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Equivariant Embeddings of Strongly Pseudoconvex CR Manifolds

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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 gen- eral 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 equivari- ant 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; !) be a K¨ahlermanifold with a Lie group G acting on Z through 3 holomorphic transformations leaving ! invariant. Assume that there exists a momentum map µ: Z ! g∗ for the action and assume that 0 2 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 mani- folds 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 transver- sal 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 pseu- doconvex 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 com- pact, 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 f[z; x] 2 C × X=K j jzj < 1g 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 equiv- ariant extension of X. Then there exists a HC-representation V and a m HC-equivariant holomorphic embedding Φ: Y ! C nf0g × V such that m Φ: X ! C × V is an embedding. 5 Note that for G = fIdg, 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 2 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 di- mension (2n; d) with n ≥ 2.
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