Differential Geometry III

Home , Tautological bundle

Technische Universitat¨ Berlin Institut fur¨ Mathematik

—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

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 particular 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 considered. The Picard group will be

holo. LB L over M with deg L = d and holo. structure ∂¯ Pic (M) := . d {holom. equivalence}

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 = Γ = Γ , 2g ∼ g where Γ is an integer lattice in R = C . We will show that in fact Picd is algebraic, 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 functions.

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 {i, j, k} ⊂ H = R ⊕ R3 .

We can express any quaternion q ∈ H as

q = w1˙ + a ,

where 1 is the multiplicative unit element of H (i.e. 1p = p1 = p for all p ∈ H), w ∈ R and a ∈ R3. For two purely imaginary quaternions a, b ∈ 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 ) ≥ 2π2. 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 ) = 4π. 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 ) ∈ {1, 4, 6, 8, 9, 10, 11, . . .} π = (N \ {2, 3, 5, 7})π .

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 manifold M with J ∈ ΓEnd(TM) such that J2 = −I.

If M is almost complex, then dim M = 2n and all tangent spaces are complex vector spaces by setting (α + iβ)X := αX + βJX .

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

 z    1    .   n  .  ∼ n CP \ H∞ = C   z1,..., zn ∈ C = C  zn     1   z    1    .    .  H∞ = C   z1,..., zn ∈ C .  zn     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), ∀X ∈ 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 ∈ 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 complex 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 = J|Tp 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 ∈ M.˜ Then g−1{q} ⊂ M is a complex submanifold.

Example 1.8. Let f : Cn → C be a complex polynomial. Then the set

n M := {p ∈ C | f (p) = 0, dp f 6= 0} is a complex submanifold of Cn—a complex hypersurface. As a concrete example,

M = {(x, y) ∈ C2 | y2 = x(x − 1)(2 − x)}

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˜ := {C(x, y, z) | y2z + x3 − x2z + 2xz2 = 0} is a torus, M˜ ∩ C2 = M and 3 M˜ ∩ H∞ = {(C(x, y, z) | x = 0)} . More generally: n {C(z0,..., zn) ∈ CP | p(z0,... zn) = 0}, 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 = {(x, y) ∈ C2 | y2 = −x3} is the image of C under

γ : C → C2, t 7→ γ(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 ∇J = 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 ∈ 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 : Ω (M) → Ω (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 βk := 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. If M is compact and oriented, then Hk M =∼ Hn−k M (Poincare´ duality). To see this, consider [ω] with harmonic representative α, then [ω] 7→ [∗α] is the desired isomorphsim.

3. H0Sn =∼ R =∼ HnSn and HkSn = 0 for 1 ≤ k ≤ n − 1.

We aim to answear the question of which compact complex manifolds M can be holomorphically embedded in some CPn. A necessary condition that we have already derived is that M admits a Kahler¨ metric.

Definition 1.21. (Kählerform) Let (M, J, h., .i) be a hermitian manifold. Then define the Kählerform σ ∈ Ω2(M) by

σ(X, Y) = hJX, Yi .

First we see that σ is well deﬁned as from J∗ = −J we get

σ(X, Y) = hJX, Yi = − hX, JYi = −σ(X, Y) .

Further σ is non-degenerate, i.e.

σ(X, Y) = 0 for all Y ∈ ΓTM ⇒ JX = 0 ⇒ X = 0 .

Example 1.22. Consider M = R2 = C with complex structure

0 −1 J = 1 0 and standard metric on R2. Then

x y x −y x y σ 1 , 1 = 1 , 2 = det 1 , 1 x2 y2 x2 y1 x2 y2 hence the Kahler¨ form σ on R2 is the determinant.

Theorem 1.23. Let V be a vector space with 2n = dim V < ∞ and σ : V × V → R skew-symmetric and non-degenerate. Then there is a basis (u1, v1,..., un, vn) of V such that we have a correspondence

0 −1 0 −1 σ ↔ diag ,..., . 1 0 1 0

In particular for σ = dx1 ∧ dy1 + ... + dxn ∧ dyn we get σ ∧ ... ∧ σ = n!dx1 ∧ dy1 ∧ ... ∧ dxn ∧ dyn 6= 0

5 Proof. Pick 0 6= u1 ∈ V. As σ is non-degenerate there is v1 ∈ V such that ⊥σ ⊥σ σ(u1, v1) = 1. Pick 0 6= u2 ∈ span {u1, v1} and v2 ∈ span {u1, v1} such that σ(u2, v2) = 1. Inductively the claim follows.

Theorem 1.24. If (M, J, h., .i) is K¨ahler, then the K¨ahlerform is closed, i.e. dσ = 0.

Proof. Fix p ∈ M and choose Xˆ , Yˆ, Zˆ ∈ Tp M and extend these to vector ﬁelds X, Y, Z on M such that Xp = Xˆ , Yp = Yˆ and Zp = Zˆ and ∇XY = ∇Y X = ∇X Z = ... = 0. Then [X, Y]p = [X, Z]p = ... = 0 and

dσ(Xˆ , Yˆ, Zˆ ) = (Xσ(Y, Z) + Yσ(Z, X) + Zσ(X, Y))p

= (hJ∇XY, Zi + hJY, ∇X Zi + hJ∇Y Z, Xi

+ hJZ, ∇Y Xi + hJ∇ZX, Yi + hJX, ∇ZYi)p = 0

Deﬁnition 1.25. (symplectic manifold) A symplectic manifold is a manifold M together with a non-degenerate and closed σ ∈ Ω2(M).

Remark 1.26. This means that every Kahler¨ manifold, which naturally comes with a Kahler¨ form, is in fact a symplectic manifold.

Theorem 1.27. If M is a compact, oriented K¨ahlermanifold, then σ is not exact.

Remark 1.28. In terms of cohomology this is [σ] 6= 0.

Proof. Assume σ = dα for some α ∈ Ω1(M), then

d(α ∧ σ ∧ ... ∧ σ) = σ ∧ ... ∧ σ = n!detM . | {z } | {z } n−1 times n times

Then Z Z 1 vol(M) = detM = n! d(α ∧ σ ∧ ... ∧ σ) = 0 M M

Corollary 1.29. If a manifold M is K¨ahler, then H2(M) 6= {0}.

Remark 1.30. That means that a cohomology has non-trivial second cohomology.

C2 \ {0} Fix λ ∈ R, λ > 1 and deﬁne M = ∼ where x ∼ y :⇔ y = λnx for some n ∈ Z.

6 Introduction

Then M is diffeomorphic to S1 × S3. The K¨unnethformula (1923) states that if ∗ 0 n [α1],..., [αm] is a basis of H (M) := H (M) ⊕ ... ⊕ H (M) and [β1],..., [βr] is a ba- ∗( ˜ ) = ∧ [ ] sis of H M then we can deﬁne γij : αi βj which are closed and γij i=1,...,m; j=1,...,r form a basis of H∗(M × N) =∼ H∗(M) ⊕ H∗(N).

Corollary 1.31. H2(S1 × S3) = {0} so the complex manifold S1 × S3 is not K¨ahler. In particular it is not realizable in any CPn hence not algebraic.

Over CPn we have the tautological line bundle deﬁned by

LCψ = Cψ for all points Cψ ∈ CPn. Note that the ﬁber at Cψ istself is a complex 1-dimensional vector space. The (Fubini-Study) metric on CPn uses the standard metric on Cn+1. Hence we get a connection ∇ on L whose curvature form is given by

R(X, Y)ψ = −σ(X, Y)Jψ .

In other words: the curvature form is given by the Kahler¨ form.

7 Note that: Z Z ˜ ∗ 1 ˜ ∗ 1 ∗ deg ( f ◦ f ) L = 2π ( f ◦ f ) σ = 2π f σ˜ ∈ Z M M where we used that the pullback of the curvature equals the curvature of the pullback (cf. DGII).

Deﬁnition 1.32. (integral forms)

Z 1. Let M˜ be a manifold, then [ω] ∈ H1(M˜ ) is called integral if γ∗ω ∈ Z for S1 all loops γ : S1 → M.˜ Z 2. σ ∈ H2(M˜ ) is called integral if f ∗σ ∈ Z for all smooth maps f : M → M˜ M where M is a compact oriented surface.

By our computation above we yield the following theorem:

Theorem 1.33. If a K¨ahlermanifold can be isometrically embedded into CPn for some 1 n ∈ N, then 2π σ is integral where σ is the K¨ahlerform of M.

Theorem 1.34. (Kodaira embedding theorem (1964)) A complex manifold can be holomorphically embedded in some CPN for some N ∈ N if and only if it is Kählerand the Kählermetric has integral Kählerform.

8 2. Classiﬁcation of Holomorphic Line Bundles

2.1 Homology

Consider a compact oriented manifold M, then the k-th cohomology of M is given by d ker Ωk(M) →k Ωk+1(M) Hk(M) := . d im Ωk−1(M) →k−1 Ωk(M)

The k-th Betti number is respectively given by

k βk := dim H (M) < ∞ .

Deﬁnition 2.1 (k-th homology). For a compact, oriented manifold M,

K ∗ Hk(M) := H (M)

is called the k-th homology vector space of M.

Example 2.2. Let Mˆ be a compact, oriented manifold and γ : Mˆ → M smooth.

Then the map Z Hk(M) 3 [ω] 7→ γ∗ω Mˆ

9 2.1 Homology is well defined as dω = 0 and for α ∈ Ωk−1(M) it holds that Z Z γ∗(ω + dα) = γ∗ω . Mˆ Mˆ Hence, γ defines a homology class [γ] (as it defines a well defined linear map from HK(M)).

Deﬁnition 2.3 (homologous). Say γ is homologous to γ˜, denoted by γ ∼ γ˜, if for all ω ∈ Ωk(M) it holds that Z Z γ∗ω = γ˜ ∗ω . Mˆ Mˆ

2.1.1 Algebraic Topology

Any books on algebraic topology treat the topics of homology and cohomology different than we do. To proof that in fact, for our purposes these deﬁnitions coincide is the next goal. Remark 2.4. Some of the following stuff may also be found in the DGIII script from 1997/98.

Algebraic topology is usually treated in the sense of simplicial homology. A k- k+1 simplex Sk ⊂ R is deﬁned as

n k+1 o Sk := x ∈ R | x1 + ... + xk+1 = 1, xj ≥ 0 .

A map σ : Sk → M gives a k-simplex in M.

As naturally a simplex comes with an orientation, for any ω ∈ Ωk(M) we can define Z Z σ∗ω =: ω . Sk σ Remark 2.5. The following definitions will be a bit vague and should, if they are not clear, be revised in some book. An excellent introduction to this theory is given on the first 50 pages in ”Differential forms in algebraic topology” by Bott and Tu. This book is written from a rather differential geometric viewpoint. Another excellent reference (on combinatorial topology) is ”A textbook on topology” by Seifert and Threlfall (1934).

10 Classiﬁcation of Holomorphic Line Bundles

A k-chain is a map

c : Ck(M) := {all k-simplices σ in M} → Z such that c(σ) = 0 except for finitely many σ. The definition of integralion over a k-chain can be extended to Z Z ω = ∑ s(σ) ω . c σ σ We define the boundary operator as

∂ : Ck(M) → Ck−1(M) .

Then ker ∂ Hk(M, Z) := k im ∂k+1 is an abelian group. Further, without a proof, we state that

∼ βk Hk(M, Z) = Z × G , where G is a ﬁnite abelian group which in this context is also called the torsion. It turns out that ∼ βk Hk(M, R) := Hk(M, Z) ⊗ R = R .

Theorem 2.6 (de-Rham). With respect to the pairing given by integration of k-forms over k-chains it holds that K ∗ Hk(M, R) = H (M) .

2.1.2 Combinatorial Topology

Combinatorial topology is similar to simplicial homology but the attention is on the simplices of a ﬁxed triangulation, i.e. a cell-decomposition into cells that are diffeomorphic to a triangle, of a surface M.

Endow all k-simplices with a random orientation. A k-chain assigns to each k- simplex σ an integer cσ ∈ Z.

Noatation: We write (−1)σ for the simplex σ with opposing orientation. Further 100σ denotes 100 copies of σ with the original orientation.

11 2.1 Homology

Also in the combinatorial topology setup we have a boundary operator

∂ : Ck(M) → Ck−1(M) and ker ∂ ∼ k = Hk(M, Z) . im ∂k+1 We recall the deﬁnition of Z H1(M) := ker d1 3 ω 7→ ω | γ : Mˆ → M smooth, Mˆ cmpct., or., dim Mˆ = 1 γ

As Z Z ω + d f = ω γ γ we can deduce that 1 ∗ H1(M) ⊂ H (M) and we yield a pairing Z h[ω]|[γ]i = ω . γ

In order to be consistent with the rest of the world we need to proof that the theories coincide.

