Abelian Varieties over the Complex Numbers

Herbert Lange, now retired, was a professor at the department of mathematics of the University Erlangen-Nuremberg. He has published more than 130 research papers on abelian varieties, curve theory and vector bundles on algebraic curves, as well as several books.

“This is a ‘graduate text’ edition of the book Complex abelian varieties by C. Birkenhake and H. Lange, a major reference in the theory of abelian varieties … . This new edition does offer substantially more exercises …” (Jean Kieffer, Mathematical Reviews, October, 2024)“This book is addressed to a wide readership of mathematicians, physicists, students pursuing graduate, masters and higher degrees in mathematics and mathematical physics. It is devoted to some of the theory of abelian varieties. It offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry.” (Ahmed Lesfari, ZbMATH 1528.14001, 2024)“The reorganization of the topics is fine surgical work. Several portions of the original monograph are sewn in a natural way in the new book, adding examples or additional text when necessary, and re-arranging the focus to make it a more friendly introduction to the subject. Careful attention to details and the required background makes the book under review accessible to an interested reader and could be a used as textbook for a course on abelian varieties.” (Felipe Zaldivar, MAA Reviews, June 18, 2023)

• 1. Line Bundles on Complex Tori.- 2 Abelian Varieties.- 3 Moduli Spaces.- 4 Jacobian Varieties.- 5 Main Examples of Abelian Varieties.- 6 The Fourier Transform for Sheaves and Cycles.- 7 Introduction to the Hodge Conjecture for Abelian Varieties.