# Torsion on Jacobians and Teichmuller dynamics

In this post, I will describe a special case of a theorem about torsion points on Jacobians and Teichmuller dynamics which is proved in this paper. Namely, I will describe a different proof of a theorem of Moeller that for Teichmuller curves, zeroes of the holomorphic ${1}$-form must land in the torsion of (a factor of) the Jacobian. Some familiarity with Teichmuller dynamics will be assumed, as in the survey of Zorich.

Theorem 1 Suppose ${E\rightarrow B}$ is a holomorphic fibration, with fiber a Kobayashi-hyperbolic manifold ${F}$. Then the fibration is locally flat, i.e. the gluing maps on overlapping local charts must be (locally) constant.
Remark 1 The requirement that the fibration be holomorphic means that locally it is holomorphically isomorphic to ${U\times F}$. For example, if ${{\mathbb B}=\{(z,w)\in {\mathbb C}^2 \rvert |z|^2+|w|^2<1\}}$ is the unit ball in ${{\mathbb C}^2}$, the natural projection ${{\mathbb B}\rightarrow {\mathbb D}}$ to the unit disk in ${{\mathbb C}}$ is not a holomorphically trivial fibration. One can see this using the maximum principle (see the MO post for details).