1 Theorem 2.7. The pairing between H (M) and H1(M) as deﬁned above is non- degenerate.

Z Proof. Let ω = 0 for all ω ∈ Ω1(M) with dω = 0, then by deﬁnition [γ] = 0. If γ Z contrarily ω = 0 for all collections of loops γ, then ω is exact by the Poincare-´ γ Lemma, i.e. [ω] = 0.

So we can deduce that 1 dim H1(M) = dim H (M) because ∼ 1 ∗ H1(M) = H (M) .

Now we give a proof that our deﬁnition of H1(M) is consistent with the combinatorial deﬁnition.

12 Classiﬁcation of Holomorphic Line Bundles proof of the consistency of the theory: Consider a closed 1-chain c of a triangulation of M. Then ∂c = 0 which means that at each vertex there is the same number (counted with mulitplicity) of edges incoming as there are outgoing. At each vertex match each incoming edge to an outgoing edge. (We think of edges e with c(E) > 1 as separated copies) Doing so yields a collection of discrete closed edge loops. Smooth each of these (by a slow-down parametrization) to yield a smooth

γ : Mˆ → M .

Then for all ω ∈ ker d1 we have Z Z ω = ω . c γ

So for a discussion of 1-homology we can replace 1-chains by smooth maps from 1-dimensional oriented, compact manifolds. The good news is that this also works for 2-homology. The way to see this is the following:

Again split multiple triangles into singly counting copies and match adjacent triangles with one that induces the opposed orientation on the shared edge.

In an octahedron, 12 triangles are meeting at the centered vertex. At each of the 6 edges we know how to connect triangles, hence we get 3 surfaces passing through this vertex. Smooth the resulting triangulated surfaces by their parametrization to yield smooth γ : Mˆ → M.

13 2.2 The Picard Group Pic(M)

The bad news is that starting with 3-homology this method doesn’t work anymore — the results are only pseudomanifolds. Rumors are (i.e. we have no reference Z ∗ yet) that not every element of Hk(M, Z) ⊗ R can be realized as ω 7→ γ ω for Mˆ some smooth γ : Mˆ → M with Mˆ a k-dimensional compact, oriented manifold.

2.2 The Picard Group Pic(M)

Let V and W be ﬁnite-dimensional complex vector spaces. Then

V ⊗ W = {β : V∗ × W∗ → C | C − bilinear}.

Further for v ∈ V and w ∈ W we set

v ⊗ w(α, η) = α(v)η(w).

Note: The way we deﬁned the tensor product, it is not symmetric! Neverthe- less, there is a canonical isomorphism V ⊗ W → W ⊗ V (switching arguments). Moreover, given a third complex vector space U there are canonical isomorphisms (U ⊗ V) ⊗ W = U ⊗ (V ⊗ W) (leaving slots empty). Thus we can write U ⊗ V ⊗ W.

Let M be a compact Riemann surface. Then

Pic(M) = {isomorphism classes of holomorphic line bundles(L, ∂¯) over M}.

˜ Lemma 2.8. Let (L, ∂¯ L) and (L˜ , ∂¯ L) be holomorphic line bundles. Then there is a unique ˜ holomorphic structure ∂¯ L⊗L on L ⊗ L˜ such that

˜ ˜ ∂¯ L⊗L(ψ ⊗ ϕ) = (∂¯ Lψ) ⊗ ϕ + ψ ⊗ (∂¯ L ϕ) .

Proof. Left as an exercise.

Theorem 2.9. The tensor product deﬁnes a multiplication

Pic(M) × Pic(M) → Pic(M)

and turns Pic(M) in an abelian group with identity element (M × C, ∂¯ on functions) and inverse element given by L−1 = L∗.

Proof. Left as an exercise.

Remark 2.10. By deﬁnition we have for ω ∈ ΓL−1 that

−1 ¯ L ¯ ¯ L ∂X ω ψ = ∂X(ω(ψ)) + ω ∂Xψ , where ∂¯ is the usual holomorphic structure on C∞ functions.

14 Classiﬁcation of Holomorphic Line Bundles

Theorem 2.11. The map deg: Pic(M) → Z is a surjective homomorphism of abelian groups.

Deﬁnition 2.12. For d ∈ Z we deﬁne

−1 Picd(M) := deg (d) ,

Jac(M) := Pic0(M) .

So we can focus on Jac in order to put more structure in Picd.

We have

Picd(M) ⊗ Picd˜(M) = Picd+d˜(M) and Jac(M) ⊂ Pic(M) .

Thus Jac(M) acts on Picd(M). One easily checks that this actions is free and tran- sitive. Further, for ﬁxed (L, ∂¯) with deg L = d the map

0 0 Jac → Picd, L 7→ L ⊗ L is bijective. Here, L0 is an arbitrary but ﬁxed point in Jac(M).

So we can focus on Jac(M) in order to put more structure on Picd(M).

Let (L, ∂¯) be a holomorphic line bundle over M. Then there is a hermitian metric h., .i on L. Remember, by a hermitian metric we mean here a real-valued ﬁber metric with respect to which J is orthogonal.

Theorem 2.13. There is a unique complex connection ∇ on (L, ∂¯) which is metric with respect to h., .i and which satisﬁes

¯ 00 1 ∂ = ∇ = 2 (∇ − J ∗ ∇) .

15 2.2 The Picard Group Pic(M)

Proof. Let ∇˜ be a metric connection on (L, h., .i). Then all other metric connections are of the form ∇ = ∇˜ + ηJ for η ∈ Ω1(M). Computing the ∂¯-operators gives 00 ˆ 00 1 ˆ 00 00 ∇Xψ = ∇Xψ + 2 (η(X)Jψ + Jη(JX)Jψ) = ∇Xψ + JηXψ where we used that for X ∈ Γ(TM) and ψ ∈ ΓL it holds that

00 1 ∇Xψ = 2 (∇Xψ + J∇JXψ) . On the other hand all holomorphic structures on L arise of the form

∂¯ = ∇ˆ 00 + ω where ω ∈ ΓKL¯ . The goal is to choose η such that

Jη00 = ω and 00 1 1 Jη = J 2 (η − J ∗ η) = 2 ∗ η + Jη . Hence the only (therefore unique) choice is

η := 2 Im ω which then leads to 1 Re ω = ∗ η . 2

Theorem 2.14. Let (L, ∂¯) be a holomorphic line bundle of degree d and σ ∈ Ω2(M) Z with σ = 2πd. Then there is a hermitian metric h., .i on L (unique up to constant M scale) such that the corresponding metric ∇ with ∇00 = ∂¯ has curvature R∇ = −σJ hence Ω = σ as curvature 2-form.

Proof. We start with some hermitian metric h., .i and its associated ∇ with ∇00 = ∂¯. Then, if we change the metric to be e2uh., .i, we obtain a new adapted connection ∇˜ . Since J is parallel with respect to both connections ∇ and ∇˜ we have that

∇˜ = ∇ + ω, ω ∈ Ω1(M; C) .

∼ Furthermore if we ask for h., .i to be metric,

2u ∼ X(e hψ, ψi) = 2h∇Xψ, ψi + hψ, ψi(du − α)(X) implies that α = du .

16 Classiﬁcation of Holomorphic Line Bundles

Moreover, from ∇˜ 00 = ∂¯ follows that ω ∈ ΓKL. Thus

ω = 2(du)0 = 2∂u .

The curvatures of ∇ and ∇˜ are thus related by

R˜ = R + 2d(∂u) = R + d ∗ duJ .

So, if R = −ΩJ, we want to solve

d ∗ du = Ω − σ, Z which is solvable due to the elliptic theorem, since Ω − σ = 0. The obtained M solution is unique upt to an additive constant.

2.3 Homology of Surfaces

Recall: We have seen that a closed 1-chain is the same as a map γ : Mˆ → M of a 1-dimensional compact oriented manifold, i.e. a ﬁnite union of oriented circles, into M. Then two 1-chains γ and γ˜ are called homologous, if Z Z γ ∼ γ˜ :⇐⇒ ω = ω for all closed ω ∈ Ω1 M . γ γ˜

Theorem 2.15. Let M˜ be a compact surface with boundary Mˆ = ∂M˜ and f : M˜ → M be smooth. Then the 1-chain γ = f |Mˆ is null-homologous, i.e. γ ∼ 0.

Proof. That follows directly from Stokes’ theorem as for ω ∈ Ω1(M) with dω = 0 we have Z Z Z Z ω = γ∗ω = d(γ∗ω) = γ∗(dω) = 0 . γ Mˆ M˜ M˜

Deﬁnition 2.16. Let γ, γ˜ : Mˆ → M. Then γ is called homotopic to γ˜, if there is a ˆ smooth map f : [0, 1] × M → M with γ = f{0}×Mˆ and γ˜ = f{1}×Mˆ

Intuition: Homologous means: Transformable into each other by a homotopy with ﬁnitely many reconnection events.

Theorem 2.17. Let M be connected. Then every collection of loops γ : Mˆ → M is homologous to nγ˜ where n ∈ Z where γ˜ is a smooth embedding.

Proof. By the transversality theorem we can homotope γ slightly so that it becomes an immersion with transversal self-intersections.

17 2.3 Homology of Surfaces

The self-intersections can be resolved, so we can assume that γ is an embedding.

Then γ divides M into several components M1,..., Mn, all of which are compact surfaces with boundary. Each boundary ∂Mi is a union of components of γ (with suitable orientation).

Now build an oriented multi-graph with vertices {M1,..., Mn} and an oriented edge from Mi to Mj, if Mi is adjacent to Mj across a component of γ which has Mi to its left. Now we reduce the number of γ components as follows: If there is a vertex with either two incoming or two outgoing edges, then the corresponding γ components can be joined in such a way that it stays embedded.

18 Classiﬁcation of Holomorphic Line Bundles

Finally we obtain a graph that has only vertices with one incoming or one outgoing edge—in which case the corresponding γ-component bounds a surface and is hence null-homologous, so these edges can be deleted—or exactly one incoming and one outgoing edge. Thus we end up with a graph which is a union of cy- cles. Since M is connected there is only one cycle. Thus all components of γ are homologous to one of them, say γ˜.

We know: For ω ∈ Ω1 M with dω = 0, Z ω = 0 for all closed 1-chains γ ⇐⇒ [ω] = 0, i.e. ω is exact . γ

1 ∗ Thus H1 M spans (H M) and hence, by de-Rhams theorem, there are γ1,..., γ2g 1 ∗ such that [γ1],..., [γ2g] form a basis (over R) of (H M) . Going further in this direction we would see that there are embeddings

1 γ1,..., γ2g : S → M such that

1 ∗ H1 M = H (M) = {n1[γ1] + ··· + n2g[γ2g] | n1,..., n2g ∈ Z} .

Deﬁnition 2.18 (dual lattice). The set Z Γ = {[ω] ∈ H1 M | ω ∈ 2πZ for all 1-chains γ} . γ

is called the dual lattice.

19 2.3 Homology of Surfaces

Motivation: f : M → S1 ⊂ C, then ω = hd f , i f i is closed. Γ is is the set of of all ω that come from such f .

Let M be a compact Riemann surface and L and L˜ holomorphic line bundles over M. Then

L ∼holo. L˜ :⇔ ∃ holom $ ∈ ΓHom(L, L˜ ) without zeros Last time we showed that:

1. It is sufﬁcient to look at the case of deg L = 0.

2. Every L with deg L = 0 has a unique connection that is ﬂat, metric, complex and satisﬁes ∇00 = ∂¯.

Remark 2.19. Note that in this context ∇ is called metric if there is a metric h., .i on L such that ∇ h., .i = 0.

As it is our goal to classify holomorphic line bundles up to holomorphic isomorphisms it sufﬁces to consider ﬂat metric connections and bundles of degree 0.

Example 2.20. The trivial bundle L = M × C with ∇ = d and standard metric h·, ·i has the desired properties.

Any other metric bundle L˜ with degL˜ = 0 is isomorphic to (L, h·, ·i). The connection on L˜ is then given by ∇˜ = ∇ + ωJ.

This yields for the curvature tensor

˜ R∇ = R∇ − dωJ.

˜ As both connections are assumed to be ﬂat, i.e. R∇ = R∇ = 0), this implies

−dω = 0 hence ω must be closed. So given a flat connection ∇ all other flat connections can be obtained by appropriately adding a closed 1-form ω. The question is now if we can state another condition on ω so that the resulting ∇˜ is flat.

