CR extensions with a classical Several Complex Variables point of view

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CR extensions with a classical Several Complex Variables point of view

August Peter Brådalen Sonne

Master’s Thesis, Spring 2018

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This master’s thesis is submitted under the master’s programmeMathematics, with programme option Mathematics, at the Department of Mathematics, University of Oslo. The scope of the thesis is 60 credits.

The front page depicts a section of the root system of the exceptional Lie group E₈, projected into the plane. Lie groups were invented by the Norwegian mathematician Sophus Lie (1842–1899) to express symmetries in differential equations and today they play a central role in various parts of mathematics.

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Acknowledgements

I would like to give a special thanks to Berit Stensønes for her eternal pa- tience and for always pushing me in the right direction.

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1 Introduction

This thesis is dedicated to developing the theory required in order to prove a theorem first proven by Trépreau in [13], stating that if M ⊂ Cⁿ is a real hypersurface. Then all function f: M → C satisfying the tangential Cauchy-Riemann equations extend holomorphically to one side ofM, if and only if M is not pseudoconvex from each side.

In chapter 1 we start off by recalling some basic properties of holomorphic functions in several complex variables as well as the basic theory on domains of holomorphy following the presentations in [5] and [11]. We then introduce some of the basic theory of envelopes of holomorphy of schlicht domains. Inspired by an example in [4] of a domain which has no envelope of holomorphy we introduce the Hartogs hull operator which leads to a useful tool for showing that certain domains are schlicht.

In chapter 2 we study real embedded submanifolds in Cⁿ loosely following a book the presentation in [1]. We start off by introducing the complex tangent spaces of a real embedded submanifold for which we offer several alternate descriptions. Inspired by an example of a submanifold for which the dimension of the complex tangent space depends on the point we then introduce the concept of an embedded CR-submanifold. The embedded CR-submanifolds may be thought of as the class of embedded submanifolds for which we may define Cauchy-Riemann equations. Important examples of embedded CR-submanifolds include the real hypersurfaces as well as the complex submanifolds. Finally we define the concept of a generic CR-submanifold and show that all such submanifolds may be brought on a particularly simple form by a local biholomorphic change of variables.

In chapter 3 we introduce the Lie bracket for vector fields overC²-smooth embedded submanifolds ofCⁿwhich in combination with Frobenius theorem gives a necessary and sufficient condition for the existence of a submanifold with a prescribed tangent bundle structure. We then introduce the concept of a Levi-flatCR-submanifold and we show that the Levi-flatness of a generic CR-submanifold is equivalent to the vanishing of a generalized Levi form.

In chapter 4 we introduce the concept of a CR-function which generalizes the concept of a holomorphic function. Expanding upon the proof in [1] we establish the Baouendi-Treves approximation theorem for embedded CR-submanifolds. This result states that ifM ⊂Cⁿis aC²-smooth generic CR-submanifold, then each point p∈M is contained in an open neighborhood ω ⊂M where allCR-functions may be uniformly approximated by a sequence of entire holomorphic functions. We then introduce the concept of an analytic disk attached to M and we show that the approximating functions from Baouendi-Treves approximation theorem in fact converge to be holomorphic on the (possibly empty) interior of all such analytic disks locally attached toM.

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In chapter 5 we study an important paper by Hill-Taiani [7] on the existence of analytic disks attached to a given C²-smooth generic CR- submanifold. We start off by recalling some elementary properties of har- monic functions. We then introduce the Hilbert transform on the unit circle which we use in order to formulate the Bishop equation. This functional equation gives a necessary and sufficient condition for the existence of (sufficiently regular) analytic disks attached to a generic C²-smooth generic CR-submanifold. By a modification of the proof given in [7] of Privalov’s theorem we show that the Hilbert transform is a linear continuous operator between Banach spaces. In combination with the implicit function theorem for Banach spaces this leads to the important existence theorem for solutions to the Bishop equation. We then introduce a result by Hill-Taiani which shows how the existence of solutions to the Bishop equation may be used in order to find a simple sufficient condition for the existence of a family of analytic disks attached to a givenC⁵-smooth genericCR-submanifold, so that the disks foliate a higher-dimensional submanifold. Finally we mention how this result may be used in order to prove a weak version of Trépreau’s theorem, and we suggest how one may proceed in order to give a proof of Trépreau’s theorem for hypersurfaces inC².

2 Holomorphic functions and domains of holomor- phy

In this chapter we recall some basic properties holomorphic functions and domains of holomorphy. The results and proofs of the results in the first two sections are well-known and may be found in [5] or [11]. In section three we introduce the concept of the envelope of holomorphy for a connected domain Ω ⊂ Cⁿ as the “maximal” domain Ω˜ ⊃ Ω where all holomorphic functions f: Ω→ Cextend holomorphically. Inspired by an example in [4]

of a domain which has no envelope of holomorphy we introduce the concept of a schlicht domain. Finally we introduce the Hartogs hull operator which gives a way of constructing the envelope of holomorphy of a schlicht domain, as well as a way of showing that certains domains are schlicht.

2.1 Holomorphic functions in several complex variables Definition 1. We define the Wirtinger partial derivatives in Cⁿ as the operators defined for1≤j≤nby

∂

∂z_j = 1 2

∂

∂x_j −i ∂

∂y_j

, ∂

∂z_j = 1 2

∂

∂x_j +i ∂

∂y_j

. (1)

Definition 2. If f: Ω → C is a differentiable function defined in an open

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setΩ⊂Cⁿ, then we define the differential of f to be the one-form df =

n

X

j=1

∂f

∂xj

dxj+ ∂f

∂yj

dyj. (2)

Introducing the complex one-forms dz_j = dx_j+idy_j and d¯z_j we obtain the following.

