Complex Abelian Varieties

Preface......Page ac_0001.djvu
Contents......Page ae_0001.djvu
Introduction......Page ak_0001.djvu
Notation......Page ao_0001.djvu
1.1. Complex Tori......Page aq_0001.djvu
1.2. Homomorphisms......Page as_0001.djvu
1.3. Cohomology of Complex Tori......Page aw_0001.djvu
1.4. The Hodge Decomposition......Page ay_0001.djvu
1.5. Exercises and Further Results......Page be_0001.djvu
2. Line Bundles on Complex Tori......Page bg_0001.djvu
2.1. Line Bundles on Complex Tori......Page bh_0001.djvu
2.2. The Appell-Humbert Theorem......Page bm_0001.djvu
2.3. Canonical Factors......Page bp_0001.djvu
2.4. The Dual Complex Torus......Page br_0001.djvu
2.5. The Poincare Bundle......Page bu_0001.djvu
2.6. Exercises and Further Results......Page by_0001.djvu
3. Cohomology of Line Bundles......Page cc_0001.djvu
3.1. Characteristics......Page cd_0001.djvu
3.2. Theta Functions......Page cg_0001.djvu
3.3. The Positive Semidefinite Case......Page cl_0001.djvu
3.4. The Vanishing Theorem......Page cn_0001.djvu
3.5. Cohomology of Line Bundles......Page cs_0001.djvu
3.6. The Riemann-Roch Theorem......Page cv_0001.djvu
3.7. Exercises and Further Results......Page cx_0001.djvu
4. Abelian Varieties......Page da_0001.djvu
4.1. Polarized Abelian Varieties......Page db_0001.djvu
4.2. The Riemann Relations......Page de_0001.djvu
4.3. The Decomposition Theorem......Page df_0001.djvu
4.4. The Gauss Map......Page dm_0001.djvu
4.5. Projective Embeddings......Page dp_0001.djvu
4.6. Symmetric Line Bundles......Page dt_0001.djvu
4.7. Symmetric Divisors......Page dx_0001.djvu
4.8. Kummer Varieties......Page ec_0001.djvu
4.9. Morphisms into Abelian Varieties......Page ef_0001.djvu
4.10. The Pontryagin Product......Page eh_0001.djvu
4.11. Homological Versus Numerical Equivalence......Page ek_0001.djvu
4.12. Exercises and Further Results......Page eo_0001.djvu
5. Endomorphisms of Abelian Varieties......Page es_0001.djvu
5.1. The Rosati Involution......Page et_0001.djvu
5.2. Polarizations......Page ey_0001.djvu
5.3. Norm-Endomorphisms and Symmetric Idempotents......Page fb_0001.djvu
5.4. Endomorphisms Associated to Cycles......Page fh_0001.djvu
5.5. The Endomorphism Algebra of a Simple Abelian Variety......Page fk_0001.djvu
5.6. Exercises and Further Results......Page ft_0001.djvu
6. Theta and Heisenberg Groups......Page fy_0001.djvu
6.1. Theta Groups......Page fz_0001.djvu
6.2. Theta Groups under Homomorphisms......Page gc_0001.djvu
6.3. The Commutator Map......Page ge_0001.djvu
6.4. The Canonical Representation of the Theta Group......Page gg_0001.djvu
6.5. The lsogeny Theorem......Page gj_0001.djvu
6.6. Heisenberg Groups and Theta Structures......Page gm_0001.djvu
6.7. The Schrodinger Representation......Page gr_0001.djvu
6.8. The lsogeny Theorem for Finite Theta Functions......Page gt_0001.djvu
6.9. Symmetric Theta Structures......Page gw_0001.djvu
6.10. Exercises and Further Results......Page hb_0001.djvu
7. Equations for Abelian Varieties......Page hg_0001.djvu
7.1. The Multiplication Formula......Page hh_0001.djvu
7.2. Surjectivity of the Multiplication Map......Page hl_0001.djvu
7.3. Projective Normality......Page ho_0001.djvu
7.4. The Ideal of an Abelian Variety in ℙ_N......Page hr_0001.djvu
7.5. Riemann\'s Theta Relations......Page hx_0001.djvu
7.6. Cubic Theta Relations......Page hz_0001.djvu
7.7. Exercises and Further Results......Page ie_0001.djvu
8. Moduli......Page ik_0001.djvu
8.1. The Siegel Upper Half Space......Page il_0001.djvu
8.2. The Analytic Moduli Space......Page io_0001.djvu
8.3. Level Structures......Page ir_0001.djvu
