Pic Zero and the Poincare Bundle

For any scheme X, \mbox{Pic}(X) is the group of line bundles on X.  In the case of an abelian variety A, there is a particularly nice subgroup \mbox{Pic}^0(A).

If L \in \mbox{Pic}^0 (A) then,

  1. m^*L\otimes p_1^*L^{-1} \otimes p_2^*L^{-1} is trivial, in particular f,g\colon S \to A \Rightarrow (f+g)^*L \cong f^*L \otimes g^*L
  2. There is a surjection with finite kernel (an isogeney) \phi\colon A \to \mbox{Pic}^0(A)
  3. If M is a line bundle on A\times S s.t. there exists s_0 \in S with M|_{A\times s_0} \in \mbox{Pic}^0(A) then M|_{A\times s} \in \mbox{Pic}^0(A) for all s \in S.

Some big results

The above claims follow pretty easily from two pretty big results:

Thm: (See-saw priciple) Let X be a complete variety, and L a line bundle on X\times S with trivial restriction on all the fibers: L|_{X\times s} \cong \mathcal{O}_X.  Then L \cong p_2^*N for some line bundle N on S.

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pf(sketch): Set L_s = L|_{X\times s}.  The significant part of the proof is an application of cohomology with base change (see page 20-21 or J.Milne’s abel. var. notes).  The condition on the fibers of L gives that p_{2*}L is a locally free sheaf and in particular invertible.  Now there is always a natural map

\alpha\colon p_2^*p_{2*}L \to L

Set X_s = X \times s \cong X.  For any open set U \subset X_s, one can compute

p_2^*p_{2*}L_s(U) = \mathcal{O}_X(U) \otimes_{k(s)} p_2^{-1}p_{2*}L_s(U) = \mathcal{O}_X(U) \otimes \Gamma(X, L_s)

Using L_s \cong \mathcal{O}_X and using that for a complete variety there are only constant global section (see post) I conclude p_2^*p_{2*}L_s \to L_s is an isomorphism.  Its an isomorphism on fibers, and in particular an iso for any point t \in X_s, i.e. for every p \in X\times S the map on stalks is an iso after modding out by the maximal ideal and a Nakayama’s lemma argument shows it was already an isomorphism.   So \alpha is an isomorphism (since it is on stalks). QED.

Thm (of the cube): Let X,Y complete and Z another variety and L a line bundle on X\times Y \times Z.  If there exists x,y,z such that the restriction of L to x \times Y\times Z ,X \times y \times Z, X \times Y \times z is trivial, then L is trivial.

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The proof is not too bad.  A rough summary: first show it holds when X is a complete curve (this uses that the Jacobian of a curve is a certain moduli space and the rigidity lemma).

For the general case, let L be the line bundle on X \times Y \times Z.  Use a complete curve C to connect any x' \in X to the distinguished x \in X (see this post).  So have

$latex C \times Y \times Z \xrightarrow{i} X \times Y

\times Z \xrightarrow{p} Z \times Z$

By the case of a curve, i^*L is trivial. This shows for every (x,z) the restriction of L to x \times Y \times z is trivial, i.e. trivial on the fibers of p.  By the see-saw theorem it follows L \cong p^*N, but N \cong L|_{X \times y \times Z} which is trivial, hence L is trivial.

Corollaries

Let p,q,r denote the three projections A \times A \times A \to A.  Then

Cor1: For any line bundle L on A, the line bundle

\begin{array}{rl} pqr(L):= & (p+q+r)^*L \\ \mbox{} & \otimes (p+q)^*L^{-1}\otimes(p+r)^*L^{-1}\otimes(q+r)^*L^{-1} \\ \mbox{} & \otimes p^*L\otimes q^*L \otimes r^*L \end{array}

is trivial on A \times A \times A.

Cor2: For morphisms f,g,h \colon S \to A the line bundle

\begin{array}{rl} fgh(L):= & (f+g+h)^*L \\ \mbox{} & \otimes (f+g)^*L^{-1}\otimes(f+h)^*L^{-1}\otimes(g+h)^*L^{-1} \\ \mbox{} & \otimes f^*L\otimes g^*L \otimes h^*L \end{array}

on S is trivial.

pf: fgh(L) = (f,g,h)^{-1}pqr(L), where (f,g,h) \colon S \to A\times A \times A.QED.