Theorem 2.1. Let f: Ω→Cbe a differentiable function defined in an open setΩ⊂Cⁿ, then df =∂f+∂f, where

∂f=

n

X

j=1

∂f

∂z_j dzj, ∂f =

n

X

j=1

∂f

∂¯z_j d¯zj. (3) We may regard the one-forms dzj and dzj as maps dzj:Cⁿ → C and dzj:Cⁿby lettingdzj(z1, . . . , zn) =zj and lettingdzj(z1, . . . , zn) =zj. In this case it is easy to show that the map∂f:Cⁿ→CisC-linear, while the map∂f:Cⁿ→Cis antilinear.

Definition 3. Letf: Ω→Cbe a continuous function defined on an open set Ω⊂Cⁿ. We say that the function is holomorphic if∂f = 0, or equivalently if we for each 1≤j≤n and each p∈Ωhave that

∂f

∂z_j(p) = 0. (4)

We shall refer to this last system of differential equations as the Cauchy- Riemann equations.

It should be noted that the condition that f: Ω → C is continuous is not required. This follows from Hartog’s theorem on separate analyticity which states that a function f: Ω → C is holomorphic if and only if it is holomorphic in each variable separately.

It is easy to see that the collection of holomorphic functions on a domain has a natural ring structure.

Definition 4. Let Ω⊂Cⁿ be a domain, then we define O(Ω)to be the ring of holomorphic functions f: Ω→C.

Definition 5. We define the polydisk centered at p ∈Cⁿ of polyradius r ∈ Rⁿ+ as the set

∆ⁿ(p, r) ={z∈Cⁿ | |z_j−p_j|< r_j for 1≤j≤n}. (5) In the special case where p = 0 and where r = 1 we shall refer to ∆ⁿ(0,1) as the unit polydisk in Cⁿ. In addition we define the distinguished boundary of the polydisk ∆ⁿ(p, r) as the set

Γⁿ(p, r) ={z∈Cⁿ | |z_j −p_j|=r_j for all 1≤j≤n}. (6)

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In the case where n = 1 we see that the unit polydisk ∆¹(0,1) ⊂ C is simply the open unit disk. We shall denote the open unit disk byD and we shall denote its boundary byS¹.

By applying the single-variable Cauchy integral formula in each variable separately one easily obtains the following.

Theorem 2.2. (The Cauchy integral formula) Letf: Ω→Cbe a holomorphic function defined in an open set Ω ⊂ Cⁿ and let ∆ⁿ(p, r) ⊂⊂ Ω be a relatively compact polydisk. If z∈∆ⁿ(p, r) then

f(z) = 1

2πi nZ

ζ∈Γⁿ(p,r)

f(ζ)

(ζ_n−z₁) · · · (ζ_n−z_n)dζ. (7) By differentiating the above expression one easily shows that any holomorphic function isC^∞-smooth.

Conversely if ∆ⁿ(p, r) is a polydisk and if f: ∆ⁿ(p, r) → C is a continuous function.Then the Cauchy-Riemann equations that the function F: ∆ⁿ(p, r)→Cgiven by

F(z) = 1

2πi nZ

ζ∈Γⁿ(p,r)

f(ζ)

(ζ1−z1) · · · (ζn−zn)dζ, (8) is holomorphic. This easily implies the following.

Theorem 2.3. Let Ω⊂Cⁿ be an open set and let {f_j}_j∈

N be a sequence of holomorphic functions f_j: Ω → C. If the functions converge uniformly on compacts to a function f: Ω→C, then f is holomorphic.

By a similar argument as in the one-variable case involving the Cauchy integral formula (Theorem2.2) one easily shows the following.

Theorem 2.4. Let f: Ω→Cbe a holomorphic function defined in an open set Ω ⊂ Cⁿ. If ∆ⁿ(p, r) ⊂⊂ Ω is a polydisk relatively compact in Ω, then the power series

F(z) = X

α∈Nⁿ0

D^αf(p)

α1!· · ·αn!(z₁−p₁)^α¹ · · · (z_n−p_n)^αⁿ, (9) converges uniformly to f in ∆ⁿ(p, r).

The power series expansion of holomorphic functions easily implies the identity theorem.

Theorem 2.5. (The identity theorem) Let f: Ω → C be a holomorphic function defined in an open connected setΩ⊂Cⁿ. Iff vanishes on an open setU ⊂Ω, then f is identically zero.

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The concept of holomorphicity extends readily to vector-valued functions.

Definition 6. Let f: Ω→C^m be a continuous function defined in an open set Ω⊂Cⁿ. We say that f is holomorphic if for each 1 ≤j ≤m we have that the function f_j: Ω→C is holomorphic.

It is convenient to introduce the holomorphic equivalent of a diffeomor- phism.

Definition 7. Let f: Ω→Cⁿ be a holomorphic function defined in an open setΩ⊂Cⁿ. If f is invertible with a holomorphic inverse, then we say that f is biholomorphic.

