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 , we have a period matrix that encodes the complex integration theory on the surface.

### Abelian Varieties

November 19, 2012

## Understanding Integration III: Jacobians

Posted by Charles Siegel under Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, CurvesLeave a Comment

October 1, 2012

## Understanding Integration I: Riemann Surfaces

Posted by Charles Siegel under Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves[5] Comments

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.

November 19, 2010

## Prym Varieties

Posted by Charles Siegel under Abelian Varieties, AG From the Beginning, Algebraic Geometry, Cohomology, Curves, Examples, Hodge Theory, MaBloWriMoLeave a Comment

Let be an unramified double cover, where is geneus . Then has genus by the Riemann-Hurwitz formula. Now, encodes lots of information about the geometry of , 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.

November 3, 2010

## Theta Characteristics and Quadrics in Characteristic Two

Posted by Charles Siegel under Abelian Varieties, AG From the Beginning, Algebraic Geometry, Curves, MaBloWriMoLeave a Comment

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.

November 2, 2010

## Subvarieties of Jacobians

Posted by Charles Siegel under Abelian Varieties, AG From the Beginning, Algebraic Geometry, CurvesLeave a Comment

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

November 1, 2010

## Jacobians of Curves

Posted by Charles Siegel under Abelian Varieties, AG From the Beginning, Algebraic Geometry, Curves, Hodge Theory, MaBloWriMoLeave a Comment

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!

July 1, 2010

## The Schottky Problem (ICTP)

Posted by Charles Siegel under Abelian Varieties, Conferences, Curves, Hodge Theory, ICTP Summer School, Moduli of Curves1 Comment

April 19, 2010

## Riemann’s Bilinear Relations

Posted by Charles Siegel under Abelian Varieties, Algebraic Geometry[2] Comments

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).

September 10, 2009

## B-N-R Part 5: Spectral Curves

Posted by Charles Siegel under Abelian Varieties, Big Theorems, Curves, Vector Bundles[4] Comments

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.

September 9, 2009

## B-N-R Part 4: Prym Varieties

Posted by Charles Siegel under Abelian Varieties, Big Theorems, Curves, Vector BundlesLeave a Comment