Definition 2.21 (flat hermitian bundle). The bundle L˜ is called trivial as a flat hermitian bundle if and only if there is a nowhere vanishing ψ ∈ ΓL˜ with |ψ| = 1 and ∇˜ ψ = 0 .

Let ϕ be deﬁned as ϕp = (p, 1) then we have ∇ϕ = 0 and |ϕ| = 1.

20 Classiﬁcation of Holomorphic Line Bundles

Let ψ ∈ γˆ ∗L be parallel with respect to γ∗(∇˜ ). Then we deﬁne an angle function f : [0, 2π] → R such that i f (t) ψt = e = ϕˆt, where ϕˆ = γˆ ∗ ϕ. Since ψ is parallel, we obtain:

∗ ∗ ∗ J f (t) 0 = (γˆ ∇˜ ∂ ψ = (γˆ ∇ + γˆ ωJ) ∂ (e )ϕˆ) ∂t ∂t = J f 0(t)eJ f (t) ϕˆ − Jω(γ0(t))eJ f (t) ϕˆ = JeJ f (t) ϕˆ( f 0(t) − ω(γ0(t))) ⇔ f 0(t) = ω(γ0(t))

Hence we choose f 0(t) = ω(γ0(t)). Thus -as in physics-, ω describes the angular speed.

Definition 2.22. Let ∇˜ be a flat metric connection on a complex line bundle over a Riemann surface M and γ : S1 → M a closed curve. Let ψ be a parallel section of ∗ 1 γˆ L. Then h(γ) ∈ S defined by ψ2π = h(γ)ψ0 is called the monodromy of ∇˜ along γ.

Theorem 2.23. If ∇˜ = ∇ − ωJ, where ∇ is the trivial connection, we have:

J R ω h(γ) = e γ .

Proof. We have

J f (2π) J f (2π) J( f (2π)− f (0)) ψ2π = e ϕˆ2π = e ϕˆ0 = e ψ0 .

This yields for the monodromy

J( f (2π)− f (0)) J R 2π f 0 h(γ) = e = e 0 .

Theorem 2.24. Two ﬂat hermitian line bundles over a Riemann surface M are isomorphic if and only if they have the same monodromy over every closed curve γ

21 2.3 Homology of Surfaces

Proof. We know that L is trivial as a ﬂat metric line bundle if and only if L has a nowhere vanishing parallel section. If L has a parallel section ψ, we know that ψ ∈ Γγˆ ∗L is parallel with respect to γ∗(∇˜ ) for any closed curve γ : S1 → M. The calculation above yields that ω has a potential f . Stokes theorem yields that Z ω = 0 γ for all the desired curves γ. Hence we have by theorem above that h(γ) = 1 . If conversely h(γ) = 1 for all γ : S1 → M we have that Z ω = 0. γ

Now construct as in the proof of exactness of 1-forms a parallel section. Finally we have to build a bridge to arbitrary bundles. In order to do this note that the bundles L and L˜ are isomorphic. It sufﬁces to show that for the trivial bundle we have h(γ) = 1 for all desired curves γ. Then we use that two arbitrary bundles L and L˜ are isomorphic if and only if L ⊗ L˜ is trivial.

Theorem 2.25. Let (L, ∇) be a ﬂat hermitian line bundle over M. Let γ, γ˜ : S1 → M. If γ ∼ γ˜ then h(γ) = h(γ˜)

Proof. This follows directly from theorem 2.22, because integrals over homolo- geous curves are equal.

Recall that the dual lattice was deﬁned to be Z Γ = {[ω] ∈ H1 M | ω ∈ 2πZ for all closed curves γ} . γ

Γ is an additive subgroup of H2(M) =∼ R2g that is isomorphic to Z2g. Now consider f : M → S1 with d f = ωJ f . Here we have ω ∈ Γ as integrating would give some winding number.

22 Classiﬁcation of Holomorphic Line Bundles

Theorem 2.26. Let ∇, ∇˜ be ﬂat metric connections on (L, h·, ·i), with ∇˜ = ∇ + ωJ, where ω ∈ Ω1(M) and dω = 0. Then L˜ ∼ L if and only if [ω] ∈ Γ.

Summary: It was our goal to classify holomorphic line bundles. In order to describe Pic for a bundle of degree d, we have seen that it suffices to consider Jac. For any bundle (L, ∂¯) ∈ Jac we found a flat metric connection ∇ with ∇00 = ∂¯. The last theorem 1 H (M) 2g yields that Jac(M) can be identified with Γ = T , so it suffices to study the 2g-dimensional torus.

Corollary 2.27. If M is diffeomorphic to S2 with H1(M) = {0}, then for each integer d ∈ Z, there is - up to isomorphism - exactly one holomorphic line bundle L with deg(L) = d.

In particular the bundle L is holomorphically trivial for deg(L) = 0. How do other line bundles over S2 look like? The Riemann–Roch theorem states that

h0(L) − h0(KL−1) = d + 1 − g . |{z} =0

Furthermore we obtain:

deg(K) = deg(T∗ M) = −χ(M) = 2g − 2 = −2 .

If we would have d < 0, then it follows that h0(L) = 0. This can be seen as follows: A holomorphic section ψ ∈ Γ(L) might have zeros but no poles. The Poincare-Hopf´ Index theorem yields that deg(L) must be positive. If we have d ≥ 0, we obtain that deg(KL−1) ≤ −2 < 0. Therefore we have h0(KL−1) = 0 and h0(L) = d + 1.

2.4 Holomorphic maps into CPn

For a compact Riemann surface M let f : M → CP1 =∼ S2 =∼ C ∪ {∞} be holomorphic. Then this is the same as if f : M → C is meromorphic.

23 2.4 Holomorphic maps into CPn

Recall that we have CPn = { 1-dimensional subspaces Cψ ⊂ Cn | ψ 6= 0} We want to show that CPn is a complex manifold. To do this consider the open set   z   1   .  n Uj = C ·  .  | zj 6= 0 ⊂ CP    zn 

Note that it suffices to consider the Uj sets, because we have to find for every p ∈ CPn a neighborhood diffeomorphic to Cn and for every p ∈ CPn we can always find a neighborhood, where at leat one component is not zero. Then we can define a coordinate chart  z1  zj  .   .   .     zj−1  z1 n−1  zj  . ϕ : U : j → C : ϕ (Cψ) =  z  , where ψ =  .  . j j  j+1   .   zj    zn  .   .  zn zj

Another way is to represent Cψ ∈ Uj by   w1  .   .    z  wj−1 1   1 .  1  =  .    ψ  .  w  j  j+1 zn  .   .  wn Thus the coordinate changes must be holomorphic, hence CPn is a complex manifold. A common view is:   w     w    1   1    .     .   n  .   .  CP = C   | w1,..., wn−1 ∈ C ∪ C   | w1,..., wn−1 ∈ C .  wn−1   wn−1       1   0  | {z } =∼CPn−2 In this case CPn−2 could be considered as a ”hyperplane at inﬁnity”. In the following we want to study the tautological line bundle Lˆ over CPn−1: We deﬁne the tautological bundle as: Lˆ = {([p], ψ) ∈ CPn−1 × Cn | ψ ∈ [p] = {λ · p | λ ∈ C}} such that every Lp gets its own zero vector

24 Classiﬁcation of Holomorphic Line Bundles

A nowhere vanishing section ψˆ ∈ ΓLˆ is of the form ψˆ p = (p, ψ(p)) with ψ : CPn−1 → Cn such that

π n CPn−1 C \{0}

id ψ

CPn−1 is commutative. We want to define ∂¯ on Lˆ such that ∂¯ψˆ = 0 if and only if ψ is holomorphic. In order to do this we define suitable sets of functions first.

Deﬁnition 2.28. (basepoint-free linear system) Let M be a complex manifold and L a complex line bundle over M. Then a basepont-free linear system is a ﬁnite dimensional complex linear subspace V ⊂ ΓL such that

1. For every p ∈ M there is ψ ∈ V with ψp 6= 0 ψ 2. For ψ, ϕ ∈ V the function : M \ ϕ−1({0}) → C is holomorphic. ϕ

Theorem 2.29. If V ⊂ Γ(L) is a basepoint-free linear systen with n = dim(V) > 0.

a) Then there is a unique holomorphic structure ∂¯ on L such that all ψ ∈ V are holomorphic.

b) The map

∗ ∼ n−1 ∗ f : M → P(V ) = CP : p 7→ {ψ 7→ α(ψp) | α ∈ Lp}

is holomorphic.

c) The map f is linearly full, i.e f (M) is not contained in any hyperplane.

25 2.4 Holomorphic maps into CPn

Remark 2.30. Here P(V∗) is the projective dual space. This means that we consider for ϕ, α ∈ V∗ the equivalence relation

ϕ ∼ α :⇔ ϕ = λα, λ ∈ C \ {0} .

∗ ∗ V ∗ ∼ n ∗ ∼ n−1 Then we set P(V ) = ∼. Since V = C , we obtain P(V ) = CP

Proof. a) Let ϕ ∈ V be chosen in such a way that ϕp 6= 0. Near p all ψ ∈ Γ(L) are representable as ψ = λϕ for some function λ : M → C. If we consider sections ψ ∈ V, we especially see that

ψ : M \ ϕ−1({0}) → C ϕ is holomorphic around p. If we now choose a non-vanishing section ψ ∈ V, then there is a ∂¯-operator such that ∂ψ¯ = 0 Therefore we obtain

0 = ∂ψ¯ := ∂¯(λϕ˜) = ∂λ¯ · ϕ + λ · ∂ϕ¯ = 0

Hence ∂ϕ¯ = 0 for ϕ ∈ V, thus ∂¯ does not depend on the concrete ϕ. Especially we can choose a basis ψ1,..., ψn of V. This yields that ∂¯ψ˜ = 0 for all ψ˜ ∈ V. b) and c) are left as an exercise.

Deﬁnition 2.31 (Kodaira embedding). Although f : M → CPn−1 =∼ P(V∗) might not be an embedding, we call f the Kodaira-embedding of V.

We apply this to M = CPn−1, L = Lˆ ∗ and the set V = {αˆ | α ∈ Cn∗}. Here we deﬁne for α ∈ (Cn)∗ the map αˆ ∈ Γ(Lˆ ∗) as

αˆ : (p, ψ ) 7→ αˆ p(p, ψ) := α(ψ). |{z} ∈Cn

26 Classiﬁcation of Holomorphic Line Bundles

By the last theorem we obtain a ∂¯-operator on Lˆ ∗ such that all αˆ ∈ V are holomorphic. If we have a ∂¯-operator on a general vector bundle L, we can deﬁne a ∂¯-operator on the dual bundle L∗. For this consider the pairing

h·, ·i : ΓL∗ × ΓL → C : hα, ψi := α(ψ).

Our operator on the dual bundle should satisfy the product rule with respect to this pairing. Hence we deﬁne

∂α¯ (ψ) = h∂α¯ , ψi = ∂¯hα, ψi − hα, ∂ψ¯ i.

Thus our ∂¯-operator from above on Lˆ ∗ yields a ∂¯-operator on Lˆ .

Theorem 2.32. Let ψˆ ∈ Γ(Lˆ ) with ψˆ p = (p, (ψ1(p),..., ψn(p))). Further choose ∗ α1,..., αn ∈ Lˆ , such that αj (z1,..., zn) := zj. Then by deﬁnition we have that hαˆ j, ψˆi = ψj and it holds that ψˆ is holomorphic if and only if the functions ψ1,..., ψn : CPn → C are all holomorphic.

Proof. Exercise

Now let M be a compact Riemann surface

If f : M → CPn−1 is holomorphic (a Kodaira embedding for instance). Consider the pullback of the dual of the tautological bundle

L = f ∗(Lˆ ∗) .

The bundle L has a ∂¯ such that f ∗αˆ is holomorphic, i.e ∂¯( f ∗αˆ ) = 0, for all holomorphic αˆ ∈ Γ(Lˆ ∗).

Recall that H0(L) = {ψ ∈ ΓL | ∂ψ¯ = 0}. Now for our purposes we deﬁne the n-dimensional linear system V ⊂ H0(L) by

∗ n∗ Vf = { f αˆ | α ∈ C }

27 2.4 Holomorphic maps into CPn

Deﬁnition 2.33. (Degree of f ) Let f : M → CPn−1 be holomorphic. Then

deg f := deg L = deg f ∗(Lˆ ∗) .