2.2 Domains of holomorphy

For any bounded domainΩ⊂Cwe may find a sequence of points{z_j}_j∈

N⊂ Ωwhich clusters at each boundary point. By the Weierstrass product theorem this suggests the existence of a holomorphic function f: Ω→ Cwhich tends to zero on the boundary. Since holomorphic functions inC have iso- lated zeroes this implies thatf does not extend to be holomorphic past the boundary. This suggests that there exist no strictly larger domainΩ˜ ⊃Ωso that we may identifyO(Ω) =O( ˜Ω). As we shall see the situation is radically different in higher dimensions. There exist bounded domains Ω(Ω˜ ⊂Cⁿ for each n≥2 so thatO(Ω) =O( ˜Ω).

If Ω⊂Cⁿ is a bounded domain, then a sufficient condition for the non- existence of a domain Ω˜ ) Ω so that O(Ω) = O( ˜Ω) is the existence of a holomorphic function f_p: Ω → C for each p ∈ ∂Ω so that the function f_p does not extend to be holomorphic past the point p. This motivates the following definition.

Definition 8.LetΩ⊂Cⁿbe an open set and letf: Ω→Cbe a holomorphic function. We say that f is completely singular at a point p ∈ ∂Ω if there exists no holomorphic function f˜: U → C defined in an open connected neighborhood U ⊂Cⁿ of p which agrees with f onU ∩Ω.

Note that iff: Ω→Cis not completely singular at a pointp∈M, then there exists an open connected neighborhoodU ⊂Cⁿofpand a holomorphic functionf˜:U → C which agrees with f on the open connected set Ω∩U. By the identity theorem for holomorphic functions this implies the existence of a local holomorphic extension off past the point p.

Definition 9. Let Ω⊂Cⁿ be an open set, we say thatΩ is a weak domain of holomorphy if there for each point p∈∂Ωexists a holomorphic function f: Ω→Cwhich is completely singular at p.

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By an earlier remark we see that a domain Ω⊂Cⁿ is a weak domain of holomorphy if there for each boundary point p ∈∂Ω exists a holomorphic functionf: Ω→Cwhich does not extend to be holomorphic past the point p.

If Ω ⊂ Cⁿ is a convex open set and if p ∈ ∂Ω. Then there exists a real-valued linear function f:Cⁿ→Rof the form

f(x1+iy1, . . . , xn+iyn) =

n

X

j=1

ajxj+bjyj, (10) so that f(p) = 0 and so that f(z) < 0 for all z ∈Ω. This may be used in order to prove the following.

Theorem 2.6. Let Ω⊂Cⁿ be a convex open set. ThenΩis a weak domain of holomorphy.

Definition 10. Let Ω ⊂ Cⁿ be an open set, we say that Ω is a domain of holomorphy if there exists a holomorphic function f: Ω → C which is completely singular at each boundary point p∈∂Ω.

It is clear that any domain of holomorphy is a weak domain of holomorphy. In fact one can show that the converse also holds; any weak domain of holomorphy is necessarily a domain of holomorphy.

Definition 11. Let Ω⊂Cⁿ be an open set, we say that Ωis a local domain of holomorphy if each point p ∈ ∂Ω is contained in an open neighborhood U ⊂Cⁿ so thatΩ∩U is a domain of holomorphy.

Definition 12. We define a Euclidean Hartogs figure to be a pair(∆ⁿ(0,1), H) where H⊂∆ⁿ(0,1)is a set of the form

H={z∈∆ⁿ(0,1)| rn<|z_n|} ∪ {z∈∆ⁿ(0,1) | |z_j|< rj for 1≤j≤n−1}. (11) For some positive real numbersr1, . . . rn satisfying 0< rj <1.

Theorem 2.7. Let n ≥ 2, let (∆ⁿ(0,1), H) be a Euclidean Hartogs figure and letf:H→C be a holomorphic function. Then there exists a holomorphic functionf˜: ∆ⁿ(0,1)→Cwhich extends f.

Proof. Since(∆ⁿ(0,1), H)is a Euclidean Hartogs figure there exist real numbersr₁, . . . , r_nsatisfying 0< r_j <1 so that

H ={z∈∆ⁿ(0,1) | |r_n<|z_n| or |z_j|< rj for 1≤j≤n−1}. (12)

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Define = min1≤j≤nr_j and pick some real number so that r_n < r < 1.

Now let Dr = {z∈∆ⁿ(0,1) | |r_n|< r}. Define the function f^∗:Dr → C by letting

f^∗(z) = 1 2πi

Z

|ζ|=r

f(z1, . . . , zn−1, ζ)

ζ−z_n dζ. (13)

Differentiating under the integral we easily verify that f^∗ is holomorphic.

Moreover by the single-variable Cauchy-Integral formula it is clear that f^∗(z) = f(z) for all points z ∈ ∆(0, ). It follows by the identity principle thatf^∗ and f agree on their common domain of definition. This implies that the function

f˜(z) =

(f^∗(z) for z∈Dr

f(z) for z∈H (14)

defines a holomorphic extension off toDr∪H= ∆ⁿ(0,1).

The above theorem suggests that if(∆ⁿ(0,1), H)is a Euclidean Hartogs figure withn≥2, then H is not a domain of holomorphy.

Definition 13. Let (∆ⁿ(0,1), H) be a Euclidean Hartogs figure and let F: ∆ⁿ(0,1) → F(∆ⁿ(0,1)) be a biholomorphism. Then we refer to the pair( ˜∆ⁿ,H) = (F(∆˜ ⁿ(0,1)), F(H)) as a general Hartogs figure.

