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Complex Analysis on Riemann Surfaces Contents 1 Complex Analysis on Riemann Surfaces Math 213b | Harvard University C. McMullen Contents 1 Introduction . 1 2 Maps between Riemann surfaces . 14 3 Sheaves and analytic continuation . 28 4 Algebraic functions . 35 5 Holomorphic and harmonic forms . 45 6 Cohomology of sheaves . 61 7 Cohomology on a Riemann surface . 72 8 Riemann-Roch . 77 9 The Mittag–Leffler problems . 84 10 Serre duality . 88 11 Maps to projective space . 92 12 The canonical map . 101 13 Line bundles . 110 14 Curves and their Jacobians . 119 15 Hyperbolic geometry . 137 16 Uniformization . 147 A Appendix: Problems . 149 1 Introduction Scope of relations to other fields. 1. Topology: genus, manifolds. Algebraic topology, intersection form on 1 R H (X; Z), α ^ β. 3 2. 3-manifolds. (a) Knot theory of singularities. (b) Isometries of H and 3 3 Aut Cb. (c) Deformations of M and @M . 3. 4-manifolds. (M; !) studied by introducing J and then pseudo-holomorphic curves. 1 4. Differential geometry: every Riemann surface carries a conformal met- ric of constant curvature. Einstein metrics, uniformization in higher dimensions. String theory. 5. Complex geometry: Sheaf theory; several complex variables; Hodge theory. 6. Algebraic geometry: compact Riemann surfaces are the same as alge- braic curves. Intrinsic point of view: x2 +y2 = 1, x = 1, y2 = x2(x+1) are all `the same' curve. Moduli of curves. π1(Mg) is the mapping class group. 7. Arithmetic geometry: Genus g ≥ 2 implies X(Q) is finite. Other extreme: solutions of polynomials; C is an algebraically closed field. 8. Lie groups and homogeneous spaces. We can write X = H=Γ, and ∼ g M1 = H= SL2(Z). The Jacobian Jac(X) = C =Λ determines an ele- ment of Hg= Sp2g(Z): arithmetic quotients of bounded domains. 9. Number theory. Automorphic forms and theta functions. Let q = P n2 exp(πiz) on H. Then f(q) = q is an automorphic form and k P n f(q) = an(k)q where an(k) is the number of ways to represent n as a sum of k ordered squares. 10. Dynamics. Unimodal maps exceedingly rich, can be studied by com- plexification: Mandelbrot set, Feigenbaum constant, etc. Billiards can be studied via Riemann surfaces. The geodesic flow on a hyperbolic surface is an excellent concrete example of a chaotic (ergodic, mixing) dynamical system. Definition and examples of Riemann surfaces. A Riemann surface is a connected, Hausdorff topological space X equipped with an open covering Ui and a collection of homeomorphisms fi : Ui ! C such that there exist analytic maps gij satisfying fi = gij ◦ fj on Uij = Ui \ Uj. We put the definition into this form because it suggest that (gij) is a sort of 1-coboundary of the 0-chain (fi). This equation will recur in the definition of sheaf cohomology. 2 The simply{connected examples. A deep theorem assets that every simply{connected Riemann surface is isomorphic to Cb, C or H. Let us consider each of these in turn. The Riemann sphere. The simplest compact Riemann surface is Cb = C[1 with charts U1 = C and U2 = Cb −f0g with f1(z) = z and f2(z) = 1=z. ∼ 1 2 Alternatively, Cb = P is the space of lines in C : 1 2 ∗ P = (C − 0)=C : The isomorphism is given by [Z0 : Z1] 7! z = Z1=Z0. 1 Note that we have many natural (inverse) charts C ! P given by f(t) = [ct + d; at + b]. The image omits a=c. Transitions between these charts are given by M¨obiustransformations. Theorem 1.1 The automorphism group of Cb is given by Aut(Cb) = SL2(C)=(±I): 2 Corollary 1.2 Every automorphism lifts to a linear map on C of deter- minant 1, well–defined up to sign. This group is also denoted PSL2(C). One can also regard the full automor- ∗ phism group as PGL2(C) = GL2(C)=C . Proof. For the proof, first normalize so that g 2 Aut(Cb) fixes z = 1. Then from the fact that g is analytic and injective at infinity one finds jg(z)j = O(jzj) on C, and hence g is a linear polynomial. n Projective space in general. It is also true that Aut CP is isomorphic to PGLn+1(C). The proof is more subtle; for example, when n = 2 it uses intersection theory to show that if L and L0 are lines, and g(L) meets L0 in two points or more, then g(L) = L0. Lie subgroups of SL2(C). The connected subgroups of SL2(C) correspond to various geometric features of the Riemann sphere. They can be classified by building up from the 1{parameter subgroups, which are easily described (as hyperbolic, parabolic or elliptic). We will get to know these subgroups 3 in examples below. Up to conjugacy, the most important ones are: G = SL2(C); K = SU(2); ∗ A = C ; N = C; B = AN; 0 ∼ G = SL2(R) = SU(1; 1); 0 ∗ A = R ; 0 N = R; B0 = A0N 0: In the list above, A and A0 are represented by diagonal matrices and N and N 0 are unipotent groups. Exercise: find the connected subgroups missing from the list above, and show your resulting list is complete. 