Equivariant Embeddings of Strongly Pseudoconvex CR Manifolds

Equivariant embeddings of strongly pseudoconvex CR manifolds

A thesis submitted for the degree of Doctor of Mathematics

Kevin Fritsch October 2019

Reviewed by Prof. Dr. Peter Heinzner Prof. Dr. George Marinescu Prof. Dr. Frank Loose

Contents

1 Transversal group actions on CR manifolds 8 1.1 Basic notions and examples ...... 8 1.2 Quotients of transversal G-actions ...... 11 1.3 Complexiﬁcation of transversal G-actions ...... 16

2 Pseudoconvexity 30 2.1 Pseudoconvexity in CR geometry ...... 30 2.2 Pseudoconvexity in complex geometry ...... 36 2.3 Boundaries of pseudoconvex domains ...... 39

3 Equivariant Embeddings 41 3.1 Semidirect Products ...... 41 3.2 Sheaf-Theory ...... 41 3.3 Equivariant embeddings ...... 50

4 Line Bundles 55 4.1 CR Line bundles ...... 55 4.2 Projective Embeddings ...... 56 4.3 G-ﬁnite functions ...... 62 Introduction

An important and much studied question in CR geometry is whether an abstract CR manifold can be realised, locally or even globally, as a CR n submanifold of C . There have been many results on this topic over the years. For example, n an analytic CR manifold can always be locally embedded into C [AH72] and globally embedded into a complex manifold [AF79]. n For global embeddings into C , one needs additional conditions to ensure the existence of suﬃciently many CR functions, as is the case in complex geometry. Considering from now on smooth CR manifolds, we have the theorem of Boutet de Monvel, which states that every compact, strongly pseudoconvex CR manifold of dimension greater than 3 and CR codimen- n sion 1 may be embedded into C [dM75].

In dimension 3, however, the situation is much more complicated. In fact, even with the assumptions above, embeddings almost never exist [Ros64], [BE90]. Even worse, an example of Nirenberg shows that the ∂b-equation may fail to have local solutions [Nir74].

Before discussing more specialised methods, we should mention the general result of Kohn, which states that the existence of a global embedding for compact, strongly pseudoconvex CR manifolds is equivalent to the Kohn- Laplacian having closed range [Koh86]. This is, however, a somewhat diﬃ- cult condition to verify. One may therefore look for another way to ensure a reasonable behaviour of the CR manifold. Baouendi, Rothschild and Treves considered transversal (local) group actions on CR manifolds and proved a local embedding result [BRT85], using the additional structure given by the action to construct good coordinates. For transversal R-actions, Lempert then proved the global embeddability for compact, strongly pseudoconvex CR manifolds in dimension 3 [Lem92].

Deepening the topic of transversal group actions, there have been recent results on CR manifolds with transversal S1-action. Hsiao, Li and Mari- nescu have proved an equivariant Kodaira embedding theorem, assuming the existence of a positive, rigid CR line bundle [HLM17] and an equivariant embedding theorem for compact, strongly pseudoconvex CR manifolds was proved by Herrmann, Hsiao and Li [HHL17].

In fact, transversal group actions on CR manifolds are also interesting from the viewpoint of transformation theory, as the following observation shows. Let (Z, ω) be a K¨ahlermanifold with a Lie group G acting on Z through

3 holomorphic transformations leaving ω invariant. Assume that there exists a momentum map µ: Z → g∗ for the action and assume that 0 ∈ g∗ is a regular value. Then the zero level M := µ−1(0) is a CR manifold with transversal, CR group action of G (see section 1.1, example 4). The manifold M plays an important role in geometric invariant theory. In particular, the quotient space M/G is of high interest.

In this thesis, we will examine transversal group actions on CR manifolds for arbitrary CR codimension. In particular, we are going to study the topic from the viewpoint of transformation theory and complex geometry.

In more detail, we will start by showing that CR manifolds with transversal group actions can be globally embedded into a complex manifold. We have Theorem 1.9. Let X be a CR manifold with proper, transversal CR action of a Lie group G. Assume that G is a subgroup of its universal complex- iﬁcation GC. Then there exists a complex manifold Z with a holomorphic GC-action and an equivariant CR embedding Φ: X → Z. Every G-equivariant CR map f : X → Y into a complex manifold Y with holomorphic GC-action extends to a unique GC-equivariant map F : Z → Y such that F ◦ Φ = f. Because of the universality condition, the manifold Z is unique up to GC- equivariant isomorphisms. We call Z the universal equivariant extension of X. Note that we do not assume X to be compact here.

We will then examine the quotient space X/G and generalise a statement of Loose (see [Loo00, Theorem 1.1]). In particular, we show Theorem 1.11. Let X be a CR manifold with proper, transversal CR action of a Lie group G. Assume that G is a subgroup of its universal complexiﬁca- tion GC. Then X/G is a complex space such that the sheaf of holomorphic functions is given by the sheaf of G-invariant CR functions on X. If Y is a CR manifold with transversal CR action of a Lie group G and X is a G-invariant hypersurface of Y , then X is a CR manifold with almost transversal G-action (see section 2.1, example 4). In section 2, we will generalise the notion of strong pseudoconvexity to CR manifolds with almost transversal G-action. In [OV07], Ornea and Verbitsky showed that a compact, strongly pseudoconvex CR manifold can be realised as the S1-bundle in a positive line bundle. We will proof a similar result in the generalised setting and begin by showing the following

4 Corollary 2.4. Let K be a compact, connected Lie group. Let X be a compact, strongly CR-pseudoconvex CR manifold with almost transversal, locally free K-action. Assume that there exists a K-transversal CR vectorﬁeld on X. Then there exists a CR S1-action on X which commutes with the K- action and the vectorﬁeld induced by this action is strongly pseudoconvex.

This induces a transversal, locally free CR S1 × K-action on X. We also 1 −1 have an S × K-action on C × X via ((s, k), (z, x)) 7→ (zs , skx), which is transversal and locally free. Using Theorem 1.11, we see that the quotients

S1 1 C × (X/K) := (C × X)/(S × K) and X/(S1 × K) are complex spaces. Since the actions are locally free, all isotropy groups are ﬁnite and we will see that both spaces are actually orbifolds. S1 1 The map C × (X/K) → X/(S × K) is an orbifold line bundle, in which X/K may be embedded via [x] 7→ [1, x] as the S1-bundle. We will then show

Theorem 2.6. Let X be a compact CR manifold with transversal, locally free, CR S1 × K-action such that the vectorﬁeld T induced by the S1-action is strongly pseudoconvex. Then the set

S1 2 {[z, x] ∈ C × X/K | |z| < 1} is strongly pseudoconvex.

Those results will be used as a motivation for the embedding results developed in section 3. In more detail, we will generalise the above setting to consider a Lie 1 group of the form H = G o S acting on a CR manifold X with transversal, proper CR action such that the quotient X/G is compact. If one assumes 0 0 that the isotropy groups are only in the G-direction, i.e. Hx < Gx for all S1 x in X, then the quotient C × X/G is a complex space and we deﬁne X to be strongly pseudoconvex if the set deﬁned in Theorem 2.6 is strongly pseudoconvex. We will then additionally assume that HC is complex reductive and use equivariant sheaf theory and methods from complex geometry to show the main result of this thesis.

Theorem 3.12. Let X be strongly pseudoconvex and Y the universal equivariant extension of X. Then there exists a HC-representation V and a m HC-equivariant holomorphic embedding Φ: Y → C \{0} × V such that m Φ: X → C × V is an embedding.

5 Note that for G = {Id}, we may use the results from section 2 and get an embedding result for compact, strongly pseudoconvex CR manifolds with transversal, CR R-action, similar to Lemperts result. We are also able to reproduce the equivariant embedding result in [HHL17] for transversal S1-actions. We will then use the methods established to proof a projective embedding result. If LK → X/K is a holomorphic line bundle, then we may pull back the transition functions to CR functions on open subsets of X, which induces a CR line bundle L → X. Corollary 4.6. Let X be a compact CR manifold with transversal action of a compact Lie group K. Assume that there exists a weakly negative line bundle LK → X/K and let L → X be the induced bundle. Then there exists −k a natural number k, ﬁnitely many CR sections si ∈ Γ(X,L ) such that W = span(si) is K-invariant and the map

∗ X 7→ P(W ) y 7→ [s 7→ s(y)] is a K-equivariant CR embedding. The theorems proved in this thesis will therefore be able to generate embedding results comparable to those in [Lem92], [HLM17] and [HHL17].

Note that there exists a work of Herrmann, Hsiao and myself with the following embedding result. Theorem [FHH18]. Let X be a compact, orientable CR manifold of dimension (2n, d) with n ≥ 2. Let K be a compact Lie group with almost transversal, locally free CR action on X. If X is strongly pseudoconvex, then there exists a K-equivariant CR embedding of X into a K-representation. Note that we do not assume the existence of a transversal CR vectorﬁeld, but there are several other restrictions. The methods used in [FHH18] are fundamentally diﬀerent from those in this thesis.

We will repeat the most important deﬁnitions and facts about CR manifolds, considered from the viewpoint of complex geometry. This thesis should therefore be easily understandable for someone with little knowledge of CR manifolds. We will, however, assume that the reader has basic knowledge of Lie groups and Lie group actions.

Acknowledgements

First, i would like to thank my supervisor Prof. Dr. Peter Heinzner for his continuous support and guidance during my research, the many helpful

6 remarks and discussions, and making the work on my thesis a generally enjoyable time. Secondly, i thank Dr. Chin-Yu Hsiao for the very helpful discussions on CR manifolds and his hospitality during my research visits. I am very grateful to Dr. Hendrik Herrmann for his mathematical insights, but also for being a good friend. Next, i thank my colleagues at the Ruhr-Universit¨atBochum for always being ready to talk about all kinds of mathematical problems. Last but not least, i want to thank my family and friends for always supporting me and giving me the opportunity to pursue my own goals.

7 1 Transversal group actions on CR manifolds

1.1 Basic notions and examples We start by giving a short introduction into the theory of CR manifolds.

If X is a real manifold, we denote by CTX = C ⊗R TX the complexiﬁed tangent bundle. ∞ We denote the smooth sections of TX and CTX by Γ (X,TX) and ∞ Γ (X, CTX), respectively. The Poisson bracket can be extended to a C- bilinear map

∞ ∞ ∞ [·, ·]:Γ (U, CTX) × Γ (U, CTX) → Γ (U, CTX). on every open subset U of X. If W is a subspace of CTxX, we denote by W = {V | V ∈ W } the complex conjugated subspace.

Deﬁnition Let X be a smooth manifold equipped with a smooth, complex 1,0 subbundle T X of the complexiﬁed tangent bundle CTX such that 1. T 1,0X ∩ T 1,0X = {0}

2.[Γ ∞(U, T 1,0X), Γ∞(U, T 1,0X)] ⊂ Γ∞(U, T 1,0X),

1,0 for every open subset U of X. Let n := dimCT X, d := dimRX − 2n. We write T 0,1X := T 1,0X and say that (X,T 1,0X) is a CR manifold of dimension (2n, d) with CR codimension d.

Note that CR stands for Cauchy-Riemann.

Deﬁnition Let X and Y be CR manifolds. We say that Y is a CR submanifold of X if Y can be realised as a smooth submanifold of X such that

1,0 1,0 T Y = CTY ∩ T X. If f : X → Y is a smooth map between manifolds and df : TX → TY its C diﬀerential, then we may extend df to a map d f : CTX → CTY , which is C-linear on every ﬁbre of CTX, by setting C dx f(V + iW ) := dxf(V ) + idxf(W )

C C for V,W ∈ TxX. Taking W ∈ CTxX, we have d f(W ) = d f(W ). Deﬁnition Let X, Y be CR manifolds. A smooth map f : X → Y is called a CR map if

dC f(T 1,0X) ⊂ T 1,0Y

8 We will introduce some basic notation for Lie group actions. We will denote Lie groups with capital letters and its Lie algebra with the corresponding small gothic letter, e.g. G and g or K and k.

Definition Let X be a smooth manifold with smooth action of a Lie group G. Every element ξ ∈ g defines a vectorfield ξX on X via d ξX (x) := exp(tξ)x. dt 0

We call ξX the fundamental vectorﬁeld of ξ on X. For x ∈ X, we deﬁne g(x) := ξX (x) | ξ ∈ g .

We say that the action of G on X is locally free if ξX (x) 6= 0 for every ξ ∈ g\{0} and x ∈ X.

Note that if the orbit Gx is a submanifold, then g(x) is the tangent space of Gx at x. We write Cg(x) for the C-linear span of g(x) in CTxX. Since ∼ g(x) is totally real in CTxX, we have Cg(x) = C ⊗R g(x). Let us consider some examples.

Example1 Let Z be a real manifold with an almost complex structure J. Deﬁne

1,0 T Z := {V − iJV ∈ CTZ | V ∈ TZ}. Note that T 1,0Z is the eigenspace of J to the eigenvalue i and T 0,1Z is the eigenspace to −i. Let U be an open subset of Z and A, B ∈ Γ∞(U, T Z), then

[A − iJA, B − iJB] ∈ Γ∞(U, T 1,0Z) ⇔[A, B] − i[A, JB] − i[JA, B] − [JA, JB] ∈ Γ∞(U, T 1,0Z) ⇔[A, JB] + [JA, B] = J([A, B] − [JA, JB]) ⇔J[A, JB] + J[JA, B] + [A, B] − [JA, JB] = 0.

Which shows that [Γ∞(U, T 1,0Z), Γ∞(U, T 1,0Z)] ⊂ Γ∞(U, T 1,0Z) if and only if the Nijenhuis tensor vanishes. The Newlander-Nirenberg Theorem [NN57] then gives that this is the case if and only if the almost complex structure J is integrable. This shows that every complex manifold is a CR manifold of CR codimension 0 and T 1,0Z is the holomorphic tangent space. The CR maps between two complex manifolds are exactly the holomorphic maps.

9 On the other hand, given a CR manifold X of CR codimension 0, one 1,0 0,1 gets CTX = T X ⊕ T X and can deﬁne an almost complex structure J on X via J(V + W ) := iV − iW for V,W ∈ T 1,0X. The above computation then shows that J is integrable and X is a complex manifold.

Example2 Let X be a real submanifold of a complex manifold Z with complex structure J. Deﬁne Wx := TxX ∩ JTxX and assume that dim Wx is constant, hence W deﬁnes a smooth subbundle of the tangent bundle. Then 1,0 1,0 T X = {V − iJV | V ∈ W } = T Z ∩ CTX 1,0 is involutive because T Z and CTX are, therefore X is a CR submanifold of Z. In particular, if Z is a complex manifold of complex dimension n and X is a real hypersurface in Z, then X is a CR manifold of dimension (2(n−1), 1). This is seen from the equation

dim(TxX ∩ JTxX) = dim(TxX + JTxX) − dimTxX − dimJTxX and the fact that TxX + JTxX = TxZ. Every restriction of a holomorphic map on Z to X is a CR map, but not every CR map extends to a holomorphic map.

Example3 Let Z be a complex manifold and L → Z a holomorphic line bundle, equipped with a metric h. Then the circle bundle X ⊂ L is a CR submanifold of L, because it is a real hypersurface. Note that L is equipped with a holomorphic S1-action which leaves X 1 1,0 0,1 invariant and fulﬁls CTxX = Cs (x) ⊕ Tx X ⊕ Tx X.

Example4 Let (Z, ω) be a K¨ahlermanifold, G a Lie group acting on Z leaving ω invariant. Assume that there exists a momentum map µ: Z → g∗ such that 0 is a regular value. Deﬁne the G-invariant submanifold M := µ−1(0) and denote by J the ⊥ω complex structure of Z. Using ker dxµ = (g(x)) , we see g(x)∩Jg(x) = {0} for all x ∈ M and T M ∩ JT M = (g(x) ⊕ Jg(x))⊥ω . Since 0 is a regular value, this is of constant dimension. We conclude that M is a CR submanifold of Z. Note that in this case, we have 1,0 0,1 CTxM = Cg(x) ⊕ Tx M ⊕ Tx M

10 These examples motivate the study of group actions on CR manifolds. We get the following central deﬁnition.

Deﬁnition Let X be a CR manifold and G be a Lie group. We call an action of G on X a CR action if G acts by CR automorphisms. The action is called transversal if

1,0 0,1 Cg(x) ⊕ Tx X ⊕ Tx X = CTxX for every x ∈ X.

Let X be a manifold with action of a Lie group G and Φ: G × X → X the action map. We abbreviate Φ(g, x) as gx and if Φg : X → X is the action map for a ﬁxed g ∈ G, we write dg for dΦg.

Let us revisit example 4 again. In the theory of transformation groups, the space M plays an important role. In particular, the quotient M/G is of very high interest. In many cases, M/G carries the structure of a complex space [HL94] [Amm97], but little qualitative statements can be made.

1.2 Quotients of transversal G-actions With the motivation of the last section in mind, we want to study quotient spaces of CR manifolds with transversal group actions. Our ﬁrst goal is to show that for transversal, proper and free actions on CR manifolds, the quotient space is a complex manifold. To do this, we will need to study the smooth case ﬁrst, in particular, we need to study slices.

Let H be a Lie group and G a closed subgroup of H. Let X be a smooth manifold with G-action. We have a G-action on H × X via

G × (H × X) 7→ H × X (g, (h, x)) 7→ (hg−1, gx).

Since the G-action on H is proper and free, the G-action on H ×X is proper and free as well, hence (H × X)/G is a smooth manifold. We will discuss the manifold structure in a moment. For (h, x) ∈ H × X, we denote by [h, x] the corresponding equivalence class in (H × X)/G.

Deﬁnition We call H ×G X := (H × X)/G the twisted product of H and X over G.

