do not asymptotically approach the Arakelov bound for smooth, non-hyperelliptic, or small base genus cases.

Teichmüller curves do not attain the maximal Lyapunov exponent ratio.

Constructed examples demonstrate limitations of fibrations approaching the Arakelov equality.

Abstract

The sum of Lyapunov exponents $L_f$ of a semi-stable fibration is the ratio of the degree of the Hodge bundle by the Euler characteristic of the base. This ratio is bounded from above by the Arakelov inequality. Sheng-Li Tan showed that for fiber genus $g\geq 2$ the Arakelov equality is never attained. We investigate whether there are sequences of fibrations approaching asymptotically the Arakelov bound. The answer turns out to be no, if the fibration is smooth, or non-hyperelliptic, or has a small base genus. Moreover, we construct examples of semi-stable fibrations showing that Teichm\"uller curves are not attaining the maximal possible value of $L_f$.