Cor3. For all a,b \in A and L a line bundle there is an isomorphism

t_{a+b}^*L \otimes L \cong t_a^*L \otimes t_b^*L

In particular, a \mapsto t_a^*L \otimes L^{-1} is a homomorphism.

pf: apply the previous corollary to id, a, b \colon A \to A , where a \colon A \to A sends everything to a.  The last statement follows by multiplying the above by L^{-2}  QED.

Given a line bundle L , define \phi_L \colon A \to \mbox{Pic}(A) by \phi_L(a) = t_a^*L \otimes L^{-1}.  The following properties are readily checked:

  1. \phi_L(a+b) = \phi_L(a)\otimes \phi_L(b) =: \phi_L(a) + \phi_L(b)
  2. \phi_{L\otimes M}(a) = \phi_L(a)\otimes \phi_M(a) =: (\phi_L + \phi_M)(a)
  3. \phi_{t_a^*L \otimes L^{-1}} = 0

For the last statement it suffices to show \phi_L = \phi_{t_b^*L}:

\phi_{t_b^*L}(a) = t_a^*(t_b^*L) \otimes t_b^*L^{-1}

= t_{a+b}^*L \otimes t_b^*L^{-1} = t_a^*L \otimes L^{-1} \otimes t_b^*L \otimes t_b^*L^{-1}

= t_a^*L \otimes L^{-1} = \phi_L(a)

Finally, define \mbox{Pic}^0(A) = \{L \in \mbox{Pic}(A) | \phi_L = 0\}.  Set \Lambda(L) = m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}. Note that by 3 above, \phi_L lands in \mbox{Pic}^0.  Now here are some more properties that are not that hard to prove

  1. L \in \mbox{Pic}^0(A) iff \Lambda(L) is trivial (see-saw principle)
  2. f,g \colon S \to A \Rightarrow (f+g)^*L \cong f^*L \otimes g^*L
  3. [n]^*L \cong L^n
  4. L^n \cong \mathcal{O}_A \Rightarrow L \in \mbox{Pic}^0  because 0 = \phi_{L^n}(a) = \phi_L(na) and use divisibility.
  5. For K(L) = \ker \phi_L, if K(L) is finite, then \phi_L \colon A \to \mbox{Pic}^0(A) is surjective. (uses Leray Spectral sequence)

For the last bullet point in the introduction, namely If M is a line bundle on A\times S s.t. there exists s_0 \in S with M|_{A\times s_0} \in \mbox{Pic}^0(A) then M|_{A\times s} \in \mbox{Pic}^0(A) for all s \in S.  The trick is to consider the augmented \Lambda(L) on A\times A \times S and use the theorem of the cube.

Dual Variety and the Poincare Bundle

From the discussion above, for L such that K(L) is finite, the scheme \hat A := A/K(L) can be constructed.  Its called the dual abelian variety.  There is a line bundle \mathcal{P} on A \times \hat A called the Poincare bundle that has the following property.  For every \alpha \in \hat A, \mathcal{P}_{\alpha} \cong \alpha, and \mathcal{P}_{o \times \hat A} is trivial.  Furthermore, it is universal for this: if L is a line bundle on A \times S such that L_s \in \mbox{Pic}^0(A) and trivial on the identity on A, then there is a morphism f \colon S \to \hat A such that L \cong (id, f)^*\mathcal{P}.  The most popular way to construct \mathcal{P} is to give decent data for \Lambda(L) on A \times A and the morphism A \times A \xrightarrow{(id, \phi_L)} A \times \hat A, see page 78 of Mumford Abelian varieties for details.

Its not hard to check in the case of \Lambda(L) on A \times A, the morphism A \to \hat A should in fact be \phi_L \colon A \to \hat A.  I probably should have said this earlier, but  the good L to choose for \phi_L are the ample line bundles, any ample one will do.

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