Technische Universitat¨ Berlin Institut fur¨ Mathematik Differential Geometry III —Complex and Quaternionic Line Bundles — Ulrich Pinkall Lecture notes Felix Knoppel¨ ,Oliver Gross and Theo Braune July 23, 2019 Table of Contents 1 Introduction1 2 Classification of Holomorphic Line Bundles9 2.1 Homology..................................9 2.1.1 Algebraic Topology......................... 10 2.1.2 Combinatorial Topology...................... 11 2.2 The Picard Group Pic(M) ......................... 14 2.3 Homology of Surfaces........................... 17 2.4 Holomorphic maps into CPn ....................... 23 2.5 Line Bundles over S2 ............................ 30 2.6 Clifford’s Theorem............................. 32 3 The Riemann-Roch Theorem 38 3.1 The Mittag–Leffler Theorem........................ 38 3.2 Complete proof of Riemann–Roch.................... 41 3.3 Teichmuller¨ Space for Tori......................... 51 3.4 Spin Bundles................................. 54 3.5 Quaternions................................. 55 I Preface The aim of this course is to cover three plans, i.e. ranges of subjects in chronolog- ical order. The background knowledge of the course participants may differ a lot, therefore there will be a quick review on topics of complex Analysis — in partic- ular Kahler¨ manifolds. Everyone should keep those in the back of their minds as they will become handy throughout the lecture. For the main course of the lecture the plans will be the following: Plan A: To start we aim to give a classification of holomorphic line bundles. Among others, some buzzwords for this topics will be Jacobian variety, Picard group, Abel map, ... Therefore, a compact, complex Riemann surface is M is consid- ered. The Picard group will be holo. LB L over M with deg L = d and holo. structure ¶¯ Pic (M) := . d fholom. equivalenceg In particular, the Jacobian variety will arise from this as Jac(M) := Pic0(M) . Each Picard group will naturally have the structure of a complex manifold g 2g 2g C R T = G = G , 2g ∼ g where G is an integer lattice in R = C . We will show that in fact Picd is al- gebraic, hence holomorphically embeddable in some CPN. Moreover we will see that Pic1 comes with a particularly nice line bundle whose holomorphic sections are called theta functions. We will then show, that all meromorphic functions on M can naturally be expressed as quotients of these theta func- tions. Plan B: The second part of the lecture aims to ”repair” an attempt to proof the Riemann-Roch theorem in the Complex Analysis II lecture of the preceding semester which turned out to be incomplete. Therefore The Mittag-leffler theorem and the Riemann-Roch theorem and its consequences will be tackled. II TABLE OF CONTENTS Plan C: If the remaining times allows us, we will introduce theory of quaternionic line bundles. Considering a conformal immersion f : M ! R3 of a Riemann surface (M, J) we can identify the euclidean three-space with the space of purely imaginary quaternions R3 = span fi, j, kg ⊂ H = R ⊕ R3 . We can express any quaternion q 2 H as q = w1˙ + a , where 1 is the multiplicative unit element of H (i.e. 1p = p1 = p for all p 2 H), w 2 R and a 2 R3. For two purely imaginary quaternions a, b 2 R3 we have the following rule of multiplication: ab˙ = − ha, bi + a × b . Using this quaternionic multiplication and the normal field N : M ! S2 ⊂ R3 of M we have N d f = d f ◦ J , or equivalently ∗d f = −N d f . This can be used to treat so called Willmore surfaces. Consider d ∗ d f = −H d f ^ d f , where H is the mean curvature. Then the Willmore functional is defined as Z W( f ) H2 det , M 1 where det = 2 d f ^ ∗d f . The (now solved) Willmore conjecture states that for every immersion of a torus f : T ! R3 it holds that W( f ) ≥ 2p2. This minimum is attained by the common parameterization of the torus with radii q R = (2) and r = 1. For spheres it is W( f ) = 4p. A Willmore surface is now defined as a critical point of W. A Willmore sphere is thus considered as an immersion f : S2 ! R3. The following interesting theorems hold: Theorem 0.1. There are Willmore spheres with W( f ) 2 f1, 4, 6, 8, 9, 10, 11, . .g p = (N n f2, 3, 5, 7g)p . Theorem 0.2 (Hopf). An immersion f : S2 ! R3 with H = const. is a round sphere. III TABLE OF CONTENTS IV 1. Introduction Definition 1.1. (Almost complex manifold) An almost complex manifold is a mani- fold M with J 2 GEnd(TM) such that J2 = −I. If M is almost complex, then dim M = 2n and all tangent spaces are complex vector spaces by setting (a + ib)X := aX + bJX . Example 1.2. 