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Section 1.7 Line Bundles and the Dual Abelian Variety (Angus)

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Subsection 1.7.1 Introduction

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Meta-goal.

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Understand line bundles on abelian varieties.

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Setup.

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A an abelian variety /k\text{.}

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Last time.

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For L a line bundle on A we get a map

\begin{align*} \phi_L\colon A(K) \amp\to \Pic(A)\\ a\amp\mapsto t_a^* L\otimes L^{-1} \end{align*}

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where

\begin{equation*} \Pic(A) = \{\text{line bundles on } A\}/\sim\text{.} \end{equation*}

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This a is a group homomorphism (by the theorem of the square 1.5.15). We define

\begin{equation*} K(L)(k) = \ker(\phi_L) = \{a\in A(k) : t_a^* L \simeq L\}\text{.} \end{equation*}
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Today.

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We are going to package these into a big map

\begin{align*} \phi\colon \Pic(A)\amp\to \Hom(A(k), \Pic(A))\\ L \amp\mapsto \phi_L\text{.} \end{align*}
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1.
\begin{align*} \phi_{L\otimes M}(a) \amp = t_a^*(L\otimes M) \otimes(L\otimes M)^{-1}\\ \amp = t_a^*L\otimes L^{-1} t_a^*M\otimes M^{-1}\\ \amp = \phi_L\otimes \phi_M \end{align*}
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2.
\begin{align*} \phi_{t_b^*L}(a) \amp = t_a^*(t_b^*L) \otimes(t_b^*L)^{-1}\\ \amp = t_{a+b}^*L \otimes(t_b^*L)^{-1}\\ \amp = t_{a}^*L \otimes t_b^*L \otimes L^{-1} \otimes (t_b^*L)^{-1}\\ \amp = \phi_L(a) \end{align*}

by the theorem of the square 1.5.15

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Definition 1.7.2.
\begin{align*} \Pic^0(A) \amp = \ker(\phi)\\ \amp = \{ L \in \Pic(A) : \phi_L = 0\}\\ \amp = \{ L \in \Pic(A) : t_a^* L\simeq L \ \forall a\in A(k)\}\\ \amp = \{\text{translation invariant line bundles}\}/\sim \end{align*}
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Goals.

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Study \Pic^0(A)\text{,} give it an abelian variety structure, solve a moduli problem, demonstrate some duality.

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Subsection 1.7.2 Aside: alternate description of \Pic^0(A)

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Definition 1.7.3. Algebraic Equivalence.

Two line bundles L_1,L_2 on an abelian variety are algebraically equivalent if there exists a variety Y with line bundle L on A\times Y and points y_1y_2 \in Y s.t. L|_{A\times\{y_1\}} \simeq L_1, L|_{A\times\{y_2\}} \simeq L_2\text{.}

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Remark 1.7.4.

This looks like homotopy.

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Subsection 1.7.3 See-Saws

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Remark 1.7.7.

In fact M = p_{2*}L\text{.}

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Subsection 1.7.4 Properties of \Pic^0 A

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1.
\begin{equation*} (m^*L\otimes (p_1^*l)^{-1} \otimes (p_2^*l)^{-1})|_{A\times\{a\}} = t_a^*L \otimes L^{-1} = \sheaf O_A \end{equation*}
\begin{equation*} (m^*L\otimes (p_1^*l)^{-1} \otimes (p_2^*l)^{-1})|_{\{a\}\times A} = t_a^*L \otimes L^{-1} = \sheaf O_A \end{equation*}

by see-saw 1.7.6 whole thing is trivial on \(A\times A\text{.}\)

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2.
\begin{equation*} (f+g)^*L\cong (f\times g)^* m^*L \cong (f\times g)^*(p_1^*L \otimes p_2^*L) \cong f^*L \otimes g^*L \end{equation*}
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3.

Induction of 3.

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4.
\begin{equation*} \phi_{\phi_L(a)} = \phi_{t^*_a L}\otimes L^{-1} = \phi _{t_a^*L}\otimes L^{-1} = \phi_L \otimes \phi_{L^{-1}} = 0 \end{equation*}

