6 1. THE CLASSICAL THEORY: PART I is a lattice. The pictures are δ2 δ1 π2 π1 Here δ1 ↔ π1 and δ2 ↔ π2 give a basis for H1(X, Z). The complex plane C = {z = x + iy} is oriented by dx ∧ dy = ( i 2 ) dz ∧ d¯ 0. We choose generators π1,π2 for Λ with π1 ∧ π2 0, and then the intersection number δ1 · δ2 = +1. We set VZ = H1(X, Z), V = VZ ⊗ Q = H1(X, Q) and denote by Q : V ⊗ V → Q Q(v, v ) = −Q(v , v) the cup-product, which via Poincar´ e duality H1(X, Q) ∼ H1(X, Q) is the intersec- tion form. We have H1(X, C) ∼ H1 DR (X) = closed 1-forms ψ modulo exact 1-forms ψ=dζ H1(X, Z)∗ ⊗ C and it may be shown that HDR(X) 1 ∼ spanC {dz, d¯} . The pairing of cohomology and homology is given by periods πi = δi dz and Π = ( π2 π1 ) is the period matrix (note the order of the πi’s). Using the basis for H1(X, C) dual to the basis δ1,δ2 for H1(X, C), we have H1(X, C) ∼ C2 = column vectors dz = Π. We may scale C by z → λz, and then Π = λΠ so that the period matrix should be thought of as point in P1 with homogeneous coordinates [ z0 z1 ]. By scaling, we may ∼ ∈ ∈
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