8.3.1. Level D-Structure......Page is_0001.djvu
8.3.2. Generalized Level n-Structure......Page it_0001.djvu
8.3.3. Decomposition of the Lattice......Page iu_0001.djvu
8.4. The Theta Transformation Formula, Preliminary Version......Page iv_0001.djvu
8.5. Classical Theta Functions......Page ix_0001.djvu
8.6. The Theta Transformation Formula, Final Version......Page jc_0001.djvu
8.7. The Universal Family......Page je_0001.djvu
8.8. The Action of the Symplectic Group......Page jh_0001.djvu
8.9. Orthogonal Level D-Structures......Page jj_0001.djvu
8.10. The Embedding of A_D(D)_0 into Projective Space......Page jk_0001.djvu
8.11. Exercises and Further Results......Page jo_0001.djvu
9. Moduli Spaces of Abelian Varieties with Endomorphism Structure......Page js_0001.djvu
9.1. Abelian Varieties with Endomorphism Structure......Page ju_0001.djvu
9.2. Abelian Varieties with Real Multiplication......Page jv_0001.djvu
9.3. Some Notation......Page ka_0001.djvu
9.4. Totally Indefinite Quaternion Multiplication......Page kd_0001.djvu
9.5. Totally Definite Quaternion Multiplication......Page kg_0001.djvu
9.6. Families of Abelian Varieties with Complex Multiplication......Page kl_0001.djvu
9.7. Group Actions on H_r,s and H_m......Page kq_0001.djvu
9.8. Shimura Varieties......Page kt_0001.djvu
9.9. The Endomorphism Algebra of a General Member......Page kx_0001.djvu
9.10. Exercises and Further Results......Page lc_0001.djvu
10. Abelian Surfaces......Page le_0001.djvu
10.1. Preliminaries......Page lf_0001.djvu
10.2. The 16₆-Configuration of the Kummer Surface......Page li_0001.djvu
10.3. An Equation for the Kummer Surface......Page ln_0001.djvu
10.4. Reider\'s Theorem......Page lq_0001.djvu
10.5. Polarizations of Type (1,4)......Page lx_0001.djvu
10.6. Products of Elliptic Curves......Page mb_0001.djvu
10.7. The Coble Hypersurface of a Principally Polarized Abelian Surface......Page mf_0001.djvu
10.8. Exercises and Further Results......Page mh_0001.djvu
11. Jacobian Varieties......Page mm_0001.djvu
11.1. Definition of the Jacobian Variety......Page mn_0001.djvu
11.2. The Theta Divisor......Page mt_0001.djvu
11.2.2. The Singularity Locus of Θ......Page mw_0001.djvu
11.3. The Poincare Bundles for a Curve C......Page my_0001.djvu
11.4. The Universal Property......Page nb_0001.djvu
11.5. Correspondences of Curves......Page ne_0001.djvu
11.6. Endomorphisms Associated to Curves and Divisors......Page ng_0001.djvu
11.7. Examples of Jacobians......Page ni_0001.djvu
11.8. The Criterion of Matsusaka-Ran......Page nm_0001.djvu
11.9. Trisecants of the Kummer Variety......Page np_0001.djvu
11.10. Fay\'s Trisecant Identity......Page ns_0001.djvu
11.11. Albanese and Picard Varieties......Page ny_0001.djvu
11.12. Exercises and Further Results......Page oe_0001.djvu
12. Prym Varieties......Page oi_0001.djvu
12.1. Abelian Subvarieties of a Principally Polarized Abelian Variety......Page oj_0001.djvu
12.2. Prym-Tyurin Varieties......Page on_0001.djvu
12.3. Prym Varieties......Page or_0001.djvu
12.4. Topological Construction of Prym Varieties......Page ot_0001.djvu
12.5. The Abel-Prym Map......Page ox_0001.djvu
12.6. The Theta Divisor of a Prym Variety......Page pa_0001.djvu
12.7. Recillas\' Theorem......Page pe_0001.djvu
12.8. Donagi\'s Tetragonal Construction......Page ph_0001.djvu
12.9. Kanev\'s Criterion......Page pn_0001.djvu
12.10. The Schottky-Jung Relations......Page ps_0001.djvu
12.11. Exercises and Further Results......Page pz_0001.djvu
13. Automorphisms......Page qe_0001.djvu
13.1. Fixed-Point Formulas......Page qf_0001.djvu