If \(H^0(A,L) \ne 0\text{,}\) we would have a nontrivial section \(s\) of \(L\) then \(\lb -1\rb ^* s\) is a nontrivial section of \(\lb-1\rb^* L = L^{-1}\text{.}\) But if both \(L\) and \(L^{-1}\) have a nontrivial section then \(L \cong \sheaf O_A\text{.}\) So since \(L\) is nontrivial \(H^0(A,L) = 0\text{.}\) Now assume \(H^i(A,L) = 0\) for all \(i\lt j\text{.}\) Consider

\begin{gather*} A\xrightarrow{\id \times 0} A\times A \xrightarrow m A\\ a\mapsto (a,0)\mapsto a \end{gather*}

this gives

\begin{equation*} H^j(A, L) \to H^j(A\times A, m^*L) \to H^j(A,L) \end{equation*}

which composes to the identity.

\begin{equation*} H^j(A\times A, m^*L) = H^j(A\times A, p_1^*L \otimes p_2^*L) = \bigoplus_{i=0}^j H^i(A,L) \otimes H^{j-i}(A,L) \end{equation*}

by Künneth. The RHS is 0 by the inductive hypothesis. So the identity on \(H^j(A,L)\) factors through 0, hence the group is 0.

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We now think of \phi_L as a map \phi_L \colon A(k) \to \Pic^0(A) with kernel K(L) (k)\text{.}

Idea is to study

\begin{equation*} \Lambda (L) = m^* L \otimes (p_1^* L)^{-1} \otimes (p_2^* L)^{-1}\text{.} \end{equation*}

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Given an ample line bundle L on A we now have an isomophism of groups

\begin{equation*} A(k)/K(L)(k) \cong \Pic^0(A) \end{equation*}

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the LHS allows us to put an abelian variety structure on \Pic^0(A)\text{.}

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Subsection 1.7.5 The Dual Abelian Variety

(Sketch) (characteristic 0) Cover \(A\) by affine opens \(U_i = \Spec R_i\) such that for all \(a \in A\) the orbit \(K(L)a \subseteq U_i\) for some \(i\text{.}\) We can do this because abelian varieties are projective. Then we say \(U_i / K(L) = \Spec(R^{K(L)}_i)\) then glue. (details in Mumford, II sec, 6 appendix). Since we are in characteristic 0, the quotient scheme is in fact a variety.

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Definition 1.7.13. Dual abelian varieties.

The dual abelian variety is

\begin{equation*} \hat A = A/K(L)\text{.} \end{equation*}
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Remark 1.7.14.
  • \begin{equation*} \hat A(K) = \Pic^0(A) \end{equation*}
  • We have an isogeny
    \begin{equation*} \phi_L\colon A \to \hat A\text{.} \end{equation*}
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Subsection 1.7.6 Dual morphisms

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Let f\colon A\to B be a homomorphism of abelian varieties. Let \sheaf P_A,\sheaf P_B be the Poincaré bundles on A and B\text{.} Consider M= (F\times \id_{\hat B})^* \sheaf P_B on A\times \hat B\text{,} then

  1. \begin{equation*} M|_{A\times \{x\}} \in \Pic^0(A) \end{equation*}
  2. \begin{equation*} M|_{\{0\} \times \hat B} = 0 \end{equation*}

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thus by the universal property we get a unique morphism

\begin{equation*} \hat f\colon \hat B \to \hat A \end{equation*}

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satisfying

\begin{equation*} (\id_A\times \hat f)^* \sheaf P_A = (f\times \id_{\hat B})^*\sheaf P_B\text{.} \end{equation*}
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Definition 1.7.16. Dual morphisms.

\hat f as above is called the dual morphism.

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Remark 1.7.17.
  • \begin{equation*} \hat f\colon \hat B = \Pic^0(B) \to \hat A(k) = \Pic^0(A) \end{equation*}
    \begin{equation*} L\mapsto f^*L \end{equation*}
  • \begin{equation*} \hat{\lb n_A\rb} = [n_{\hat A}] \end{equation*}

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Consider the Poincaré bundle \sheaf P_{\hat A} on \hat A \times \hat{\hat{A}}\text{,} now think of \sheaf P_A as living on \hat A \times A\text{.} By the universal property of \sheaf P_{\hat A} get a unique morphism

\begin{equation*} \operatorname{can}_A\colon A\to\hat{\hat A}\text{.} \end{equation*}
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Definition 1.7.21. Symmetric morphisms, (principal) polarizations.

A morphism f\colon A \to \hat A is symmetric if f = \hat f\circ \operatorname{can}_A

A polarization is a symmetric isogeny f\colon A \to \hat A s.t. f= \phi_L for some ample line bundle L on A\text{.}

A principal polarization is a polarization of degree 1, i.e. an isomorphism.

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Remark 1.7.22.

Elliptic curves always admit principal polarization.

If one wishes to mimic the theory of elliptic curves, one should study principally polarized abelian varieties.