11 Note that p: H ×G X → H/G is the bundle associated with the principle bundle H → H/G with typical ﬁbre X. We may identify X with the ﬁbre p−1(G) via x 7→ [1, x], then every H-orbit in H ×G X intersects X in exactly one G-orbit. Additionally, the isotropy group of H in a point x ∈ X is equal to the isotropy of G in x.

Definition Let H be a Lie group, X a smooth manifold with H-action and x ∈ X. Let S be a submanifold of X which contains x and is invariant under Hx Hx. We say that S is a slice for the H-action at x if the map H × S → X, [g, s] 7→ gs is a diffeomorphism onto an open subset of X and S is Hx- equivariantly diffeomorphic to an open subset of a Hx-representation. If H is a complex Lie-group acting holomorphically on a complex manifold X and S is a complex submanifold as above such the maps above are biholomorphic, then we call S a holomorphic slice.

Note that in the literature, slices are generally not required to be equivariantly diﬀeomorphic to an open subset of a representation. For embedding purposes however, it is reasonable to impose this condition.

If S is a Slice through x, then in every point s ∈ S, the isotropy group Hs of H is contained in Hx. If all isotropy groups Hs are of the same dimension, then every Hs has to contain the connected component of the identity 0 0 of Hx, which we will denote by Hx. In this case, Hx does not act on S.

Let X be a smooth manifold with proper action of a Lie group G. Fix x ∈ X, then Gx is compact and we find a neighbourhood basis of x consisting of Gx-invariant sets. Take open, Gx-invariant neighbourhoods U of x and V of 0 ∈ TxX with a diffeomorphism f : U → V such that dxf = id. We may average f over G and get F := R d Φ−1 ◦ f ◦ Φ dg, where Φ is the action x Gx x g g map. Then F is Gx-equivariant with dxF = id, hence we may assume that F is a diffeomorphism after shrinking U and V . Given a Gx-invariant complement W to g(x) in TxX, we may use F to find a Gx-invariant submanifold S of X with TxS = W . Using [Pal60, Sec- tion 2.2, Lemma], we conclude that, after possibly shrinking S, it is a Slice.

We now want to discuss the manifold structure on X/G. Given a point x ∈ X, we find a smooth slice S and the map ΦS : G × S → X,(g, s) 7→ gs is a diffeomorphism onto an open subset US of X. Let π : X → X/G be the quotient map. Then π : S → π(US) is bijective onto the open subset π(US). We may identify S with an open subset of a vectorspace and define a chart for π(US). If T is another slice with UT ∩ US 6= ∅, then the change of charts from S −1 to T is given by pr ◦ Φ and is therefore smooth. Here, pr denotes 2 T S∩UT 2

12 the projection on the second component.

For CR manifolds with transversal group actions however, it is not gen- 1,0 0,1 erally possible to find a slice S such that CTsS = Ts X ⊕ Ts X in every point s ∈ S. In fact, if we find such a Slice S through x ∈ X, then there exists a neighbourhood U of x which is diffeomorphic to G ×Gx S and S is a CR submanifold of X. Gx Gx We have the bundle map p: G × S → G/Gx and for [1, s] ∈ G × S, C 1,0 0,1 we get ker d[1,s]p = CT[1,s]S = T[1,s]X ⊕ T[1,s]X. But since the G-action is C 1,0 0,1 CR and p is a G-equivariant map, we have ker dy p = Ty X ⊕ Ty X for every y ∈ G ×Gx S. S C But the bundle defined by K := yker dy p is involutive, i.e. for two smooth sections V,W ∈ Γ(G ×Gx S, K), we have

dC p([V,W ]) = [dC p(V ), dC p(W )] = 0, hence [V,W ] is a section into K, as well. But then T 1,0X ⊕ T 0,1X is involutive, as well. If the CR manifold is strongly CR-pseudoconvex (see section 2.1), this is not possible. We will see later that S3, with the standard S1-action, is a CR manifold with transversal S1-action and a counterexample for the existence of CR slices in the above sense.

We therefore need to use another method to construct a CR structure on the quotient space. Lemma 1.1. Let X,Y be smooth manifolds and π : X → Y a surjective submersion. Let E be a smooth, complex subbundle of CTX. For every C y ∈ Y , let Fy be a complex subspace of CTyY such that dx π : Ex → Fπ(x) is S an isomorphism for every x ∈ X. Then F = y Fy is a smooth subbundle of CTY and for every x ∈ X, there exist open neighbourhoods U of x and Ω of π(x) with π(U) = Ω, such that for every smooth section W ∈ Γ(Ω,F ), there exists a smooth, π-related section V ∈ Γ(U, E), i.e. we have dC π◦V = W ◦π.

n n Proof. Denote by B the ball with radius 1 in R . Let x ∈ X, then we find a neighbourhood which is diffeomorphic to Bn × Bd and the image in Y under π is diffeomorphic to Bn, where π is the projection. We may also assume that there exist smooth sections n d Vi ∈ Γ(B × B ,E), i = 1, .., k, such that Vi(p) form a complex basis for Ep in every point p ∈ Bn × Bd. d n n d Now fix w ∈ B and set sw : B → B × B , z 7→ (z, w). Then define w w C sections Wi in CTY via Wi (z) = d(z,w)π(Vi(sw(z))). w Since dπ : E → F is an isomorphism in every point, the Wi (z) define a d n basis for Fw for every w ∈ B and z ∈ B . Since s, π and the Vi are smooth, n d w the map B × B → CTY ,(z, w) 7→ Wi (z) is smooth.

13 S In particular, the space F = w Fw deﬁnes a smooth subbundle of CTY .

Now let W ∈ Γ(Bn,F ) be a smooth section. For w ∈ Bd, we write P w w w W (z) = i fi (z)Wi (z) for complex-valued functions fi . We may con- w struct an inner product hw on F by deﬁning the Wi to be an orthonormal w basis. Then hw is smooth in w and the fi are just the orthogonal projection w of W onto Wi and are therefore also smooth in w.

P w Now we can construct a section of E via V (z, w) := i fi (z)Vi(z, w), which is a smooth section. We have C d(z,w)π(V (z, w)) X w C = fi (z)d(z,w)π(Vi(z, w)) i X w w = fi (z)Wi (z) i =W (π(z, w)).

Theorem 1.2. Let X be a CR manifold with a proper, free and transversal CR action of a Lie group G. Then X/G is a complex manifold such that the sheaf of holomorphic functions on X/G is given by the sheaf of G-invariant CR functions on X. Proof. Let X be a CR manifold of dimension (2n, d). Since the action is free, the group G has dimension d and X/G is a smooth manifold of dimension 2n. The quotient map π : X → X/G is a surjective submersion. C The kernel is given by ker dxπ = g(x) and ker dx π = Cg(x).

C −1 1,0 Now deﬁne the subspace Fy := {dx π(V ) | x ∈ π (y),V ∈ Tx X} of CTy(X/G). The action of G on X is CR and transversal and the map dπ is G- C 1,0 invariant, therefore the map dx π : Tx X → Fπ(x) is an isomorphism for S every x ∈ X. Using Lemma 1.1, the space F = x Fπ(x) deﬁnes a smooth, complex subbundle of CT (X/G).

We want to show that F deﬁnes a CR structure on X/G. For this, let 1,0 C C C V,W ∈ Tx X, then dx π(V ) = dx π(W ) = dx π(W ) implies V = W and V = W = 0. For an open subset U ⊂ X/G and V ∈ Γ∞(U, F ), we use Lemma 1.1 ∞ 1,0 and ﬁnd a smooth section V0 ∈ Γ (U0,T X) with π(U0) = U, such that C dx πV0 = V (π(x)). Then C C C V (π(x))(f) = d f(V (π(x))) = d f(d π(V0(x))) = V0(x)(f ◦ π),

14 ∞ or alternatively (V f) ◦ π = V0(f ◦ π). For another W ∈ Γ (U, F ) and W0 as above, this then implies

[V,W ](π(x))(f) = V (W (f))(π(x)) − W (V (f))(π(x))

= V0(W0(f ◦ π))(x) − W0(V0(f ◦ π))(x)

= [V0,W0](x)(f ◦ π) C = d π([V0,W0](x))(f).

C Hence [V,W ](π(x)) = d π([V0,W0](x)) and [V,W ](π(x)) ∈ Fπ(x).

We get that F deﬁnes a CR structure on X/G of CR codimension 0, hence X/G is a complex manifold (see section 1.1, example 1).

Now a function f : X/G → C is holomorphic if and only if it fulﬁls C 1,0 1,0 C 1,0 1,0 d f(T (X/G)) ⊂ T C. This is equivalent to d (f ◦ π)(T X) ⊂ T C, i.e. f ◦ π is a G-invariant CR map on X.

Note that the case for free actions was already known due to Loose [Loo00, Theorem 1.1]. The result of Loose is more general in the sense that 1,0 0,1 he only assumes the action to fulﬁl Cg(x) ∩ Tx X ⊕ Tx X = {0} and then shows that the quotient is a CR manifold. The above proof does however cover this setting, as well. We will proof a more general version later.

Now let Y be a G-invariant closed submanifold of X. Then the natural map Y/G → X/G is a closed topological embedding. The space TxY is Gx-invariant in TxX, hence we ﬁnd a Gx-invariant subspace of TxX transversal to TxY . From the construction of slices we discussed, we ﬁnd a slice SY for the G-action on Y such that SY is a submanifold of a slice SX for the G-action on X. This shows that Y/G is a submanifold of X/G.

Corollary 1.3. Let X be a CR manifold of dimension (2n, d) with proper, free and transversal CR action of a Lie group G. Let π : X → X/G be the quotient map and Y be a closed, G-invariant CR submanifold of dimension (2n0, d0) of X. C 1,0 C 1,0 Then d π(T Y ) = d π(CTY ) ∩ T (X/G), i.e. π(Y ) = Y/G is a closed CR manifold of dimension (2n0, d0 − d). Proof. First note that π(Y ) is closed if π−1(π(Y )) = Y is closed, which is the case by assumption. C 1,0 C 1,0 It only remains to show that d π(T Y ) = d π(CTY )∩T (X/G). By construction, π is a CR map. Hence the inclusion from the left to the right is obvious.

15 C 1,0 C For V ∈ d π(CTyY )∩T (X/G), we ﬁnd V0 ∈ CTyY with V = dy π(V0) C 1,0 and V = dy π(W0) for W0 ∈ Ty X by construction of the CR structure on X/G. C But then W0 −V0 ∈ ker dy π = Cg(y) ⊂ CTyY , because Y is G-invariant. 1,0 1,0 We conclude W0 ∈ CTyY ∩ Ty X = Ty Y .

1.3 Complexification of transversal G-actions We want to use the quotient theorem to proof a first embedding theorem. For this, we need to consider universal complexifications first.

Deﬁnition Let G be a real Lie group. A complex Lie group GC, together with a continuous morphism Φ: G → GC is called the universal com- plexiﬁcation of G if for every complex Lie group H and every continuous homomorphism f : G → H, there exists a unique holomorphic morphism F : GC → H such that F ◦ Φ = f.

The universal complexiﬁcation always exists and is unique [Hoc69, Sec- tion XVII.5]. We will from now on denote by GC the universal complexiﬁ- cation of G. The map Φ does not need to be injective. However, if G can be realised as a subgroup of some complex group H, then there exists a map F : GC → H such that F ◦ Φ is the inclusion of G into H. In particular, Φ is injective.

From now on, let G be a subgroup of its universal complexiﬁcation GC. This implies that G is a totally real, closed subgroup of GC [Hei93, §1 Proposition]. Let X be a CR manifold with a transversal, locally free CR action of G. Then the G-action on GC is proper and free, hence GC ×G X is a smooth manifold. We equip GC ×G X with a complex structure as follows. Consider the CR manifold GC × X with CR structure

T 1,0(GC × X) := T 1,0GC ⊕ T 1,0X.

We ﬁrst show that the G-action on GC ×X is transversal. For this, let ξ ∈ g and (h, x) ∈ GC × X. −1 The ﬂow of ξGC×X is given by Φt(h, x) = (h exp(tξ) , exp(tξ)x), this implies that

C ξGC×X (h, x) = ξGC (h) + ξX (x) ∈ ThG × TxX.

1,0 C 0,1 C 1,0 0,1 If ξGC×X (h, x) ∈ T (G ×X)⊕T (G ×X), then ξX (x) ∈ T X ⊕T X. Since the G-action on X is transversal, we conclude ξX (x) = 0. By assumption, the action is locally free and therefore ξ = 0. We may apply Theorem 1.2 and see that GC ×G X is a complex manifold.

16 We have a GC-action on GC ×G X via

η : GC × (GC ×G X) → (GC ×G X) (g, [h, x]) 7→ [gh, x].

From the commuting diagram

η˜ GC × (GC × X) GC × X

C C G C G G × (G × X) η G × X we see that η is holomorphic, since it is a CR map.

We formalise what we mean by a CR embedding.

Deﬁnition Let X, Y be CR manifolds. A map Φ: X → Y is called a CR embedding if it is a smooth embedding of X into Y and

C 1,0 C 1,0 d Φ(T X) = d Φ(CTX) ∩ T Y, i.e. Φ(X) is a CR submanifold of Y . We require all our embeddings to be closed.

Note that in the case where X is of CR codimension 1, it suﬃces that Φ is an embedding and a CR map. For higher CR codimension however, this is not true.

We will proof the following general Lemma, which we will use again later.

Lemma 1.4. Let G be a closed subgroup of H. Let H0 be a closed subgroup of H, deﬁne the subgroup G0 := G ∩ H0 and assume that the natural map G/G0 → H/H0 is closed. Let X be a smooth manifold with H0-action, then the map Φ: G ×G0 X → H ×H0 X is an embedding.

G Proof. We denote by [g, x] the equivalence classes in G× 0 X and by [g, x]H H the equivalence classes in H × 0 X, then Φ[g, x] = [g, x]H .

We start by showing that Φ is injective. −1 Let [g0, x0]H = [g, x]H , then there exists an h ∈ H0 such that g0h = g. −1 We get h = g g0 ∈ G ∩ H0 = G0 and [g0, x0] = [g, x].

Now we show that Φ is an immersion. For this, consider the commuting diagram

17 Φ˜ G × X H × X

π πH Φ G ×G0 X H ×H0 X.

Now ker dΦ = dπ(ker d(πH ◦ Φ)).˜ But the kernel of dπH in a point (h, x) is exactly the tangent space of the H0-orbit. We have G ∩ H0 = G0 and the kernel of d(πH ◦ Φ)˜ then has to be the tangent space to the G0-orbit, which gets mapped to 0 under dπ.

It remains to show that the map is closed. For this, take a sequence (gn, xn) ∈ G × X such that [gn, xn]H converges. H The map p: H × 0 X → H/H0 is continuous, hence p([gn, xn]H ) converges in H/H0. Since the map G/G0 → H/H0 is closed, we conclude that the image of gn in G/G0 converges, i.e. there exists a sequence hn ∈ G0 such that gnhn → g ∈ G. −1 −1 We have [gn, xn]H = [gnhnhn , xn]H = [gnhn, hn xn]H , which is conver- −1 gent. We therefore ﬁnd a sequence cn ∈ H0 such that gnhncn → h and −1 cnhn xn → x for h ∈ H, x ∈ X. But the H-action on itself is proper and gnhn converges, hence we may −1 assume that cn, and therefore hn xn converge. This implies that [gn, xn] is a convergent sequence in G ×G0 X and Φ is closed.

If a CR manifold X may be embedded into a complex manifold Z, it is an interesting question whether the CR functions on X may (locally) be extended to holomorphic functions on Z. If H is a complex Lie group acting on a complex manifold Z, we say that the action is holomorphic if the action map H × Z → Z is holomorphic.

Deﬁnition Let G be a Lie group and X a CR manifold with transversal action of G. Let Z be a complex manifold with holomorphic GC-action and Φ: X → Z a G-equivariant CR map. We say that (Z, Φ) is the universal equivariant extension of X if for every complex manifold Y with holomorphic GC-action and every G-equivariant CR map f : X → Y , there exists a unique GC-equivariant holomorphic map F : Z → Y with f = F ◦ Φ.

Note that if an universal equivariant extension exists, it is unique up to a GC-equivariant biholomorphic map. To see this, assume that (Z, Φ) and (Z,ˆ Φ)ˆ are two universal equivariant extensions of X. Using the universal property of Z, there exists a GC-equivariant holomorphic map F : Z → Zˆ such that Φˆ = F ◦ Φ. With the same argument, we ﬁnd Fˆ : Zˆ → Z with Φ = Fˆ ◦ Φ.ˆ In particular, Fˆ ◦ F ◦ Φ = Fˆ ◦ Φˆ = Φ = idZ ◦ Φ and because the

18 extension of functions is unique, we conclude Fˆ ◦ F = idZ . Repeating this argument, we conclude that F and Fˆ are inverse to each other.

Theorem 1.5. Let G be a subgroup of its universal complexiﬁcation GC. Let X be a CR manifold with transversal, locally free CR action of G. Then the map Φ: X → GC ×G X, x 7→ [1, x] is a G-equivariant CR embedding of X into the complex manifold GC ×G X. Furthermore, for every G-invariant open subset U ⊂ X, we have that (GCΦ(U), Φ) is the universal equivariant extension of U. In particular, the manifold (GC ×G X, Φ) is the universal equivariant extension of X.

Proof. We may write X = G ×G X and then apply Lemma 1.4 for H = GC, H0 = G and G0 = G. This gives that Φ is an embedding.

Consider the CR manifold G × X with the trivial structure on G. Since 1,0 G is totally real in GC, we get CTG ∩ T GC = {0} and the natural map G × X → GC × X is a CR embedding. Then Φ: X → G × X → GC ×G X is a CR embedding because of Corol- lary 1.3.

