Abstract
These lecture notes present, in outline, the theory of abelian varieties over the complex numbers. They focus mainly on the analytic side of the subject. In the first section we prove some basic results on complex tori. The second section is devoted to a discussion of isogenics. The third section (the longest) describes the necessary and sufficient conditions that a complex torus must satisfy in order to be isomorphic to an abelian variety. In the fourth section we describe the construction of the dual abelian variety and the concluding two sections discuss polarizations and the moduli space of principally polarized abelian varieties. Proofs for the most part are omitted or only sketched. Details can be found in [SW] or [L-A] (see the list of references at the end of this chapter). For the algebraic-geometric study of abelian varieties over arbitrary fields, the reader is referred to [M-AV] and to the articles of J. S. Milne in this volume.
(Notes by F. O. McGuinness, Fordham University)
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Annotated References
Griffiths, P. and Harris, J. Principles of Algebraic Geometry. Wiley: New York, 1978. Chapter 2 (Riemann Surfaces and Algebraic Curves) contains a discussion of Jacobians and abelian varieties from the complex analytic point of view.
Igusa, J. Theta Functions. Springer-Verlag: New York, 1972. This book contains an extensive treatment of both the analytic and algebraic aspects of the theory of theta functions. Chapter IV discusses the equations defining abelian varieties under the projective embedding obtained from theta functions.
Lang, S. Introduction to Algebraic and Abelian Functions, 2nd edn. Springer-Verlag: New York, 1982. Chapters VI, VII, VIII, X are relevant for the content of these lectures. Chapter IX constructs abelian manifolds whose algebra of endomorphisms contains a quaternion algebra.
Lang, S. Complex Multiplication. Springer-Verlag: New York, 1983. A complete exposition of the theory of complex multiplication of abelian varieties.
Mumford, D. Abelian Varieties, Oxford University Press: Oxford, 1970. (2nd edn 1974). Chapter I treats the analytic theory using line bundles instead of divisors. Most of the book gives a general treatment of abelian varieties from the point of view of schemes.
Mumford, D. Curves and Their Jacobians. University of Michigan Press: Ann Arbor, 1975. An excellent survey of algebraic curves and their Jacobians.
Mumford, D. Tata Lectures on Theta, I. Birkhäuser-Verlag: Basle, 1983. A more down-to-earth treatment of theta functions than [I].
Robert, A. Introduction aux Variétés Abéliennes Complexes, Enseign. Math., 28 (1982), 91–137. A self-contained discussion of the criterion for a complex torus to be projectively embeddable.
Shafarevich, I. R. Basic Algebraic Geometry, Springer-Verlag: New York, 1977. Part 3 contains an excellent treatment of complex analytic manifolds.
Shimura, G., and Taniyama, Y. Complex Multiplication of Abelian Varieties, Mathematical Society of Japan: Tokyo, 1961. The original treatment of the subject of the title, due to Shimura, Taniyama and Weil.
Siegel, C. L. Topics in Complex Function Theory, Vol. III. Wiley: New York, 1972. An exposition of the classical approach to Riemann surfaces and Abelian integrals.
Swinnerton-Dyer, H. P. F. Analytic Theory of Abelian Varieties. Cambridge University Press: Cambridge, 1974. A treatment of abelian manifolds close to these lectures.
Weil, A. Introduction à l’Étude des Variétés Kählériennes. Hermann: Paris, 1971. Weil’s approach to the analytic theory, based on Hodge theory.
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Rosen, M. (1986). Abelian Varieties over ℂ. In: Cornell, G., Silverman, J.H. (eds) Arithmetic Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8655-1_4
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