Complex Abelian Varieties

• Notation.- 1. Complex Tori.- 2. Line Bundles on Complex Tori.- 3. Cohomology of Line Bundles.- 4. Abelian Varieties.- 5. Endomorphisms of Abelian Varieties.- 6. Theta and Heisenberg Groups.- 7. Equations for Abelian Varieties.- 8. Moduli.- 9. Moduli Spaces of Abelian Varieties with Endomorphism Structure.- 10. Abelian Surfaces.- 11. Jacobian Varieties.- 12. Prym Varieties.- 13. Automorphisms.- 14. Vector bundles on Abelian Varieties.- 15. Further Results on Line Bundles an the Theta Divisor.- 16. Cycles on Abelian varieties.- 17. The Hodge Conjecture for General Abelian and Jacobian Varieties.- A. Algebraic Varieties and Complex Analytic Spaces.- B. Line Bundles and Factors of Automorphy.- C. Some Algebraic Geometric Results.- C.1 Some Properties of ?-Divisors.- C.2 The Kodaira Dimension.- C.3 Vanishing Theorems.- C.6 Some Results from Intersection Theory.- C.7 Adjoint Ideals.- D. Derived Categories.- D.1 Definition and First Properties.- D.2 Derived Functors.- D.3 The Grothendieck-Riemann-Roch Theorem.- E. Moduli Spaces of Sheaves.- F. Abelian Schemes.- F.1 Abelian Schemes and the Poincar´e Bundle.- F.2 Relative Fourier Functor.- F.3 The Relative Jacobian.- Glossary of Notation.

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