The extension theorem for Euclidean Hartogs figures extends easily to general Hartogs figures.

Theorem 2.8. Let n≥ 2, let ( ˜∆ⁿ,H)˜ be a general Hartogs figure and let f: ˜H → C be a holomorphic function. Then there exists a holomorphic functionf˜: ˜∆ⁿ→C which extends f.

Proof. Since ( ˜∆ⁿ,H)˜ is a general Hartogs figure there exists a Euclidean Hartogs figure (∆ⁿ(0,1), H) and a biholomorphic function F: ∆ⁿ(0,1) → Cⁿ so that F(H) = ˜H and so that F(∆ⁿ(0,1)) = ˜∆ⁿ. It follows that the function G = (f ◦F)|_H:H → C is a holomorphic function which by Theorem2.7 has a holomorphic extension G: ∆˜ ⁿ(0,1)→ C. Now consider the holomorphic functionf˜=G◦F⁻¹: ˜∆ⁿ→C, thenf|˜_H_˜ = (G◦F⁻¹)|_H_˜ = f. It follows thatf˜is a holomorphic extension of f.

Definition 14. Let Ω ⊂ Cⁿ be an open set. We say that Ω is Hartogs- pseudoconvex if for each general Hartogs figure ( ˜∆ⁿ,H)˜ with H˜ ⊂ Ω we have that ∆˜ⁿ⊂Ω.

It is easy to see that a domain of holomorphy Ω ⊂ Cⁿ is necessarily Hartogs-pseudoconvex. If this were not the case then there would exist a general Hartogs figure ( ˜∆ⁿ,H)˜ so that H˜ ⊂ Ω while ∆˜ⁿ ( Ω. Letting p∈∂Ω∩H˜ and applying Theorem 2.7this would imply that any holomorphic functionf: Ω→Cwould extend past p.

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We shall soon see that domains of holomorphy may also be characterized by a form of disk convexity. We first introduce the concept of an analytic disk.

Definition 15. If ψ: D → Cⁿ is a continuous map which is holomorphic onD, then we refer toψ as an analytic disk. We shall refer to the mapψ|_S1

as the boundary of the analytic disk.

We shall frequently identify an analytic disk ψ:D→ Cⁿ with its image ψ D

⊂Cⁿ. Similarly we shall oftentimes refer to the image ψ(S¹) as the boundary of the analytic disk.

Definition 16. Let ψ: [0,1]×D→Cⁿ be a continuous function so that for each t∈[0,1]the map ψt:z7→ψ(t, z) is an analytic disk. Then ψ is called a continuous family of analytic disks.

Definition 17. Let Ω ⊂ Cⁿ be an open set. We say that Ω has the disk property if for each continuous family of analytic disks ψ: [0,1]×D→ Cⁿ with

ψ {0} ×D

∪ψ [0,1]×S¹

⊂Ω (15)

it follows that

ψ [0,1]×D

⊂Ω. (16)

It should be mentioned that there exist several (equivalent) definitions of the disk property. The above definition is taken from [5].

Definition 18. Let Ω⊂Cⁿ be an open set and let f: Ω→R∪ {−∞} be a function. We say that f is upper semicontinuous if for each c ∈ R the set {z∈Ω | f(z)< c} is open.

It is obvious that any continuous function is necessarily upper semicontinuous. In addition one can show that if f: Ω → R∪ {−∞} is upper semicontinuous then for each compactK ⊂Ωwe have that

−∞ ≤ Z

K

fdλ <∞. (17)

Definition 19. Let Ω⊂Cbe an open set. An upper semicontinuous function f: Ω→ R∪ {−∞} is called subharmonic if for each point z∈Ω there exists someρ >0 so that for all 0< r≤ρ one has

f(z)≤ 1 2π

Z 2π 0

f(z+re^iθ) dθ. (18) It is not hard to show that subharmonic functions satisfy the maximum principle.

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Theorem 2.9. Let Ω ⊂ Cⁿ be a connected open set and let f: Ω → R∪ {−∞} be a subharmonic function. If w∈Ωis a point so that f(w)≥f(z) for allz∈Ω, then f is constant.

The following result gives a simple way of classifying sufficiently regular subharmonic functions.

Theorem 2.10. Let Ω ⊂ Cⁿ be an open set and let f: Ω → R be a C²- smooth function. The functionf is subharmonic if and only if its Laplacian satisfies∆f ≥0.

The natural extension of subharmonicity to several variables is plurisub- harmonicity.

Definition 20. Let Ω ⊂ Cⁿ be an open set. An upper semicontinuous functionf: Ω→R∪ {−∞}is called plurisubharmonic if for eachz∈Ωand for each w ∈ Cⁿ one has that the function ζ 7→ f(z+ζw) is subharmonic wherever it is defined.

We are now ready to define the concept of pseudoconvexity.

Definition 21. Let Ω⊂Cⁿ be an open set and let d_∂Ω: Ω → R∪ {∞} be the distance function

d_∂Ω(z) = inf

w∈∂Ωkz−wk

Cⁿ. (19)

We say thatΩ is pseudoconvex if the function −logdΩ: Ω →R∪ {−∞} is plurisubharmonic.

It should be noted that the pseudoconvexity of a domainΩ⊂Cⁿ is typically defined by the existence of a plurisubharmonic exhaustion function.

These two definitions can however be shown to be entirely equivalent.

