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 gge 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 Teichmüller curves are not attaining the maximal possible value of L_f.
Highlights
The closure of a curve in Mg can equivalently be regarded as a semi-stable fibration f : X → C of genus g
In the present paper we propose to study a new invariant for semi-stable fibrations, namely the sum of non-negative Lyapunov exponents L f
If f is a fibration coming from a Teichmüller curve generated by a flat surface (X, ω), the slope λ f and the speed L f are related by λf
Summary
The closure of a curve in Mg can equivalently be regarded as a semi-stable fibration f : X → C of genus g. Lyapunov exponents originate from dynamical systems and have been brought to the study of the geometry of the moduli space of curves through the connection with billiards and the SL2(R)-action on the moduli space of flat surfaces They measure the growth rate of cohomology classes of a flat bundle when parallel transported along a geodesic flow. Theorem 1.2 Let f : X → C be a semi-stable fibration of genus g ≥ 2 with s singular fibers and let m ∈ N be a number such that. For g ≥ 6, we improve a result of [30] to get lower bounds for the slope of non-hyperelliptic fibrations of genus g, which implies strict upper bounds for the speed.
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