It remains to show the universality condition. First note that if U is open and G-invariant in X, then GCU is open in GC ×G X, hence it is a complex manifold. We will assume U = X here, the general statement is proved analogously. Let Y be a complex manifold with holomorphic GC-action and f : X → Y a G-equivariant CR map. We have GC · Φ(X) = GC ×G X by construction of GC ×G X and for x ∈ X, we get GCx ∩ Φ(X) = Gx. This implies that F : GC ×G X → Y ,[h, x] 7→ hf(Φ(x)) is well-deﬁned. Consider the following commuting diagram id ×f GC × X GC × Y

π η

GC ×G X Y F where η is the action map and π the projection onto the quotient. Now F is holomorphic if and only if F ◦ π is a CR map. But this is the case because id × f and η are CR.

Requiring the action of G to be locally free is a somewhat strong condition. We want to extend our result to omit this assumption. This will require some preparation.

19 Let G be a subgroup of its universal complexiﬁcation GC and X be a CR manifold with proper, transversal CR action of G.

We will use the following notation. For a Lie group G, we will denote by G0 the connected component of the identity, which is a normal Lie-subgroup of G.

Let S be a slice at x ∈ X and L = Gx. Because all G-orbits are of the 0 same dimension, all isotropy groups Gs for s ∈ S have to contain L . Hence L0 does not act on S, we get an L/L0-action on S and the L-orbits on S are ﬁnite. Since L is compact, we have LC = Lexp(il) and every connected component of LC intersects L. We conclude that LC/(L0)C = L/L0 and get an LC-action on S as a ﬁnite group. Because of [Hei93, §3 Corollary 1], we see that LC is a closed complex subgroup of GC. For Ω = G ×L S, set

ΩC := GC ×LC S.

Since the LC–action on GC is proper and free, ΩC is a smooth manifold. Note that for s ∈ S, we get (Ls)C = (LC)s. Hence for g ∈ G and [g, s] ∈ GC ×LC S, we conclude

C C −1 C −1 −1 C C (G )[g,s] = g(L )sg = g(Ls) g = (gLsg ) = (G[g,s]) .

We call ΩC the extension of the slice S or just a slice extension around x.

Note that this is a diﬀerent situation than in Theorem 1.5. In general, we can not identify ΩC with GC ×G Ω. An example to see this is given by choosing for X a complex manifold with trivial G-action. In this case GC ×G X =∼ GC/G×X, which is in general not complex. On the other hand, every open subset Ω of X is a slice and ΩC = GC ×GC Ω =∼ Ω, which is of course complex. If the G-action on X is locally free, then L is ﬁnite and LC = L. In this case, GC ×G (G ×L S) = GC ×L S = GC ×LC S.

We will now proof the analogous version of Theorem 1.5 and begin with the following Lemma.

Lemma 1.6. The natural map Φ: G ×L S → GC ×LC S is an embedding.

Proof. The idea of the proof is to apply Lemma 1.4 for H = GC, H0 = LC and H = L. Note that it is not yet obvious why LC ∩ G = L.

20 From [Hei93, §1 Proposition] we get the existence of an involutive anti- holomorphic homomorphism Θ on GC such that G is the ﬁxed point set of the involution Θ. Let θ be the corresponding involution on gC, then l is ﬁxed under θ and lC is invariant. Indeed, because Θ is anti-holomorphic, we get θ|l = 1 and θ|il = −1. Since L is compact, we have LC = Lexp(il). Now take l ∈ LC and write l = kexp(P ) for k ∈ L and P ∈ il. We compute lΘ(l)−1 = kexp(P )Θ(kexp(P ))−1 = kexp(P )Θ(exp(P ))−1k−1 = kexp(P )exp(−θ(P ))k−1 = kexp(2P )k−1. We conclude that l ∈ G if and only if lΘ(l)−1 = Id, which is then equivalent to exp(P ) = Id, hence P = 0. This implies G ∩ LC = L.

It remains to show that the map G/L → GC/LC is closed. For this, let gn be a sequence in G such that the image of gn in GC/LC converges. We ﬁnd a sequence ln ∈ LC such that gnln → g ∈ GC. Write ln = knexp(Pn) with kn ∈ L and Pn ∈ il. Then −1 −1 −1 −1 −1 lngn Θ(lngn ) = lnΘ(ln) = knexp(2Pn)kn , using the same computation as above. Since L is compact, we may assume that kn and therefore exp(Pn) converges. This implies that ln and therefore gn is a convergent sequence. Since G is closed in GC, the sequence gn converges in G.

Our results analogous to Theorem 1.5 are the following. Theorem 1.7. Let G be a subgroup of its universal complexiﬁcation GC. Let X be a CR manifold with proper, transversal CR action of G and ΩC a slice extension around x ∈ X. Then the map Φ:Ω → ΩC is a CR embedding of Ω into the complex manifold ΩC. Furthermore, for every G-invariant open subset U ⊂ Ω, we have that (GCΦ(U), Φ) is the universal equivariant extension of U. In particular, the manifold (ΩC, Φ) is the universal equivariant extension of Ω. Proof. Set Ω = G ×L S and ΩC = GC ×LC S as above. We begin by showing that ΩC is a complex manifold. Consider the map η : GC × Ω → ΩC (h, x) 7→ hΦ(x). We have the following commutative diagram

21 η˜ GC × G × S GC × S

id × πG πGC η GC × (G ×L S) GC ×LC S

whereη ˜(h, g, s) = (hg, s) and η(h, [g, s]) = [hg, s]. Sinceη ˜ and πGC are both surjective submersions, η is a surjective submersion, as well.

LC We denote by [h, s]C the elements of GC × S. −1 L The ﬁbre η ([1, s0]C ) consists of the elements (h, [g, s]) ∈ GC ×(G× S) with [hg, s]C = [1, s0]C . In particular, this means that there exists an l ∈ LC such that ls = s0. But since the LC-orbits on S are equal to the L-orbits, we conclude that we may actually choose l to be in L. Replacing g by gl, we see that

−1 η ([1, s0]C ) = {(h, [g, s0]) | [hg, s0]C = [1, s0]C }.

C This condition is equivalent to the existence of an l ∈ (G )[1,s0] such that hgl−1 = 1 ⇔ hg = l ⇔ h = lg−1.

We have shown that

−1 −1 C η ([1, s0]C ) = {(lg , [g, s0]) | l ∈ (G[1,s0]) , g ∈ G}.

Now let h0 ∈ GC. Since η is GC-equivariant, we get

−1 −1 C η ([h0, s0]) = {(h0lg , [g, s0]) | l ∈ (G[1,s0]) , g ∈ G}. Since η is a submersion, every ﬁbre of η is a submanifold with tangent space equal to the kernel of dη. We conclude that in a point (h0, y) ∈ GC × Ω, we get

C ker d(h0,y)η = {(dh0(ξ − µ), µΩ(y)) | ξ ∈ (gy) , µ ∈ g}, where (gy)C denotes the Lie algebra of the isotropy group (GC)y. Now let W ∈ ker dC η ∩ T 1,0(GC × Ω). Since the G-action on Ω is transversal, we (h0,y) C conclude that W is of the form W = (d h0(ξ), 0) for ξ ∈ C(gy)C. Deﬁne K := ker dC η ∩ T 1,0 (GC × Ω) (h0,y) (h0,y) (h0,y) C 1,0 C = {(d h0(ξ), 0) | ξ ∈ T1 (Gy) } and

C 0,1 C K(h0,y) = {(d h0(ξ), 0) | ξ ∈ T1 (Gy) },

22 then

ker dC η ∩ (T 1,0(GC × Ω) ⊕ T 0,1(GC × Ω)) = K ⊕ K . (h0,y) (h0,y) (h0,y) Since all isotropy groups of the G-action on Ω are of the same dimension, we see that the dimension of Kx does not depend on x ∈ GC × Ω, hence it deﬁnes a complex subbundle of T 1,0(GC × Ω). Choosing a riemannian metric on GC × Ω and extending it to a hermitian metric on CT (GC × Ω), we ﬁnd a complex subbundle E of T 1,0(GC×Ω) such that E⊕K = T 1,0(GC×Ω).

For x ∈ GC × Ω, deﬁne

C 1,0 C Fx := {dx η(V ) | V ∈ Tx (G × Ω)} and for y ∈ ΩC, set [ Fy := Fx. η(x)=y

C Note that d η deﬁnes an isomorphism between Ex and Fx in every point. −1 We want to show that Fy = Fx for every x ∈ η (y). The map η is invariant under the CR G-action on GC × Ω, deﬁned by −1 (g, (h, x)) 7→ (hg , gx). This shows that F[h,gx] = F[hg,x] for g ∈ G.

It therefore suﬃces to show F(h,[1,s]) = F(h0,[1,s0]) if η(h, [1, s]) = η(h0, [1, s0]). But the latter implies that [h, s]C = [h0, s0]C . In particular, there exists an l ∈ LC such that ls = s0 and since the LC-orbits on S are equal to the

L-orbits, we ﬁnd l0 ∈ L with l0s = s0. We have F[h0,[1,l0s]] = F[h0l0,[1,s]] and −1 C may assume that s = s0, hence hh0 ∈ (G[1,s]) . C It therefore remains to proof that F(l,[1,s]) = F(1,[1,s]) for all l ∈ (G[1,s]) . Deﬁne Ls = G[1,s] < L, then we have F(l,[1,s]) = F(1,[1,s]) for all l ∈ Ls. C C Note that [1, s] ∈ Ω implies (G[1,s]) = (G )[1,s]. C Because η is equivariant, we conclude F(l,[1,s]) = d l(F(1,[1,s])) and need C C C to show that d l(F(1,[1,s])) = F(1,[1,s]) for l ∈ Ls . But now CTη(1,[1,s])Ω is C a holomorphic Ls -representation and F(1,[1,s]) is a complex subspace which C is invariant under Ls. Therefore it is invariant under Ls , which was to show.

C We see that dx η induces an isomorphism between Ex and Fη(x) = Fx, S C hence Lemma 1.1 states that F = x Fη(x) is a subbundle of CT Ω .

C C C C C Now let dx η(V ), dx η(W ) ∈ Fx and assume dx η(V ) = dx η(W ) = dx η(W ). We conclude V − W ∈ Kx ⊕ Kx and get V − W = V0 − W 0 with V0 ∈ Kx C C and W 0 ∈ Kx. But then V0 = V , W0 = W and dx η(V ) = dx η(W ) = 0. One now sees that F is an involutive bundle in exactly the same way as in Theorem 1.2.

23 The real dimension of Fx is the dimension of Ex, which is 2n + 2d, if X is a CR manifold of dimension (2n, d). But the real dimension of ΩC = GC/(L0)C ×N S is also 2n + 2d, hence F deﬁnes a CR structure T 1,0ΩC of CR codimension 0 and ΩC is a complex manifold.

Lemma 1.6 states that Φ is a smooth embedding, let us check that it is a CR embedding. Note that Φ0 :Ω → GC × Ω, x 7→ (1, x) is a CR embedding with Φ = η ◦ Φ0. Since Φ is by construction a CR map, we have C 1,0 C 1,0 d Φ(T Ω) ⊂ (d Φ(CT Ω) ∩ T ΩC).

C 1,0 C Let y ∈ Ω and W ∈ dy Φ(CTyΩ) ∩ Tη(1,y)Ω . By construction of the CR C 1,0 C C structure, W = d(1,y)η(W0) for W0 ∈ T(1,y)(G × Ω) and W = dy Φ(V0) for V0 ∈ CTyΩ. C C C Now W0 −dy Φ0(V0) ∈ ker d(1,y)η = {(ξ −µ, µΩ(y)) | ξ ∈ C(g )y, µ ∈ Cg}. C C We ﬁnd ξ ∈ C(g )y, µ ∈ Cg such that W0 = (ξ − µ, µΩ(y) + dy Φ0(V0)). But 1,0 C C 1,0 C W0 ∈ T(1,y)(G ×Ω) and since g is totally real in g , we get Cg∩T1 G = {0} 1,0 and µ = 0. Because Φ0 is a CR embedding, this implies V0 ∈ Ty Ω, which was to show.

Now let us consider the universality condition. We will again assume U = Ω. Let Y be a complex manifold with holomorphic GC-action and f :Ω → Y a G-equivariant CR map. As in Theorem 1.5, we deﬁne F :ΩC → Y , hx 7→ hf(x) for h ∈ GC, x ∈ Ω. For this to be well-deﬁned, we need to check that h ∈ (GC)x implies C C C h ∈ (G )f(x). Since x ∈ Ω, we have (G )x = (Gx) and because the action of C C C C C G on Y is holomorphic, Gx ⊂ (G )f(x) implies (G )x = (Gx) ⊂ (G )f(x). We have the commuting diagram id ×f GC × Ω GC × Y

η η˜

ΩC Y F whereη ˜ is the action map on Y . By construction of T 1,0ΩC, the map F is holomorphic if F ◦ η is CR, but this follows from the diagram.

We want to give a global version of this local statement. For this, we consider the union over all slice extensions and identify the overlapping parts. We formalise this as follows. C Around every x ∈ X, there exists a slice extension Ωx . We may cover C X with countably many sets Ωi with extensions Ωi for i ∈ N.

24 For i, j ∈ N with Ωi ∩ Ωj 6= ∅, deﬁne the open subset

C C C Ωij := G · (Ωi ∩ Ωj) ⊂ Ωi .

Then the identity map Ωi ∩Ωj → Ωi ∩Ωj extends to a unique GC-equivariant C C holomorphic map ϕji :Ωij 7→ Ωji, using Theorem 1.7. Because of the uniqueness, we get

ϕii = id C (1) Ωi and

ϕkj ◦ ϕji = ϕki (2)

C C on G (Ωi ∩ Ωj ∩ Ωk) ⊂ Ωj . This also implies

−1 ϕij = ϕji . (3)

Now deﬁne . [ C Z := Ωi ∼, i∈N

C C where x ∈ Ωi and y ∈ Ωj are equivalent if Ωi ∩ Ωj 6= ∅ and ϕji(x) = y. Because of (1), (2) and (3), this does indeed define an equivalence relation. Define Z := S ΩC to be the disjoint union over the ΩC, the quotient 0 i∈N i i map π : Z0 → Z and equip Z with the quotient topology. Lemma 1.8. The space Z is hausdorff and second countable.

Proof. We will ﬁrst proof some general topological properties. Note that a set U ⊂ Z0 is open if an only if its intersection with every connected C component is open, i.e. if U ∩ Ωi is open for every i ∈ N. C −1 C Fix i ∈ N and let U ⊂ Ωi be open. Then π (π(U))∩Ωj = ϕji(U ∩Ωij), which shows that π−1(π(U)) is open. C −1 −1 Let U, V ⊂ Ωi be disjoint and assume that π (π(U)) and π (π(V )) C are not disjoint. That means that there exists a j and y ∈ Ωj such that C C y ∈ ϕji(U ∩Ωij)∩ϕji(V ∩Ωij). Since ϕij is injective, we conclude U ∩V 6= ∅, which is a contradiction.

Let us now proof that Z is hausdorﬀ. Without loss of generality, we take C C x ∈ Ω1 , y ∈ Ω2 such that x is not equivalent to y and want to separate π(x) and π(y) by open subsets. We distinguish several cases.

C C Case 1: x ∈ Ω12. We may assume x ∈ Ω2 without changing π(x). But then we can separate π(x) and π(y) using the general statements we proved

25 C above, using that Ω2 is hausdorﬀ.

C Case 2: y ∈ Ω21. This is analogous to case 1.

C C C Case 3: x∈ / Ω12 and y∈ / Ω21. First assume that x ∈ Ω1 ⊂ Ω1 and C y ∈ Ω2 ⊂ Ω2 . Let ι1 and ι2 be the inclusions of Ω1 and Ω2 into X as open subsets. Then ι1(x) and ι2(y) can not be in the same G-orbit in X, since C C ι1(Ω1\Ω12) and ι2(Ω2\Ω21) are disjoint G-invariant subsets containing x and y, respectively. Using the existence of Slices for the G-action on X, we find open, disjoint, G-invariant neighbourhoods Ux and Uy of ι1(x) and ι2(y) such that −1 Ux ⊂ ι1(Ω1) and ι2(Uy) ⊂ Ω2. Define the open subsets Wx := GC · ι (Ux) C −1 C C and Wy := G · ι (Uy) of Ω1 and Ω2 . We claim that π(Wx) and π(Wy) C are disjoint. Assume that this is not the case. Then we find x0 ∈ Wx ∩ Ω12 C C and y0 ∈ Wy ∩ Ω21 such that x0 and y0 are equivalent. There exists h ∈ G −1 such that hx0 ∈ Ω1 ∩ Wx = ι (Ux) and since ϕ21(Ω1) ⊂ Ω2, we conclude −1 hy0 ∈ Ω2 ∩ Wy = ι (Uy). But hx0 and hy0 are equivalent if and only if ι1(hx0) = ι2(hy0), but then Ux and Uy are not disjoint, which is a contradiction.

C C C C C Now let x ∈ Ω1 \Ω12 and y ∈ Ω2 \Ω21 be arbitrary. We ﬁnd hx, hy ∈ G such that hxx ∈ Ω1 and hyy ∈ Ω2. The sets Whxx and Whyy we constructed C are G -invariant, hence π(Whxx) and π(Whyy) separate π(x) and π(y), as well. This concludes the proof that Z is hausdorﬀ.

