Introduction to Complex Abelian Varieties

Juan Carlos Naranjo
Dept Matemàtiques i Informàtica
Universitat de Barcelona (España)
 
Exercises
 
Exercise 1. Let
\[A\]
a matrix with
\[g\]
rows and
\[2g\]
columns and complex entries. Prove that
\[A\]
is a period matrix (the columns generate a lattice) iff the
\[2g\times 2g\]
matrix obtaining adding below to
\[A\]
the conjugate of the rows is non- singular.
Exercise 2. a) Let
\[\mathbb{T}=V/\Gamma\]
be a complex torus and let
\[X\subset \mathbb{T}\]
a subtorus of dimension
\[h\]
, then there exists
\[\Gamma'\subset \Gamma\]
a subgroup of rank
\[2h\]
such that
\[W:=\Gamma'\otimes_{\mathbb{Z}}\mathbb{R}\]
is stable by multiplication by
\[i\]
, and
\[X=W/\Gamma'\]
 b) Identify
\[V\cong \mathbb{C}^2\]
with
\[\mathbb{R}^4\]
, and let
\[M\in GL_4(\mathbb{R})\]
an invertible matrix with real entries. We attach to
\[M\]
a lattice
\[\Gamma_M\]
generated by the columns of
\[M\]
. So there is a bijection between the set of 2-dimensional complex tori and
\[\mathcal{U}:=GL_2(\mathbb{R})\subset \mathbb{R}^4\]
. Prove that there are countably many proper algebraic subvarieties
\[\mathcal{X}_n\subset \mathcal{U}\]
such that for all
\[M\in \mathcal{U}\setminus \bigcup_n \mathcal{X}_n\]
the torus
\[\mathbb{C}^2/\Gamma_M\]
is simple.
Exercise 3. Let
\[C\]
be a smooth projective curve and consider an Abel-Jacobi map
\[i:C\hookrightarrow JC\]
. Using Inversion Jacobi Theorem prove that the pull-back map
\[i^*:JC^{\vee}\to JC\]
is the inverse of the polarization map
\[\phi_{\Theta_C}:JC\to JC^{\vee}\]
. (Hint: Inversion Jacobi Theorem can be stated as follows: fix
\[p\in C\]
giving the embedding of
\[C\]
in
\[JC\]
and let
\[\xi\in Pic^{g-1}(C)\]
be a line bundle such that
\[\Theta_{-\xi}^{can}=\Theta\]
, Then
\[i^*(\Theta)\cong\omega_C\otimes \xi^{-1}(p)\]
). 
Exercise 4. Let
\[\pi:D\to C\]
be a double covering of projective smooth curves (possibly ramified). Define the norm map
\[N_{m_{\pi}}:JD\to JC\]
as
\[N_{m_{\pi}}(\mathcal{O}_D(\Sigma n_lx_l))=\mathcal{O_C(\Sigma n_l\pi(x_l))}\]
. Prove that the dual map of
\[N_{m_{\pi}}\]
is the pull-back
\[\pi^*:JC\to JD\]
(up to identifying
\[JC=JC^{\vee}\]
and
\[JD=JD^{\vee}\]
by means of the polarizations). 
Exercise 5. Let
\[\pi:D\to C\]
a covering of smooth curves of degree
\[d\geq 2\]
, suppose
\[g(C)\geq 1\]
. Let
\[P:=Ker(N_{m_{\pi}}:JD\mapsto JC)^{0}\]
. Let
\[\Theta\]
the canonical polarization of
\[JD\]
and let
\[\Theta_P\]
be its restriction to
\[P\]
. Then, the following are equivalent: 
a) 
\[\pi_{*}:H_1(D, \mathbb{Z})\to H_1(C, \mathbb{Z})\]
surjective. 
b) 
\[\pi^{*}:JC\to JD\]
is injective. 
c) 
\[Ker(N_{m_{\pi}})\]
is connected. 
Exercise 6. We assume the following result: 
 See-saw Lemma: Let
\[L\in Pic(X\times Y)\]
be a line bundle in a product of two smooth projective varieties. Assume the two conditions
a) 
\[L|_{X\times \{y\}}\]
is isomorphic to 
\[\mathcal{O}_X\]
for all 
\[y\in Y\]
.
b) 
\[L|_{\{x_0\}\times Y}\cong \mathcal{O}_Y\]
Then 
\[L\cong \mathcal{O}_{X\times Y}\]
Prove: 
a) The Poincaré bundle
\[\mathcal{P}\]
on
\[A\times A^{\vee}\]
(A abelian variety) is the only one with the properties:
\[\mathcal{P}|_{A\times\{\alpha\}}\cong \alpha\]
,
\[\forall \alpha \in A^{\vee}\]
and
\[\mathcal{P}|_{\{0\}\times A^{\vee}}\cong \mathcal{O}_{A^{\vee}}\]
b) Let
\[L\in Pic(A)\]
an ample line bundle on an abelian variety
\[A\]
and let
\[\phi_L:A\to A^{\vee}\]
be the isogeny attached to
\[L\]
(sending
\[a\]
to
\[t_a^{*}(L)\otimes L^{-1}\]
). Prove the isomorphism: 
\[(1\times \phi_L)^{*}(\mathcal{P})\cong m^{*}(L)\otimes \pi_1^{*}(L^{-1})\otimes \pi_2^{*}(L^{-1}),\]
where 
\[m:A\times A\to A\]
is the addition and 
\[\pi_i\]
are the projections on each factor.
c) Let 
\[L\in Pic^{0}(A)\]
. Prove that 
\[m^{*}(L)\cong \pi_1^{*}(L)\otimes \pi_2^{*}(L).\]
 
d) Show that 
\[Pic^{0}(A\times B)\cong Pic^{0}(A)\times Pic^{0}(B)\]