This, like the previous and probably the next few posts, is an attempt at understanding some of the ideas in Mumford’s Abelian Varieties.

Let ${A}$ be an abelian variety. Last time, I described a formula which allowed us to express the pull-back ${n_A^* \mathcal{L}}$ of a line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$ as

$\displaystyle n_A^*\mathcal{L} \simeq \mathcal{L}^{(n^2 + n)/2} \otimes (-1)^* \mathcal{L}^{(n^2 - n)/2}.$

This was a special case of the so-called “theorem of the cube,” which allowed us to express, for three morphisms ${f, g, h: X \rightarrow A}$, the pull-back

$\displaystyle (f + g + h)^* \mathcal{L}$

in terms of the various pull-backs of partial sums. Namely, we had the formula

$\displaystyle (f + g+ h)^* \mathcal{L} \simeq (f+g)^* \mathcal{L} \otimes (f + h)^* \mathcal{L} \otimes (g + h)^* \mathcal{L} \otimes f^* \mathcal{L}^{-1} \otimes g^* \mathcal{L}^{-1} \otimes h^{*} \mathcal{L}^{-1} .$

In this formula, we take ${f = \mathrm{Id} = 1_A}$ and ${g, h}$ constant maps at ${x,y \in A}$, respectively. Let ${T_x, T_y}$, etc. denote the translation maps on ${A}$. Then ${T_x, T_y, T_{x+y}}$ are the relevant sums, and we have

$\displaystyle T_{x+y}^* \mathcal{L} \simeq T_x^* \mathcal{L} \otimes T_y^* \mathcal{L} \otimes \mathcal{L}^{-1}.$

We get the theorem of the square:

Theorem 1 (Theorem of the square)Let ${A}$ be an abelian variety, ${\mathcal{L} \in \mathrm{Pic}(A)}$, and ${x, y \in A}$ points. Then

$\displaystyle T_{x+y}^* \mathcal{L} \otimes \mathcal{L} \simeq T_x^* \mathcal{L} \otimes T_y^* \mathcal{L}.$

In the rest of this post, I’ll describe some applications of this result, for instance the fact that abelian varieties are projective.  (more…)

1. Introduction

Let ${A}$ be an abelian variety over an algebraically closed field ${k}$, of dimension ${g}$. One of the basic tools in analyzing the properties of ${A}$ is the study of line bundles on ${A}$. It’s a little non-intuitive to me why this is the case, so I’m going to try to motivate the topic.

Given ${A}$, we are interested in questions of the following form: What is the structure of the ${n}$-torsion points ${A[n]}$? To compute ${|A[n]|}$, we are reduced to computing the degree of multiplication by ${n}$,

$\displaystyle n_A: A \rightarrow A$

(which is a morphism of varieties). In fact, we will show that ${n_A}$ is a finite flat morphism, and determine the degree of ${n_A}$, which is thus the cardinality of the fiber over ${0}$. The determination will be done by analyzing how ${n_A^*}$ acts on line bundles. For a symmetric line bundle ${\mathcal{L}}$ over ${A}$, one can prove the crucial formula

$\displaystyle n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2},$

and comparing the Hilbert polynomials of ${\mathcal{L}}$ and ${n_A^* \mathcal{L}}$, one can get as a result

$\displaystyle \deg n_A = n^{2g} .$

Another way of phrasing this deduction is the following. By the Hirzebruch-Riemann-Roch formula and the parallelizability of an abelian variety, we have

$\displaystyle \chi(\mathcal{L}) = \frac{c_1(\mathcal{L})^g}{g!}$

for any line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$. Consequently, in view of the asserted formula ${n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2}}$, we find for any line bundle ${\mathcal{L}}$:

$\displaystyle \chi(n_A^* \mathcal{L}) = n^{2g} \chi(\mathcal{L}).$

Choosing ${\mathcal{L}}$ to be a high power of a very ample line bundle, we will have ${\chi(\mathcal{L}) \neq 0}$. Now we can appeal to the following result:

Theorem 1 Let ${X}$ be a proper scheme over a field. Let ${G}$ be a finite group scheme, and let ${\pi: P \rightarrow X}$ be a ${G}$-torsor. Then if ${\mathcal{F}}$ is a coherent sheaf on ${X}$, we have

$\displaystyle \chi(\pi^* \mathcal{F}) = (\deg \pi) \chi(\mathcal{F}).$

It follows from this result that ${\deg n_A = n^{2g}}$. For ${n}$ prime to the characteristic, the morphism ${n_A}$ can be seen to be separable, and it follows as a result there are ${n^{2g}}$ points of ${n}$-torsion on ${A}$. (more…)

So, now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one another. Namely, we want to say that, given a complex vector bundle, we can compute the Chern classes in de Rham cohomology by picking a connection — any connection — on it,  computing the curvature, and then applying various polynomials.

We shall start by warming up with a special case, of a line bundle, where the algebra needed is easier. Let ${M}$ be a smooth manifold, ${L \rightarrow M}$ a complex line bundle. Let ${\nabla}$ be a connection on ${L}$, and let ${\Theta}$ be the curvature.

Thus, ${\Theta}$ is a global section of ${\mathcal{A}^2 \otimes \hom(L, L)}$; but since ${L}$ is a line bundle, this bundle is canonically identified with ${\mathcal{A}^2}$. (Recall the notation that $\mathcal{A}^k$ is the bundle (or sheaf) of smooth $k$-forms on the manifold $M$.)

Proposition 1 (Chern-Weil for line bundles) ${\Theta}$ is a closed form, and the image in ${ H^2(M; \mathbb{C})}$ is ${2\pi i}$ times the first Chern class of the line bundle ${L}$. (more…)