C Now let us show that Z is second countable. Every Ωi is a complex α manifold and therefore second countable, let Ui be a countable basis of the C topology for Ωi . α Then the π(Ui ) are open and form a countable (taken over i and α) basis of the topology for Z, which we see as follows. For every z ∈ Z, z = π(z0), C −1 C z0 ∈ Ωi and a neighbourhood U of z, we have that π (U)∩Ωi is open. We α −1 C α α −1 ﬁnd a Ui ⊂ π (U) ∩ Ωi with z ∈ Ui , then π(Ui ) ⊂ π(π (U)) = U.

C C C Since the Ωi are complex manifolds and π :Ωi → π(Ωi ) is a homeo- morphism, they define charts for Z. Because the ϕji are biholomorphic, this gives Z the structure of a complex manifold. C C C The G -action on Z is holomorphic because the G -actions on the Ωi C are holomorphic. Consider the maps Φi :Ωi → Ωi . We define Φ: X → Z C x 7→ π(Φi(x)), if x ∈ Ωi . By construction, this is a well-defined CR injective immersion of X into Z. We summarise our results in the following Theorem.

26 Theorem 1.9. Let G be a subgroup of its universal complexiﬁcation and X a CR manifold with proper, transversal CR G-action. Then the map Φ: X → Z is a G-equivariant CR embedding of X into the complex manifold Z. Furthermore, for every G-invariant open subset U ⊂ X, we have that (GCΦ(U), Φ) is the universal equivariant extension of U. In particular, the manifold (Z, Φ) is the universal equivariant extension of X.

Proof. Let us check that Φ is a closed map. Let xn be a sequence in X such C that Φ(xn) converges against z ∈ Z. Then z is contained in some Ωi and C we get z ∈ Ωi because Ωi is closed in Ωi . We have proved that Φ is a CR embedding, because the Φi are CR embeddings.

It remains to proof the universality condition. We will again assume U = X and take a G-equivariant CR map f : X → Y . For every x ∈ X, C C the map f extends to a G -equivariant CR map Fi :Ωi → Y and we define C F : Z → Y via F (y) = Fi(y) if y ∈ Ωi . We need to show that this is well- C C C defined. Consider Ωi and Ωj with Ωi ∩ Ωj 6= ∅. But then Fi :Ωij → Y and C Fj ◦ ϕji :Ωij → Y are both extensions of the same function f on Ωi ∩ Ωj, hence Fi = Fj ◦ ϕji and F is well-defined. We now want to consider the quotient X/G and show that it has the structure of a complex space. Since the action of GC on Z is generally not proper, we need to proof the existence of slices. Note that in general, S fails to be a complex submanifold of GC ×LC S. In fact, the Slice S is a complex submanifold if and only if it is a CR LC 1,0 0,1 submanifold of GC × S such that T S ⊕ T S = CTS. But since S is a submanifold of Ω, which is a CR submanifold of GC ×LC S, this implies 1,0 0,1 that Tx Ω ⊕ Tx Ω = CTxS in every point x ∈ S. In section 1.2, we have already seen that such Slices do not exist in general. Note that the argument in section 1.2 also shows that S does not have to contain any complex submanifolds. We therefore need to use another method to construct holomorphic slices, which will use the notion of K- orbit convex sets.

Deﬁnition Let Z be a complex manifold with an action of a complex reductive group KC, where K is a maximal compact subgroup of KC. An open subset W ⊂ Z is called K-orbit convex if W is K-invariant and for every z ∈ W and ξ ∈ k, we get that {t ∈ R | exp(itξ)z ∈ W } is connected.

If Z, Y are complex manifolds with KC-action, W ⊂ Z is K-orbit convex and f : W → Y is a K-equivariant map, then we may deﬁne F : KCW → Y via F (kz) := kf(z) for k ∈ KC and z ∈ W . We check that this is well- deﬁned, i.e. kf(z) = f(kz) for all k ∈ KC and z ∈ W such that kz ∈ W .

27 We have KC = K exp(ik) and since f is K-equivariant, we only need to check the above for h = exp(iξ) with ξ ∈ k. Fix z ∈ W , ξ ∈ k and define Ω = {w ∈ C | exp(wξ)z ∈ W }. Then Ω is an open neighbourhood of R. Define ϕ:Ω → Y , w 7→ exp(−wξ)f(exp(wξ)z). Then ϕ is holomorphic and constant on R. Since W is K-orbit convex, we get that iR ∩ Ω is connected, hence ϕ is constant on iR ∩ Ω, which was to show. Proposition 1.10. Let G be a subgroup of its universal complexification LC GC. Let x ∈ X, L = Gx and ΩC = GC × S a slice extension around x. Then there exists a holomorphic slice SC for the GC-action on ΩC. Proof. The group GC is a Stein manifold [Hei93, §1 Proposition] and LC is complex reductive, hence GC/LC is a Stein manifold [Mat60]. We have an LC-action on GC/LC via the multiplication from the left. Since LC is complex reductive and 1 · LC ∈ GC/LC is a fixed point, we find an LC- invariant neighbourhood U of 1 · LC and a LC-equivariant embedding of U onto an open subset UV of an LC-representation V [Sno82, Theorem 5.2 and Remark 5.4]. We get that L acts by unitary transformations on V for a suitable inner ξ d 2 product and µ (z) := dt 0||exp(itξ)z|| defines a momentum map for the L-action. Since 1 · LC ∈ UV is an L-fixed point, we see µ(1 · LC) = 0. From [HS07, Theorem 13.7], we conclude that 1 · LC ∈ UV has an L-orbit convex neighbourhood basis.

Via the LC-equivariant quotient map GC/(L0)C → GC/LC, every L- orbit convex neighbourhood W of 1 · LC ∈ GC/LC deﬁnes an L-orbit convex neighbourhood W˜ of 1 · (L0)C ∈ GC/(L0)C which is invariant under the LC-action from the right. We claim that, if W˜ is as above and W0 is an LC-invariant neighbourhood of x ∈ S, then W˜ × W0 deﬁnes an L-orbit convex neighbourhood of x in GC ×LC S. To prove the claim, we see that the set is L-invariant and if ξ ∈ l and [w, s] ∈ W˜ × W0, then exp(itξ)[w, s] is in W˜ × W0 if and only if there exists −1 an l ∈ LC with exp(itξ)wl ∈ W˜ and ls ∈ W0 but since W˜ and W0 are LC- invariant, this is equivalent to exp(itξ)w ∈ W˜ and s ∈ W0. The claim then follows from the L-orbit convexity of W˜ . We conclude that x ∈ GC ×LC S has an L-orbit convex neighbourhood basis.

Now take some L-orbit convex neighbourhood W1 of x and an open L-invariant neighbourhood W2 of 0 ∈ TxΩC with an L-equivariant biholomorphic map ϕ: W1 → W2. We may extend ϕ to a LC-equivariant map Φ: LCW1 → LCW2. Apply [HS07, Theorem 13.7] on the origin in TxΩC and ﬁnd some L-orbit −1 convex neighbourhood W˜ 2 ⊂ W2. Then extend ϕ to a LC-equivariant

28 function Φ:˜ LCW˜ 2 → LCW1. But because Φ ◦ Φ˜ = Id on W˜ 2, we see Φ ◦ Φ˜ = Id on LCW˜ 2, because of equivariance. The same argument applies to Φ˜ ◦ Φ: LCΦ(˜ W˜ 2) → LCW˜ 2 and we have found LC-invariant open neighbourhoods U1 of x and U2 of 0 ∈ TxΩC with an LC-equivariant biholomorphic map Φ: U1 → U2.

By choosing an LC-invariant subspace of TxΩC which is perpendicular to lC(x), we ﬁnd an LC-invariant complex submanifold SC of ΩC through x LC such that the induced map η : GC × SC → ΩC is an immersion in [1, x].

We ﬁnd some L-invariant neighbourhood W˜ of x in ΩC and an L- LC equivariant holomorphic mapη ˜: W˜ → GC × SC with η ◦ η˜ = Id. C This implies thatη ˜|S∩W˜ is an L -equivariant map, which extends to a C C ˜ G -equivariant mapη ¯ on the open set G ·(W ∩S). Therefore η◦η¯|S∩W˜ = Id and η ◦ η¯ = Id on GC · (W˜ ∩ S), as follows directly from the equivariance of η andη ¯. One also getsη ˜ =η ¯ on the open set (GC(W˜ ∩ S)) ∩ W˜ , henceη ¯ is holomorphic. We have shown that η is biholomorphic after possibly shrinking SC .

Remark If Z is the universal equivariant extension of X as in Theorem 1.9, then the Proposition also states the existence of slices on Z. If the G-action on X is locally free, then the universal equivariant extension is isomorphic to GC ×G X and we also get the existence of slices on this space.

Theorem 1.11. Let G be a closed subgroup of its universal complexiﬁcation and X a CR manifold with transversal, proper CR G-action. Then X/G is a complex space, such that the the sheaf of holomorphic functions on X/G is given by the sheaf of G-invariant CR functions on X.

Proof. Take x ∈ X with a slice extension ΩC. We then get Ω/G = ΩC/GC and Proposition 1.10 gives ΩC/GC = SC /LC, where SC may be realised as an open subset of an LC-representation. Because LC is complex reductive, SC /LC is an open subset of an aﬃne variety, giving X/G the structure of a complex space. Because of the universality condition, every G-invariant CR map on Ω extends to a unique GC-invariant holomorphic function on ΩC.

Remark Because LC acts as a ﬁnite group on SC , we have actually shown that X/G is a complex orbifold.

29 2 Pseudoconvexity

2.1 Pseudoconvexity in CR geometry An important question in CR geometry is whether a CR manifold X may be n realised as a CR submanifold of C or not. As in complex geometry, there are additional conditions we need to impose on X to ensure the existence of such an embedding. The theorem of Boutet de Monvel, which we will state shortly, gives a satisfying answer for compact CR manifolds of dimension greater than 5 and CR codimension equal to 1.

In this section, we introduce the notion of CR pseudoconvexity for CR codimension equal to one and generalise it for higher codimension. Let us start with a basic example to motivate the topic.

3 2 Consider S ⊂ C as a CR submanifold. Then the CR structure is spanned by the complex vector field Z := z ∂ − z ∂ . Note that we find 2 ∂z1 1 ∂z2 a vector field given by X := iz ∂ + iz ∂ − iz ∂ − iz ∂ such that 1 ∂z1 2 ∂z2 1 ∂z1 2 ∂z2 3 CX ⊕ CZ ⊕ CZ = CTS . The vector field X is in fact induced by the standard S1-action on S3, d 2 3 −1 i.e. X(z) = dt 0exp(it)z. Define ρ = ||z|| , then S = ρ (1). We get c d ρ = (∂ − ∂¯)ρ = z1dz1 + z2dz2 − z1dz1 − z2dz2, hence dcρ(X) = 2i and

c dd ρ = −2dz1 ∧ dz1 − 2dz2 ∧ dz2. 1 c 3 The 1-form α = 2i d ρ is an intrinsic object of S , which will lead us to the abstract deﬁnition of strong CR-pseudoconvexity.

Let X be a CR manifold of dimension 2n + 1 with CR structure T 1,0X. 1,0 0,1 A vectorﬁeld T is called transversal if CT (x)⊕Tx X ⊕Tx X = CTxX.

Assume now that X is orientable. Choose a riemannian metric on X and let L be the real line bundle over X given by the orthogonal complement to TX ∩ (T 1,0X ⊕ T 0,1X) in TX. Since TX ∩ (T 1,0X ⊕ T 0,1X) is a complex vector bundle over X, it is orientable. Since X is orientable, the first Stiefel-Whitney classes of TX and TX ∩ (T 1,0X ⊕ T 0,1X) vanish. Since the first Stiefel-Whitney class is additive, we conclude that L is orientable, hence trivial (on Stiefel-Whitney classes, see [MS74]). This shows that there always exist transversal vectorfields if X is orientable. In fact, the above argument shows that X is orientable if and only if there exist transversal vectorfields.

30 Let T be a transversal vectorfield on X. We may define a 1-form on X via α(T ) = 1 and α(T 1,0X ⊕ T 0,1X) = 0, which we will call the pro- 1,0 1 jection onto T . For V,W ∈ T X, we define ω(V,W ) := 2i dα(V, W ), by extending dα to a C-bilinear map on every fibre of CTX. By definition, ω is a sesquilinear form on T 1,0X and it is also hermitian because 1 1 1 2i dα(W, V ) = − 2i dα(V,W ) = 2i dα(V, W ). 3 2 1 c In the case of S ⊂ C , we have α = 2i d ρ and 1 1 dα = − ddcρ 2i 4 1 1 = dz ∧ dz + dz ∧ dz 2 1 1 2 2 2 Definition Let T be a transversal vectorfield on X, the form α the projection onto T and ω be the induced hermitian sesquilinearform. We say that T is strongly pseudoconvex if ω is positive definite. We call X strongly CR-pseudoconvex if there exists a strongly pseudoconvex vectorfield T on X.

Note that what we call strong CR-pseudoconvexity is usually just called strong pseudoconvexity.

Remark For V,W ∈ C∞(X,T 1,0X), we get

dα(V, W ) = V α(W ) − W α(V ) − α([V, W ]) = −α([V, W ]).

This shows that for V,W ∈ T 1,0X, the expression α([V, W ]) is well-deﬁned since it does not depend on the extension of V,W to sections. An equivalent notion of pseudoconvexity would be to consider β = −α instead of α and deﬁne strong pseudoconvexity to be the positivity of the 1 form 2i β([V, W ]). The most prominent result in the theory of CR embeddings is arguably the following theorem by Boutet de Monvel [dM75] .

Theorem [Boutet de Monvel] Let X be a compact CR manifold of dimension (2n, 1), with 2n + 1 ≥ 5, which is strongly CR-pseudoconvex. Then m there exists a CR embedding X → C . We will give a more general deﬁnition of strong CR-pseudoconvexity and translate it to complex geometry.

Deﬁnition Let X be a CR manifold of dimension (2n, d) with an action of a Lie group G. We say that the action of G on X is almost transversal if dim g(x) = d − 1 for all x ∈ X and

1,0 0,1 g(x) ∩ (Tx X ⊕ Tx X) = {0}.

31 We will shortly see that CR manifolds with almost transversal group actions appear naturally as invariant hypersurfaces in CR manifolds with transversal group action (see example 4).

Deﬁnition Let X be a CR manifold with an almost transversal action of a Lie group G. A vectorﬁeld T is called G-transversal if

1,0 0,1 CT (x) ⊕ Cg(x) ⊕ Tx X ⊕ Tx X = CTxX.

Let X be a CR manifold with almost transversal, locally free action of a Lie group G. Choose again a riemannian metric on X and define the line 1,0 0,1 bundle L to be the orthogonal complement to g(x)⊕TxX ∩(Tx X ⊕Tx X). S Because the bundle x g(x) has a trivialisation, we may repeat the argument from above and always find G-transversal vectorfields if X is orientable.

Let T be a G-transversal vectorfield. We again define α by α(T ) = 1 1,0 0,1 and α(Ck ⊕ T X ⊕ T X) = 0 to be the projection onto T , which extends the definition of strong CR-pseudoconvexity to CR manifolds with almost transversal, locally free Lie group actions.

Now assume that T is a strongly pseudoconvex vectorﬁeld on X with α being the projection on T . Let S be another G-transversal vectorﬁeld. We get that α(S) is pointwise non-vanishing, therefore β := α(S)−1 · α is the projection onto S. But then 1 1 1 1 dβ(V,W ) = − β([V,W ]) = − α(S)−1α([V,W ]) = α(S)−1 · dα(V,W ). 2i 2i 2i 2i We have proved the following lemma.

Lemma 2.1. Let X be a CR manifold with almost transversal G-action. Then X is strongly CR-pseudoconvex if and only if there exists a G-transversal vectorﬁeld and for every G-transversal vectorﬁeld T and every connected component X0 of X, either T or −T is strongly pseudoconvex on X0.

Definition We say that a vectorfield T on a CR manifold X is a CR vectorfield if the flow of T acts by CR automorphisms.

We have already seen that S3 is a strongly pseudoconvex CR manifold with transversal CR vectorﬁeld. In general, however, CR manifolds do not need to admit CR vectorﬁelds. We will discuss this at a later point (see Theorem 2.7). Let us consider some examples.

3 2 Example1 We generalise the example of S ⊂ C . n Let D ⊂ C be a relatively compact, strongly pseudoconvex domain with smooth boundary X = ∂D. Then X is an orientable CR manifold, hence we

32 find a non-vanishing transversal vectorfield T with α being the projection onto T . Around every point x ∈ X, we find a neighbourhood U of x and a strictly plurisubharmonic function ρ: U → R with ∂D ∩ U = {ρ = 0} and D ∩ U = {ρ < 0}. n c If J is the complex structure on C , we get d ρ(TX ∩ JTX) = 0 and since ρ is real-valued, we conclude dcρ = −dcρ. 1 c 1 Then β := 2i d ρ is a one-form on X such that V,W 7→ 2i dβ(V, W ) defines a positive definite hermitian form on T 1,0X. We have β(T ) 6= 0 pointwise, hence α = β(T )−1β and the calculation before Lemma 2.1 shows 1 that 2i dα is definite in every point of X. 1 We conclude that 2i dα is positive or negative definite on every connected 1 component of X, hence 2i dα is positive definite everywhere after possibly changing the sign of T on connected components of X.

n Example2 In example 1, we may replace C with any complex manifold and the statement stays the same. In particular, if L → Z is a weakly negative line bundle (see Section 4.2) and X is the S1-bundle of L, then X is a CR manifold with transversal CR S1-action, which induces a strongly pseudoconvex vectorﬁeld T .

