## 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
$2g\times 2g$
$A$
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
$m:A\times A\to A$
$\pi_i$
$L\in Pic^{0}(A)$
$m^{*}(L)\cong \pi_1^{*}(L)\otimes \pi_2^{*}(L).$
$Pic^{0}(A\times B)\cong Pic^{0}(A)\times Pic^{0}(B)$