Now given a nonzero linear form α ∈ Cn∗ we deﬁne a hyperplane in the projective space by H = {Cψ ∈ CPn−1 | α(ψ) = 0}. It turns out that

∗ #{p ∈ M | f (p) ∈ H} = #{p ∈ M | ( f α)p = 0} = deg L

Note that the intersection points are counted with multiplicity.      x  Example 2.34. Let f (M) = C y ∈ CP2 | y2w = xw2 − x3 .  z  x Then C y ∈ f (M) if and only if y2 = x − x3 z

Let M be a Riemann surface of genus g and L → M a holomorphic line bundle of degree d. Set n := dim H0(L) = h0(L). We know

d < 0 ⇒ n = 0 n = 1 if L is trivial d = 0 ⇒ n = 0 if L is not trivial

By the Riemann-Roch theorem,

n − h0(KL−1) = d + 1 − g

28 Classiﬁcation of Holomorphic Line Bundles from which we get that n ≥ d + 1 − g. where we use that h0(KL−1) ≥ 0. Furthermore, deg(KL−1) = 2g − 2 − d, thus h0(KL−1) = 0 for d ≥ 2g − 1. So if h0(KL−1) = 0, which is d = 2g − 2, then h0(L) ∈ {0, 1}.

So, e.g. for g = 3 we get the following picture:

Theorem 2.35. n ≤ d + 1, for d ≥ −1.

Proof.d = 0 is okay. For p1 ∈ M and deﬁne 0 U1 = {ψ ∈ H (L) | ψp1 = 0} then dim U1 ≥ n − 1 .

Pick p2 ∈ M with p1 6= p2 and deﬁne

U2 = {ψ ∈ U1 | ψp2 = 0}.

Then dim U2 ≥ n − 2. If we continue we get dim Un−1 ≥ 1. But all ψ ∈ Un−1 have (n − 1) zeros p1,..., pn−1. Thus d > n − 1.

Hence h0(KL−1) ≤ 2g − 1 − d for 2g − 2 − d ≥ −1 ⇔ d ≤ 2g − 1 , and hence n ≤ d + 1 − g + 2g − 1 − d = g .

29 2.5 Line Bundles over S2

Theorem 2.36. If g = 0 and d ≥ −1 then n = d + 1.

For g = 0 and arbitrary d ∈ Z there is a unique holomorphic line bundle of degree d and d + 1 for d ≥ 0 dim H0(L) = 0 for d < 0

Corollary 2.37 (Uniformization of Riemann surfaces of genus zero). g = 0 if and only if there is a holomorphic diffeomorphism f : M → CP1.

Proof. Take a holomorphic line bundle L of degree 1. Then dim H0(L) = 2 and the Kodaira embedding yields a bijective (deg L = 1) holomorphic map, i.e. a holomorphic diffeomorphism.

From this we get the following

Theorem 2.38. There is a holomorphic line bundle such that d = 1, n = 2 ⇔ g = 0.

Line bundles over surfaces of genus zero:

Without loss of generality M = CP1. Let Lˆ denote the tautological line bundle and L˜ = Lˆ −1. Then, for α ∈ (C2)∗ we have αˆ ∈ H0(L˜ ). For ψˆ = (p, ψ) ∈ Lˆ , αˆ (ψˆ) = α(ψ) . Each αˆ , α 6= 0 has a unique zero. Thus deg L˜ = 1 and so dim H0(L˜ ) = 2, by Riemann–Roch. Hence H0(L˜ ) = {αˆ | α ∈ (C2)∗} . So if now deg L = d, then L =∼ L˜ d, whose ﬁbers consist of homogeneous polynomials of degree d. Thus holomorphic sections are restrictions of homogeneous polynomials.

2.5 Line Bundles over S2

Up to biholomorphic diffeomorphism there is only one compact surface of genus zero: CP1 =∼ S2 ⊂ R3 with JX = N × X , where N denotes the Gauss map.

30 Classiﬁcation of Holomorphic Line Bundles

There is only one holomorphic line bundle L for a given degree

deg L = d ∈ Z .

If Lˆ is the tautological bundle of CP1 then

L =∼ (Lˆ −1)⊗d .

We know that dim H0(L) = d + 1 .

Let P : C2 → C be a homogeneous polynomial of degree d. For example, if d = 2, then P(λψ) = λ2P(ψ). Such P are in one-to-one correspondence with polynomials P˜ : C → C of degree d: P˜(z) = P(z, 1). Let

Vd := {all such homogenous P of degree d} then dim Vd = d + 1 .

1 For every P ∈ Vd and q ∈ CP the map

αq : Lˆ q → C , αq((q, ψ)) = P(ψ) satisﬁes d αq(λψ) = λ α(ψ) . ˆ −1 Thus αq can be viewed as an element of Lq .

All these αq are clearly holomorphic and since

0 − dim Vd = d + 1 = H (Lˆ d) all holomorphic sections of Lˆ −d come from homogeneous polynomials. A basis of H0(Lˆ −d) is given by wd, zwd−1,..., zd−1w, zd .

The Kodaira embedding γ : S2 → CPd = P(H0(Lˆ −d)∗) is given as follows:

1 CP 3 q 7→ C(αq 7→ hψ | αqi) .

For q 6= ∞ = C(1, 0) ∈ CP1 we can write q = C(z, 1) with z ∈ C. For q = C(z, q), we deﬁne γ(q) = C(P 7→ P(z, 1)) ∗ d d−1 d−1 d So in the basis of Vq dual to w , zw ,..., z w, z , we have

γ(z, 1) = C(1, z, z2,..., zd)

—the rational normal curve in CP1.

In afﬁne coordinates γ looks as follows

γˆ(z) = (z, z2,..., zd) .

31 2.6 Clifford’s Theorem

Deﬁnition 2.39. Given a non-constant holomorphic γ : S2 → CPn of degree d the image of γ is called a rational curve in CPn.

Such a rational curve is the same thing as a linear subspace U ⊂ H0(Lˆ −d), where −d ∼ ∗ −1 Lˆ = γ (Lˆ CPn ) (i.e. the degree of γ and the degree of the bundle ﬁt together). Since the inclusion U ,→ H0(L) is injective, we get that the dual map H0(L)∗ → U∗ is surjective and hence the following commuting diagram:

P(H0(L)∗)

γd π

∗ CP1 P(U ) γKodaira

n Consider γ˜ : C → C given by γ˜(z) = (P1(z)/Q1(z),..., Pn(z)/Qn(z)). Then γ is the afﬁne coordinate of   P1(z)Q2(z) ··· Qn(z)  Q (z)P (z) ··· Q (z)   1 2 n  ψ(z) =  .  ,  .  Q1(z)Q2(z) ··· Pn(z)) which is polynomial (in each of its components). Thus     a11 ... a1d 1  . .   .  ψ(z) =  . .   .  . d an1 ... and z

For each d there is a natural representation A 7→ Aˆ of SL(2, C) on Vd given by Aˆ(P) = P ◦ A...?

2.6 Clifford’s Theorem

Let L be a holomorphic line bundle over M, d = deg L, n = h0(L), L˜ = KL−1. Then

d˜ = deg(KL−1) = deg T∗ M − deg L = 2g − 2 − d .

Thus d + d˜ = g − 1 , 2

32 Classiﬁcation of Holomorphic Line Bundles or equivalently, [d − (g − 1)] = −[d˜− (g − 1)] .

symmetric

0 d g − 1 d˜ 2g − 2 = deg K

By Riemann–Roch, we get

d−(g−1) d˜−(g−1) d−(g−1) d˜−(g−1) n − n˜ = d + 1 − g = 2 − 2 ⇔ n − 2 = n˜ − 2 . Moreover, if n, n˜ ≥ 0, then n − [d − (g − 1)] ≥ 0 , thus n ≥ d + 1 − g .

Theorem 2.40 (Clifford’s Theorem). Suppose n > 0 and n > d − (g − 1). Then

d n ≤ + 1 . 2

In order to show this we ﬁrst show three seemingly unrelated theorems.

Theorem 2.41. Let M0, M1 ⊂ M be compact complex submanifolds of a K¨ahlermanifold with positive sectional curvature, dim M0 + dim M1 ≥ dim M. Then

M0 ∩ M1 6= ∅ .

33 2.6 Clifford’s Theorem

Before we prove Theorem 2.41 let us recall the ﬁrst variational formula of length:

Let γ : (−ε, ε) × [0, L] → M, γt(s) = γ(t, s) , 0 be a smooth variation of the curve γ0 parametrized by arclength, |γ0| = 1. Then with = ∂γ Y ∂t (0,s) we have d Z L ( ) = h 0 i L − h 00i t=0L γt Y, γ0 0 Y, γ0 . dt 0 Proof. Since ∇ is torsion-free we have

0 · 0 · 0 0 0 0 0 |γt| = h(γ ) , γ i/|γ | = hγ˙ , γ i/|γ | .

Thus d Z L Z L ( ) = h 0 0 i = h 0 i − h 00i t=0L γt Y , γ0 Y, γ0 Y, γ0 dt 0 0 Since the distance d : M × M → [0, ∞) is a continuous function it has a minimum L on a compact set. Thus for two compact submanifolds M0, M1 ⊂ M of a complete Riemannian manifold there is always a geodesic γ0 connecting them γ0(0) ∈ M0, γ0(L) ∈ M1 which realizes this distance. Since γ0 was a Minimum d of the distance, we have dt |0L(γt) = 0 for all variations γ of γ0 with γt(0) ∈ M0 and γt(L) ∈ M1 for all t ∈ (−ε, ε). Since γ0 is a geodesic, we obtain from the ﬁrst variational formula 0 0 0 = hY(L), γ0(L)i − hY(0), γ0(0)i . Since we can achieve variations with arbitrary vectors Y(0), Y(L) we conclude that 0 ( ) 0 ( ) γ0 0 is perpendicular to Tγ0(0) M0 and γ0 1 is perpendicular to Tγ0(1) M1.

Lemma 2.42. Let M0, M1 ⊂ M be two compact submanifolds of a complete Rieman- nian manifold and γ : (−ε, ε) × [0, L] → M be a variation of the shortest curve γ0 connecting M0 to M1 such that η0(t) := γt(0) ∈ M0 and η1(t) := γt(L) ∈ M1. Then

2 d 00 0 00 0 L(γt) = hη (0), γ (L)i − hη (0), γ (0)i dt2 t=0 1 0 0 0 L Z L ˆ ˆ 0 ˆ 00 0 ˆ 0 ˆ + Y, Y − hY + R(γ0, Y)γ0, Yi , 0 0

where Yˆ denotes the normal part of the variational vector ﬁeld of γ.

34 Classiﬁcation of Holomorphic Line Bundles

0 · 0 0 0 0 0 0 Proof. As before, we have |γt| = hγ˙ , γ i/|γ | = hγ˙ , γ /|γ |i. Thus

D 0 E D 0 E | 0|·· = ( 0)· γ + 0 ( γ )· γt γ˙ , |γ0| γ˙ , |γ0| D 0 E D 0 · 0 E = ( 0)· γ + 0 1 ( 0)· − h(γ ) ,γ i 0 γ˙ , |γ0| γ˙ , |γ0| γ hγ0γ0i γ D 0 E D E = ( 0)· γ + 0 1 ( 0)⊥ γ˙ , |γ0| γ˙ , |γ0| γ˙ , where (.)⊥ denotes the projection to the normal space of γ. Furthermore,

((γ˙ )0)· = R(γ˙ , γ0)γ˙ + γ¨ 0 .

So we get for t = 0 0 ·· 0 0 ˙ 0 0 0 0 ⊥ γt = R(Y, γ )Y, γ0 + Y , γ0 + Y , (Y ) . 00 ⊥ 0 0 ⊥ Note that, since γ0 = 0, we have (Y ) = (Y ) and by the symmetries of the curvature tensor 0 0 ⊥ 0 ⊥ 0 R(Y, γ0)Y, γ0 = R(Y , γ0)Y , γ0 . Thus, d2 Z L ( ) = ( ⊥ 0) ⊥ 0 + ( ⊥)0 ( ⊥)0 + ˙ 0 0 2 t=0L γt R Y , γ Y , γ0 Y , Y Y , γ0 dt 0 Z L 0 ⊥ ⊥ 00 ⊥ 0 0 ⊥ ⊥ 0 ˙ 0 = − Y , (Y ) + R(Y , γ )γ0 + (Y ), (Y ) + Y, γ0 . 0

Furthermore, let M˜ be a Riemannian manifold and f : M → M˜ be an immersion. Then M is itself becomes a Riemannian manifold and the Levi-Civita connections of M and M˜ are related by the second fundamental form

∗ ∗ α ∈ Γ(TM ⊗ TM ⊗ ⊥ f M˜ ) , where ⊥ f M˜ ⊂ TM˜ denotes the normal bundle of f ,

⊥ (⊥ f M˜ )p = d f (Tp M) For X, Y ∈ ΓTM, ∗ ( f ∇˜ )Xd f (Y) = d f (∇XY) + α(X, Y) . We leave it as an exercise to to check that α is a symmetric tensor. If now γ : I → M is a smooth curve and γ˜ = f ◦ γ. Then

γ˜ 00 = d f (γ00) + α(γ0, γ0) .