Example3 We study an example for non-embeddable CR manifolds. Because the involutivity condition is always fulfilled if T 1,0X is one- 3 3 dimensional, we have that every section W : S → CTS , such that [W, W ] 3 is point-wise not contained in spanC(W, W ), gives S the structure of a 3 1,0 3 strongly pseudoconvex CR manifold (S ,TW S ). 3 Indeed, define T (x) = i[W, W ](x) ∈ TxS , with α the projection onto T . 1 1 1 Then 2i dα(W, W ) = − 2i α([W, W ]) = 2 . Fix such a section W , then for a sufficiently small, smooth function 3 Φ Φ: S → R, the section W = W + ΦW defines a strongly pseudoconvex CR structure as well. It can now be shown that non-embeddability of 3 1,0 3 3 1,0 3 (S ,TW Φ S ) is generic in the sense that the functions Φ for which (S ,TW Φ S ) n ∞ 3 is not embeddable into C are dense in C (S , R) [BE90, Theorem 4.57].

Example4 Let G be a Lie group and M a CR manifold of dimension (2n, d) with transversal CR action of G. Let X be a G-invariant hypersurface of M, then X is a CR manifold of dimension (2(n − 1), d + 1) with almost 1,0 0,1 transversal G-action. To see this, deﬁne Wx = (Tx M ⊕ Tx M) ∩ TxM, then

dimWx + dimTxX = dim(TxX + Wx) + dim(TxX ∩ Wx).

Since X is G-invariant, TxX does contain g(x) and TxX + Wx = TxM. One then sees that dim(TxX ∩ Wx) = 2n − 1.

33 Now every Wx has a complex structure, which we denote by Jx (deﬁned analogously to section 1.1, Example 1). Deﬁne Vx := TxX ∩ Wx, then

dim(Vx ∩ JxVx) = dim Vx + dim JxVx − dim(Vx + JxVx), hence dim(Vx ∩ JxVx) = 2(n − 1).

In [OV07], it was shown that a compact CR manifold of CR codimension 1 with strongly pseudoconvex CR vectorﬁeld admits a transversal CR S1-action and can be realised as the S1-bundle in a positive holomorphic line bundle. We will generalise the result to higher codimension, using comparable methods.

Let X be a CR manifold with almost transversal action of a compact Lie group K. Let T be a K-transversal vectorﬁeld and α the projection onto T . 1,0 0,1 The K-action leaves the bundle Ck(x) ⊕ Tx X ⊕ Tx X invariant, hence for k ∈ K, we get α(dxk(T (x))) 6= 0. We say that K preserves the orientation of T if α(dxk(T (x))) > 0 for all k ∈ K and x ∈ X. Note that if K is connected, this is always the case.

Lemma 2.2. Let X be a CR manifold with almost transversal K-action. Let T be a K-transversal CR vectorﬁeld such that K preserves the orientation of T . Then there exists a K-transversal CR vectorﬁeld T˜ which is K-invariant. If T is strongly pseudoconvex, then so is T˜.

Proof. We average over K: Z T˜(x) := dk(T (k−1x))dk. K Let α be the projection onto T , then we get Z α(T˜) = α(dkT (k−1x))dk. K

Because K preserves the orientation of T , we conclude α(T˜) > 0. In particular, we get that T˜ is K-transversal. ˜ T Let us check that T is a CR vectorfield. If Φt denotes the flow of T , then the flow of T˜ is given by Z T˜ T −1 Φt (x) = k(Φt (k x))dk. K

1,0 If V ∈ Tx X, then Z C T˜ C C T C −1 dx Φt (V ) = d k(d Φt (d k (V )))dk. K

34 But since the K-action is CR and T is a CR vectorﬁeld, we see that T˜ is CR. For strong pseudoconvexity, we use Lemma 2.1 and the discussion pre- ceding it.

Theorem 2.3. Let X be a compact CR manifold with almost transversal, locally free K-action. Let T be a strongly pseudoconvex K-invariant CR vectorfield. Then there exists a strongly pseudoconvex K-invariant CR vectorfield T˜ such that the flow of T˜ defines an S1-action.

Proof. Let α be the projection onto T . We will deﬁne a riemannian metric d on X. The map k → k(x), ξ 7→ dt 0exp(tξ)x is an isomorphism for every x. Therefore, we may ﬁx an inner product h on k and get an inner product on every k(x).

A vector V ∈ T X can be decomposed as V = V + V + V + V , where x T k C C V ∈ T 1,0X, V ∈ T 0,1X, V ∈ T (x) and V ∈ k(x). Deﬁne a riemannian C x C x T R k metric via 1 1 g(V,U) = α(V )α(U ) + h(V ,U ) + dα(U , V ) − dα(U , V ) T T k k 2i C C 2i C C

1 1,0 Since 2i dα(UC, V C) is positive deﬁnite on T X, we conclude that 1 dα(U , V ) = − 1 dα(V ,U ) is negative deﬁnite on T 0,1X. 2i C C 2i C C

T Denote by Φt the flow of T . Since T is K-invariant and CR, the flow T 1,0 0,1 Φt commutes with the K-action and leaves the bundles T X and T X T invariant. This implies that α is invariant under Φt and that the flow acts by isometries with respect to g.

The isometry group Iso(X) of (X, g) acts proper on X, i.e. the map Iso(X)×X → X ×X,(ϕ, x) 7→ (ϕ(x), x) is proper. But since X is compact, we conclude that Iso(X) is compact. Consider the closed subgroup U of Iso(X) given by the CR isometries which commute with the K-action. T Now Φt deﬁnes a subgroup G of U. If G is closed, the result follows immediately. Assume that G is not closed, then its closure is a torus U0 in U.

Consider the smooth map

f : u0 × X → R d (ξ, x) 7→ α( exp(tξ)x). dt 0

If ξ0 deﬁnes the vectorﬁeld T , then we get fξ0 (x) = 1. Since X is compact, we obtain f > 0 on Ω × X, where Ω is a neighbourhood of ξ0.

35 −1 Now consider the one-form αξ := fξ · α for ξ ∈ Ω. We see αξ0 = α and 1 1,0 2i dαξ0 defines a positive form on T X. Using again that X is compact, we 1 conclude that 2i dαξ defines a positive form for all ξ in some neighbourhood of ξ0. The result now follows because the vectorfields defining S1-actions are dense in U0. As a conclusion of this section, we have proved the following

Corollary 2.4. Let K be a compact, connected Lie group. Let X be a compact, strongly CR-pseudoconvex CR manifold with almost transversal, locally free K-action. Assume that there exists a K-transversal CR vectorﬁeld on X. Then there exists a CR S1-action on X which commutes with the K- action and the vectorﬁeld induced by this action is strongly pseudoconvex.

2.2 Pseudoconvexity in complex geometry In this short section, we will deﬁne what we understand to be a strictly plurisubharmonic function on a complex space and proof some properties we will be using later.

Definition Let Z be a complex space and ρ: Z → R a continuous function. We call ρ plurisubharmonic if for every holomorphic map f : ∆ → Z from the unit disk ∆ into Z, the composition ρ ◦ f is subharmonic on ∆. We say that ρ is strictly plurisubharmonic, if around every point in Z, n n we find a local model A ⊂ C for Z and a neighbourhood U of A in C such that for every sufficiently small C2-function h on U, the function h + ρ is still plurisubharmonic on U ∩ A.

In [FN80], it is shown that every plurisubharmonic function is the restriction of a plurisubharmonic function, defined on an open neighbourhood n of a local model A ⊂ C . n If ϕ is a strictly plurisubharmonic function and x ∈ A ⊂ C a local model, we may define ρ to be the squared distance to x, which is a strictly plurisubharmonic function. For a small t ∈ R, ϕ − tρ is plurisubharmonic on A, hence it extends to a plurisubharmonic function Φ on an open neighbourhood. But then Φ + tρ is strictly plurisubharmonic and its restriction to A is ϕ. We conclude that every strictly plurisubharmonic function is the restriction of a strictly plurisubharmonic function, defined on an open neighbourhood of a local model.

Lemma 2.5. Let Z be a complex manifold, H a ﬁnite group acting on Z. Let ρ: Z → R be a smooth, H-invariant strictly plurisubharmonic function. Then the induced map ρ: Z/H → R is strictly plurisubharmonic.

36 Proof. From the smooth Slice theorem, we conclude that there exists an open and dense subset W of Z/H such that the quotient map π : π−1(W ) → W is locally biholomorphic. Therefore, ρ is plurisubharmonic on W and using [GR56, Satz 3], we get that ρ is plurisubharmonic on Z/H. m Now let A ⊂ C be a local model for Z/H. Take some open neigh- m bourhood U of A in C such that U ∩ A is relatively compact. Then −1 π (U) is relatively compact in Z, we therefore get that ||π|π−1(U)||2 is 2 bounded, where ||g||2 = supx||g(x)|| + supx||dg(x)|| + supx||d g(x)||. Given a twice continuously diﬀerentiable function f : U → R, we obtain that ||(f ◦ π)|π−1(U)||2 ≤ c||f|U ||2 for some constant c independent of f.

This shows that for suﬃciently small functions f on U, the function ρ + (f ◦ π) is still plurisubharmonic, hence ρ + f is plurisubharmonic, using the same argument as before.

Note that the statement of Lemma 2.5 is not an equivalence. To see ∗ this, consider the Z2-action on C as the subgroup {−1, 1} < C . Then the 2 quotient C/Z2 is isomorphic to C with quotient map given by z 7→ z . 4 The map ρ(z) := |z| is not strictly plurisubharmonic on C, since the diﬀerential in 0 vanishes, but the induced map ρ on the quotient is given by |z|2, which is strictly plurisubharmonic.

Deﬁnition Let Z be a complex space and U ⊂ Z an open subset. We say that U is strongly pseudoconvex if for every point x ∈ ∂U there exists a neighbourhood Ω of x and a strictly plurisubharmonic function ϕ:Ω → R such that U ∩ Ω = {ϕ < 0} and ∂U ∩ Ω = {ϕ = 0}.

Now let X be a compact CR manifold with transversal, locally free CR action of S1 × K (compare Theorem 2.3). Denote H = S1 × K and consider Y := HC ×H X, which is a complex manifold (Theorem 1.5). From Proposition 1.10, we conclude the existence of holomorphic slices on HC ×H X. 1 Now consider the CR manifold (C × KC) × X with S × K-action given by ((s, k), (z, h, x)) 7→ (zs−1, hk−1, skx). This action is CR, transversal and locally free, therefore Z := ((C × KC) × X)/H is a complex space (Theorem S1 1.11). With the same argument, Z/KC = C × X/K is a complex space, as well. The natural map Y → Z is an open holomorphic injective immersion.

The following theorem will be the main motivation for our further study. It establishes a link between CR geometry and complex geometry.

Theorem 2.6. Let X be a compact CR manifold with transversal, locally free, CR S1 × K-action such that the vectorﬁeld T induced by the S1-action

37 is strongly pseudoconvex. Then the set

S1 2 W := {[z, x] ∈ C × X/K = Z/KC | |z| < 1} is strongly pseudoconvex in Z/KC.

2 Proof. Deﬁne the map h: Z/KC → R,[z, x] 7→ |z| , then W = {h < 1}. Let U be the preimage of W under Z → Z/KC. We may pull back h to a function on Z, which we will still denote by h, then U = {h < 1}. Embed X as a CR submanifold into Y via x 7→ [1, 1, x] and extend T to ∗ a C × KC-invariant vectorﬁeld on Y . ∗ S1×K Now ∂U = {[1, g, x] ∈ (C × KC) × X} = KCX is a hypersurface, hence CR submanifold of Y with transversal S1-action. Let α be the projection onto T on ∂U.

Take y0 ∈ ∂W and let π : Z → Z/KC be the quotient map. We then ﬁnd an x0 ∈ X such that π(x0) = y0. Let S be a holomorphic slice through L x0 for the HC-action and Ω = HC × S, where L is the ﬁnite isotropy of C H in x0. We may assume that S is biholomorphic to a ball and CTx0 S = 1,0 0,1 Tx0 X ⊕ Tx0 X. ∗ L Let [w, g, y] be the coordinates of (C ×KC)× S and take some smooth function ϕ on Ω such that h[w, g, y] · eϕ[w,g,y] = |w|2. By construction, ϕ is HC-invariant and Ω ∩ ∂U is given by Φ[w, g, y] := |w|2 − eϕ[w,g,y] = 0.

∂ ∗ Let ∂w be the complex vector ﬁeld induced be the C -component in Ω, let ( ∂ ) be a basis for T 1,0S and extend this to a basis ( ∂ , ∂ ) ∂zj j=1,...,m ∂w ∂zj j=1,...,n for T 1,0Ω. This is always possible after shrinking Ω. We want to compute the CR structure of ∂U ∩ Ω in terms of the above basis and are therefore considering the equation X ∂ ∂ X ∂ 0 = dC Φ( a + b ) = − a eϕ ϕ + bw. j ∂z ∂w j ∂z j j j j

∂ w ϕ ∂ ∂ This implies that Z = + 2 e ϕ gives a basis for the CR structure. j ∂zj |w| ∂zj ∂w ∂ Rewriting ∂w in Polar coordinates (r, θ) gives ∂ 1 w ∂ 1 w ∂ = − i . ∂w 2 |w| ∂r 2 |w|2 ∂θ

∂ 1 ∂ ∂ 1 ∂ ∂ eϕ We get Z = + ϕ − i ϕ , using that 2 = 1 on the j ∂zj 2|w| ∂zj ∂r 2 ∂zj ∂θ |w| boundary of U. ∂ On ∂U, the projection α is given by α( ∂θ ) = α(T ) = 1 and α(Zj) = α(Zj) = 0, therefore X 1 ∂ X 1 ∂ α = dθ + i ϕdz − i ϕdz . 2 ∂z j 2 ∂z j j j j j

38 1,0 0,1 1,0 Now CTx0 S = Tx0 X ⊕ Tx0 X, hence for v, w ∈ Tx0 X, we have the equation 1 X 1 ∂2 dα(v, w) = − ϕ(v, w), 2i 2 ∂z ∂z ij j i which implies that −ϕ is strictly plurisubharmonic in a neighbourhood of x0 in S. After shrinking S, we may assume that −ϕ is strictly plurisubharmonic on S. ∗ L We restrict ϕ to an L-invariant function on S, write Ω/KC = C × S and

∗ L 2 ϕ(y) W ∩ (Ω/KC) = {[w, y] ∈ C × S | |w| < e }.

∗ ∗ L Deﬁne the map πL : C × S → C × S, then we get −1 2 ϕ(y) πL (W ) = {(w, y) | |w| < e } and compute |w|2 |w|2 < eϕ(y) ⇔ < 1 ⇔ log(|w|2) − ϕ(y) < 0. eϕ(y) Because S is biholomorphic to a ball, −ϕ is strictly plurisubharmonic and −1 −1 πL (W ) is a Hartogs domain over S, we conclude that πL (W ) is strongly pseudoconvex. Using Lemma 2.5 then gives the result.

Remark If we assume that the S1-action on X is free, the above theorem ∗ actually proofs that the line bundle Z/KC → Y/(C ×KC) is weakly negative (see Section 4.2).

2.3 Boundaries of pseudoconvex domains To conclude this section, we want to consider strongly pseudoconvex do- n mains in C and see that the existence of a CR vectorﬁeld on the boundary implies that the domain has some symmetry.

n Theorem 2.7. Let D ⊂ C be a relatively compact domain with smooth, connected boundary X := ∂D. Assume that there exists a non-vanishing CR vectorﬁeld on X. Then Aut (D) is not trivial.

Proof. Let T be a non-vanishing CR vectorfield on X with flow ϕt. We will show that, for sufficiently small t, we may extend ϕt to an automorphism of the domain D. From [Sal10, Theorem 1.4.1], we conclude that we may extend ϕt to a n n continuous map Φt : D → C such that Φt : D → C is holomorphic. It is not obvious, however, that Φt(D) = D. 0 n We may define a norm on C (D, C ) via ||f|| := supz∈D|f(z)|. Because of the maximum principle, we conclude that for fixed t0, t1, the map Φt0 − Φt1 attains its maximum on the boundary X.

39 This implies ||Φt0 − Φt1 || = supz∈∂D|ϕt0 − ϕt1 | and the map t 7→ Φt is continuous with respect to the induced topology. Using the maximum principle on Φ0 − Id, we conclude that Φ0 is the identity. Since D is compact, we conclude that there exists an R > 0 such that for all t with |t| < R, the map Φt is injective on D. The map Φt = ϕt : X → X is bijective for every t, hence Φt(D) ⊂ D for all t with |t| < R. But now Φt ◦ Φ−t is well-deﬁned for all |t| < R with Φt ◦ Φ−t − Id = 0 on X. Using the maximum principle, we conclude Φt ◦ Φ−t = Id on D. A result from Greene and Krantz [GK82, Theorem 1.10] shows that n for strongly pseudoconvex domains of C , n ≥ 2, with smooth boundary, having a trivial automorphism group is a generic property. One should think of this result as symmetry being easy to destroy by perturbations, but hard to generate. We see that in this situation, the non-existence of CR vectorﬁelds is generic.

40 3 Equivariant Embeddings

3.1 Semidirect Products Let H be a subgroup of its universal complexiﬁcation HC. Assume that there exist closed subgroups G and S of H, where G is normal and H is the semi-direct product H = G o S. We have a map GC → HC and denote by GC its image in HC.

Lemma 3.1. The group GC is normal in HC.

C C C Proof. Let NHC (G ) be the normaliser of G in H . We want to show C C H ⊂ NHC (G ). Note that the Lie-Algebra of G is the complexiﬁcation of g. Now we see that the adjoint action of H on the Lie-Algebra hC of HC leaves g and therefore gC invariant. We conclude that (GC )0 is invariant under the adjoint action of H. But C S C 0 C G = g∈G g(G ) , which shows that G is invariant under the adjoint action of H on HC. C C Now NHC (G ) is a closed, complex subgroup of H which contains H, hence it is equal to HC.

