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.