2 The round sphere. The round metric on Cb is given by 2jdzj=(1 + jzj ) and preserved exactly by ( ! ) a b SU(2) = : jaj2 + jbj2 = 1 =∼ S3: −b a 1 2 Alternatively, if we represent points zi in P by unit vectors vi in C , then the spherical distance satisfies: 2 2 cos (d(z1; z2)=2) = jhv1; v2ij : The factor of 1=2 is not unexpected: for real projective space, the map 1 2 1 1 S ⊂ R − f0g ! RP = S =(±1) doubles angles as well. We can compute that in the round metric Z x 2 ds −1 d = d(0; x) = 2 = 2 tan (x); (1.1) 0 1 + s so x = tan(d=2). (This is the familiar transformation from calculus that turns any rational function of sin(θ) and cos(θ) into the integral of an ordi- nary rational function.) To check verify consistency with the cosine formula, 4 p 2 assume the latter, and set v1 = (1; 0) and v2 = (1; x)= 1 + x ; then 1 1 cos2(d=2) = hv ; v i2 = = 1 2 1 + x2 1 + tan2(d=2) using the identity 1 + tan2 θ = sec2 θ; so tan(d=2) = x. The Euclidean plane. Let B = AN ⊂ PSL2(C) denote the solvable group ∗ of affine maps z 7! az + b, a 2 C , b 2 C. To study the Euclidean plane we first note: Theorem 1.3 The automorphism group of C is isomorphic to B. Proof. Given g 2 Aut(C), apply Riemann's removable singularities theo- rem to 1=g(1=z) to show g extends to an element of Aut(Cb). ∼ Since B is the stabilizer of 1 2 Cb, we have Cb = G=B. Here B is called a Borel subgroup because the associated homogeneous space is a projective variety. The hyperbolic plane. Finally we consider the upper half-plane H = fz : Im(z) > 0g. By Schwarz reflection we easily find: Theorem 1.4 The automorphism group of H is isomorphic to P SL2(R). Note that PSL2(R) preserves not only the compactified real axis Rb, but also its orientation. Miraculously, the full automorphism of H preserves the hyperbolic metric jdzj=y. The main point is to check the case z 7! −1=z. The geodesics are circles perpendicular to the boundary. The imaginary axis is parameterized at unit speed by z = ies; the circle jzj = 1, by z = tanh(s) + i sech(s): (Recall tanh2(s) + sech2(s) = 1.) ∼ On the unit disk ∆ = H the automorphism group becomes ( ! ) a b SU(1; 1) = : jaj2 − jbj2 = 1 ; b a ∼ As a 3-manifold, SU(1; 1) = SL2(R) is homeomorphic to a solid torus, since it double covers the unit tangent bundle of ∆. Thus SL2(C) retracts 5 onto SU(2) (as is well-known) but not onto SU(1; 1), since they have different homotopy types. The reason is that the Gram{Schmidt process fails for indefinite metrics, because a null vector cannot be normalized. Hyperbolic distance in ∆. The unit disk ∆ can be interpreted as the 2 2 2 2 space of positive lines in C for the Hermitian form kZk = jZ0j − jZ1j . (Positivity implies jzj = jZ1=Z0j < 1.) As in the case of the sphere, we can use the underlying inner product to compute distances in the hyperbolic metric on the disk. We find: 2 2 cosh (d(z1; z2)=2) = jhv1; v2ij : Here it is useful to recall that t 7! (cosh t; sinh t) sweeps out (one sheet of) 1;1 2 2 the unit `circle' in R defined by x − y = 1, just as (cos t; sin t) sweeps 2 out the unit circle in R . Analogous to (1.1), we have, for x 2 (−1; 1), Z x d = d(0; x) = 2 ds=(1 − s2) = 2 tanh−1(x); 0 or equivalently x = tanh(d=2). Aside: H as the moduli space of triangles. We can think of H as the space of marked triangles (with ordered vertices) up to similarity (allowing reversal of orientation). The triangle attached to τ 2 H is the one with vertices (0; 1; τ). The `modular group' S3 acts by reflections, in the line x = 1=2 and the circles of radius 1 centered at 0 and 1. It is generated by τ 7! 1=τ and τ 7! 1 − τ. Note that the fixed loci of these maps are geodesics, and that the special point where they come together corresponds to an equilateral triangle, and that each line corresponds to isosceles triangles.
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