### Abelian Varieties

Now, we’re going to talk a bit about the geometry of the periods, which were completely analytic in nature.  As we mentioned, for a compact Riemann surface $X$, we have a period matrix $\Omega$ that encodes the complex integration theory on the surface.

I’m back! And now, posting from Kavli IPMU in Japan.  Now, I’m going to start a series on theta functions, Jacobians, Pryms, and abelian varieties more generally, hopefully with some applications, with my goal being at least one post a week, and eventually establishing a regular posting schedule again.  But today, we’ll start with basics, something that should be completely understandable to graduate students and advanced undergraduates.

Let ${\pi:\tilde{C}\rightarrow C}$ be an unramified double cover, where ${C}$ is geneus ${g}$. Then ${\tilde{C}}$ has genus ${2g-1}$ by the Riemann-Hurwitz formula. Now, ${J(C)}$ encodes lots of information about the geometry of ${C}$, especially with the additional data of the theta divisor. It turns out that for double covers, there’s an abelian variety that contains a lot of this data.

Last time we defined theta characteristics as square roots of the canonical bundle.  Today, we’re going to analyze the notion a bit, and relate them to quadrics in characteristic two.

For this whole post, we’ll take $C$ to be a curve and $J=J(C)$ the Jacobian of the curve.  We’re going to construct several special subvarieties (not special in any technical sense, though) of $J$, which encode a great deal of geometric information about $C$.

As promised in the last post, I’m making another go at MaBloWriMo…maybe others will as well.  I don’t know if I’m going to have a coherent topic over the course of this month, but I’ll be starting with abelian varieties associated to curves, so expect me to talk about generalizations of Prym varieties eventually (unless I get distracted by something else along the way).  Today, the basic case: Jacobians!

These are my notes, and are only a rough approximation of the actual talk:

This post begins my series on some classical geometry of curves and abelian varieties. We’ll start with some talk of line bundles and polarizations on abelian varieties in general, and the first big theorem I’m really targeting is the Torelli theorem (going to go with Andreotti’s proof, though once some other stuff is covered, might reprove it another way or two) but I might get distracted by other things along the way. We’ll see how this goes. Posting will be sporadic at best, and most of this material is, in more detailed form, going to appear in my master’s thesis (hopefully).

Today we’re back to some material from the first post in this series, and going to prove an actual theorem about vector bundles.  Next time, we’ll be getting into the heart of the paper, and that may be my last post on B-N-R.

The last post was on the generalities of Abelian varieties, and constructing a map.  This time, we’re going to do it for a specific one, and the maps involved will all be useful later.  We start out with $\pi:Y\to X$ a finite morphism of curves.

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