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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
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Abelian Varieties over the Complex Numbers - A Graduate Course (Paperback, 1st ed. 2023)
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Abelian Varieties over the Complex Numbers - A Graduate Course (Paperback, 1st ed. 2023)
Series: Grundlehren Text Editions
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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.
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