Theorem 2.43. If M˜ is K¨ahler, M complex and f : M → M˜ a holomorphic immersion. Then, for X, Y ∈ ΓTM,

α(X, JY) = Jα(X, Y) = α(JX, Y) .

35 2.6 Clifford’s Theorem

Proof. We have

∗ ∗ d f (∇X JY) + α(X, JY) = ( f ∇˜ )Xd f (JY) = ( f ∇˜ )X Jd f (Y) ∗ = J( f ∇˜ )Xd f (Y) = Jd f (∇XY) + Jα(X, Y) .

Comparing normal parts yields the ﬁrst equality. The second follows using the symmetry of α.

Corollary 2.44. In the above situation we have, α(JX, JX) = −α(X, X). In particular, the mean curvature of M is zero and thus M is a minimal surface.

Proof of Theorem 2.41. Let γ denote the shortest curve connecting M0 to M1. As- sume that L(γ) = L > 0, i.e. M0 ∩ M1 6= ∅. 0 0 Then we know that N0 := γ (0) is a normal vector of M0 and N1 := γ (L) is a normal vector of M1. Let

P : Tγ(0) M → Tγ(L) M denote the parallel transport along γ. Since ∇ is metric and complex, P is orthogonal and a complex linear isomorphism. Since γ is a geodesic, γ0 is parallel and hence

P(N0) = N1 .

Now, let us consider U = P(Tγ(0) M0) and V = Tγ(L) M1. Then, since

dim U + dim V ≥ m = dim M and ⊥ 0 6= N1 ∈ (U + V) we have

1 ≤ dim(U + V)⊥ = m − (dim U + dim V − dim(U ∩ V)) ≤ dim(U ∩ V) .

So there is a parallel vector ﬁeld Y along γ such that Y(0) is tangent to M0 and Y(L) is tangent to M1. If we now vary γ such that γ˙ = Y and apply Lemma 2.42, we get Z L 0 0 0 ≤ hη¨1(L), γ (L)i − hη¨0(0), γ (0)i − Kspan{Y,γ0} . 0 Since K > 0, we have then

0 0 0 0 0 < hη¨1(L), γ (L)i − hη¨0(0), γ (0)i = hα(YL, YL), γ (L)i − hα(Y0, Y0), γ (0)i .

Since U and V were complex, JY has the same properties as Y, but α(JY, JY) = −α(Y, Y) by the last corollary. This yields the above inequality with opposite sign, which is a contraction.

36 Classiﬁcation of Holomorphic Line Bundles

Theorem 2.45. Let U, V, W be ﬁnite-dimensional complex vector spaces, such that dim U + dim V ≤ dim W, and ∗ : U × V → W bilinear, such that u ∗ v = 0 ⇒ u = 0 or v = 0. Then dim W ≤ dim U + dim V − 1 .

Proof. Let dim U = r, dim V = s, dim W = t and deﬁne the Segr´eembedding by

σ : CPr−1 × CPs−1 → CPrs−1, ([ψ], [ϕ]) 7→ [ψ ⊗ ϕ] ∈ P(U ⊗ V) .

We leave as an exercise to show that σ deﬁnes a holomorphic embedding. Now choose bases ψ1,..., ψr of U and ϕ1,..., ϕs of V and deﬁne a linear map

f : U ⊗ V → W by f (ψi ⊗ ϕj) = ψi ∗ ϕj .

Then dim ker f = rs − dim im f ≥ rs − t . Then M˜ := P(ker f ) ⊂ CPrs−1 is of dimension ≥ rs − t − 1. Furthermore, we have that Mˆ = im σ is of dimension r + s − 2.

Suppose now that t < r + s − 1, i.e. t ≤ r + s − 2. Then

dim M˜ + dim Mˆ ≥ r + s − 2 + rs − t − 1 ≥ rs − 1 = dim CPrs−1 .

Thus, by Theorem 2.41, Mˆ ∩ M˜ 6= ∅ and hence there are 0 6= ψ ∈ U and 0 6= ϕ ∈ V with ψ ∗ ϕ = f (ψ ⊗ ϕ) = 0, which is a contradiction to the ”zero-divisor free” property of ∗.

Theorem 2.46. h0(L) > 0, h0(L˜ ) > 0 ⇒ h0(L ⊗ L˜ ) ≥ h0(L) + h0(L˜ ) − 1

Proof. Apply the last theorem to the map

∗ : H0(L) × H0(L˜ ) → H0(L ⊗ L˜ ) deﬁned by the tensor product.

Proof of Theorem 2.40. Assume that h0(L), h0(L˜ ) > 0, where L˜ = KL−1. Then, by Theorem 2.46 and the ”teen” Riemann-Roch theorem,

g = h0(K) ≥ n + n˜ −1 = 2n − d + g − 2 , |{z} n−d+g−1 hence 2n ≤ d + 2 which ﬁnally leads to n ≤ d/2 + 1 .

37 3. The Riemann-Roch Theorem

3.1 The Mittag–Lefﬂer Theorem

Let M Riemannian surface and L a holomorphic line bundle over M.

∈ ∈ ( | ) Deﬁnition 3.1. Let p1,..., pn M pairwise distinct, then ψ Γ L M\{p1,...,pn} is called a section with poles at p1,..., pn, if

1. there is a smooth section ω ∈ Γ(KL¯ ) such that ∂ψ¯ = ω, and

2. for each j ∈ {1, . . . , n} and every metric on L we have either ψ is smoothly extendable to pj or lim |ψp| = ∞. p→pj

Theorem 3.2. For ω ∈ Γ(KL¯ ). Then the ∂¯-problem ∂ψ¯ = ω is locally solvable.

Proof. Without loss of generality (by gluing) we can assume that L a bundle such that KL−1 is of negative degree and thus H0(KL−1) = {0}. In particular, ω ∈ H0(KL−1)⊥, thus ∂ψ¯ = ω by the elliptic theorem.

Theorem 3.3. If ψ is a section of L with poles {p1,..., pn}, then for each j ∈ {1 . . . , n}

there is a neighborhood Uj of pj, a meromorphic section ψ of L|Uj with pole at pj and a

smooth ϕj ∈ Γ(L|Uj ) such that

ψ|Uj = ψj + ϕj .

Proof. On some neighborhood Uj of pj the previous theorem yields a solution ϕj ∈

Γ(L|Uj ) of ∂ϕ¯ j = ω . = | − ¯ = Then ψj ψ Uj\{pj} ϕj satisﬁes ∂ψj 0 and either ψj is smoothly extendable to pj or lim |ψj,p| = ∞. p→pj

38 The Riemann-Roch Theorem

If ω is a section with poles of K and p ∈ M, then we can deﬁne the residue Respω as follows: If z : U → C is a chart at p, then Z Respω := lim ω . r→0 |z|=r

It is quite clear that Respω is well-deﬁned. We leave it as an exercise.

Deﬁnition 3.4. Two sections ψ, ψ˜ with a pole at p are said to have the same principle part at p, ψ˜ − ψ can be smoothly extended to p.

Theorem 3.5 (Mittag–Lefﬂer). Let L be a holomorphic line bundle over a compact Riemann surface M and ψ a section of L with poles at p1,..., pn. Then the following are equivalent:

(1) There is a smooth ϕ ∈ Γ(L) such that ψ − ϕ is meromorhic.

(2) For all ω ∈ H0(KL−1), n ∑ Resphω, ψi = 0 . j=1

Remark 3.6. In particular the Mittag–Lefﬂer problem, to ﬁnd meromorphic section for a given principle part, is solvable if and only if the second condition holds.

Example 3.7. Consider L = S2 × C. Then meromorphic sections are just meromorhic functions. From deg(KL−1) = −2 and so the Mittag–Lefﬂer problem is always solvable (partial fractions).

Proof of Mittag–Lefﬂer. Choose disjoint disk neighborhoods Uj of pj and deﬁne M0 = M \ (∪jUj). After adding a smooth section to ψ we may assume that

∂ψ¯ = 0 . Uj\{pj}

∈ ( ¯ ) | = ¯ Let η Γ KL be the unique section with η M\{p1,...,pn} ∂ψ. Now, condition (1) is equivalent to

∃ϕ ∈ ΓL : ∂ϕ¯ = η which, by the elliptic theorem, is equivalent to Z ω ∧ η = 0 M for all ω ∈ H0(KL−1). Furthermore, under the usual identiﬁcation

KK¯ 3 σ ↔ σˆ ∈ Λ2 M

39 3.1 The Mittag–Lefﬂer Theorem

1 where σˆ (X, Y) = σX(Y) − σY(X) . we have for any ω ∈ Γ(KL−1) and ψ ∈ ΓL

dhω|ψi = h∂ω¯ |ψi + hω ∧ ∂ψ¯ i .

Applied to our situation, where ω is holomorphic, this yields Z Z Z Z Z hω ∧ ηi = hω ∧ ηi = hω ∧ ∂ϕ¯ i = dhω ∧ ϕi = hω ∧ ϕi M M0 M0 M0 ∂M0

Theorem 3.8. For ω ∈ Γ(KL−1) and ψ ∈ ΓL we have that ∼ d hω|ψi = ∂ω¯ ψ − hω ∧ ∂ψ¯ i .

Remark 3.9. Note that hω|ψi is a C-valued 1-form, hence taking the exterior derivative makes sense. Moreover h· ∧ ·i prescribes which product is taken to ”multiply” values of the two distinct bundles, that is the pairing h·|·i.

Proof. Locally, on U ⊂ M open choose X ∈ Γ(TM) with [X, JX] = 0 (X holomorphic). Then the complex connection ∇U on TU with ∇U X = 0 is torsion-free. Choose a connection ∇ on L with ∇00 = ∂¯. Then it is sufﬁcient to verify

dhω|ψi(X, JX) = XhωJX|ψi − (JX)hωX|ψi

= h∇XωJX|ψi + hωJX|∇Xψi − h∇JXωX|ψi − hωX|∇JXψi

= h(∇Xω)JX|ψi − h(∇JXω)X|ψi + hωJX|∇Xψi − hωX|∇JXψi

= h(∇Xω)JX + J(∇JXω)JX|ψi − hω ∧ ∇ψi(X, JX)

= h2(∂¯ Xω)JX|ψi − hω ∧ ∂ψ¯ i(X, JX) = (h(∂ω¯ )|ψi − hω ∧ ∂ψ¯ i)(X, JX) .

Theorem 3.10 (Mittag–Lefﬂer). Let L be a holomorphic line bundle over a compact Riemann surface M and ψ ∈ ΓL a section with poles at p1,..., pn. Then there is a meromorphic section ψ˜ of L such that ψ˜ ≡ ψ mod ΓL if and only if

n h | i = ∀ ∈ 0( −1) ∑ Respj ω ψ 0, ω H KL . j=1

Proof. Without loss of generality there are disks U1,..., Un around p1,..., pn such ¯ | = = \ (∪n ) ∈ ( ¯ ) | = ¯ that ∂ψ Uj\{pj} 0. Deﬁne M0 M j=1Uj , η Γ KL , η M\{p1,...,pn} ∂ψ, so

1This identiﬁcation has been introduced in section 7 of the ”Complex Analysis II” script from WS18/19

40 The Riemann-Roch Theorem

η|Uj = 0. Now suppose there is ϕ ∈ ΓL such that ψ − ϕ is meromorphic. Then η − ∂ϕ¯ = 0, so the ∂¯-problem ∂ϕ¯ = η has a solution. Conversely, if ∂ϕ¯ = η, then ψ − ϕ is meromorphic. So we have to prove: ∂ϕ¯ = η solvable if and only if Z Z Z Z 0 = hω|ηi = hω|ηi = hω|∂ψ¯ i = − hω|ψi M M0 M0 ∂M0 n Z n = h | i = h | i ∑ ω ψ 2πi ∑ Respj ω ψ , j=1 Uj j=1 where the last equality used that ψ is holomorphic on Uj \{pj}.

3.2 Complete proof of Riemann–Roch

Let M be a compact Riemann surface, p ∈ M. Then there is a holomorphic line bundle L(p) such that there is ϕ ∈ H0(L(p)) with a single simple zero at p—the skyscraper bundle2. Up to holomorphic isomorphism the bundle L(p) is unique: If L˜ is another such holomorphic line bundle then the quotient of the corresponding sections is a function with removable singularity at p, i.e. L(p)L˜ −1 is holomorphically trivial.

