BUNTES Line Bundles and the Dual Abelian Variety (Angus)

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

Subsection 1.7.1 Introduction

Meta-goal.

Understand line bundles on abelian varieties.

Setup.

\(A\) an abelian variety \(/k\text{.}\)

Last time.

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*}

where

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

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*}

Today.

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*}

Proposition 1.7.1.

\(\phi\) is a group homomorphism
\begin{equation*} \phi_{t_a^* L} = \phi_L \end{equation*}

Proof.

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*}

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

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*}

Goals.

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

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

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. \(L|_{A\times\{y_1\}} \simeq L_1, L|_{A\times\{y_2\}} \simeq L_2\text{.}\)

Remark 1.7.4.

This looks like homotopy.

Proposition 1.7.5.

\begin{equation*} \Pic^0(A) = \{\text{line bundles which are alg. equiv to } \sheaf O_A\} \end{equation*}

Proof.

[81].

Subsection 1.7.3 See-Saws

Theorem 1.7.6. See-saw theorem.

Let \(X,T\) be varieties \(X\) complete, let \(L\) be a line bundle on \(X\times T\text{,}\) let \(T_1 = \{t\in T : L|_{X\times\{t\}} \text{ is trivial}\}\) then \(T_1\) is closed in \(T\text{.}\) Further let \(p_2\colon X\times T_1 \to T_1\text{,}\) then \(L|_{X\times T_1} \cong p^*_2 M\) for some line bundle \(M\) on \(T_1\text{.}\)

Remark 1.7.7.

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

Corollary 1.7.8. that no one states/only Milne.

Let X, T be as above and let \(L, M\) be line bundles on \(X\times T\) s.t.

\begin{equation*} L|_{X\times \{t\}} \cong M|_{X\times \{t\}} \forall t\in T \end{equation*}

\begin{equation*} L|_{\{t\}\times X} \cong M|_{\{t\}\times X} \text{ for some } x\in X \end{equation*}

then \(L\cong M\text{.}\)

Subsection 1.7.4 Properties of \(\Pic^0 A\)

Lemma 1.7.9.

\(L\in \Pic^0(A)\) and \(m,p_1,p_2\colon A\times A \to A\)

\begin{equation*} m^* L \cong p_1^* L\otimes p_2^* L \end{equation*}
Given \(f,g \colon X \to A\)
\begin{equation*} (f+g)^* L \cong f^* L \otimes g^* L \end{equation*}
\begin{equation*} [n]^* L \cong L^{\otimes n} \end{equation*}
\begin{equation*} \phi_L(A(k)) \subseteq \Pic^0(A) \end{equation*}
for \(L \in \Pic (A)\text{.}\)

Proof.

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 \(A\times A\text{.}\)

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*}

3.

Induction of 3.

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*}

Proposition 1.7.10.

If \(L\) is nontrivial in \(\Pic^0(A)\) then \(H^i(A,L) = 0\) \(\forall i\text{.}\)

Proof.

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.

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

Theorem 1.7.11.

If \(K(L)(k)\) is finite then \(\phi_L\) is surjective.

Proof.

Idea is to study

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

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*}

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

Subsection 1.7.5 The Dual Abelian Variety

Theorem 1.7.12.

Let \(A\) be an abelian variety and \(L\) an ample line bundle on \(A\text{,}\) then the quotient scheme \(A/K(L)\) exists and is an abelian variety of the same dimension as \(A\text{.}\)

Proof.

(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.

Definition 1.7.13. Dual abelian varieties.

The dual abelian variety is

\begin{equation*} \hat A = A/K(L)\text{.} \end{equation*}

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*}

Theorem 1.7.15.

There is a unique line bundle \(\sheaf P\) on \(A\times \hat A\) called the Poincaré bundle such that

\begin{equation*} \sheaf P|_{A\times \{x\}} \in \Pic^0(A) \text{ for all }x\in \hat A \end{equation*}
\begin{equation*} \sheaf P|_{0\times \hat A} = 0 \end{equation*}
If \(Z\) is a scheme with a line bundle \(R\) on \(A\times Z\) satisfying 1., 2., there exists a unique
\begin{equation*} f\colon Z\to \hat A \end{equation*}
s.t.
\begin{equation*} (\id\times f)^* \sheaf P = R\text{.} \end{equation*}

That is \((\hat A, \sheaf P)\) represents the functor

\begin{equation*} Z \mapsto \left\{ L\in \Pic (A\times Z) : \substack{ L|_{A\times \{z\}} \in \Pic^0(A) \forall z\in Z \\L|_{ 0 \times Z } = 0}\right\}/\sim\text{.} \end{equation*}

Subsection 1.7.6 Dual morphisms

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

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

thus by the universal property we get a unique morphism

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

satisfying

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

Definition 1.7.16. Dual morphisms.

\(\hat f\) as above is called the dual morphism.

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*}

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*}

Theorem 1.7.18.

\(\operatorname{can}_A\) is an isomorphism.

Lemma 1.7.19.

\begin{equation*} \phi_{f^*L} = \hat f\circ \phi_L\circ f\text{.} \end{equation*}

Proposition 1.7.20.

If \(f\colon A \to B\) is an isogeny, then \(\hat f\colon \hat B \to \hat A\) is an isogeny. Further if \(N = \ker f\text{,}\) then \(\hat N = \ker \hat f\) is the Cartier dual of \(N\text{.}\)

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.

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.