1 C ∗ Theorem 3.2. Assume that S = S . Then we get HC = G o C . Proof. We have shown that GC is normal in HC, hence HC/GC is a complex ∗ C Lie group. Consider the Lie group morphism p: C → HC/G . From the universality of HC and GC, we conclude that H/G → HC/GC is the universal complexiﬁcation of H/G. Since H/G is isomorphic to S1, we ∗ ∗ 1 conclude that p is a group morphism C → C which is injective on S , hence it is an isomorphism.

1 ∼ 1 Assume now H = G o S . For h ∈ H, we denote by hS ∈ H/G = S the image of h in the quotient. In the same manner, for h ∈ HC, denote C ∼ ∗ by hC ∈ HC/G = C the image of h in the quotient. Note that the maps h 7→ hS and h 7→ hC are group morphisms.

3.2 Sheaf-Theory Instead of considering compact CR manifolds with K×S1-action, we want to take a more general approach. From now on, we will consider the following situation. Let H be a subgroup of its universal complexiﬁcation HC, G a closed 1 and normal subgroup of H and assume H = G o S . We have seen that C ∗ HC = G o C . Let X be a CR manifold with transversal, proper CR action of H and 0 assume that Hx < Gx for every point x ∈ X. To explain the last property, we proof the following lemma.

41 Lemma 3.3. Let T be the vectorﬁeld induced by the S1-action. Then 1,0 0,1 0 CT (x) ⊕ Cg(x) ⊕ Tx X ⊕ Tx X = CTxX if and only if Hx < Gx for every x ∈ X.

0 1 Proof. Assume that Hx < Gx. This implies that the S -action is locally free and T does not vanish. Now assume that T (x) ∈ g(x), then there ex- 1 ists ξ ∈ g such that ξX (x) = −T (x). If T is generated by η ∈ T1S , then 0 ξ +η ∈ hx deﬁnes a one-dimensional subgroup of Hx, which is not contained in Gx, which is a contradiction.

1,0 0,1 Now assume that CT (x)⊕Cg(x)⊕Tx X ⊕Tx X = CTxX. If we assume 0 1 that Hx is not a subgroup of Gx for some x ∈ X, then we ﬁnd η ∈ T1S , η 6= 0 and ξ ∈ g such that (η + ξ)X (x) = 0. But then ηX (x) = −ξX (x) and since T has to be non-vanishing, ηX (x) and −ξX (x) are non-vanishing. But then T (x) ∈ g(x), which is a contradiction.

0 0 Note that the condition Hx < Gx is always satisﬁed if the H-action is locally free.

∗ We have an H-action on the CR manifold C × X given by

∗ ∗ H × (C × X) → C × X −1 (h, (z, x)) 7→ (zhS , hx). This action is proper and CR, we will show that it is transversal. ∗ ∗ Take ξ ∈ T1C = C, µ ∈ g and (z, x) ∈ C × X. 1,0 ∗ 0,1 ∗ Then (ξ+µ)C∗×X (z, x) ∈ T (C ×X)⊕T (C ×X) implies (ξ+µ)X (x) = 0. 0 0 But because Hx ⊂ Gx, we conclude ξ = 0 and (ξ + µ)C∗×X (z, x) = 0. Counting dimensions, we see that the action is transversal. Theorem 1.11 shows that

∗ ∼ ∗ S1 Y˜ := (C × X)/H = C × (X/G) is a complex space. Now we deﬁne

∼ S1 Z := (C × X)/H = C × (X/G), where the H-action on C × X is deﬁned as above. In the same manner as before, we see that Z is a complex space and the map Y˜ → Z is holomorphic.

We may embed X into its universal equivariant extension Y as in Theo- C −1 rem 1.9. We have an H -action on C × Y given by (h, (w, z)) 7→ (whC , hz) and show that the natural map Z = (C × X)/H → (C × Y )/HC is biholomorphic.

42 It is suﬃcient to consider a slice neighbourhood Ω = H ×Hx S of X and show that the map ϕ:(C×Ω)/H → (C×ΩC)/HC is biholomorphic. But the −1 HC H inverse of ϕ is given by ϕ :(C × (HC × x S))/HC → (C × (H × x S))/H with [z, [h, s]] 7→ [zhC , [1, s]]. ∗ C We see that Z = (C×Y )/HC = C×C Y/G and with the same argument ∗ C Y˜ = (C × Y )/HC = Y/G .

Deﬁnition We say that X is strongly pseudoconvex if the set

S1 Z∆ := {[w, z] ∈ C × X/G | |w| < 1} is strongly pseudoconvex in Z.

This definition is of course motivated by Theorem 2.6. Let us consider some standard examples for strongly pseudoconvex CR manifolds. Let K be a compact, connected Lie group and X a compact, strongly CR pseudoconvex CR manifold with almost transversal, locally free K-action. Assume there exists a K-transversal CR vectorfield on X. Then from Corol- lary 2.4 and Theorem 2.6, we see that X admits a S1 × K-action such that X is strongly pseudoconvex in the above sense. In particular, if K = {Id} and X is a compact, strongly CR pseudoconvex CR manifold with transversal, CR R-action, then X fulfils the above requirements. This special case was, however, already known [OV07]. If X is CR strongly pseudoconvex, has a transversal, CR S1-action and T is the vectorfield induced by this action, then on every connected component of X, either T or −T is strongly pseudoconvex. Since s 7→ s−1 is an automorphism of S1, we may change the S1-action and get that T is strongly pseudoconvex on every connected component of X. Theorem 2.6 then states that X is strongly pseudoconvex in the above sense.

Let us now get back to the general case. The following construction, including Corollary 3.6, is an idea by Grauert, see [Gra62, §3.2]. We will reformulate it here, adjusted to our purposes.

Deﬁnition Let M be a set with S1-action, U ⊂ M a subset and V a vectorspace. We say that a map f : U → V is of order d ∈ Z if for all s ∈ S1 and m ∈ U such that sm ∈ U, we have f(sm) = sdf(m).

Let X be a CR manifold with transversal, proper CR action of H and 0 assume Hx < Gx for every x ∈ X. Let Y be the universal equivariant ∗ C extension of X and Z = C ×C Y/G . C H Consider the complex space X/H = Y/H with structure sheaf OY . Given a coherent analytic sheaf GY over Y/HC, we deﬁne a sheaf GZ over

43 Z as follows. The holomorphic map p: Z → Y/HC gives the structure sheaf H OZ of Z the structure of an OY -module. Deﬁne

Z Y G (U) = G (p(U)) ⊗ H OZ (U). OY (p(U))

Now GZ is coherent and analytic on Z because GY is coherent and analytic on Y/HC.

The map Y/HC → Z, y 7→ [0, y] is an embedding of Y/HC as the ∗ Y analytic set of C -fixed points. Define the coherent sheaf OZ of holomorphic functions on Z which vanish on Y/HC. We have the (non-analytic) subsheaf Y C Y OZ,d of functions vanishing on Y/H which are of order d. The sheaves OZ Y C and OZ,d define analytic sheaves on Y/H via

Y Y −1 OY (U) := OZ (p (U)) and

Y Y −1 OY,d(U) := OZ,d(p (U)).

Y Let G be a coherent analytic sheaf over Y/HC. Fix some d0 ∈ N and deﬁne the map

M q C Y Y q Z A≤d0 : H (Y/H , G ⊗ OY,d) → H (Z, G ), d≤d0 where Hq denotes the q-th Cechˇ cohomology, as follows. For every open subset U of Y/HC and every d, we have the natural map

Y Y Y −1 Z −1 Ad,U : G (U) ⊗ OY,d(U) → G (U) ⊗ OZ (p (U)) = G (p (U)).

C q Y Y Given coverings Ud of Y/H and cochains fd ∈ C (Ud, G ⊗OY,d) for d ≤ d0, we may reﬁne the Ud and assume that Ud = V for some common reﬁnement.

For some q-simplex σ = (Vi1 , ..., Viq ) and |σ| := Vi1 ∩ ... ∩ Viq , we then deﬁne X X A≤d0 ( fd)(σ) = Ad,|σ| fd(σ). d d This then extends to a well-deﬁned map on the cohomology groups.

We now want to deﬁne a map

q q C Y Y Bd : H (Z, GZ ) → H (Y/H , G ⊗ OY,d).

Let U ⊂ Z be open with U ∩ Y/HC 6= ∅ and g : U → C a holomorphic function. Remember that Z = (C × Y )/HC.

44 Let π : C × Y → (C × Y )/HC be the quotient map, then g := g ◦ π gives rise to a HC-invariant holomorphic function on π−1(U). Consider the set −1 U0 = {(0, w) ∈ π (U)}, take (0, w0) ∈ U0 and develop g into a power series P d g = d z fd,w0 (w) in some connected neighbourhood of (0, w0) of the form

Bw0 × Ωw0 .

If (0, w1) ∈ U0 is another point such that Bw1 × Ωw1 ∩ Bw0 × Ωw0 6= ∅, we compare the power series and see fd,w0 = fd,w1 on the intersection. We may therefore assume that f is defined on Ω := S B × Ω and set d (0,w)∈U0 w w d gd(z, w) := z fd(w) on Ω. C Now the gd are actually H -invariant, which we will see as follows. Set C g˜d(hx) := gd(x) for h ∈ H , x ∈ Ω, we need to show that this is well-defined. To see this, fix some h ∈ HC and observe that every connected component of hΩ intersects U0. For (0, w0) ∈ U0, we consider the power series expansions of g and g ◦ h−1, which have to coincide because g is HC-invariant. For ∗ c ∈ C , we compute −1 −1 d −1 gd(h (cz, w)) = gd(czhC , h w) = c gd(zhC , h w).

−1 Comparing coefficients in the power series expansions then gives gd = gd◦h on Ω ∩ hΩ andg ˜d is well-defined. We may therefore assume that the sets Bw ×Ωw are HC-invariant. Since the g are of order d, they may be extended to functions on S ×Ω . d (0,w)∈U0 C w −1 C Now the gd define holomorphic functions gd on p (p(U∩Y/H )) of degree d.

For every open subset U ⊂ Z with U ∩ Y/HC 6= ∅, we have a map

Z Y C Y C Bd,U : G (U) → G (p(U ∩ Y/H )) ⊗ OY,d(p(U ∩ Y/H )) deﬁned by

X X p(U) B ( ξ ⊗ g ) := res (ξ ) ⊗ g , d,U k k p(U∩Y/HC) k k,d k k where gk,d is the map we constructed above and res denotes the restriction map. If V is an open subset of U with V ∩ Y/HC 6= ∅, then V ∩ Y/HC is a subset of U ∩ Y/HC and from the above construction, we see

B (resU (g)) = resp(U∩Y/HC)B (g). (4) d,V V p(V ∩Y/HC) d,U

Note that by construction, we also have Bd,p−1(U) ◦ Ad,U = Id for every open set U ⊂ Y/HC.

Now we want to deﬁne Bd on the cohomology groups. Given a covering V of Z, we may deﬁne a new covering VY of Y/HC Y C C q Z via Vi := p(Vi ∩ Y/H ), if Vi ∩ Y/H 6= ∅. For a cochain g ∈ C (V, G )

45 and a q-simplex σY for VY , with corresponding simplex σ for V, we deﬁne Y Bd(g)(σ ) := Bd,|σ|(g(σ)). q−1 Z Now if G ∈ C (V, G ) is a cochain, σ is a q-simplex for V and ∂jσ is the corresponding (q − 1)-simplex by omitting the j-th set, then we have

X j |∂j σ| Bd,|σ|(δG(σ)) = Bd,|σ| (−1) res|σ| (G(∂jσ)) j X p(|∂ σ|∩Y/HC) = (−1)jres j (B (G(∂ σ))). p(|σ|∩Y/HC) d,|∂j σ| j j

From this computation and equation (4), we conclude that the map

q Z q C Y Y Bd : H (Z, G ) → H (Y/H , G ⊗ OY,d), is well-deﬁned, since it maps cocycles to cocycles and coboundaries to coboundaries.

Lemma 3.4. For every d0, the map A≤d0 is injective. Proof. We have the map B : Hq(Z, GZ ) → L Hq(Y/HC, GY ⊗ OY ), ≤d0 d≤d0 Y,d which is just the sum of the Bd and by construction, we get B≤d0 ◦A≤d0 = Id.

We conclude that A≤d0 is injective. The following Proposition was proved by Grauert [Gra58, Proposition 4] for the structure sheaf on complex manifolds. The general version is mostly analogous, we will repeat the proof given there and adjust it if necessary.

Proposition 3.5. Let Z be a complex space and Ω ⊂ Z a strongly pseudoconvex, relatively compact set. Let G be a coherent analytic sheaf on Z. Then the complex vectorspaces Hq(Ω, G) are ﬁnite-dimensional for q > 0.

Proof. Let x ∈ ∂Ω, then we find an open cover Ui of Ω such that x is ˆ S ˜ contained in the open set U := U0\ i6=0 Ui. Let U be a neighbourhood of x which is relatively compact in Uˆ. We may choose the Ui to be Stein spaces and assume that, if Ui ∩∂Ω 6= ∅, then Ui ∩ Ω is given by ϕi < 0 for some strictly plurisubharmonic function on Ui. Using [FN80, Theorem 5.4], we conclude that Wi := Ui ∩ Ω is a Stein space for all i. One may also choose U0 small enough such that it may be embedded into n an open subset A of C . Let ϕ0 be the strictly plurisubharmonic function on U0 such that U0 ∩ Ω = {ϕ0 < 0}. Take a twice differentiable function f S ˜ on A such that f = 0 on U0 ∩ i6=0 Ui and f > 0 on U. We may choose f sufficiently small such that ϕ0 + f is still strictly plurisubharmonic on U0. We conclude that we find a strongly pseudoconvex, relatively compact set Ω1 ⊂ Z with Ω ⊂ Ω1 such that Ω1 ∩ Ui = Ω ∩ Ui for i 6= 0 and x ∈ Ω1. Furthermore, Ui ∩ Ω1 is a Stein space for every i.

46 1 1 1 Define Wi := Ui ∩ Ω, Wi := Ui ∩ Ω1 and let W = (Wi), W = (Wi ) 1 1 be the covers, then Wi ∩ Wj = Wi ∩ Wj for i 6= j. In particular, this q q 1 q implies Z (Wi, G) = Z (Wi , G) for q > 0, where Z denotes the cocycles. Set W := W ∩ ... ∩ W and W 1 := W 1 ∩ ... ∩ W 1 , i1,...,ip i1 ip i1,....,ip i1 ip then Hq(W , G) = Hq(W 1 , G) = 0 for q > 0 because W and i1,...,ip i1,...,ip i1,...,ip W 1 are Stein and G is a coherent analytic sheaf. We may apply the i1,...,ip q q q 1 q Leray theorem and get H (Wi, G) = H (Ω, G), H (Wi , G) = H (Ω1, G) and q q the restriction map r : H (Ω1, G) → H (Ω, G) is surjective. k Since Ω is relatively compact, we find finitely many covers Ui such that the U˜ k cover ∂Ω. We may repeat the above argument finitely many times to find a relatively compact, strongly pseudoconvex set Ω0 such that Ω is relatively compact in Ω0 and the restriction map r : Hq(Ω0, G) → Hq(Ω, G) is surjective for q > 0.

0 0 Now take two coverings Ui and Ui of Ω such that Ui is relatively compact 0 0 0 in Ui . We may choose the Ui and Ui small enough such that Ui ∩ Ω, Ui ∩ Ω , 0 0 0 Ui ∩ Ω and Ui ∩ Ω are Stein spaces and we have exact sheaf sequences

mi fi ni O | 0 −→O | 0 → G| 0 → 0. Z Ui Z Ui Ui 0 0 0 Deﬁne Wi := Ui ∩ Ω, Wi := Ui ∩ Ω and denote by W = (Wi) and 0 0 W = (Wi ) the corresponding coverings. We deﬁne the linear maps

µ, η : Zq(W0, G) ⊕ Cq−1(W, G) → Zq(W, G) µ(z ⊕ c) = r(z) + δ(c) η(z ⊕ c) = −r(z), where r is the restriction and δ is the coboundary operator. The Leray theorem again implies Hq(W0, G) = Hq(Ω0, G) and Hq(W, G) = Hq(Ω, G), hence µ is surjective, as we have seen before.

The Wi1,...,ip and Wi1,...,ip may be exhausted by countably many compact sets, which gives the space of holomorphic functions on Wi1,...,ip and

Wi1,...,ip the structure of Fr´echet spaces such that the induced topology is the topology of compact convergence. 0 Now since Wi is relatively compact in Wi , it follows from the theorem n 0 n of Montel that the restriction map r : OZ (Wi ) → OZ (Wi) is a compact operator. Consider the commuting diagram

0 n 0 π n 0 m 0 OZ (Wi ) OZ (Wi )/fi(OZ (Wi ))

r r˜

On (W ) On (W )/f (Om(W )). Z i π Z i i Z i

47 Since π and π0 are open and continuous maps, we conclude thatr ˜ is a compact operator, as well. This implies that η is a compact operator. We now use that if µ, η : F1 → F2 are linear operators between Fréchet spaces such that µ is continuous and surjective and η is a compact operator, then F2/(µ + η)(F1) is finite-dimensional [Ser54, Corollary]. Applied to our setting, this gives that Hq(W, G) = Hq(Ω, G) is finite-dimensional.

Corollary 3.6. Let X/G be compact, X be strongly pseudoconvex and GY a coherent analytic sheaf on Y/HC. Then for every q > 0, there exists a d0 such that q C Y Y H (Y/H , G ⊗ OY,d) = 0 for all d ≥ d0.