0 Lemma 3.11. There is ψ ∈ H (L) with ψp 6= 0 if and only if there is no meromorphic section ω of KL−1 with a single simple pole at p.

Proof.

”⇒”: If there would be such ψ and ω then hψ|ωi would have a single simple pole at p, which contradicts the fact that the residues of a meromorphic 1-form sum up to zero.

”⇐”: Suppose all ψ would vanish at p and η is a section of Γ(KL−1) has a single simple pole at p. Then Resphη|ψi = 0 for all holomorphic ψ. So by the Mittag–Lefﬂer theorem there is a meromorphic ω with ω ≡ η mod Γ(KL−1).

Remark 3.12. Note, that in contrast to the approach we tackled the theorem of Riemann–Roch in Complex Analysis II where we used triangulations, here no 0 such thing is needed. That e.g. dimC H (K) is ﬁnite just comes out of the elliptic theory. In some sense the zeros and poles of meromorphic sections provide natural discretizations.

0 As dimC H (K) < ∞ we deﬁne the genus of a compact Riemann surface as follows:

2Note that in contrast to the notation that is used in the Complex Analysis II script (there (p) is the skyscraper bundle) here L(p) is the skyscraper bundle, i.e. using the ”old” notation L(p) := (M × C) ⊗ (p).

41 3.2 Complete proof of Riemann–Roch

Deﬁnition 3.13. The genus g of a compact Riemann surface is deﬁned to be the dimension of H0(K): 0 g := dimC H (K) .

Theorem 3.14. Let L be a holomorphic line bundle over a compact Riemann surface M and p ∈ M. Then the following are equivalent:

(1)h 0(L ⊗ L(p)) > h0(L) .

(2) There is a meromorphic section ψ of L with a single simple pole at p.

0 (3) There is ψ˜ ∈ H (L ⊗ L(p)) with ψ˜ p 6= 0.

Proof. Consider the injection H0(L) ,→ H0(L ⊗ L(p)) given by

ψ 7→ ψ ⊗ ϕ where ϕ ∈ H0(L(p)) denotes the famous section of L(p) (holomorphic with single simple zero at p). The image of this map equals ker(ψ˜ 7→ ψ˜ p). In particular,

h0(L ⊗ L(p)) − h0(L) ∈ {0, 1} .

So the equivalence of (1) and (3) is clear. We prove (2)⇔(3): Let ψ be a meromorphic section of L with a single simple pole at p. Then ψ ⊗ ϕ ∈ H0(L ⊗ L(p)) 0 with (ψ ⊗ ϕ)p 6= 0. Conversely, if ψ˜ ∈ H (L ⊗ L(p)), ψ˜ p 6= 0, then ψ = ψ˜/ϕ is meromorphic with single simple pole at p.

0 From Mittag–Lefﬂer we got as a corollary that there is ψ ∈ H (L) with ψp 6= 0 if and only if there is no meromorphic section ω of KL−1 with a single simple pole at p. This together with the last theorem yields the following

Theorem 3.15. The following are equivalent:

(1)h 0(KL−1) > h0(KL−1 ⊗ L(−p)).

(2) There is a meromorphic section ω of KL−1 ⊗ L(−p) with a single simple pole at p.

0 (3) not[There is ψ˜ ∈ H (L ⊗ L(p)) with ψ˜ p 6= 0].

(4) not[h0(L ⊗ L(p)) > h0(L)].

Boolean algebra-wise, the equivalence of (1) and (4) yields in particular that

h0(L ⊗ L(p)) − h0(L) = 1 − (h0(KL−1) − h0(KL−1 ⊗ L(−p))) .

42 The Riemann-Roch Theorem

Thus

h0(L ⊗ L(p))−h0(KL−1 ⊗ L(−p)) − deg (L ⊗ L(p)) = h0(L ⊗ L(p)) − h0(KL−1 ⊗ L(−p)) − deg L − 1 0 0 −1 = h (L) − h (KL ) − deg L =: nL .

In particular,

0 0 nK = h (K) − h (M × C) − deg K = g − 1 − deg K and 0 0 nM×C = h (M × C) − h (K) − deg (M × C) = 1 − g .

Theorem 3.16. Every holomorphic line bundle has a meromorphic section.

Proof. Let L be a holomorphic line bundle over M, p ∈ M. For large ` the bundle L˜ = L ⊗ L(p)` satisﬁes deg (KL˜ −1) < 0. Thus h0(KL˜ −1) = 0 and so

0 ˜ ˜ nL = nL˜ = h (L) − deg L .

Hence h0(L˜ ) > 0 for large `.

Finally, we are ready to prove the theorem of Riemann–Roch.

Theorem 3.17 (Riemann–Roch). For every holomorphic line bundle L over a compact Riemann surface M we have

nL = nM×C = 1 − g . h i h0(L) − h0(KL−1) = deg L − g + 1

Proof. Choose a meromorphic section and succesively remove all poles and zeros by tensoring in suitable powers of point bundles and use that nL does not change when tensoring point bundles into L. In the end we obtain the trivial bundle.

Theorem 3.18. deg K = 2g − 2

Proof.g − 1 − deg K = nK = 1 − g.

Let M be a Riemann surface of genus g, γ : M → CPm−1 be holomorphic (not ∗ −1 m−1 contained in any hyperplane) and set L := γ LTaut, where LTaut → CP denotes the tautological line bundle. Let d := deg L = deg γ and n := h0(L). Then we have n ≥ m.

43 3.2 Complete proof of Riemann–Roch

Furthermore, from Riemann-Roch, we get n − h0(KL−1) = d + 1 − g and thus n ≥ d + 1 − g. Clifford’s theorem then yields

n = d + 1 − g or n ≤ d/2 + 1 .

Interesting Case: m ≥ 2, so n ≥ 2. Note that d equals the number of points in γ(M) ∩ H (conted with multiplicity), where H denotes a projective hyperplane.

Theorem 3.19. If d = 1 and m ≥ 2, then g = 0, m = 2 and γ is a diffeomorphism.

”The only holomorphic curves in CPm−1 of degree 1 are the projective lines.”

Proof. From Clifford’s theorem follows

2 − g = n ≥ 2 ⇒ g = 0, n = 2 ⇒ m = 2 or 3 n ≤ 2 ⇒ n ≤ 1 ⇒ m ≤ 1 (a contradiction) From d = 1 then follows that every ψ ∈ H0(L) has exactly one zero. Thus #γ−1({p}) = 1 for all points p ∈ CP1.

A Riemann surface is called hyperelliptic if it comes together with a holomorphic involution

Theorem 3.20. If d = 2, then either

• m = 2 (and M is called hyperelliptic), or

• m = 3 and g = 0 (and γ is a called a conic section).

Proof. Assume that m 6= 2. We want to prove that m = 3 and g = 0. By Clifford’s theorem, if n ≤ 2, then m ≤ 2. But then m = 2—a contradiction. Thus n = 3 − g. If now g > 0, then n ≤ 2—again a contradiction. So g = 0. Hence n = 3. Thus, as 2 ≤ m ≤ n and m 6= 2, we get m = 3.

Theorem 3.21. If d = 3 and m ≥ 3, then either

• g = 0 and m ∈ {3, 4}, or

• g = 1 and m = 3.

3 Proof. Clifford’s theorem yields either m ≤ n ≤ 2 + 1 ⇒ m ≤ 2— a contradiction— or m ≤ n = 4 − g, which then splits in two cases: If g = 0, we get m ∈ {3, 4}. If g = 1 , we get m = 3.

44 The Riemann-Roch Theorem

Recall: Any rational curve (deﬁned on CP1) is a projection of the rational normal curve. For d = 3 the rational normal curve γ : CP1 → CP3 is given by

γ(C(z, w)) = C(z3, z2w, zw2, w3) .

Corollary 3.22. Cubic (d = 3) plane curves (m = 3) are either rational (g = 0 n = 4 γ is a projected normal curve) or elliptic g = 1. ; ; Proof. This follows fro the theorem before with d = m = 3.

Remark 3.23. If γ is rational, then for n = 4, γ is a projected rational normal curve.

Remark 3.24. A good reference on algebraic curves are

• Brieskorn and Knorrer¨ ”Algebraische Kurven”

• Wieleitner ” Spezielle ebene Kurven”

• or Wieleitner ”Ebene algebraische Kurven”

Example 3.25 (Steiner cycloid). A parametrization is given by

2 cos(ϕ) + cos(−2ϕ) x 2 sin(ϕ) + sin(−2ϕ)  =: y 1 z

We make a coordinate change t := eiϕ, z := x + iy, w := x − iy. Then  + 1  2t t2 C 1 = C¯ 7→ C 2 ( ) = C 1 2 = ( ) ∈ C3 γ : P P , γ t 2 t + t  : ψ t 1 We compute 2t3 + 1 ψ(t) = C 2t + t4  t2

45 3.2 Complete proof of Riemann–Roch

t t For p = ∈ CP1, so ∈ Ltaut, we homogenize ψ by s s p

2t3s + s4 t f : Ltaut → γ∗L˜ taut, f = 2ts3 + t4 s   t2s2

t Hence f ( ) = ψ(t). At each point (and in each component), f is a homoge- 1 neous polynomial of deg = 4. t If f = 0, then by the third component of f , we get that t = 0 or s = 0. By the s ﬁrst, respectively second component, it then follows that t = s = 0 in both cases. ⊗4 So f is a bundle isomorphism between Ltaut and γ∗L˜ taut . We know that ⊗4 deg(Ltaut) = −1, hence deg(γ∗L˜ taut ) = −4. So we can conclude that deg(γ) = −4 Example 3.26. The space of lines in CP2 = P(C3) is P((C3)∗). Again we consider the Steiner cycloid

2t3 + 1 ψ(t) = 2t + t4  . t2

In order to ﬁnd the tangent, we consider the derivative3

 6t2   3t2  0 ψ = 2 + 4t3 (=) 1 + 2t3 . 2t t

For two vectors ψ, ϕ, v ∈ C3 we can deﬁne ∗ ψ ∧ ϕ ∈ C3 by (ψ ∧ ϕ)(v) := det(ψ, ϕ, v) . Using the interior product ι one can write this as

ψ ∧ ϕ = ιψ,ϕ det .

In our case of C3 this corresponds to taking the cross-product. We check that

 2t2 + t5 − t2 − 2t5   t2 − t5  0 (ψ ∧ ψ )(t) =  3t4 − 2t4 − t  =  t4 − t  2t3 + 4t6 + 1 + 2t3 − 6t3 − 3t6 t6 − 2t3 + 1  −t2   −t2  = (t3 − 1)  t  (=)  t  . t3 − 1 t3 − 1 Hence, we deﬁne that the dual curve has degree 3 if and only if the original curve is of class 3. 3(=) is to be understood as ”up to a (possibly t-dependent) factor”.

46 The Riemann-Roch Theorem

Remark 3.27. Note that even at the caps, the tangents are well deﬁned.

Now let M be of genus 1 and M z : M → Γ be a holomorphic diffeomorphism where Γ = {na + bm | n, m ∈ Z}. Let γ : M → 2 CP be of degree 3 and pick an arbitrary p1 ∈ M. We deﬁne a holomorphic map

τp1 : M → M by the condition that p2 ∈ M gets mapped to the point such that

γ(p1), γ(p2), γ(τp1 (p2)) are collinear.

Note that this is well deﬁned as deg γ = 3, hence every line intersects γ(M) exactly 2 = thrice. Obviously, τp1 is an involution, i.e. τp1 idM which makes it a holomorphic involution.

We have z as a complex coordinate. Either this involution comes from a euclidean motion, that is

z(τp1 (p2)) = z(p2) + cp1 where 2cp1 ∈ Γ , or

z(τp1 (p2)) = −z(p2) + cp1 .

We note that if span {γ(p1), γ(p2)} is tangent to γ(M) at γ(p2) then p2 is a ﬁxed point of τp1 . Thus τp1 cannot be a translation! Therefore,

γ(p1), γ(p2), γ(p3) are collinear ⇔ z(p1) + z(p3) = cp1

⇔ z(p1) + z(p2) + z(p3) = bp1 ∈ C

By construction the latter equation has to be symmetric in p1, p2 and p3. In particular bp1 is independent of p2 and p3, hence — by symmetry — b is independent of p1, p2 and p3.

47 3.2 Complete proof of Riemann–Roch

This means that there is a constant b ∈ C such that γ(p1), γ(p2), γ(p3) are collinear b if and only if z(p1) + z(p2) + z(p3) = b. By the change of variables z 7→ z − 3 we obtain the statement

γ(p1), γ(p2), γ(p3) are collinear ⇔ z(p1) + z(p2) + z(p3) = 0 .

