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 A1 Q , 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.
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