Now, we continue our tour through Beauville, Narasimhan, Ramanan.  We’ve talked about Twisted Endomorphisms and we’ve talked about the Generalized Theta Divisor on the Moduli Space of Vector Bundles.  So today we’ll talk a bit about Abelian Varieties, Principal Polarizations, and we’ll prove some lemmas.  This one’s going to be a bit longer than the last few.

We start with (A,\Theta) a principally polarized abelian variety.  As for what that is, an abelian variety is an irreducible projective algebraic group.  Pick \Theta a divisor.  Then we have a map A\to \mathrm{Pic}^0(A) by a\mapsto t_a^*\Theta-\Theta, where t_a:A\to A is given by t_a(x)=x+a.  If this map is an isogeny (that is, a surjective finite morphism) it is called a polarization.  What we want is a principal polarization, so we require that this map be an isomorphism.

The reason people care about principally polarized abelian varieties (ppav’s) is that this is a natural generalization of the notion of the Jacobian of a curve.  The pair consisting of a Jacobian and its Theta Divisor is an example of a ppav.  An important and interesting problem is to determine which ppav’s are Jacobians.

Anyway, we take N\subset A an abelian subvariety, so it’s a subgroup variety of A, and is irreducible.  Then we define the orthogonal, P, to be the abelian subvariety given by \{a\in A|t_a^*\Theta|_N=\Theta|_N\}.  It is fairly easy to check that P\to A/N and N\to A/P are both isogenies.  Now, we define \Theta_N and \Theta_P to be the restrictions of \Theta to N and P.  We can define some rational maps now.  There’s a straightforward one \phi:P\to |\Theta_P|^* which is given by the linear system |\Theta_P|, but also, we can define \psi:P\to |\Theta_N| by p\mapsto t_p^*(\Theta)|_N.

Proposition: There is a canonical isomorphism \iota:|\Theta_P|^*\to |\Theta_N| such that \iota\circ\phi=\psi.

Proof: Look at the isogeny \pi:N\times P\to A by summing points.  (To see that it and the earlier isogenies are isogenies, look at the Lie algebras, and it follows fairly easily).  We want to start by comparing \pi^*\Theta with \pi_N^*\Theta_N\otimes \pi_P^*\Theta_P.  To do so, we will use:

Theorem of the Square: Let A,B be abelian varieties and L\in \mathrm{Pic}(A\times B).  Then, if for all a\in A,b\in B, we have L|_{\{a\}\times B} and L|_{A\times\{b\}} are trivial, then L is trivial.

This theorem isn’t very hard to prove, and shows up in many guises depending on what you’re looking to do.  In our case, we’re going to apply this theorem to \pi_N^*\Theta_N\otimes \pi_P^*\Theta_P\otimes (\pi^*\Theta)^{-1} to show that the two bundles we’re looking at are, in fact, isomorphic.  Doing so consists of playing with the definition of the orthogonal abelian subvariety, and showing that restricting to N, both become \Theta_N and similarly for P.

So now that \pi^*\Theta\cong \pi_N^*\Theta_N\otimes \pi_P^*\Theta_P, we take the canonical section of \Theta, and it’s pullback is then an element of \Gamma(N,\Theta_N)\otimes \Gamma(P,\Theta_P), and so induces a map \iota:\Gamma(P,\Theta_P)^*\to \Gamma(N,\Theta_N).  We claim that this map is the isomorphism we seek.

Though I’m not going to go into careful details with defining the group involved, the structure of the remainder of the argument is as follows: our section is the pullback of an effective divisor, and so it is nonzero.  So we now need to show that it is nondegenerate.  The section turns out to be equivalent to giving a map between irreps of the group, and the group has a unique finite dimensional irrep: \Gamma(\Theta_P)\otimes \Gamma(\Theta_N).  As the divisor is nonzero, its an isomorphism of irreps, and so is an isomorphism.

To actually see that this isomorphism is the one we want, we note that \phi is characterized by invariance properties, and that \iota^{-1}\circ\psi has the same properties, so they are equal.  \Box.

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