It can be shown that iff, g: Ω→R∪ {−∞}are plurisubharmonic functions, then so ismax{f, g}: Ω→R∪ {−∞}. This implies the following.

Theorem 2.11. LetΩ₁ ⊂CⁿandΩ₂ ⊂Cⁿbe pseudoconvex domains. Then Ω₁∩Ω₂ is pseudoconvex.

The famed solution to the Levi problem gives us several ways of verifying that a domain is a domain of holomorphy.

Theorem 2.12. (The solution to the Levi problem) Let Ω⊂Cⁿ be an open set, then the following are equivalent.

(i) Ωis a weak domain of holomorphy.

(ii) Ωis a domain of holomorphy.

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(iii) Ωis a local domain of holomorphy.

(iv) Ωis Hartogs pseudoconvex.

(v) Ωhas the disk property.

(vi) Ωis pseudoconvex.

For open setsΩ⊂CⁿwithC²-smooth boundary we may define domains of holomorphy by a simple differential criterion. Ifp∈∂Ω, then there exists a C²-smooth function r: U →R defined in an open neighborhoodU ⊂Cⁿ ofp so that

(i) ∂Ω∩U ={z∈U | r(z) = 0}, (ii) Ω∩U ={z∈U | r(z)<0}, (iii) dr6= 0 inU.

We refer to any such function as an oriented locally defined defining function for Ω at p. Furthermore we define the holomorphic tangent space at p as the subspaceTp^(1,0)∂Ω⊂Cⁿ given by

T_p^(1,0)∂Ω =







a∈Cⁿ |

n

X

j=1

aj

∂r

∂z_j(p) = 0







. (20)

It can be shown that this subspace is a complex(n−1)-dimensional subspace which is independent of the function r.

Definition 22. Let Ω ⊂Cⁿ be an open set with C²-smooth boundary. We say that Ω is Levi-pseudoconvex if there for each point p ∈ ∂Ω exists an oriented locally defined defining function r: U → R for Ω at p so that the Levi form Lpr:Tp^(1,0)∂Ω→R given by

L_pr(a₁, . . . a_n) =

n

X

j=1

X

k=1

∂²r

∂zj∂zk

(p)a_ja_j, (21) is non-negative.

It turns out that the condition thatLpr≥0is independent of the choice of functionr. In fact the following holds.

Theorem 2.13. LetΩ⊂Cⁿbe an open set withC²-smooth boundary, then Ωis pseudoconvex if and only if it is Levi-pseudoconvex.

Note that by the solution to the Levi problem this gives a simple way of verifying whether aC²-smooth domain is a domain of holomorphy.

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2.3 Envelopes of holomorphy

In Theorem2.7we saw that if(∆ⁿ(0,1), H)is a Euclidean figure withn≥2, then all holomorphic functions f:H → C admit a holomorphic extension f˜: ∆ⁿ(0,1) → C. This suggests the identification O(H) = O(∆ⁿ(0,1)).

Furthermore, since ∆ⁿ(0,1) is convex it follows from Theorem 2.6 that it is pseudoconvex. This implies that there exist no strictly larger domains Ω˜ )∆ⁿ(0,1) withO( ˜Ω) =O(H). In this sense we see that∆ⁿ(0,1)is the

“maximal” extension domain ofH. We make the following definition.

Definition 23. Let Ω ⊂ Cⁿ be a connected domain. Suppose that there exists a pseudoconvex connected set Ω˜ ⊃Ω so thatO(Ω) =O( ˜Ω). Then we refer toΩ˜ as the envelope of holomorphy of Ω.

Since the envelope of holomorphy of a domain is supposed to represent the maximal extension domain we should require that it is unique. The following theorem shows that this is indeed the case.

Theorem 2.14. Let Ω⊂Cⁿ be a connected domain with envelopes of holomorphy Ω˜₁ andΩ˜₂, then Ω˜₁ = ˜Ω₂.

Proof. If Ω⊂ Cⁿ is a connected domain with envelopes of holomorphy Ω˜1

and Ω˜₂. Then we may for each holomorphic function f: Ω→ C find holomorphic functions f˜₁: ˜Ω₁ → C and f˜₂: ˜Ω₂ → C which agree with f on Ω.

SinceΩ˜1 andΩ˜2are connected sets containingΩwe see thatΩ˜1∪Ω˜2 is connected. It follows by an application of the identity theorem (Theorem2.5) that the functionf˜: ˜Ω₁∪Ω˜₂ →C given by

f˜(z) =

(f˜1(z) forz∈Ω˜1

f˜₂(z) forz∈Ω˜₂ (22) is a well-defined holomorphic function. This construction suggests the iden- tificationO( ˜Ω₁) =O( ˜Ω₂) =O( ˜Ω₁∪Ω˜₂).

Now if Ω˜1 6= ˜Ω2, then there exists (after a possible relabeling) some point p∈∂Ω˜₁∩Ω˜₂. The above construction suggests that all holomorphic functions on Ω˜₁ extend to be holomorphic on the connected set Ω˜₁ ∪Ω˜₂ which implies that the domain Ω˜1 is not a weak domain of holomorphy.

By the solution to the Levi problem (Theorem 2.12) this contradicts the pseudoconvexity ofΩ˜₁. We must therefore conclude that Ω˜₁ = ˜Ω₂.

