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AuthorLange, Herbert. author
TitleComplex Abelian Varieties [electronic resource] / by Herbert Lange, Christina Birkenhake
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1992
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Descript VIII, 435 p. online resource


Abelian varieties are special examples of projective varieties. As such theycan be described by a set of homogeneous polynomial equations. The theory ofabelian 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


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 -- A Algebraic Varieties and Complex Analytic Space -- B Line Bundles and Factors of Automorphy -- Bibliograpy -- Glossary of Notation

Mathematics Algebraic geometry Number theory Mathematics Algebraic Geometry Number Theory


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