Compact oriented surface are classiﬁed (up to diffeomorphism) by their genus g ∈ N.

Deﬁnition 3.28. Let (M, J) and (M˜ , J˜) be compact Riemann surfaces of genus g. Then (M, J) ∼ (M˜ , J˜) if and only if there is a holomorphic diffeomorphism f : M → M.˜ This is an equivalence relation and the set of equivalence classes is called the Riemann moduli space Mg.

1 We have seen that M0 = {CP }. How does Mg look for g > 0. To study Mg we can assume that M = M˜ , so the problem becomes to classify J’s on M. We say J ∼ J˜ if there is a diffeomorphism f : M → M intertwining J and J˜, i.e. J ◦ d f = d f ◦ J˜. The set of orientation preserving diffeomorphisms f : M → M is denoted by Diffﬁ(M) and is a group which acts on

M˜ = {J ∈ ΓEnd(TM)|J2 = −I, X, JX positively oriented for all X ∈ TM, X 6= 0}. via f .J = d f ◦ J ◦ d f −1. 2 For J ∈ Γ(Tp M) with J = −I we have trJ = 0 (Cayley-Hamilton). Thus J ∈ 2 sl(Tp M). For A ∈ sl(R ),

a b a b A2 = = (a2 + bc) I = −(det A)I. c −a c −a | {z } quad. form on sl with (++-)

Thus A2 = hA, AiI deﬁnes a Lorentz metric on sl(TM) and the complex structures are a hyperbolic plane bundle.

48 The Riemann-Roch Theorem

Given a reference complex structure J on M we can write any other J˜ as J˜ = λ(J + Q), where Q ⊥ J with respect to the Lorentz metric on sl(TM). Then

1 −I = J˜2 = λ2(−I + JQ + QJ +Q2) λ2 = . | {z } 1 − hQ, Qi =0 ;

Imagine M˜ is an ’inﬁnite-dimensional manifold’: (−ε, ε) → M˜ , t 7→ J(t) is called smooth if (−ε, ε) × M → End(TM), (t, p) 7→ J(t)p is smooth. For smooth J,

˙ d J = dt |t=0 J(t) ∈ ΓEnd(TM) .

2 From J − −I we get JJ˙ + JJ˙ = 0. So J˙ ∈ ΓEnd−(TM).

−1 Deﬁnition 3.29. TJM˜ := ΓEnd−(TM) = Γ(KK¯ ).

For g > 0 there are J’s for which there is a diffeomorphism f : M → M such that f .J = J, i.e. f is holomorphic with respect to J. the group action is not free, i.e. it is not true that f ∈ Diffﬁ(M), J ∈ M˜ , f .J = J ⇒;f = idM.

Deﬁnition 3.30. Let M and M˜ be manifolds. Two maps g0, g1 : M → M˜ are called homotopic if there is a homotopy between g0, g1, i.e. smooth map f : [0, 1] × M → M˜ such that g0 = f (0, .) and g1 = f (1, .).

Deﬁnition 3.31. Let M and M˜ be manifolds. Two diffeomorphisms g0, g1 : M → M˜ are called isotopic if there is a homotopy such that ft = f ( f ,.) is a diffeomorphism for all t.

Deﬁnition 3.32. Let M be a manifold, p ∈ M. Then deﬁne

homotopy classes with ﬁxed endpoints of smooth π (M) := . 1 γ : [0, 1] → M, γ(0) = p = γ(1)

For [γ], [γ˜] we deﬁne γ ∗ gamma˜ to be the concatenation of γ and γ˜, where we possibly have to slow down the speed to zero at the concatenation point to make this curve smooth again.

The deﬁnition of π1(M) depended on p. Choosing a different point p˜ results in an isomorphic group π˜ 1(M). But the isomorphism might not be unique.

A smooth map f : M → M˜ induces a group homomorphism f∗ : π1 M → π1 M˜ , well deﬁned up to conjugacy: Two group homomorphisms ρ0, ρ1 : G → H are −1 called conjugate if there is h ∈ H such that ρ1(g) = hρ0(g)h for all g ∈ G

49 3.2 Complete proof of Riemann–Roch

Theorem 3.33. let M be a compact oriented surface and f , f˜ ∈ Diffﬁ(M). Then the following are equivalent:

1. f is isotopic to f˜ .

2. f is homotopic to f˜ .

3.f ∗ is conjugate to f˜∗.

The group Diffﬁ(M) is not connected. The subgroup

Diff0(M) = { f ∈ Diffﬁ(M)| f isotopic to idM}, which is the connected component of the identity, is a normal subgroup, i.e.

−1 h ∈ Diffﬁ(M), g ∈ Diff0(M) ⇒ hgh ∈ Diff0(M) .

Deﬁnition 3.34. The quotient group MCG(M) := Diffﬁ(M)/Diff(M) is called the mapping class group.

[pic: Dehn twists are not in Diff0(M).]

Moreover,

MCG(M) = {automorphisms of π1(M)}/{inner automorphisms of π1(M)} .

MCG(M) acts on H1(M): f∗[γ] = [ f ◦ γ]. Note: For g ≥ 2 there are f ∈ Diffﬁ(M) with f∗ = idH1(M). Diffﬁ(M) acts on M˜ :

R(M) = M˜ /Diffﬁ(M), T(M) = M˜ /Diff0(M)

Furthermore, MCG(M) acts on T(M) and we have

T(M)/MCG(M) = R(M) .

˜ ¯ −1 d Recall, TJM = Γ(KK ). Let X ∈ Γ(TM), Xp = dt |t=0gt(p) with g0 = idM.

Deﬁnition 3.35. The Lie-derivative of J with respect to X is given by

d LX J := − gt∗ J . dt t=0

Exercise 3.36. Show that LXY = [X, Y].

50 The Riemann-Roch Theorem

This deﬁnition is made in such a way that the Leibnitz rule holds: For Y ∈ Γ(TM),

d then similarly LXY = − gt∗Y = [X, Y] and hence dt t=0

LX(JY) = (LX J)Y + JLXY (LX J)Y = LX(JY) − JLXY = [X, JY] − J[X, Y] .

2 ; Moreover, J = −I yields (L)XJ)J + J(LX J) = 0. ¯ 1 ¯ We know ∂Y X = 2 ([Y, X] + J[JY, X]). Thus (LX J)Y = 2J∂Y X or, shorter,

LX J = 2J∂¯X .

Theorem 3.37. J˙ ∈ TJM˜ is tangent to the Diffﬁ(M)-orbit through J if and only if J˙ ⊥ H0(K2).

Proof. J˙ ∈ TJM˜ is tangent to the Diffﬁ(M)-orbit through J ⇔ there is X ∈ Γ(TM) −1 −1 such that J˙ = LX J ⇔ J˙ ∈ Im ∂¯, where ∂¯ : Γ(K ) → Γ(KK¯ ) ⇔ ∂¯-problem ∂¯X = J˙ has a solution X ∈ K−1 ⇔ J˙ ⊥ H0(K2).

We know  0 for g = 0  dim H0(K2) = 1 for g = 1  3g − 3 for g > 2

Theorem 3.38.  0 for g = 0  dimR T(M) = 2 for g = 1  6g − 6 for g > 2

Fact: On a Riemann surface with g ≥ 2 there is a unique Riemannian metric compatible with J with curvature K = −1. Then in each homotopy class of curves there is a minimal geodesic. Cutting yields 2(g − 1) pairs of pants.

[pics]

Each of these pairs of pants by the lengths of the boundary curves. This yields 3g − 3 paramters. Furthermore, there are 3g − 3 twist parameters when gluing.

3.3 Teichm¨ullerSpace for Tori

2 Let M = T , g = 1. Then M = C/Γ, where Γ = {mγ1 + nγ2 | m, n ∈ Z} with γ1, γ2 ∈ C, det(γ1, γ2) > 0. 1 Deﬁne τ = γ2/γ1. Let ηi : S → M such that γ˜i(t) = π(tγi) = ηi(t). In this context π describes the projection map from the factor.

51 3.3 Teichm¨ullerSpace for Tori

−1 Let f ∈ Diff0(M). Then f ◦ γ˜i is homotopic to ηi. Let J˜ = f∗ J = d f ◦ J ◦ d f . 0 There is ω ∈ H (K), ωp 6= 0 for all p. Although ω is not exact we write ω = dz. Z γi = dz . η ; i where τ = γ2/γ1 is independent of ω. ω˜ = dz˜, f ∗ω˜ = ω ω˜ ∈ K˜, ∂¯˜ω˜ = 0. τ˜ = τ. ; ; Summary: Choose two homology classes [η1], [η2] with σ([η1], [η2]) = 1. Here σ is Z dual to σ˜ : H1 M × H1 M → C, σ˜ (ω, η) = ω ∧ η. Choose ω ∈ H0K and compute Z Z M τ = ω/ ω. Then Im τ > 0. τ is invariant if we change J by an element η2 η1 f ∈ Diff0(M). Conversely, every τ with Im(τ) > 0 occurs. Hence we have the following

Theorem 3.39. The Teichm¨ullerspace of compact Riemann surfaces of genus g = 1 is the upper halfplane.

Let us look at tori of revolution: All tori of revolution are the image of f (u, v) = (aeiu/a, beiv/b) for some real a, b with a2 + b2 = 1 (an isometry) under the stereo- graphic projection. So we have a biholomorphism from C/{2π(am + bn) | m, n ∈ Z}. Here we have γ1 = 2πa · i and γ2 = 2πb. A parametrization is given by  u  a cos a u ˜ 3 a sin a  f : C/Γ → S : (u, v) 7→  v  b cos b  v b sin b [SKETCH] b Hence τ = i a .

Theorem 3.40 (Garcia 1960). Every compact Riemann surface can be conformally embedded into R3.

Sketch of proof. Start with any embedding. Add shallow ripples to improve con- formality. Repeat with even more shallow ripples. Prove this converges to a C∞ map.

[Exkurs: Homotopy classes of immersions]

Action of Diffﬁ(M) instead of only Diff0(M) allows to change the lattice generators γ1, γ2 according to

γ˜1 = aγ1 + bγ2

γ˜2 = cγ1 + dγ2

52 The Riemann-Roch Theorem with a, b, c, d ∈ Z and ad − bc = 1. Then τ˜ = (aτ + b)/(cτ + d). That’s invariant under changing sign of all a, b, c and d. So the relevant group action on τ is PSL(2, Z) = SL(2, Z)/{±I}.

Lemma 3.41. Let τ ∈ C with Imτ >, 0, a, b, c, d ∈ Z with ad − bc = 1 and τ˜ = (aτ + b)/(cτ + d). Then Im τ Im τ˜ = . |cτ + d|2

Proof. A calculation yields

aτ + b aτ¯ + b 2i Im τ˜ = − cτ + d cτ¯ + d (aτ + b)(cτ¯ + d) − (aτ¯ + b)(cτ + d) = |cτ + d|2 Im (adτ + bcτ¯) = 2i |cτ + d|2 Im τ = 2i |cτ + d|2

[picture of the fundamental domain D]

We deﬁne 1 D := {τ ∈ C | Im τ > 0, |Re τ| ≤ 2 , |τ| ≥ 1} .

Theorem 3.42. The set D is the fundamental domain for the action of PSL(2, Z) on the upper halfplane H = {τ ∈ C | Im τ > 0}, i.e.

a b aτ + b a) For every τ ∈ H there is ∈ SL(2, Z) such that ∈ D c d cτ + d

a b aτ + b b) If τ, τ˜ ∈ D with τ 6= τ˜ and there is ∈ SL(2, Z) with τ˜ = , then c d cτ + d τ˜ = −τ¯ and 1 |τ| = 1 or |Re τ| = . 2

Recall: If M is a compact Riemann surface of genus g = 1, then there is a ﬂat metric h., .i compatible with g, unique up to a constant factor. Consider the torus M = C/Γ, where Γ = {mγ1 + nγ2 | m, n ∈ Z}, det(γ1, γ2) > 0, τ = γ2/γ1. In this setting we also know that Im(τ) > 0. Note that Im τ = area(h., .i). Without loss of generality we can assume that γ1 = 1 and γ2 = τ.

