"Complex Abelian Varieties" is a comprehensive study of complex abelian varieties, written by Herbert Lange and Christina Birkenhake. The book is part of the "Grundlehren der mathematischen Wissenschaften" series, edited by several prominent mathematicians. It provides an in-depth exploration of complex tori, line bundles, cohomology, theta functions, and abelian varieties. The text is structured into 12 chapters, each covering various aspects of abelian varieties, including their properties, structures, and applications. Topics range from complex tori and homomorphisms to polarized abelian varieties, endomorphisms, theta and Heisenberg groups, equations for abelian varieties, moduli spaces, and Jacobian varieties. The book also includes detailed discussions on Prym varieties, their constructions, and their properties. It covers the theory of theta functions, their transformation formulas, and their connections to moduli spaces. The text is supported by exercises at the end of each chapter, allowing readers to apply the concepts learned. The book is intended for advanced students and researchers in mathematics, particularly those working in algebraic geometry and complex analysis. It includes a bibliography, a glossary of notation, and an index for reference. The work is published by Springer-Verlag and is a significant contribution to the field of mathematics, offering a thorough and systematic treatment of complex abelian varieties."Complex Abelian Varieties" is a comprehensive study of complex abelian varieties, written by Herbert Lange and Christina Birkenhake. The book is part of the "Grundlehren der mathematischen Wissenschaften" series, edited by several prominent mathematicians. It provides an in-depth exploration of complex tori, line bundles, cohomology, theta functions, and abelian varieties. The text is structured into 12 chapters, each covering various aspects of abelian varieties, including their properties, structures, and applications. Topics range from complex tori and homomorphisms to polarized abelian varieties, endomorphisms, theta and Heisenberg groups, equations for abelian varieties, moduli spaces, and Jacobian varieties. The book also includes detailed discussions on Prym varieties, their constructions, and their properties. It covers the theory of theta functions, their transformation formulas, and their connections to moduli spaces. The text is supported by exercises at the end of each chapter, allowing readers to apply the concepts learned. The book is intended for advanced students and researchers in mathematics, particularly those working in algebraic geometry and complex analysis. It includes a bibliography, a glossary of notation, and an index for reference. The work is published by Springer-Verlag and is a significant contribution to the field of mathematics, offering a thorough and systematic treatment of complex abelian varieties.