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.

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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.

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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.

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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.

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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.

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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!

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

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