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8 1. THE CLASSICAL THEORY: PART I Definitions. (i) A Hodge structure of weight one is given by a Q-vector space V with a line V 1,0 ⊂ VC satisfying VC = V 1,0 ⊕ V 0,1 V 0,1 = V 1,0 . (ii) A polarized Hodge structure of weight one (PHS) is given by the above together with a non-degenerate form Q : V ⊗ V → Q, Q(v, v ) = −Q(v , v) satisfying the Hodge-Riemann bilinear relations Q(V 1,0 , V 1,0 ) = 0 iQ(V 1,0 , V 1,0 ) 0. In practice we will usually have V = VZ ⊗ Q. The reason for working with Q will be explained later. When dim V = 2, we may always choose a basis so that V ∼ Q2 = column vectors and Q is given by the matrix above. Then V 1,0 ∼ C is spanned by a point [ τ 1 ] ∈ PVC ∼ P1 Identification. The space of PHS’s of weight one (period domain) is given by the upper-half-plane H = {τ : Im τ 0}. The compact dual ˇ given by subspaces V 1,0 ⊂ VC satisfying Q(V 1,0 , V 1,0 ) = 0 (this is automatic in this case) is ˇ = PVC ∼ P1 where P1 = {τ-plane} ∪ ∞ = lines through the origin in C2.2 It is well known that H and P1 are homogeneous complex manifolds i.e., they are acted on transitively by Lie groups. Here are the relevant groups. Writing z = z0 z1 , w = w0 w1 and using Q to identify Λ2V with Q we have Q(z, w) = t wQz = z ∧ w and the relevant groups are Aut(VR,Q) ∼ SL2(R) for H Aut(VC,Q) ∼ SL2(C) for P1. In terms of the coordinate τ the action is the familiar one: τ → aτ + c cτ + d where ( a b c d ) ∈ SL2. This is because τ = z0/z1 and a b c d z0 z1 = az0 + bz1 cz0 + dz1 = z1 aτ + b cτ + d . 2 [CM-SP] is a general reference for period domains and their differential geometric proper- ties. A recent source is [Ca].