Complex Abelian Varieties

May be used as an introduction or reference. Covers the theory of abelian varieties over the field of complex numbers. Topics include projective embeddings of an abelian variety including their equations and geometric properties, special results of Jacobians and Prym varieties allowing applications to the theory of algebraic curves, complex tori, cohomology of line bundles, and constructions of several moduli spaces of abelian varieties with additional structure. Problems follow each chapter. Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions. The second edition contains five new chapters which present some of the most important recent result on the subject. Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. Introduction.- Complex Tori.- Line Bundles on Complex Tori.- Cohomology of Line Bundles.- Abelian Varieties.- Endomorphisms of Abelian Varieties.- Theta and Heisenberg Groups.- Equations for Abelian Varieties.- Moduli.- Moduli Spaces of Abelian Varieties with Endomorphism Structure.- Abelian Surfaces.- Jacobian Varieties.- Prym Varieties.- Automorphisms.- Vector bundles on Abelian Varieties.- Further Results on Line Bundles an the Theta Divisor.- Cycles on Abelian Varieties.- The Hodge Conjecture for General Abelian and Jacobian Varieties.- Appendix: Chapter A.Algebraic Varieties and Complex Analytic Spaces.- Chapter B.Line Bundles and Factors of Automorphy.- Chapter C.Some Algebraic Geometric Results.- D.Derived Categories.- Chapter E.Moduli Spaces of Sheaves.- Chapter F.Abelian Schemes.- Index.- Glossary of Notation Front Matter....Pages i-viii Introduction....Pages 1-4 Notation....Pages 5-5 Complex Tori....Pages 6-22 Line Bundles on Complex Tori....Pages 23-45 Cohomology of Line Bundles....Pages 46-70 Abelian Varieties....Pages 71-114 Endomorphisms of Abelian Varieties....Pages 115-146 Theta and Heisenberg Groups....Pages 147-181 Equations for Abelian Varieties....Pages 182-211 Moduli....Pages 212-246 Moduli Spaces of Abelian Varieties with Endomorphism Structure....Pages 247-287 Abelian Surfaces....Pages 288-319 Jacobian Varieties....Pages 320-364 Prym Varieties....Pages 365-408 Back Matter....Pages 409-435 This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994. This text looks at the theory of abelian varieties over the field of complex numbers. It covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference Christina Birkenhake, Herbert Lange. Rev. Of: Complex Abelian Varieties / Herbert Lange, Christina Birkenhake. 1992. Includes Bibliographical References (p. [603]-624) And Index.