In the statement of de Rham theorem, a pairing is defined as follows:
$H_i(X, \mathbb R) \times H^i_{\mathrm{de Rham}}(X) \rightarrow \mathbb R$
It is given by
$\left( \left( \sum a_i \gamma_i \right) , \omega \right) \mapsto \sum a_i \int_{\gamma_i} \omega $.
Here $a_i$ takes real values, and $\gamma_i$ are integral homology cycles, and $\omega$ is an $i$-form.
The integral given above is called the period of the above integral, and the isomorphism given by this pairing is sometimes called the period isomorphism.
Question:
Why is the above integral of a closed form over a cycle called a period?
My peeve with this terminology of "period" is that it does not agree at all with anything else I know about this word. They are the following: 1) The period of a periodic function. 2) Periods as generalizations of algebraic numbers, as integrals of algebraic(rational) expressions over domains in Euclidean spaces defined by algebraic inequalities.
Indeed, this strange use of the word "period" is used even by Ahlfors in his book on complex analysis, for integrals of holomorphic(rather, meromorphic? depending on the domain …) functions over loops.
I do not understand why on earth the word period appears in this setting of integrating on abstract manifolds. True, the cycles capture some underlying geometry of the space. But why is something a "period" when you integrate a form over a cycle? Why is the integral of a $1$-form over a line segment not a period?(Or is it also period, in some definition I am not aware of?)
Best Answer
In the case of elliptic curves, integrating some fixed holomorphic differential over the first homology gives the lattice of periods of the corresponding elliptic functions. (These elliptic functions are certain meromorphic functions on $\mathbb C$ which are "doubly periodic", i.e. are invariant under $z \mapsto z + \omega$, where $\omega$ lies in the lattice of periods obtained as above.)
This is the source of the terminology.