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