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.

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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).

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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.

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Rigidity for Isometric Actions

In this post, I will discuss some elementary results about the action of an isometry on a compact metric space. I found them as homework problems assigned by Prof. Amie Wilkinson in a Smooth Dynamics course, and although quite elementary, I found the results rather striking.

In summary, (ergodic) actions by isometries look a lot like translations on compact abelian groups. For example, orbit closures in a Riemannian manifold are always tori. Precise statements are below the fold. Continue reading