Author | Kempf, George R. author |
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Title | Complex Abelian Varieties and Theta Functions [electronic resource] / by George R. Kempf |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1991 |
Connect to | http://dx.doi.org/10.1007/978-3-642-76079-2 |
Descript | IX, 105 p. 1 illus. online resource |
1. Complex Tori -- ยง 1.1 The Definition of Complex Tori -- ยง 1.2 Hermitian Algebra -- ยง 1.3 The Invertible Sheaves on a Complex Torus -- ยง 1.4 The Structure of Pic(V/L) -- ยง 1.5 Translating Invertible Sheaves -- 2. The Existence of Sections of Sheaves -- ยง 2.1 The Sections of Invertible Sheaves (Part I) -- ยง 2.2 The Sections of Invertible Sheaves (Part II) -- ยง 2.3 Abelian Varieties and Divisors -- ยง 2.4 Projective Embeddings of Abelian Varieties -- 3. The Cohomology of Complex Tori -- ยง 3.1 The Cohomology of a Real Torus -- ยง 3.2 A Complex Torus as a Kรคhler Manifold -- ยง 3.3 The Proof of the Appel-Humbert Theorem -- ยง 3.4 A Vanishing Theorem for the Cohomology of Invertible Sheaves -- ยง 3.5 The Final Determination of the Cohomology of an Invertible Sheaf -- ยง 3.6 Examples -- 4. Groups Acting on Complete Linear Systems -- ยง 4.1 Geometric Background -- ยง 4.2 Representations of the Theta Group -- ยง4.3 The Hermitian Structure on ?(X, ?) -- ยง 4.4 The Isogeny Theorem up to a Constant -- 5. Theta Functions -- ยง 5.1 Canonical Decompositions and Bases -- ยง 5.2 The Theta Function -- ยง 5.3 The Isogeny Theorem Absolutely -- ยง 5.4 The Classical Notation -- ยง 5.5 The Length of the Theta Functions -- 6. The Algebra of the Theta Functions -- ยง 6.1 The Addition Formula -- ยง 6.2 Multiplication -- ยง 6.3 Some Bilinear Relations -- ยง 6.4 General Relations -- 7. Moduli Spaces -- ยง 7.1 Complex Structures on a Symplectic Space -- ยง 7.2 Siegel Upper-half Space -- ยง 7.3 Families of Abelian Varieties and Moduli Spaces -- ยง 7.4 Families of Ample Sheaves on a Variable Abelian Variety -- ยง 7.5 Group Actions on the Families of Sheaves -- 8. Modular Forms -- ยง 8.1 The Definition -- ยง 8.2 The Relationship Between ?โ*NA and H in the Principally Polarized Case -- ยง 8.3 Generators of the Relevant Discrete Groups -- ยง 8.4 The Relationship Between ?โ*NA and H is General -- ยง 8.5 Projective Embedding of Some Moduli Spaces -- 9. Mappings to Abelian Varieties -- ยง 9.1 Integration -- ยง 9.2 Complete Reducibility of Abelian Varieties -- ยง 9.3 The Characteristic Polynomial of an Endomorphism -- ยง 9.4 The Gauss Mapping -- 10. The Linear System D| -- ยง 10.1 When } Has No Fixed Components -- ยง 10.2 Projective Normality of D| -- ยง 10.3 The Factorization Theorem -- ยง 10.4 The General Case -- ยง 10.5 Projective Normality of D| on X/{ยฑ} -- 11. Abelian Varieties Occurring in Nature -- ยง 11.1 Hodge Structure -- ยง 11.2 The Moduli of Polarized Hodge Structure -- ยง 11.3 The Jacobian of a Riemann Surface -- ยง 11.4 Picard and Albanese Varieties for a Kรคhler Manifold -- Informal Discussions of Immediate Sources -- References