Proof. Let Z∆ be as in the deﬁnition for pseudoconvexity. Since Z∆ is a neighbourhood of Y/HC, we may use the same construction as above and get the maps

∆ M q C Y Y q Z A≤d0 : H (Y/H , G ⊗ OY,d) → H (Z∆, G ), d≤d0 and

∆ q Z q C Y Y Bd : H (Z∆, G ) → H (Y/H , G ⊗ OY,d).

We then see that A∆ is injective. ≤d0

The result then follows from Proposition 3.5.

Let V be a HC-representation. We may deﬁne a new HC-representation d structure on V via (h, v) 7→ hC hv. This is a well-deﬁned linear action because h 7→ hC is a group morphism and the HC-action on V is linear. We denote this new representation by Vd.

C If U is an open, HC-invariant subset of Y and f : U → C is a G -invariant holomorphic function of order d > 0, then we may deﬁne f˜: C × U → C, (w, x) 7→ wdf˜(x). This is then a holomorphic function which is invariant C −1 under the H -action on C × U given by (h, (w, x)) 7→ (whC , hx). We may therefore extend it to a holomorphic function f of order d on the open subset ∗ C C ×C U/G of Z. ∗ C On the other hand, if we have a holomorphic map f : C ×C U/G of ∗ C order d > 0, then we have the map U → C ×C U/G , y 7→ [1, y] and the pullback f of f along this map gives a GC -invariant holomorphic map of order d.

48 Theorem 3.7. Let X/G be compact, X strongly pseudoconvex, V a HC- representation and y ∈ X. Then there exists a natural number k0 such that for all d > k0, we have the following. Given an open, HC-invariant neighbourhood U of y in Y , an HC-equivariant map ϕ: U → V and a GC -invariant holomorphic map f : U → C of order d, then there exists a HC-equivariant holomorphic map Φ: Y → Vd with Φ(y) = f(y)ϕ(y) and dΦ(y) = d(fϕ)(y). Proof. Let π : Y → Y/HC be the projection. We deﬁne the sheaf F Y on Y/HC as

F Y (U) := {f : π−1(U) → V | f holomorphic, HC-equivariant }.

For z ∈ Y/HC we set

Y Y Fz,z(U) := {f ∈ F (U) | f(z) = 0 and df(z) = 0}.

Y C C Then Fz,z is given by the H -equivariant maps which vanish on H z of order 2. But the existence of holomorphic slices (Proposition 1.10) also implies that every HC-orbit is analytic, hence we may use [Rob86, Theorem 3.1] to see that both sheaves above are coherent analytic sheaves on Y/HC.

Using Corollary 3.6, we conclude that for every z ∈ Y/HC, there exists 1 C Y Y kz such that H (Y/H , Fz,z ⊗ OY,k) = 0 for all k ≥ kz.

Now let ϕ and f of order d ≥ kπ(y) be as in the statement of this Proposition. Then we get f as in the discussion above. We see that ϕ ⊗ f Y Y defines an element in F (π(U)) ⊗ OY,d(π(U)). Since the first cohomology groups vanish, we find a global section ˜ Y Y Φ ∈ F (π(Y )) ⊗ OY,d(π(Y )) such that ˜ Y Y Φ − ϕ ⊗ f ∈ Fπ(y),π(y)(π(U)) ⊗ OY,d(π(U)). ˜ P P We may write Φ = i ϕi ⊗ f i and Φ := i ϕi · fi defines a holomorphic, HC-equivariant map Φ: Y → Vd.

P P 0 0 0 Y But now i ϕi ⊗ f i − ϕ ⊗ f = j ϕj ⊗ f j with ϕj ∈ Fπ(y),π(y)(π(U)). We conclude

X X 0 0 ϕi(y)fi(y) − ϕ(y)f(y) = ϕj (y)fj (y) = 0, i j hence Φ(y) = ϕ(y)f(y). Using the product rule for the diﬀerential, one sees dΦ(y) = d(fϕ)(y) in the same manner.

49 Theorem 3.8. Let X/G be compact, X strongly pseudoconvex, V a HC- representation and x, y ∈ X with Hx 6= Hy. Then there exists a natural number k0 such that for all d > k0, we have the following. Given open, HC- invariant neighbourhoods Ux and Uy of x and y in Y , equivariant holomor- C phic maps ϕx : Ux → V , ϕy : Uy → V and G -invariant holomorphic maps fx : Ux → C, fy : Uy → C of order d, then there exists a HC-equivariant holomorphic map Φ: Y → Vd with Φ(x) = fx(x)ϕx(x) and Φ(y) = fy(y)ϕy(y). Proof. We deﬁne F Y as in Theorem 3.7. For z, w ∈ Y/HC, consider

Y Y Fz,w(U) := {f ∈ F (U) | f(z) = f(w) = 0}.

The proof is now analogous to the proof of Theorem 3.7

3.3 Equivariant embeddings We start by constructing local equivariant embeddings. We will use sheaf theory later to globalise these. From now on, we will assume that HC is complex reductive.

0 C Let x ∈ X and L = Hx with L ⊂ G. Then the image of LC in HC/G ∗ is a ﬁnite group N. We call |N| the order of C in x. Note that the action of LC on HC/GC from the left is exactly the action |N| C C of N and the map hC 7→ hC is an N-invariant map on H /G .

LC Proposition 3.9. Let y ∈ X, L = Hy and HC× SC be a holomorphic slice ∗ in Y . Let l be the order of C in x. Then there exists a HC-representation V LC and a HC-equivariant holomorphic injective immersion ϕ: HC × SC → V such that, for every m ∈ N, the map

C LC Φ: H × SC → Vml × V(m+1)l ml (m+1)l [h, z] 7→ (hC ϕ[h, z], hC ϕ[h, z]) is a well-deﬁned, injective equivariant immersion on some open, HC-invariant neighbourhood Um of y in Y . C For all y0 ∈ Um, the restricted map Φ: G y0 → Vml × V(m+1)l is an embedding.

LC LC Proof. We see that HC × SC ⊂ HC × TyY , the latter being an affine variety with an action of the complex reductive group HC, therefore we find an equivariant holomorphic embedding into some representation V . LC Let ϕ: HC × SC → V be the restriction of that embedding. The LC- LC orbits on SC are all finite and we conclude that all HC-orbits in HC × SC LC are also closed in HC × TyY , hence ϕ is an embedding on every HC-orbit.

50 Let Φ be deﬁned as in the statement of the Proposition. By construction of l, the map is well-deﬁned. ¯ LC We show that Φ is injective. Take [h, z], [h, z¯] ∈ HC × SC such that Φ[h, z] = Φ[h,¯ z¯]. We conclude

l(m+1) ¯l(m+1) ¯ ¯l lm hC ϕ[h, z] = hC ϕ[h, z¯] = hC hC ϕ[h, z], l ¯l hence hC = hC , because ϕ is an immersion and is therefore non-vanishing. But then ϕ[h, z] = ϕ[h,¯ z¯] and [h, z] = [h,¯ z¯] because ϕ is injective.

∗ C Now considerz ˜ := [1, z], take ξ ∈ C = T1C , ν ∈ g and v ∈ Tz˜SC , deﬁne w = ξ(˜z) + ν(˜z) + v . Then

dz˜Φ(w) = (lm · ξ · ϕ(˜z) + dz˜ϕ(w), l(m + 1) · ξ · ϕ(˜z) + dz˜ϕ(w)) and because ϕ(˜z) 6= 0 and ϕ is an immersion, one sees that Φ is an immersion inz ˜. Because Φ is equivariant, it is an immersion in some invariant neighbourhood Um of HCy.

C For ﬁxed y0 ∈ Um, we get Φ|G y0 = (c1ϕ, c2ϕ) for some constants c1 and c2, which is an embedding.

C LC C LC ml Set U := H × SC . The map H × SC → C,[h, z] 7→ hC is GC -invariant of order ml. It may therefore be extended to a holomorphic ∗ function on the open subset of Z given by C ×C U of order ml.

For ﬁxed y ∈ X we may use Proposition 3.9 and Proposition 3.7 to ﬁnd a HC-representation Vy, an HC-equivariant holomorphic map Φy : Y → Vy and a neighbourhood Uy of y, such that the map Φy : Uy → Vy is an immer- C sion on Uy, injective on HCy and Φy : G y → Vy is an embedding.

Using Proposition 3.7 again for the trivial HC-representation C and ϕ being a constant map, we may construct a GC -invariant holomorphic map m F : Y → C which does not vanish on Y and every component Fi of F is of 2 order di > 0. Because X/G is compact, we obtain infx∈X ||F (x)|| = c > 0.

Lemma 3.10. Let Φy, F be as above. Then there exist open, HC-invariant m neighbourhoods Uy of y in Y and Wy of (F, Φy)(y) in C \{0} × Vy such that (F, Φy): Uy → Wy is an embedding. The set Wy is saturated with m m respect to the quotient map C \{0} × Vy → (C \{0} × Vy)//HC and two distinct HC-orbits in (F, Φy)(Uy) may be separated by disjoint, HC-invariant neighbourhoods.

Proof. We will omit the subscript and write Φ = Φy. We will ﬁrst show that m (F, Φ): HCy → C \{0} × V is an embedding.

51 ∗ C m Let wn ∈ C , hn ∈ G such that (F, Φ)(wnhny) converges in C \{0}×V . L di We get F (wnhny) = F (wny) = wn Fi(y), which converges against a non- zero value. There exists an i such that Fi(y) 6= 0, hence wn is bounded from ∗ below and above, therefore we may assume that it converges in C . −1 But then wn Φ(wnhny) = Φ(hny) converges and since Φ is an embedding on GC y, we have proved the claim.

∗ m We get that the C -action on C \{0}, induced by the orders of the Fi, is proper and locally free, therefore we find a complex slice S around ∗ F (y) which is biholomorphic to some complex ball. Let Γ = CF (y), then ∗ Γ ∗ Ω := C × S is an open, C -invariant Stein neighbourhood of F (y) in m C \{0} and the HC-orbit through (F, Φ)(y) is closed in Ω × V . Using [Hei88, Einbettungssatz 1], we find a closed HC-equivariant embedding µ of Ω×V into some HC-representation V0. Now µ◦(F, Φ), defined on (F, Φ)−1(Ω×V ), is an immersion in y and an embedding on the HC-orbit through y. From the proof of Proposition 5.1 in [Sno82], one now finds a slice S0 around µ ◦ (F, Φ)(y) in V0 and a slice SY around y in Y such that the maps LC LC µ◦(F, Φ): SY → S0 and µ◦(F, Φ): HC× SY → HC× S0 are embeddings. Furthermore, HC · S0 is saturated with respect to the quotient map. −1 Define W0 = HC · S0, W := µ (W0) and U := HC · SY . Then (F, Φ): U → W is an embedding. Since µ:Ω×V → V0 is an embedding and Ω×V is saturated with respect m m to the quotient map C \{0} × V → (C \{0} × V )//HC, we conclude that W is saturated. Now every HC-orbit in U is closed, hence every HC-orbit in µ◦(F, Φ)(U) is closed in W0. We may separate two distinct, closed orbits in W0 by HC- invariant open neighbourhoods and the result follows. Proposition 3.11. Let X be strongly pseudoconvex and X/G compact. Let x, y ∈ X such that Hx 6= Hy. Then there exists a GC -invariant holomorphic 2 map F : Y → C of order d > 0 and open, disjoint neighbourhoods Ux and ∗ Uy of F (x) and F (y), which are invariant with respect to the diagonal C - action. x y Proof. We find slices SC and SC around x and y such that the corresponding C y C HxC x C Hy neighbourhoods Ωx = H × SC and Ωy = H × SC are disjoint. Let 2 V = C be the trivial HC-representation and 2 ϕ:Ωx ∪ Ωy → C ( e z ∈ Ω z 7→ 1 x e2 z ∈ Ωy ∗ We find a d ≥ d0 such that the orders of C in x and y divide d, therefore d the map f[h, z] = hC is well-defined on Ωx ∪ Ωy.

52 Theorem 3.8 now delivers the desired function. We may ﬁnd disjoint neighbourhoods U1 and U2 of e1 and e2 which are ∗ 2 invariant with respect to the diagonal C -action on C . Let us shortly summarise the assumptions we made to this point. We have that H is a subgroup of its universal complexiﬁcation HC, G is a 1 closed, normal subgroup of H such that H = G o S . Furthermore, we assume HC to be complex reductive. Now let X be a CR manifold with 0 0 proper, transversal CR action of H such that Hx ⊂ Gx for every x ∈ X and Y the universal equivariant extension of X. Then we proof the main result of this thesis.

Theorem 3.12. Let X be strongly pseudoconvex and X/G compact. Then there exists a HC-representation V and a HC-equivariant holomorphic m m embedding Φ: Y → C \{0}×V , such that Φ: X → C ×V is an embedding. m C Here, C is the trivial G -representation and decomposes into irreducible ∗ C -representations with positive weights.

C Proof. Let F ,Φy, Uy ⊂ Ωy and Wy be as in Lemma 3.10. Using that X/G is compact, we may cover Y be ﬁnitely many open subsets Uy with y ∈ I and consider

˜ M m M Φ := (F, Φy): Y → C × Vy, y∈I y∈I which is an immersion on Y and injective on every HC-orbit.

˜ m L ˜ m L We write V = C × y Vy and V0 = C \{0} × y Vy and show that the map Φ:˜ Y → V˜0 is proper. Let an be a sequence in Y such that Φ(˜ an) = bn converges against b ∈ V˜0. Because Y/HC = X/H is compact, we may assume that π(an) → π(a) converges, where π : Y → Y/HC. Around a, we ﬁnd a neighbourhood Uy and an open neighbourhood M W˜ y := Wy × Vx x∈I,x6=y of Φ(˜ a), such that Φ:˜ Uy → W˜ y is an embedding and W˜ y deﬁnes an open neighbourhood of b in V˜0 //HC. This implies the convergence of a subse- quence of an.

2 Since infx∈X ||F (x)|| = c > 0, we conclude that Φ:˜ X → V˜ is proper.

We also have that Φ˜ is injective on every HC-orbit and it only remains to show that we may separate points on diﬀerent orbits.

53 Lemma 3.10 states that for every w, z ∈ Uy in different orbits, we may separate Φ(˜ w) and Φ(˜ z) by HC-invariant open sets. C C S C C Because (Y/H × Y/H )/ y∈I (Uy/H × Uy/H ) is compact, we conclude that we may use Proposition 3.11 finitely many times to find a GC - ˜ 2k ˜ invariant map F : Y → C , every component Fi being of order di > 0, such that for w, z ∈ Y in different orbits, we may separate (F,˜ Φ)(˜ w) and (F,˜ Φ)(˜ z) by HC-invariant neighbourhoods.

54 4 Line Bundles

4.1 CR Line bundles We may use the methods from the previous sections to proof an equivariant embedding theorem similar to the Kodaira embedding theorem.

Let X be a compact CR manifold and K a compact Lie group with transversal CR action on X. Note that K is a subgroup of its universal complexiﬁcation KC, we may therefore apply the results from Section 1. We will use the deﬁnitions from [HLM17] for rigid, positive CR line bundles, generalised to arbitrary compact groups.

Deﬁnition Let X be a CR manifold and p: L → X a complex line bundle. We say that L is a CR line bundle if there exists a cover Ui of X over ∗ which L is trivial, where the induced transition functions gij : Ui ∩ Uj → C are CR.

Let X be a CR manifold with transversal K-action, U ⊂ X open, K- invariant and f : U → R a K-invariant CR function. For x ∈ U and a sufficiently small slice extension ΩC around x, the function f extends to a KC-invariant holomorphic function F on ΩC. We say that f is strictly K-plurisubharmonic in x if the hermitian 1,0 form on Tx X, given by V,W 7→ ∂∂F (V, W ), is positive definit. Here, ∂ and ∂ are the usual operators on the complex manifold ΩC. C C Note that for two different slice extensions Ω1 and Ω2 of the same set Ω, the identity map extends to a KC-equivariant biholomorphic map between C C Ω1 and Ω2 , from which we conclude that the definition does not depend on the expansion. We say that f is strictly K-plurisubharmonic if it is strictly K- plurisubharmonic in every point.

1,0 0,1 We have Ck(x) ⊕ Tx X ⊕ Tx X = CTxX. Fix x ∈ X and take a Slice LC 1,0 0,1 extension ΩC = KC × S around x, then CkC(x)⊕Tx X ⊕Tx X = CTxΩC. 1,0 0,1 Now Tx X ⊕ Tx X is invariant under the tangential Kx-action, therefore it is invariant under the tangential (KC)x = (Kx)C-action. From the proof of Proposition 1.10, we conclude that we ﬁnd a slice SC 1,0 0,1 in ΩC for the KC-action such that CTxSC = Tx X ⊕ Tx X. Now let f be strictly K-plurisubharmonic in x and F be the KC-invariant C extension of f onto Ω . Then ∂∂F S is positive in x, hence positive in a C neighbourhood of x in SC . After shrinking SC , we may assume that F SC is strictly plurisubharmonic in the sense of complex manifolds.

55 C C C C The quotient Ω/K is isomorphic to Ω /K = SC /Kx . But Kx acts as C a ﬁnite group on SC and we get that the induced map F : SC /Kx → R is strictly plurisubharmonic from Lemma 2.5.

Deﬁnition Let X be a CR manifold with transversal K-action. We say that a CR line bundle L → X is K-invariant if there exists a cover Ui of X over which L is trivial such that each Ui is K-invariant and the transition functions gij are K-invariant. We call such a bundle positive if there exist K-invariant positive functions hi : Ui → R with |gij|hj = hi and log(hj) is strictly K-plurisubharmonic.

