1. THE CLASSICAL THEORY: PART I 11 where Vz1,0 is the line in VC corresponding to z. If we use the group SL2(C) that preserves Q in place of GL2(C), then VC/Vz1,0 V 1,0∗ z . Thus the cotangent space Tz P1 Vz2,0 where in general we set Vn,0 = (V1,0)⊗n. The above identification ωP1 OP1(−2) is an SL2(C), but not GL2(C), equivalence of homogenous bundles. Convention. We set ω 1/2 H = V1,0. The Hodge bundle V1,0 H has an SL2(R)-invariant metric, the Hodge metric, given fibrewise by the 2nd Hodge-Riemann bilinear relation. The basic invariant of a metric is its curvature, and we have the following General fact. Let L Y be an Hermitian line bundle over a complex mani- fold Y . Then the Chern (or curvature) form is c1(L) = 1 2πi ∂∂ log s 2 where s O(L) is any non-vanishing local holomorphic section and s 2 is its length squared. Basic calculation. c1(V1,0) = 1 dx dy y2 = i (Im τ)2 . This has the following Consequence. The tangent bundle T H V0,2 has a metric dsH 2 = dx2 + dy2 y2 = 1 (Im τ)2 Re(dz d¯) of constant negative Gauss curvature. Before giving the proof we shall make a couple of observations. Any SL2(R) invariant Hermitian metric on H is conformally equivalent to dx2+ dy2 hence it is of the form h(x, y) dx2 + dy2 y2 for a positive function h(x, y). Invariance under translation τ τ + b, b R, corresponding to the subgroup ( 1 b 0 1 ), implies that h(x, y) = h(y) depends only on y. Then invariance under τ corresponding to the subgroup a1/2 0 0 a−1/2 , a 0, gives that h(y) = constant. A similar argument gives that c1(V1,0) is a constant multiple of the form above. The all important sign of the curvature K may be determined geometrically as follows: Let Γ SL2(R) be a discrete group such that Y = Γ\H is a compact
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