# 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.

# Semisimplicity of the Kontsevich-Zorich cocycle

In this post, I would like to discuss some of the ideas from my paper Semisimplicity and Rigidity of the Kontsevich-Zorich cocycle. It contains a number of somewhat disparate methods – Hodge theory, random walks and a bit of homogeneous and smooth dynamics. I will try to explain how these concepts interact and what the essential aspects are. Some familiarity with flat surfaces, as explained for example in the survey by Anton Zorich, will be assumed.

Below the fold I will discuss some of the analytic ideas from variations of Hodge structures. Then I’ll try to connect these to Teichmuller dynamics and explain how this leads to various semisimplicity results. For applications, I’ll discuss how this implies that measurable ${{\mathrm{SL}}_2{\mathbb R}}$-invariant bundles have to be real-analytic. I’ll also mention how real multiplication on factors of the Jacobians arises.

# Functions nice in each variable separately

Consider the following question. If ${f:{\mathbb R}^2\rightarrow {\mathbb R}}$ is continuous in each variable separately, need it be continuous in both? Although Cauchy (incorrectly) said it should, note it is not even obvious that it should be measurable.

A classical example is

$\displaystyle f(x,y):= \frac {xy}{x^2+y^2}$

Defining ${f(0,0)=0}$, this function is real-analytic in each variable separately, but it is not even continuous at the origin.

By contrast, in the complex-analytic situation we have the following result.

Theorem 1 (Hartogs) Suppose ${f:{\mathbb C}^{n_1}\times {\mathbb C}^{n_2}\rightarrow {\mathbb C}}$ is holomorphic in each variable separately. Then ${f}$ is holomorphic on ${{\mathbb C}^{n_1+n_2}}$.

Below the fold, I will state some more results of this type and prove some of them. Continue reading

# Descent for vector spaces

In this note, I would like to record mainly for my own benefit some basic things about how to descend algebraic data along Galois extensions of fields. I have benefited from some notes of K. Conrad available online. The exposition below is a shorter version of those.

# Holomorphic bundles with hyperbolic fibers

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 ${E\rightarrow B}$ 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).

# Ordinals (the basics)

This note is meant to collect a few basic facts about ordinals. I found that I didn’t really know how they work and this was embarrassing. I also couldn’t find any source that covered both the basics and some interesting applications, so I tried to assemble them here. Some results proved below are

• The Borel ${\sigma}$-algebra has the cardinality of the continuum
• (Sierpinski) There is a set ${A}$ in the plane such that every line intersects it in exactly two points
• (Cantor-Bendixson) Any closed subset of ${{\mathbb R}}$ is a union of a perfect set and a countable set.

In what follows, many set-theoretical subtleties are ignored. In particular, von Neumann’s construction of ordinals is not presented.