Given a positive bundle L, one can construct a metric on L by setting −1 |s|h := |s|ihi , where |s|i denotes the norm in the trivialisation on Ui. Then −1 −1 −1 |s|ihi = |gij||s|jhi = |s|jhj and the metric is well-defined. This definition is equivalent to [HLM17, Definition 2.9], see [HM12, Proposition 6.1]. If L → X is a K-invariant CR line bundle, then the transition functions gij define holomorphic functions gij on the open subset (Ui ∩Uj)/K of X/K. Those then induce a holomorphic line bundle LK → X/K. From the above discussion, we conclude the following Lemma. Lemma 4.1. Let X be a CR manifold with transversal, CR K-action and L → X a K-invariant positive CR line bundle. Then the induced line bundle LK → X/K is positive.

Deﬁnition Let X be a compact, complex manifold. A complex line bundle L → X with metric h is called weakly negative if the set {z ∈ L | h(z) < 1} is strongly pseudoconvex.

Note that every negative line bundle (in the usual sense) is weakly negative [Gra62, §3 Satz 1].

To conclude this section, we have seen that the existence of a positive, K-invariant CR-line bundle over X implies the existence of a negative, hence weakly negative line bundle over X/K.

4.2 Projective Embeddings Let X be a CR manifold with transversal K-action and assume that there exists a weakly negative line bundle LK → X/K. By pulling back the transition functions of LK to K-invariant CR functions on X, we get an induced CR line bundle L → X. There then exists a cover Ui of X of K-invariant open sets such that L is trivial over Ui.

d If L is trivial over Ui with transition functions gij, then L is trivial d over Ui with transition functions gij. If V is a complex vectorspace, then

56 V deﬁnes a trivial vector bundle over X and we have the tensor product d d d L ⊗ V . Now L ⊗ V is trivial over Ui with transition functions gij ⊗ idV . d ∼ ∼ On a trivialisation, we have L ⊗ V = C ⊗ V = V . Proposition 4.2. Let K be a compact Lie group and X a CR manifold with transversal K-action. Assume that there exists a weakly negative line bundle LK → X/K with induced CR bundle L → X. Let V be a KC-representation, x ∈ X and U a K-invariant neighbourhood of x over which L is trivial. Then there exists a natural number d such that for every K-equivariant CR map ϕ: U → V , there exists a CR section Φ: X → L−d ⊗ V such that in the trivialisation over U, the map ϕ − Φ vanishes of order 2 on the K-orbit through x. 1 Proof. Denote by XS the S -bundle of L. Then the map S1 C × XS → L [z, s] 7→ zs

S1 and the corresponding map C × XS/K → LK are isomorphisms. Because LK is weakly negative, XS is a strongly pseudoconvex CR manifold with transversal S1 × K-action.

Let k0 be as in Theorem 3.7, take d > k0 and let ϕ: U → V be a K-equivariant CR map. Via the map p: L → X, we may extend ϕ to a ∗ −1 C -invariant CR map ϕ on p (U). Let (w, y) be the coordinates of the trivialisation on p−1(U) and deﬁne d the K-invariant CR function f : C × U → C,(w, y) 7→ w . We use Theorem 1.9 and embed XS into its universal equivariant exten- ∗ sion Y . Then f and ϕ extend to C × KC-equivariant holomorphic maps on ˆ ∗ −1 U := (C × KC) · (p (U) ∩ XS) ⊂ Y . Here, we understand V as the trivial ∗ C -representation. −1 We may assume (1, x) ∈ p (U) ∩ XS after possibly shifting the trivi- ∗ alisation. Apply Theorem 3.7 and ﬁnd a C × KC-equivariant holomorphic map Φ: Y → Vd such that Φ − fϕ vanishes of order 2 in (1, x). Because ∗ of equivariance, Φ − fϕ actually vanishes of order 2 on the C × KC-orbit through (1, x).

We may restrict Φ to XS and extend it to a CR map on L of order d. Denote this map by Φ: L → V . Let Ui be a cover of K-invariant open subsets of X over which L is trivial with transition functions gij. Without loss of generality, U1 = U. −d Deﬁne a CR section Φ˜ i : Ui → L ⊗ V via Φ˜ i(y) := 1 ⊗ Φ(1, y) in the trivialisation over Ui. This does in fact give a global section Φ,˜ since the transition from Ui to Uj is given by −d −d d 1i ⊗ Φ(1i, y) = gji 1j ⊗ Φ(gji1j, y) = gji 1j ⊗ gjiΦ(1j, y) = 1j ⊗ Φ(1j, y),

57 where 1i and 1j denote 1 in the respective trivialisations. We identify L−d ⊗ V with V on U using c ⊗ v 7→ cv and conclude Φ(˜ x) = Φ(1, x) = f(1, x)ϕ(1, x) = ϕ(x). ˜ Let v ∈ TxU, then dxΦ(v) = d(1,x)Φ(0, v) = d(1,x)(fϕ)(0, v) = dxϕ(v). Because of the equivariance of Φ˜ and ϕ, we conclude that Φ˜ − ϕ vanishes of order 2 on the K-orbit through x.

Proposition 4.3. Let K be a compact Lie group and X a CR manifold with transversal K-action. Assume that there exists a weakly negative line bundle LK → X/K with induced CR bundle L → X. Let V be a KC-representation, x, y ∈ X and Ux,Uy K-invariant neighbourhoods of x and y over which L is trivial. Then there exists a natural number d such that for all K-equivariant CR maps ϕx : Ux → V and −d ϕy : Uy → V , there exists a CR section Φ: X → L ⊗ V such that in the trivialisations over Ux and Uy, the maps ϕx − Φ and ϕy − Φ vanish on the K-orbits through x and y, respectively.

Proof. This proof is analogous to the proof of Proposition 4.2, only using Theorem 3.8 instead.

k −k k Denote by L the sheaf of CR sections X → L . For x ∈ X, deﬁne Lx k to be the sheaf of CR sections vanishing in x of order 2 and Lx,y the CR sections vanishing in x and y.

Deﬁnition For a natural number k and x ∈ X, we say that X is immersible of degree k in x if the sequence

k k k k 0 → Lx(X) → L (X) → (L /Lx)(X) → 0 is exact.

Exactness of this sequence means that given a neighbourhood U of x k k and a CR section s0 : U → L , we find a globally defined CR section s ∈ L such that s − s0 vanishes of order 2 in x. We reformulate this as follows. Given a trivialising neighbourhood U of k ∗ k L around x, a value c ∈ C and α ∈ (TxX) , we find a section s ∈ L with s(x) = c and dxs = α in the trivialisation.

Deﬁnition For a natural number k and x, y ∈ X, we say that x and y are separable of degree k if the sequence

k k k k 0 → Lx,y(X) → L (X) → (L /Lx,y)(X) → 0 is exact.

58 k Exactness in this case is equivalent to ﬁnding sections sx, sy ∈ L with sx(x) 6= 0, sx(y) = 0, sy(x) = 0 and sy(y) 6= 0.

We want to use the above Propositions to construct embeddings. For this, we ﬁrst need to show that local embeddings exist. Lemma 4.4. There exists a KC-representation V such that around every point x ∈ X, there exists a K-invariant neighbourhood U of x and a K- equivariant injective immersion ϕ: U → V .

KC Proof. Take a slice extension around x of the form ΩC = KC × x SC for some holomorphic slice SC . KC This is an open subset of the aﬃne variety KC × x TxΩC, for which we ﬁnd a KC-equivariant embedding into some representation Vx. Restricting that embedding to X ∩ ΩC gives the desired map. Because X is compact, we may choose Vx independent of x. Theorem 4.5. 1. For every x ∈ X, there exists a neighbourhood U and a natural number k such that X is immersible in y of degree mk for every y ∈ U and every m ∈ N.

2. For x, y ∈ X, there exist neighbourhoods Ux, Uy of x and y and a natural number k such that x and y are separable of degree mk for all z ∈ Ux, w ∈ Uy and m ∈ N. Proof. 1) Take x ∈ X and let ϕ: U → V be the map from Lemma 4.4. By shrinking U, we may assume that L is trivial over U. Let Φ: X → L−d ⊗ V be the section obtained when applying Proposi- tion 4.2 to ϕ. Choosing a basis for V then gives a decomposition of Φ into d m := dimV sections si ∈ L (X). Because ϕ is an injective immersion on ∗ Kx, the dxsi form a generating system for (TxX) .

d Using Proposition 4.2, we also ﬁnd a section s0 ∈ L (X) with s0(x) = 1 and dxs0 = 0 in the trivialisation on U. We may now assume that si(x) = 0 for all i ≥ 1 and, because of continuity, the dysi form a generating system ∗ for (TyX) for all y in a neighbourhood U0 of x. We may also assume that s0(y) 6= 0 for all y ∈ U0 and that U0 ⊂ U.

Fix y0 ∈ U0 and set

s0(y)si(y0) s˜i(y) := si(y) − . s0(y0)

This is well-defined, fulfilss ˜i(y0) = 0 and for y0 close enough to x, we get ∗ that (˜s1, ..., s˜m) form a generating system of (Ty0 X) . ∗ Now given c ∈ C and α ∈ (TxX) , we find a sections ˆ, which is a linear combination of thes ˜i, such that dy0 sˆ = α. By adding a multiple of s0, we

59 may also arranges ˆ(y0) = c.

This shows that X is immersible in y of degree k = d for every y in a neighbourhood of X. km ∼ k m Now take m ∈ N, then L = (L ) and we get that the natural map Γ(X, (Lk)m) → Γ(X,Lkm) is surjective. This implies that if the sequence is exact for k in a point, it is also exact for km.

2) We will first see that x and y are separable for some degree d. We distinguish two cases. First we assume that x and y are in the same K-orbit. We then repeat d the argument from the first part to obtain sections s1, ..., sm ∈ L such that the map y 7→ (s1, ..., sm) is an injective immersion on Kx. But then we find a sections ˆ withs ˆ(x) 6=s ˆ(y) in a trivialisation around the K-orbit. We also get a section s0 with s0 = 1 on the K-orbit through x. In particular, we have s0(x) = s0(y) = 1 and taking linear combinations ofs ˆ and s0 shows that x and y are separable of degree d.

Second, we assume that x and y are in separate K-orbits. We may then separate them by K-invariant disjoint open subsets Ωx and Ωy over which L is trivial. Deﬁne ϕx to be the function on Ωx ∪ Ωy which is equal to one on Ωx and equal to zero on Ωy. Let ϕy be the respective map with zero and one swapped. We may then apply Proposition 4.3 and get that x and y are separable of degree d.

Let s1 be a section with s1(x) = 0 and s1(y) 6= 0, s2 a section with s2(y) = 0 and s2(x) 6= 0. For x0 near x, deﬁne

s2(z)s1(x0) s˜1(z) := s1(z) − s2(x0) and for y0 near y, deﬁne

s1(w)s2(y0) s˜2(w) := s2(w) − . s1(y0)

Thens ˜1(x0) = 0,s ˜1(y0) 6= 0,s ˜2(y0) = 0 ands ˜2(x0) 6= 0 for x0 close to x and y0 close to y.

We see that x0 and y0 are separable of degree k = d for x0 close to x and y0 close to y. Using the same argument as at the end of the ﬁrst part, we see that x0 and y0 are also separable of degree km. Corollary 4.6. Let X be a compact CR manifold with a transversal CR action of a compact Lie group K.

60 Assume that there exists a weakly negative line bundle LK → X/K. Then k there exists a natural number k and ﬁnitely many CR sections si ∈ L (X) such that, for W = span(si), we have that

∗ X → P(W ) y 7→ [s 7→ s(y)] is a CR embedding.

Proof. Because X is compact, we conclude from Theorem 4.5 that we ﬁnd a k ∈ N such that X is immersible in every point x ∈ X of degree k and x, y are separable of degree k for all x, y ∈ X.

k For a subspace W ⊂ L (X), we denote by ΦW the map defined in the corollary for this subspace. We find finitely many sections s1, ..., sm such that for every x ∈ X, there exists a i such that si(x) 6= 0. In particular, the map ΦW for W = span(si) is well-defined. We will from now on assume that the map ΦW is well-defined for every subspace, since we may always add finitely many section to ensure that it is.

Let x ∈ X be ﬁxed and take a neighbourhood U of x over which L is k U ∗ trivial. For a subspace W ⊂ L (X), we deﬁne ΦW : U → W to be given ∗ ∗ by y 7→ (s 7→ s(y)). Take a basis s1, ..., sm for W with dual basis s1, ..., sm U P ∗ U and write ΦW (y) = i αi(y)si . Then αi(y) = ΦW (y)(si) = si(y) and we U conclude that ΦW (y) = (s1(y), ..., sm(y)) in this basis.

k Now we ﬁnd sections g1, ..., gm ∈ L such that y 7→ (g1(y), ..., gm(y)) is an immersion in x in the trivialisation over U. We also get a section g0 with g0(x) 6= 0 and dxg0 = 0.

Set Wx = span(g0, ..., gm), then ΦWx (y) = [g0(y), ..., gm(y)] in the corresponding dual basis. Since g0(x) 6= 0, we ﬁnd a neighbourhood of Φ(x) on ∗ which [z0, ..., zm] 7→ (z1/z0, ..., zm/z0) is a chart for P(Wx ).

In this chart, ΦWx (y) = (g1(y)/g0(y), ..., gm(y)/g0(y)) and we conclude that ΦWx is an immersion in x.

We may repeat this process finitely many times to find a subspace W of k L (X) such that ΦW is an immersion on X. We may therefore cover X by finitely many open sets Ui such that ∗ S ΦW : Ui → P(W ) is injective. Now (X × X)/ i(Ui × Ui) is compact, S and for two points (x, y) ∈ (X × X)/ i(Ui × Ui), we may apply Theorem 4.5 to find a sections ˆ such that ΦW +Csˆ separates x and y. We only need to do this finitely many times to ensure that Φ is injective.

61 4.3 G-ﬁnite functions We want to show that the embedding from Corollary 4.6 can also be done equivariantly.

Let X be a manifold with action of a Lie group G and L → X a line bundle. Assume that L admits a G-invariant trivialisation, i.e. there exists a covering Ui of X of G-invariant subsets over which L is trivial such that the transition functions gij are G-invariant. We then have an action of G on Γ∞(X,L) via (gs)(x) = s(g−1x), which is well-defined because of the above assumption. This induces an action on Γ∞(X,L)∗ via (gλ)(s) = λ(g−1s). Note that the map defined in Corollary 4.6 is K-equivariant with respect to this action if W is K-invariant. One may try to replace W by the smallest subspace containing KW , but that does not need to be a finite-dimensional subspace.

A section s ∈ Γ∞(X,L) is said to be G-finite if the G-orbit Gs is contained in a finite-dimensional subspace of Γ∞(X,L). Now assume that X is compact and that there exists a metric h on L. Let U be a chart for X, we may then define a seminorm on Γ∞(X,L) via

X ∂ ||s|| := sup h(s(x)) + h( s(x)), U x∈U ∂x i i where ∂ denote the partial differentials in the chart U. ∂xi Note that ||s||U is finite for every s but does depend on the choice of the chart. Since X is compact, we may cover X by finitely many charts Ui, i ∈ I and define the norm

||s|| := supi||s||Ui .

The following Lemma was proven by Mostow for smooth functions on compact manifolds [Mos57, Section 7.4]. The CR version for line bundle sections is mostly analogous. Note that this was developed as part of a joint work with Hendrik Herrmann.

Lemma 4.7. Let X be a compact CR manifold with CR action of a compact group K. Let L be a K-invariant CR line bundle over X. Then the K-ﬁnite CR sections in L are dense in the set of all CR sections.

Proof. Fix a CR section s on X and ε > 0. Using that X is compact, we ε ﬁnd a neighbourhood U of 1 ∈ K such that ||s − ks|| < 2 for all k ∈ U. Let χ be a non-negative continuous function on K vanishing outside U with

62 R R K χ(k)dk = 1. For any continuous function ϕ on K, set sϕ := K ϕ(k) ks dk, ∂ which is a smooth section. Denote by j the partial differentials in Uj, then ∂xi Z Z ||sχ − s|| = || χ(k) ks dk − s|| = || χ(k)(ks − s)dk|| K K Z =supUj supx∈Uj h χ(k)(ks − s)(x)dk K Z X ∂ + h j χ(k)(ks − s)dk (x) i ∂xi K Z ≤supUj supx∈Uj χ(k)h (ks − s)(x) dk K Z X ∂ + χ(k) h j (ks − s)(x) dk K i ∂xi Z ≤ χ(k) supUj supx∈Uj h (ks − s)(x) K X ∂ + h j (ks − s)(x) dk i ∂xi Z Z ε ε = χ(k)||ks − s||dk ≤ χ(k) dk = . K K 2 2 The Theorem of Peter-Weyl states that the K-finite functions on K are R dense in the continuous functions. Set c = || K ks dk||, then we may choose ε χ˜ to be K-finite with ||χ − χ˜||sup ≤ 2c , where || · ||sup denotes the standard supremum norm. We then get Z Z ε ||sχ − sχ˜|| = || (χ − χ˜)(k) ks dk|| ≤ ||χ − χ˜||sup · || ks dk|| ≤ . K K 2 But now Z Z −1 k1sϕ = ϕ(k) k1ks dk = ϕ(k1 k) ks dk = sk1ϕ. K K

If ϕ is K-ﬁnite, then sϕ is K-ﬁnite because it depends linearly on ϕ. 1,0 We only need to check that sϕ is CR. For this, let x ∈ X and v ∈ T X, then in a trivialisation Z −1 dxsϕ(v) = ϕ(k)dk−1xs ◦ dxk (v) dk K and the result follows from the fact that the K-action is CR and s is a CR section. As a direct application of the Lemma and the discussion above, we conclude

Corollary 4.8. The sections si in Corollary 4.6 can be chosen to be K-ﬁnite and the embedding to be K-equivariant.

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