It has been known since the nineteenth century that there is a group structure on the points of the smooth cubic complex plane curve (called an elliptic curve) and that it is isomorphic to the quotient of C by a lattice. Conversely, any such quotient is an elliptic curve.

The higher-dimensional analogs are complex tori, where is a lattice in a (finite-dimensional) complex vector space V. The group structure and the analytic structure are obvious, but not all tori are algebraic. For that, we need an additional condition, which was formulated by Riemann: the existence of a positive definite Hermitian form on V whose (skew-symmetric) imaginary part is integral on . An algebraic complex torus is called an abelian variety. When this skew-symmetric form is in addition unimodular on, we say that the abelian variety is principally polarized. It contains a hypersurface uniquely determined up to translation (“the” theta divisor).

The combination of the algebraic and group structures makes the geometry of abelian varieties very rich. This is one of the reasons why it is useful to associate, whenever possible, an abelian variety (if possible principally polarized) to a given geometric situation. This can be done only in a few specific cases, and the theory of periods, mainly developed by Griffiths, constitutes a far-reaching extension.

Our aim is to present an elementary introduction to this theory. We show many examples to illustrate its diversity, with no pretense at exhaustivity, and no proofs. For those interested in pursuing this very rich subject, we refer to [Carlson et al. 2003] and its bibliography.