It is important to note that there exist connected domainsΩ⊂Cⁿwhich admit no envelope of holomorphy, an example of such a domain may be found in [4]. This can be remedied if we allow our holomorphic functions to extend

“holomorphically” to a special kind ofn-dimensional complex manifold. We shall not need to do so, instead we introduce the following definition.

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Definition 24. Let Ω⊂Cⁿ be a connected domain whose envelope of holomorphy exists. Then we refer to Ωas a schlicht domain.

We will need the following property of schlicht domains.

Theorem 2.15. Let Ω ⊂ Cⁿ be a schlicht domain with envelope of holo- morphyΩ˜ ⊂Cⁿ. Iff: ˜Ω→Cis a holomorphic function, thenf(Ω) =f( ˜Ω).

Proof. If this is not the case then there exists some point w∈Ω˜\Ωso that f(w)∈/f(Ω). Now define the holomorphic functiong: Ω→C by letting

g(z) = 1

f(z)−f(w). (23)

By the definition of the envelope of holomorphy this function has a holomorphic extension˜g: ˜Ω→C. Applying the identity theorem we see that

f(z) = 1

˜

g(z) +f(w). (24)

Evaluating this expression atz =w gives that 1/˜g(w) = 0 which is clearly absurd. We must therefore conclude thatf(Ω) =f( ˜Ω).

In general it is not a simple task to verify whether that a domain is schlicht. In order to give one way of showing that a domain is schlicht we introduce the Hartogs hull operator which is inspired by the fact that a domain is pseudoconvex if and only if it is Hartogs pseudoconvex.

Definition 25. Let Dⁿ denote the collection of connected sets in Cⁿ. We define the Hartogs hull operator T:Dⁿ → Dⁿ by lettingT(Ω) be the collection of all pointsz∈Cⁿ that are contained in some∆˜ⁿ, where ( ˜∆ⁿ,H)˜ is a general Hartogs figure with H˜ ⊂Ω.

It is easy to see that the Hartogs hull operator is well-defined. IfΩ⊂Cⁿ is a connected domain then we define a sequence of nested open connected sets

Ω =T⁰(Ω)⊂T(Ω)⊂T²(Ω)⊂ . . . ⊂T^j(Ω)⊂ . . . (25) by the recursive relation T^k+1(Ω) = T(T^k(Ω)). The importance of the Hartogs hull operator follows from the following theorem.

Theorem 2.16. Let Ω⊂Cⁿ be a connected domain and let T^∞(Ω) = [

j∈N

T^j(Ω). (26)

ThenT^∞(Ω)is a pseudoconvex connected domain. Moreover ifΩis schlicht, thenT^∞(Ω) is its envelope of holomorphy.

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Proof. We first show that ifΩ⊂Cⁿis any domain, thenT^∞(Ω)is a pseudoconvex connected set. The fact that T^∞(Ω) is a connected domain follows easily from a topological argument. To see that it is also pseudoconvex we recall from the solution to the Levi problem (Theorem 2.12) that T^∞(Ω) is pseudoconvex if and only if it is Hartogs pseudoconvex. Now if ( ˜∆ⁿ,H)˜ is a general Hartogs figure with H˜ ⊂ T^∞(Ω), then H˜ ⊂ T^k(Ω) for some k ∈ N. It follows from the definition of the Hartogs hull operator that

∆˜ⁿ⊂T^k+1(Ω)⊂T^∞(Ω).

Now suppose that Ω is a schlicht domain with envelope of holomorphy Ω, we need to show that˜ Ω =˜ T^∞(Ω). By another application of the solution to the Levi problem we see that the envelope of holomorphy Ω˜ is Hartogs pseudoconvex. SinceΩ⊂Ω˜ this implies thatT^j(Ω)⊂Ω˜ for allj∈Nwhich shows thatT^∞(Ω)⊂Ω.˜

Suppose now that this inclusion is strict, then by another application of the solution to the Levi problem we see that the set T^∞(Ω)is a domain of holomorphy contained in Ω. It follows that there exists a holomorphic˜ function f:T^∞(Ω)→ C which does not extend to be holomorphic on any Ω. But then the function˜ f|_Ω: Ω→Cis a holomorphic function which has no holomorphic extension to the envelope of holomorphy Ω˜ which contradicts the assumption that Ω˜ is the envelope of holomorphy of Ω. We must therefore conclude thatT^∞(Ω) = ˜Ω.

It is interesting to note that even for a non-schlicht domain Ω⊂Cⁿ the set T^∞(Ω) is always pseudoconvex. The reason for T^∞(Ω) not being the envelope of holomorphy of Ω must therefore stem from the fact that there exists some k∈N0 for which not all holomorphic functions f:T^k(Ω)→ C extend to be holomorphic on T^k+1(Ω). The existence of non-schlicht domains might therefore appear to contradict Theorem 2.8 which states that if ( ˜∆ⁿ,H)˜ is a general Hartogs figure with H˜ ⊂ T^k(Ω). Then each holomorphic functionf: ˜H →Chas a holomorphic extensionf˜: ˜∆ⁿ→C. This gives a local holomorphic extension of f to an open set inT^k+1(Ω), but it may fail to give a global holomorphic extension to all of T^k+1(Ω).