13.2. The Fixed-Point Set of a Finite Automorphism Group......Page qg_0001.djvu
13.3. Abelian Varieties of CM-Type......Page qk_0001.djvu
13.4. Abelian Surfaces with Finite Automorphism Group......Page qo_0001.djvu
13.5. Poincare\'s Reducibility Theorem with Automorphisms......Page qv_0001.djvu
13.6. The Group Algebra Decomposition of an Abelian Variety......Page qy_0001.djvu
13.7. Exercises and Further Results......Page rd_0001.djvu
14. Vector bundles on Abelian Varieties......Page rg_0001.djvu
14.1. Some Properties of the Poincare Bundle......Page rh_0001.djvu
14.2. The Fourier Transform for WIT-Sheaves......Page rl_0001.djvu
14.3. Some Properties of the Fourier Transform......Page rp_0001.djvu
14.4. The Dual Polarization......Page ru_0001.djvu
14.5. Application: Global Generation of Vector Bundles......Page rw_0001.djvu
14.6. Picard Sheaves......Page sa_0001.djvu
14.7. The Fourier Transform of a Complex......Page sf_0001.djvu
14.8. Vector Bundles on Abelian Surfaces......Page sk_0001.djvu
14.9. Exercises and Further Results......Page sr_0001.djvu
15. Further Results on Line Bundles an the Theta Divisor......Page su_0001.djvu
15.1. Very Ample Line Bundles on General Abelian Varieties......Page sv_0001.djvu
15.2. Syzygies of Line Bundles on Abelian Varieties......Page sz_0001.djvu
15.3. Seshadri Constants......Page td_0001.djvu
15.4. Bounds for Seshadri Constants......Page tg_0001.djvu
15.5. The Minimal Length of a Period......Page tl_0001.djvu
15.6. Seshadri Constants of Line Bundles on Abelian Surfaces......Page ts_0001.djvu
15.7. Subvarieties of Abelian Varieties......Page tw_0001.djvu
15.8. Singularities of the Theta Divisor......Page ub_0001.djvu
15.9. Exercises and Further Results......Page ug_0001.djvu
16. Cycles on Abelian varieties......Page uk_0001.djvu
16.1. Chow Groups......Page ul_0001.djvu
16.2. Correspondences......Page uo_0001.djvu
16.3. The Fourier Transform on the Chow Ring......Page uq_0001.djvu
16.4. The Fourier Transform on the Cohomology Ring......Page uv_0001.djvu
16.5. A Decomposition of Ch(X)_ℚ......Page ux_0001.djvu
16.6. The Künneth Decomposition......Page vb_0001.djvu
16.7. The Bloch Filtration of Ch₀(X)......Page vg_0001.djvu
16.8. Exercises and Further Results......Page vj_0001.djvu
17. The Hodge Conjecture for General Abelian and Jacobian Varieties......Page vm_0001.djvu
17.1. Hodge Structures and Complex Structures......Page vn_0001.djvu
17.2. Symplectic Complex Structures......Page vp_0001.djvu
17.3. The Hodge Group of an Abelian Variety......Page vr_0001.djvu
17.4. The Theorem of Mattuck......Page vw_0001.djvu
17.5. The Hodge Conjecture for a General Jacobian......Page vy_0001.djvu
17.6. Exercises and Further Results......Page wb_0001.djvu
A. Algebraic Varieties and Complex Analytic Spaces......Page we_0001.djvu
B. Line Bundles and Factors of Automorphy......Page wi_0001.djvu
C.1. Some Properties of ℚ-Divisors......Page wo_0001.djvu
C.2. The Kodaira Dimension......Page wp_0001.djvu
C.3. Vanishing Theorems......Page wq_0001.djvu
C.7. Adjoint Ideals......Page wr_0001.djvu
D.1. Definition and First Properties......Page wu_0001.djvu
D.2. Derived Functors......Page wy_0001.djvu
D.3. The Grothendieck-Riemann-Roch Theorem......Page xa_0001.djvu
E. Moduli Spaces of Sheaves......Page xc_0001.djvu
F.1. Abelian Schemes and the Poincare Bundle......Page xi_0001.djvu
F.2. Relative Fourier Functor......Page xk_0001.djvu
F.3. The Relative Jacobian......Page xm_0001.djvu
Bibliography......Page xo_0001.djvu
Glossary of Notation......Page yk_0001.djvu
Index......Page yq_0001.djvu