Grothendieck famously wrote a paper called Hodge’s general conjecture is false for trivial reasons. In this post, I would like to record, for my own benefit, some of the observations made in it.
Consider the following question. If 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
Defining , 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 is holomorphic in each variable separately. Then is holomorphic on .
Below the fold, I will state some more results of this type and prove some of them. Continue reading
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.
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).
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 -algebra has the cardinality of the continuum
- (Sierpinski) There is a set in the plane such that every line intersects it in exactly two points
- (Cantor-Bendixson) Any closed subset of 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.
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