1. THE CLASSICAL THEORY: PART I 7 normalize to have π1 = 1, so that setting τ = π2 the normalized period matrix is [ τ 1 ] where Im τ 0. τ 1 Differential forms on an n-dimensional complex manifold Y with local holo- morphic coordinates z1,...,zn are direct sums of those of type (p, q) f dzi 1 ∧ · · · ∧ dzi p p ∧ d¯j 1 ∧ · · · ∧ d¯j q q . Thus the C∞ forms of degree r on Y are ⎧ ⎨Ar(Y ⎩ ) = ⊕ p+q=r Ap,q(Y ) Aq,p(Y )= Ap,q(Y ). Setting H1,0(X) = span{dz} H0,1(X) = span{d¯} we have H1(X, C)= H1,0(X) ⊕ H0,1(X) H0,1(X) = H1,0(X). This says that the above decomposition of the 1-forms on X induces a similar decomposition in cohomology. This is true in general for a compact K¨ ahler manifold (Hodge’s theorem) and is the basic starting point for Hodge theory. A recent source is [Cat1]. From dz ∧ dz = 0 and ( i 2 ) dz ∧ d¯ 0, by using that cup-product is given in de Rham cohomology by wedge product and integration over X we have Q ( H1,0(X),H1,0(X) ) = 0 iQ ( H1,0(X), H1,0(X) ) 0. Using the above bases the matrix for Q is Q = 0 −1 1 0 and these relations are Q(Π, Π) = t ΠQΠ = 0 iQ(Π, Π) = itΠQΠ 0. For Π = [ τ 1 ] the second is just Im τ 0.

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