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 \[Pic^{0}(A\times B)\cong Pic^{0}(A)\times Pic^{0}(B)\] . |