Introduction to Complex Abelian Varieties

Juan Carlos Naranjo
Dept Matemàtiques i Informàtica
Universitat de Barcelona (España)
This short course is intended to give the basic notions of complex tori and abelian varieties and their interplay with the theory of algebraic smooth curves. The main topics are:

Part 1. Complex tori and complex abelian varieties (see, [2, 3, 5, 8]).

  • Definitions and examples. Homomorphisms. Torsion points.
  • Singular (co)homology and Hodge decomposition.
  • Picard group. Appell-Humbert Theorem.
  • Dual torus. Theorem of the square.
  • Non-degenerate line bundles, ample line bundles and polarizations.

Part 2. Jacobians (see, [1]).

  • Albanese and Picard varieties of a complex projective smooth manifold.
  • Jacobians of curves. Abel-Jacobi map.
  • Theta divisor of the Jacobian of a curve. Riemann’s parametrization Theorem and Riemann’s singularity Theorem.
  • Moduli space of abelian varieties. Torelli map and Torelli Theorem.
  • Differential of the Torelli map. Babbage-Enriques-Petri Theorem.
  • Isogenies of Jacobians.

The preliminaries for this course are (see, [4, 6, 7]):

  • Basic notions of Algebraic Geometry. Sheaves and cohomology. Divisors and Picard group.
  • Hodge theory of compact Kähler manifolds (mainly Hodge decomposition).
  • Riemann-Roch theorem for algebraic curves. Canonical map.
[1] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris: Geometry of Algebraic Curves, Volume 1, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, New York, 1985.
[2] Ch. Birkenhake and H. Lange: Complex Abelian Varieties, 2nd augmented editionSpringer-Verlag Berlin Heidelberg, 2004.
[3] O.Debarre: Tores et avriétés abeliennes complexes, EDP Sciences 1999.
[4] R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
[5] D. Mumford: Abelian varieties, Oxford University Press 1970.
[6] R. Miranda: Algebraic Curves and Riemann Surfaces, Volumen 5 de Dimacs Series in Discrete Mathematics and Theoretical Comput, Número 5 de Graduate studies in mathematics, American Mathematical Soc., 1995.
[7] Ch. A. M. Peters, J. H. M. Steenbrink: Mixed Hodge Structures, Volumen 52 de Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Springer Science & Business Media, 2008.
[8] A. Polishchuk: Abelian varieties, Theta functions and the Fourier transform, Cambrige University Press, 2002.