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 -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.
This brief note was inspired by this MathOverflow post. A definition of a Kobayashi hyperbolic manifold is available here. Informally, it says that one can’t holomorphically map arbitrarily large complex unit disks into the complex manifold. In a certain sense “most” complex manifolds are Kobayashi-hyperbolic. Below is the proof of the following theorem of Royden, proved in this paper.
Theorem 1 Suppose is a holomorphic fibration, with fiber a Kobayashi-hyperbolic manifold . 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 . For example, if is the unit ball in , the natural projection to the unit disk in is not a holomorphically trivial fibration. One can see this using the maximum principle (see the MO post for details).