12 1. THE CLASSICAL THEORY: PART I Riemann surface of genus g 2 with the metric induced from that on H. By the Gauss-Bonnet theorem 0 2 2g = χ(Y ) = 1 Y KdA = K Area(Y ) . Proof of basic calculation. We define a section s Γ(H, V1,0) by s(τ) = τ 1 Vτ,0.1 The length squared is given by s(τ) 2 = its(τ)Qs(τ) = 2y. Using for τ = x + iy ∂τ= 1 2 (∂x i∂y) ∂¯= 1 2 (∂x + i∂y) we obtain i ∂∂ = 1 (∂x 2 + ∂y)dx 2 dy. This gives i ∂∂ log s(τ) 2 = 1 dx dy y2 . Remark. There is also a SU(2)-invariant metric on OP1(−1) induced from the standard metric on C2. For this metric s(τ) 2 c = 1 + |τ|2 (the subscript c on 2 c stands for “compact”). Then we have c1(Vc,0) 1 = 1 dx dy (1 + |τ|2)2 . Thus, V1,0 H is a positive line bundle whereas V1,0 c P1 is a negative line bundle with deg OP1(−1) = P1 c1(V1,0) c = −1. This sign reversal between the SL2(R)-invariant curvature on the open domain H and the SU(2) (= compact form of SL2(C))-invariant metric on the compact dual ˇ = P1 will hold in general and is a fundamental phenomenon in Hodge theory. Above we have holomorphically trivialized V1,0 H using the section s(τ) = τ 1 . We have also noted that we have the isomorphism of SL2(R)-homogeneous line bundles ωH V2,0. Now ωH has a section and a useful fact is that under this isomorphism = s(τ)2.
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