Different occurences of the word 'period' in algebraic geometry

Question

I have come across the word 'period' in several contexts and I wonder if these notions are related.

(1) The period map and domain: Let $ \pi : X \rightarrow B $ be a proper holomorphic submersion of complex manifolds. If a fiber $ X_0 $ of $ \pi $ is (compact) Kahler, then the cohomology groups $ H^k(X_0, \mathbb{Z}) $ (modulo torsion) are Hodge structures of weight $ k $. Now if we consider a small enough neighborhood $ U $ of $ 0 \in B $, then all fibers of U are Kahler, the Hodge numbers $ h^{p,q} $ are constant, and there is a fixed isomorphism (#) $ H^k(X_b, \mathbb{C}) \cong H^k(X_0, \mathbb{C}) $ from Ehresmann's theorem. So the numbers $ a_p = \text{dim} F^pH^k(X_b, \mathbb{C}) $ are also constant. The period map then sends $ b \in U $ to the (image under (#) of the) flag $ \{F^pH^k(X_b, \mathbb{C})\}_p $ in the flag variety $ G(a_k , \ldots , a_1, H^k(X_0, \mathbb{C})) $. The period domain is the subset of the flag variety whose flags satisfy $ F^pH^k(X_0, \mathbb{C}) \oplus \overline{F^{k+1-p}H^k(X_0, \mathbb{C})} = H^k(X_0, \mathbb{C}) $.

(2) Period: A real number is a period if it is an integral $ \int_{p \ge 0} q(x_1, \ldots, x_n) dx $ where $ p $ is a polynomial with rational coefficients and $ q $ is a rational function with rational coefficients. We can allow $p,q $ to be algebraic functions, this is equivalent. So abelian integrals $ \int_0^r \frac{dx}{\sqrt {4x^3 - ax - b}} $ are periods ($ r,a,b \in \mathbb{Q} $).

(3) Period of a cohomology group: I came across this notion after reading Daniil Rudenko's question on this website. This one I do not understand at all.

So my question is: How are these notions related? Surely (2) and (3) are related as you're supposedly computing an integral in (3) as well. And I suspect (1) and (3) are related as well as you're dealing with some cohomology group.

I apologize if the question is a bit too general but I'd like to understand this.

Answer 1

The second and the third are pretty much equivalent. Indeed, "the period" in XIX century sense is essentially the same as the discrepancy between the branches of a multi-valued function, obtained as an integral of an algebraic function. If you take the Riemann surface associated with this algebraic function (that is, a branched covering of C where it is well-defined), the period becomes an integral of the holomorphic differential $fdz$ associated with this function over a closed loop, that is, an integral of a holomorphic 1-form over a closed cycle.

The modern definition of "periods" is, more or less, "the pairing between holomorphic differential forms and integral homology". However, this notion can (and often is) extended to Hodge structures, and this is related to the first notion you mention.

Define the Teichmuller space of a complex manifold as the space of all complex structures (here the complex structures are understood as endomorphisms of $TM$ satisfying $I^2=-Id$ and the integrability condition) up to isotopies: $Teich= \frac{Comp}{Diff_0}$. Never mind that this space might be horribly non-Hausdorff; the period map is naturally defined on $Teich$, and (if you are lucky) it would help to make sense even of the non-Hausdorff pathologies.

The period map is the map taking $I\in Teich$ to the Hodge structure on $H^*(M,I)$, which is understood as a point in the appropriate flag space. This map is holomorphic, and, if you are lucky, defines a biholomorphism between $Teich$ and the space of Hodge structures ("period space"). This holds, unfortunately, only for complex tori, but weaker versions of this statement are true for K3, hyperkahler manifolds, complex curves and in some other cases. These results are called "global Torelli theorems".

The local Torelli theorems are statements about the local structure of the period map, saying (in most cases) that is it locally an immersion; this is true, for instance, for Calabi-Yau manifolds.

The modern definition of periods is not entirely equivalent to the traditional, because the Hodge structure contains more data than just holomorphic forms; however, if you know the Hodge structure on cohomology, you know the pairing between the holomorphic forms and the integer homology, hence it is an extension of the XIX century notion.

If you restrict yourself to the first cohomology, the Hodge structure is a flag $H^{1,0}(M)\subset H^1(M)$; in this case the XIX century notion of "periods" coincides with the Hodge-theoretic notion.