nLab
period

General

The word ‘period’ has many meanings in mathematics, most of them coming from physics: the period of an oscillation, periods in celestial mechanics, even the period of a periodic function? comes from the intuition that the periodicity is in the time dimension. Functions on tori are periodic in two directions, say the Weierstrass functions on elliptic curves, so it is not a surprise that more involved kinds of periods came from the study of elliptic curves and then more general Riemann surfaces.

In differential geometry

The period of a closed differential form $\omega \in {\Omega }_{\mathrm{cl}}^n(X)$ over an $n$-cycle $S$ is the integral ${\int }_S\omega$.

In number theory and algebraic geometry

There is another deep notion of periods in number theory and a more specific version related to specific situations in algebraic geometry. We distinguish irrational and rational numbers; complex numbers divide into algebraic and transcendental. Periods are more general than algebraic numbers: they are those (complex) numbers which can be obtained as integrals of algebraic function?s (all of whose coefficients are also algebraic numbers) over semialgebraic sets. The periods form a subring of complex numbers bigger than the field of algebraic numbers. There are several operations which lead to new periods. In fact, if we view them abstractly, as integrals of some abstract function over an abstract semialgebraic set, then we can take unions of such sets, do partial integration and so on. There is a conjecture that there are no relations among periods except those of a short list of such obvious relations!

Periods appear in a number of situations in classical algebraic geometry. Specific matrices of periods are defined and important in the theory of algebraic functions, Hodge theory for algebraic cycles, the study of actions of motivic Galois group?s, etc. They come as generalizations of “periods of Riemann surfaces” from 19th century.

References

• M. Kontsevich, D. Zagier, Periods

• M. Kontsevich, Operads and motives in deformation quantization, Lett.Math.Phys. 48 (1999) 35–72, math.QA/9904055

• A. B. Goncharov, Periods and mixed motives, math.AG/0202154

• mathoverflow: ring of periods not a field

• Annette Huber, Stefan Müller-Stach, On the relation between Nori motives and Kontsevich periods, 1105.0865