Abelian Varieties over the Complex Numbers

A Graduate Course
2023
ISBN: 978-3-031-44446-3
Verlag: Springer International Publishing

Buch, Englisch, 384 Seiten, HC gerader Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 758 g

Reihe: Grundlehren Text Editions

This textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles.
The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the Fourier–Mukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained.
This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.

Zielgruppe

Graduate

Autoren/Hrsg.

Lange, Herbert

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis

Inhaltsverzeichnis

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.

Über Autor(innen)

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.