1. M = Cn. 2. M = CPn the complex projective space which is defined as the set of all complex 1-dimensional subspaces of Cn+1. Coordinate charts are given by 8 0z 1 9 > 1 > <> B . C => n B . C ∼ n CP n H¥ = C B C z1,..., zn 2 C = C > @znA > > > : 1 ; 8 0z 1 9 > 1 > <> B . C => B . C H¥ = C B C z1,..., zn 2 C . > @znA > > > : 0 ; where H¥ is a hyperplane at infinity. Definition 1.3. (holomorphic map) Let M and M˜ be almost complex manifolds. Then f : M ! M˜ is called holomorphic, if d f (JX) = Jd˜ f (X), 8X 2 TM . 1 Remark 1.4. Holomorphicity of a map f : M ! M˜ is equivalent to saying that dp f : Tp M ! Tf (p) M˜ is a complex linear map for all p 2 M. Definition 1.5. (complex manifold) An almost complex manifold is called complex, if there exist local holomorphic coordinate charts. The complex projective space CPn is a complex manifold. Definition 1.6. (complex submanifold) A submanifold M in a almost complex com- plex manifold M˜ is called a complex submanifold, if all tangent spaces Tp M ⊂ Tp M˜ are complex subspaces (invariant under J).˜ The restriction of J˜ to TM is called the induced complex structure. A complex submanifold has an induced almost complex structure ˜ Jp = JjTp M and the inclusion M ,! M˜ is holomorphic. Theorem 1.7. (complex implicit function theorem) Let M and M˜ be complex manifolds, g : M ! M˜ a holomorphic submersion and q 2 M.˜ Then g−1fqg ⊂ M is a complex submanifold. Example 1.8. Let f : Cn ! C be a complex polynomial. Then the set n M := fp 2 C j f (p) = 0, dp f 6= 0g is a complex submanifold of Cn—a complex hypersurface. As a concrete example, M = f(x, y) 2 C2 j y2 = x(x − 1)(2 − x)g 2 2 2 is such a hypersurface in C . If we consider it as a submanifold of CP = C [ H¥ (H¥ denotes the plane at infinity), then it forms a torus with a point removed: 2 Introduction M˜ := fC(x, y, z) j y2z + x3 − x2z + 2xz2 = 0g is a torus, M˜ \ C2 = M and 3 M˜ \ H¥ = f(C(x, y, z) j x = 0)g . More generally: n fC(z0,..., zn) 2 CP j p(z0,... zn) = 0g, where p denotes a homogeneous complex polynomial of degree k, is called an algebraic hypersurface in CPn of degree k. Definition 1.9. (algebraic variety) The intersection of finitely many algebraic hyper- surfaces of degree k1,..., k` is called an algebraic variety of degree k1 · ... · k`. Remark 1.10. If an algebraic variety is a submanifold, i.e. there are no singularities, then it is complex. Theorem 1.11 (Resolution of Singularities, Hironaka ’64). Every algebraic variety is the image of a compact complex manifold under a holomorphic map which—away from singularities—is biholomorphic. Example 1.12. The variety M = f(x, y) 2 C2 j y2 = −x3g is the image of C under g : C ! C2, t 7! g(t) = (−t2, t3) . Theorem 1.13 (Chow). Every compact complex submanifold of CPn is algebraic. Remark 1.14. Globally we can view all compact complex submanifolds of CPn as the zeroset of some polynomial. 3 Definition 1.15. (hermitian manifold) A hermitian manifold is an almost complex manifold (M, J) with a Riemannian metric h., .i such that hJX, Yi = − hX, JYi for all vector fields X and Y. Definition 1.16. A hermitian manifold is called K¨ahlerif rJ = 0. Example 1.17. CPn has a famous metric, namely the Fubini–Study metric. The sectional curvatures KE of the metric satisfy 1/4 ≤ KE ≤ 1 for all 2-planes E ⊂ Tp M for all p 2 M. Remark 1.18. A theorem from differential geometry II states that if the sectional 1 curvatures of M satisfy 4 < KE ≤ 1 then it is diffeomorphic to an n-sphere. So CPn is the manifold that is ”closest” to a sphere but not a sphere itself. Easy to see: A complex submanifold of a Kahler¨ manifold is itself Kahler.¨ Corollary 1.19. All algebraic submanifolds in CPn are K¨ahler. Proof. This follows by the observation above and Chows theorem. On de-Rham Cohomology Define the exterior derivative by k k+1 dk : W (M) ! W (M) then dk+1 ◦ dk = 0 from which we get im dk ⊂ ker dk+1 . With Hk M we denote the k-th de-Rahm cohomology, Hk M = ker dk . im dk−1 By the elliptic theorem, Hk M is a finite-dimensional vector space. The dimension k bk := dim H M is called the k-th Betti–number. Example 1.20. 1. If M is connected, then H0 M = R. 4 Introduction 2.