To examine what may go wrong we attempt to define a global holomorphic extension f˜:T^k+1(Ω)→ Cin the following way. If z∈ T^k+1(Ω) then there exists a general Hartogs figure ( ˜∆ⁿ_z,H˜_z) so that H˜_z ⊂T^k(Ω)and so thatz∈∆˜ⁿ_z. Now consider the holomorphic functionf|_H_˜

z: ˜H_z→Cand let f˜_z: ˜∆ⁿ_z →C be the holomorphic extension from Theorem 2.8. We wish to define a global holomorphic extension off by definingf˜(z) =f_z^∗(z), where

f_z^∗(z) =

(f(z) for z∈Ω,

f˜z(z) for z∈∆˜ⁿ_z. (27)

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If the functionsf_z^∗ are well-defined then by a connectedness argument and the identity principle one sees that this indeed gives a well-defined holomorphic extension. This need however not be the case, in order to show that the functions f_z^∗ are well-defined we would have to show that for all z∈T^k(Ω)∩∆˜ⁿ_z we have thatf(z) = ˜fz(z). If the setT^k(Ω)∩∆˜ⁿ_z is connected then this follows from the identity principle, but in the case where the set fails to be connected the function need not be well-defined. This is exactly what goes wrong in the example of a non-schlicht domain Ω⊂C² given in [4] where the functionf(z, w) =√

z−3has a well-defined local holomorphic extension at each point but it fails to form a global holomorphic extension.

The above idea immediately gives the following interesting result.

Theorem 2.17. Let D^∗ ⊂ Dbe a class of connected open domains with the following properties:

(i) If Ω∈ D^∗, then T(Ω)∈ D^∗.

(ii) For eachΩ∈ D^∗ and each general Hartogs figure( ˜∆ⁿ,H)˜ withH˜ ⊂Ω, we have that Ω∩∆˜ⁿ is connected.

Then each element Ω∈ D^∗ is schlicht.

Proof. LetΩ∈ D^∗, thenΩ is schlicht if and only if we for eachk∈N0 and each holomorphic function f: T^k(Ω) → C have a holomorphic extension f˜: T^k+1(Ω) → C. By the above discussion this will be the case if we can show that for any general Hartogs figure( ˜∆ⁿ,H)˜ withH˜ ⊂T^k(Ω)we have that T^k(Ω)∩∆˜ⁿ is a connected set. This follows immediately from the second property of the class D^∗.

In fact by the solution to the Levi problem we have that a domain is pseudoconvex if and only if it is pseudoconvex locally. This can be used in order to define a generalized Hartogs hull operators T: Dⁿ → Dⁿ for 0 < ≤ ∞ by only considering general Hartogs figures whose diameter is smaller than . It is not hard to show that these operators also lead to a way of constructing the envelope of holomorphy of schlicht domains.

3 Embedded submanifolds of C

ⁿ

We shall assume that the reader is familiar with the concept of real and complex differentiable manifolds (both with and without boundary), differentiable functions between such manifolds, as well as some basic knowledge of real and complex differential forms. In this chapter we introduce the concept of an embedded submanifold of Cⁿ and we study its real and complexified tangent space. We then introduce a special class of embedded submanifolds known as CR-submanifolds whose geometric tangent bundle

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contains a complex subbundle. Important examples ofCR-submanifolds include the real hypersurfaces and complex submanifolds. Many of the results in this chapter are taken from [1], but we offer several anternate, missing and expanded proofs.

3.1 Real embedded submanifolds of Cⁿ

Definition 26. Let M ⊂Cⁿ, we say that M is a C^s-smooth (1≤s≤ ∞) embedded submanifold of Cⁿ of codimension 0 ≤d≤2n if for each p ∈M there exists an open set U ⊂ Cⁿ, and a C^s-smooth function r: U → R^d which satisfies the following:

(i) M∩U ={z∈U | r(z) = 0}.

(ii) dr₁∧ · · · ∧dr_d6= 0 onU.

We refer to any such functionr as a locally defined defining function for M atp.

It can be shown that ifM ⊂Cⁿ is aC^s-smooth(1≤s≤ ∞) embedded submanifold of codimension0≤d≤2n, then M has the structure of a C^s- smooth manifold of dimension2n−dequipped with the subspace topology.

In addition we remark that the embedded submanifolds of codimensiond= 0 are exactly the open sets ofCⁿ.

Definition 27. Let p ∈Cⁿ, we define the tangent space of Cⁿ at p as the real 2n-dimensional vector space

T_pCⁿ= span_R ( ∂

∂x1

p

, ∂

∂y1

p

, . . . , ∂

∂xn

p

, ∂

∂yn

p

)

. (28) Definition 28. Let M ⊂ Cⁿ be an embedded submanifold of codimension 0≤d≤2n and letp∈M. We define the tangent space of M at p to be the real (2n−d)-dimensional vector space

TpM ={X_p∈TpCⁿ | Xprj = 0 for all 1≤j ≤d}. (29) Wherer:U →R^d is any locally defined defining function for M at p.

It can be shown that the definition of the tangent space T_pM does not depend on the choice of locally defined defining function atp. Note that we do not define the tangent space TpM to be the space of point-derivations at p. This is done in order to avoid infinite-dimensional tangent spaces.

Indeed, ifM is aC^s-smooth manifold then the space of point-derivations at pis finite-dimensional if and only ifs=∞.

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Definition 29. Let M ⊂ Cⁿ be an embedded submanifold. We define the tangent bundle of M as the set

T M = [

p∈M

{p} ×TpM. (30) For our applications it is sufficient to treat the tangent bundle T M as a set without any additional structure. With some work we could however have givenT M the structure of a manifold. This would give us a particularly elegant formulation of vector fields on M. Instead we shall settle for the following equivalent formulation.

