58 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Before we state the theorem precisely, we need to explain what makes a
function of ε a period.
9.2. According to Kontsevich and Zagier (KZ01), most or all constants
appearing in mathematics should be periods.
Definition 9.2.1. A number α ∈ C is a period if there exists an alge-
braic variety X of dimension d, a normal crossings divisor D ⊂ X, and a
form ω ∈
Ωd(X)
vanishing on D, all defined over Q; and a homology class
γ ∈ Hd(X(C),D(C)) ⊗ Q
such that
α =
γ
ω.
We are interested in periods which depend on a variable ε ∈ (0, 1). Such
families of periods arise from families of algebraic varieties over the affine
line.
Suppose we have the following data.
(1) an algebraic variety X over Q;
(2) a normal crossings divisor D ⊂ X;
(3) a Zariski open subset U ⊂ AQ,
1
defined over Q, such that U(R)
containts (0, 1).
(4) a smooth map X → U, of relative dimension d, also defined over
Q, whose restriction to D is flat.
(5) a relative d-form ω ∈
Ωd(X/U),
defined over Q, and vanishing along
D.
(6) a homology class γ ∈ Hd((X1/2(C),D1/2(C)), Q), where X1/2 and
D1/2 are the fibres of X and D over 1/2 ∈ U(R). We assume
that γ is invariant under the complex conjugation map on the pair
(X1/2(C),D1/2(C)).
Let us assume that the maps
X(C) → U(C)
D(C) → U(C)
are locally trivial fibrations. For t ∈ (0, 1) ⊂ U(R), we will let Xt(C) and
Dt(C) denote the fibre over s(t) ∈ U(R).
We can transfer the homology class γ ∈ H∗(X1/2(C),D1/2(C)) to any
fibre (Xt(C),Dt(C)) for t ∈ (0, 1). This allows us to define a function f on
(0, 1) by
f(t) =
γt
ωt.
The function f is real analytic. Further, f is real valued, because the cycle
γt ∈ Hd(X1/2(C),D1/2(C))
is invariant under complex conjugation.