53 3.4 Spin Bundles

A change of the lattice generators leads to τ˜ = (aτ + b)/(cτ + d), with a, b, c, d ∈ Z, ad − bc > 1. The Riemann surface M is called

rectangular :⇔ ∃τ˜ : Re τ˜ = 0 rhombic :⇔ ∃τ˜ : |τ˜| = 1 square :⇔ ∃τ˜ : τ˜ = i √ 1+i 3 hexagonal :⇔ ∃τ˜ : τ˜ = 2

[pic: illustration in upper half plane]

Here an example of a hexagonal torus:

[pic: hexagonal torus in 3-space (contours)]

Im τ a b Proof of Theorem (Serre). Recall that Im τ˜ = . So there is ∈ SL(2, Z) |cτ + d|2 c d such that Im τ˜ is maximal. Use transformation of the form τ 7→ τ + n, n ∈ Z, to 1 1 achieve that in addition |Re τ˜| ≤ . If we would have that |τ˜| < 1, then τˆ = − 2 τ˜ τ˜ would satisfy Im τˆ = > Im τ˜, which contradicts the maximality of Im τ˜. This |τ˜| shows a). For b) we can assume without loss of generality that Im τ˜ ≥ Im τ. So |cτ + d| ≤ 1. If d = 0, then we get 1 ≥ |cτ| ≥ |c| and thus |c| = 1. In- particular |τ = 1. Without loss of generality we can assume that c = −1. Then a b 1 b = det = 1 and τ˜ = −a − = −a − τ¯. The rest is left as an exercise. −1 0 τ

3.4 Spin Bundles

Let M be a compact Riemann surface and L be a complex line bundle over M. The Riemann–Roch paired bundle is KL−1. Then

deg(KL−1) = 2g − 2 − deg L deg L = deg KL−1 ⇔ deg L = g − 1. ; Moreover, deg L = g − 1 ⇔ L2 =∼ K. Hence there is a bilinear bundle map (., .) which map two points of L to points in K:

Lp 3 ψ, ϕ 7→ (ψ, ϕ) ∈ Kp.

Hence the elements of L can be regarded as square roots of complex linear forms— so called spinors.

Deﬁnition 3.43. A spin bundle (also called spin structure) over a Riemann surface M is a complex line bundle L → M together with a bundle isomorphism L ⊗ L → K.

54 The Riemann-Roch Theorem

Theorem 3.44. Up to isomorphism there are 22g different spin structures on a surface of genus g.

Let L be a holomorphic line bundle over a Riemann surface M, ψ, ϕ ∈ H0L, ϕ 6= 0. Then ψ = f ϕ for some meromorhic f . If now ψ˜ = aψ + bϕ, ϕ˜ = cψ + dϕ then ψ˜ aψ + bϕ a f + b f˜ = = = . ϕ˜ cψ + dϕ c f + d So f˜ = g ◦ f , where g : S2 → S2 is Mobius.¨ All meromorphic functions f arise in this way from a 2-dimensional subspace U ⊂ H1L, where L is some holomorphic line bundle.

3.5 Quaternions

Deﬁnition 3.45. An algebra is a ﬁnite-dimensional real vector space A together with a bilinear map A × A → A, a, b 7→ ab.

Deﬁnition 3.46. If A is an algebra, then 1 ∈ A is called a unit if a · 1 = 1 · a = a for all a ∈ A.

Example 3.47. A = R, C = span {1, J}, Rn×n, Cn×n are all associative algebras, i.e. (ab)c = a(bc) for all a, b, c ∈ A .

Deﬁnition 3.48. H = R4 = span {1, i, j, k} with multiplication deﬁned by the fact that 1 is a unit and

i2 = j2 = k2 = −1, ij = k = −ji, jk = i = −kj, ki = j = −ik .

Theorem 3.49. H is associative.

Proof. Deﬁne complex 2 × 2 matrices 1 0 J 0 0 −1 0 −J I = , X = , Y = , Z = . 0 1 0 −J 1 0 −J 0

2×2 Then F : H → spanR{I, X, Y, Z} ⊂ C given by F(w + xi + yj + zk) = wI + xX + yY + zZ is an algebra isomorphism, and C2×2 is associative.

55 3.5 Quaternions

By the identiﬁcation R3 =∼ Im H := span{i, j, k}, every q ∈ H can uniquely be written as q = r + v with α ∈ R and v ∈ R3. Set

Re q = α, Im q = v .

Theorem 3.50. For u, v ∈ R3 we have uv = −hu, vi + u × v.

Proof. One can check this on basis vectors. We leave it as an exercise.

p We have qq¯ = (α + v)(α − v) = α2 + hv, vi ≥ 0. Deﬁne |q| = qq¯.

Theorem 3.51. |pq| = |p||q| for all p, q ∈ H.

Proof. Note that pq = q¯p¯. Thus |pq|2 = pqpq = pqq¯p¯ = ppq¯ q¯ = |p|2|q|2.

In particular, if pq = 0, then p = 0 or q = 0. So H is a division algebra. And, if q 6= 0, then q/|p|2 is the multiplicative inverse element of q.

Deﬁnition 3.52. A quaternionic vector space is a real vector space V together with a real bilinear map V × H → V, v, λ 7→ vλ such that v1 = v and v(λµ) = (vλ)µ for all λ, µ ∈ H and v ∈ V.

Deﬁnition 3.53. If V, W are quaternionic vector spaces then f : V → W is called quaternionic linear, if it is real linear and

f (vλ) = f (v)λ, for all v ∈ V, λ ∈ H .

Example 3.54. Hn is a quaternionic vector space. The quaternionic linear maps Hn → Hm are given by matrix multiplication from the left with A ∈ Hm×n.

Consider an immersion f : M → R3, then

f conformal ⇔ d f (JX) = Nd f (X) ∀X ∈ TM ⇔ ∗d f = −Nd f .

This looks like Cauchy–Riemann with J replaced by N. Goal: Interpret f as a holomorphic section of a holomorphic quaternionic vector bundle.

Deﬁnition 3.55. A quaternionic vector bundle of rank k is a real vector bundle V of rank 4k where all ﬁbers have the structure of a quaternionic vector space, in such a way that ψ ∈ ΓV, λ ∈ C∞(M; H) implies that ψλ ∈ ΓV.

Theorem 3.56. Every quaternionic line bundle over a surface is trivial, i.e. isomorphic to M × H.

56 The Riemann-Roch Theorem

Proof. The total space of L has real dimension 6 and the graph of any section has dimension 2. So the graph of a generic section does not intersect the graph of the zero section. So there is a nowhere vanishing section, i.e. a global frame ﬁeld.

Deﬁnition 3.57. A complex quaternionic line bundle is a quaternionic line bundle L 2 together with J ∈ ΓEndH(L) such that J = −I

A complex quaternionic line bundle can also be viewed as a complex vector bundle of rank 2: Choose your favorite quaternion on S2 ⊂ R3, for example i and deﬁne E = {ψ ∈ L | Jψ = ψi} . Claim: E is of real rank 2. Right multiplication by i deﬁnes another complex structure on L and J is complex linear with respect to this structure. So there is 2 λ = α + iβ and ψ ∈ Lp, ψ 6= 0 such that Jψ = ψλ. Then −ψ = J ψ = J(ψλ) = (Jψ)λ = ψλ2. Hence λ2 = −1 and so λ = ±i. If Jψ = −ψi, then J(ψj) = −ψ(ij) = (ψj)i.

Theorem 3.58. L = E ⊕ Ej, where E = {ψ ∈ L | Jψ = ψi} is a rank 2 real subbundle. As complex (with respect to J) line bundles E and Ej are isomorphic.

Deﬁnition 3.59. deg L = degCE.

Let L be a complex quaternionic line bundle over a Riemann surface M and ∇ a quaternionic connection, i.e. ∇(ψλ) = (∇ψ)λ + ψ(dλ) for all ψ ∈ ΓL and λ ∈ C∞(M; H).

Deﬁnition 3.60. The complex quaternionic line bundles KL and KL¯ are deﬁned as

KL = {ω ∈ Λ1(M; L) | ω(JX) = Jω(X) ∀X ∈ TM} , KL¯ = {ω ∈ Λ1(M; L) | ω(JX) = −Jω(X) ∀X ∈ TM} .

Clearly: Ω1(M, L) = Γ(KL) ⊕ Γ(KL¯ ). Let M be a Riemann surface and L a quaternionic Line bundle over M. We know that L is isomorphic to M × H. Furthermore L has a ﬂat quaternionic connection ∇, i.e ∇ϕ = 0. Consider

EndH(L) = {B : Lp → Lp | B quaternionic linear }.

Let ϕ ∈ Lp be a basis vector. If we have ψ = ϕ · λ and Bϕ = ϕb for λ, b ∈ H, we obtain: Bψ = B(ϕλ) = (Bϕ)λ = bλ.

Hence B is represented by b ∈ H1×1. This yields End L =∼ M × H. One shold note that the isomorphism is in general not canonical.

57 3.5 Quaternions

Deﬁnition 3.61. A complex quaternionic line bundle L is a quaternionic line bundle together with J ∈ ΓEnd(L) such that J2 = −I

If L is a complex quaternionic line bundle, then End L = End+L ⊕ End−L, where End±L = {A ∈ End L | AJ = ±JA}. Moreover, End+L is a subalgebra bundle canonically isomorphic to M × C, √ α + −1β ←→ αI + βJ .

If A, B ∈ ΓEnd−L, then A · B ∈ ΓEnd+L. Similarly, if A ∈ ΓEnd±L and B ∈ ΓEnd∓L, then A · B ∈ ΓEnd−L.

Theorem 3.62. If ∇ is a quaternionic connection on a complex quaternionic line bundle L over a manifold M, then there is a unique complex connection ∇ˆ on L (∇ˆ J = 0) and 1 η ∈ Ω (M; End−L) such that ∇ = ∇ˆ + η .

Proof. We have ∇Jψ = (∇J)ψ + J∇ψ. Thus ∇ψ = −J(∇J)ψ − J∇(Jψ). Hence

1 1 ∇ = 2 (∇ − J∇J) + 2 (∇ + J∇J) . | {z } | {z } =:∇ˆ J(∇J)=:η

The types are clear. We only have to check that ∇ˆ is a quaternionic connection. Let λ ∈ C∞(M; H), then

∇ˆ (ψλ)−(∇ˆ ψ)λ − ψdλ 1 1 = 2 ((∇ψ)λ + ψdλ − J ∇(Jψλ) ) − 2 (∇ψ − J(∇Jψ)λ − ψdλ) = 0 . | {z } ∇(Jψ)λ+Jψdλ

Example 3.63. Consider f : M → R3 with unit normal N. Then d f (JX) = N d f (X). Where L = M × H and Jψ = Nψ. HEnce we have ∇ = d.

[Picture of the map]

Theorem 3.64. If M is an almost complex manifold and ∇ a quaternionic connection on a complex quaternionic line bundle L → M, then ∇ can be split uniquely as

∇ = ∇0 + ∇00 ,

where 0 1 00 1 ∇ = 2 (∇ + J ∗ ∇), ∇ = 2 (∇ − J ∗ ∇) .

Remark 3.65. This is similar to a ∂¯-operator, but

∇0 : ΓL → ΓKL; ∇00 : ΓL → ΓKL¯

58 The Riemann-Roch Theorem

Combining the two splittings we ﬁnd that

∇ = ∂ + A + ∂¯ + Q , where ∂ : ΓL → Γ(KL) and ∂¯ : ΓL → Γ(KL¯ ) both commute with J, A ∈ Γ(KEnd−L) and Q ∈ Γ(K¯End−L).

Deﬁnition 3.66. D : ΓL → Γ(KL¯ ) is called a holomorphic structure on a complex quaternionic line bundle L over an almost complex manifold M, if

1 D(ψλ) = (Dψ)λ + 2 (ψdλ − J ∗ dλ) .

Exercise 3.67. D is an elliptic operator.

Let V be a quaternionic vector space, α ∈ V∗, i.e. hα|ψλi = hα|ψiλ. We turn V∗ into a quaternionic vector space by setting for λ ∈ H

hαλ|ψi = λ¯ hα|ψi .

Theorem 3.68. (L, J, D) holomorphic quaternionic line bundle over a Riemann surface M of genus g, H0L := ker D, h0L := dim H0L. Then

h0L − h0(KL¯ ) = deg L + 1 − g .

Back to surfaces in 3-space: Let f : M → R3 be a conformal immersion, We split

1 1 ∇J = (∇J − J ∗ ∇J) + (∇J + J ∗ ∇J), 2 2 ˆ ¯ 1 such that ∇ = ∂ + ∂, 2 η = A + Q. Then J∇J = η = 2(A + Q), ∇J = −2(JA + JQ) = 2(∗Q − ∗A) .

If dN = d f (HI + q), q ∈ ΓEnd−(TM) = Γ(KTM¯ ) is called Hopf-differential—the trace-free part of the shape operator—and H is the mean curvature. A comparison then yields that −2 ∗ A = HI and 2 ∗ Q = d f ◦ q.