Definition 30. Let M ⊂ Cⁿ be an embedded submanifold. A C^s-smooth (0≤s≤ ∞) vector field on M is a mapX:M →T M of the form X(p) = (p, Xp). Where

Xp =

n

X

j=1

aj(p) ∂

∂xj

p

+bj(p) ∂

∂yj

p

∈TpM, (31) has coefficientsa_j(p) and b_j(p) depending C^s-smoothly on p.

We shall frequently identify a vector field X: M →T M with the associated projection p7→Xp.

Definition 31. LetV be a real vector space. We define the complexification of V as the complex vector space

CV =C⊗V. (32) In other words the complexification CV is exactly the complex vector space we obtain by extendingV to be a vector space over the complex numbers. It is clear that ifV is finite-dimensional thendim_RV = dim_CCV.

In order to better understand the complexified tangent spaces ofCⁿ we recall the Wirtinger partial differential operators defined for1≤j≤nby

∂

∂zj

= 1 2

∂

∂xj

−i ∂

∂yj

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∂

∂zj

= 1 2

∂

∂xj

+i ∂

∂yj

. (34)

The Wirtinger partial differential operators give an alternate way of de- scribing the complexified tangent spaces ofCⁿ.

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Theorem 3.1. Let p∈Cⁿ then

CT_pCⁿ= span_C ( ∂

∂x₁ p

, ∂

∂y₁ p

, . . . , ∂

∂x_n p

, ∂

∂y_n p

)

, (35) or equivalently

CT_pCⁿ= span_C ( ∂

∂z1

p

, ∂

∂z1

p

, . . . , ∂

∂zn

p

, ∂

∂zn

p

)

. (36) The concept of a tangent bundle of an embedded submanifold easily generalizes to complexified tangent spaces.

Definition 32. Let M ⊂ Cⁿ be an embedded submanifold, we define the complexified tangent bundle of M as the set

CT M = [

p∈M

{p} ×CTpM. (37) Definition 33. Let M ⊂ Cⁿ be an embedded submanifold. A C^s-smooth (0≤s≤ ∞) complex vector field on M is a map X:M →T M of the form X(p) = (p, X_p). Where X_p ∈CT_pM, and where

X_p =

n

X

j=1

a_j(p) ∂

∂xj

p

+b_j(p) ∂

∂yj

p

, (38)

has coefficientsa_j(p) and b_j(p) depending C^s-smoothly on p.

As with real vector fields we shall typically identify a complex vector fieldX:M →T M with the associated projectionp7→X_p.

It is easy to verify that a map X:M → T M is a C^s-smooth complex vector field if and only if it is of the formX(p) = (p, X_p), where

X_p =

n

X

j=1

c_j(p) ∂

∂z_j p

+d_j ∂

∂z_j p

∈CT_pM, (39) has coefficientsc_j(p) and d_j(p) depending C^s-smoothly onp.

Using the Wirtinger partial derivatives we introduce the concept of a conjugate tangent vector.

Definition 34. Let p∈Cⁿ and letXp ∈CTpCⁿ be given by Xp =

n

X

j=1

aj

∂

∂z_j p

+bj

∂

∂z_j p

. (40)

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We define the conjugate tangent vector X_p ∈CT_pCⁿ by X_p =

n

X

j=1

b_j ∂

∂z_j p

+a_j ∂

∂z_j p

. (41)

It is trivial to show that conjugation is an antilinear and involutive op- eration. In this sense we see that conjugation of tangent vectors behaves exactly as the usual conjugation of complex numbers. With some more work we shall show that if M ⊂ Cⁿ is an embedded submanifold and p ∈ M.

Then for anyX_p ∈CT_pM we also have thatX_p ∈CT_pM. In order to show this we first prove the following theorem which is important in its own right.

Theorem 3.2. Let M ⊂Cⁿ be an embedded submanifold and letp∈M. If r:U →R^d is a locally defined defining function for M at p, then

CT_pM ={X_p∈CT_pCⁿ | X_pr_j = 0 for all 1≤j≤d}. (42) Proof. Letr:U →R^dbe a locally defined defining function forMatp. First suppose that Xp ∈ CTpM, then Xp = ajYp +ibjYp for some real tangent vector Yp ∈ TpM. It follows trivially that Xprj = 0 for all 1 ≤ j ≤ d.

Conversely suppose thatX_p ∈CT_pCⁿis such thatX_pr_j = 0for all1≤j≤d.

Writing

X_p=

n

X

k=1

(a_k+ib_k) ∂

∂xk

p

+ (c_k+id_k) ∂

∂yk

p

(43) for real numbersak, bk, ckanddkwe get thatXp =Yp+iZp. WhereYp, Zp ∈ T_pCⁿ are real tangent vectors of Cⁿ atpgiven by

Yp =

n

X

k=1

a_k ∂

∂x_k p

+c_k ∂

∂y_k p

, (44)

and

Z_p=

n

X

k=1

b_k ∂

∂x_k p

+d_k ∂

∂y_k p

. (45)

By assumptionXprj =Yprj+iZprj = 0for all 1≤j≤d, this implies that Y_p, Z_p ∈T_pM. It follows that X_p=Y_p+iZ_p ∈CT_pM.

Using this theorem we can show that complexified tangent spaces are closed under conjugation.

Theorem 3.3. Let M ⊂Cⁿ be an embedded submanifold and letp∈M. If Xp ∈CTpM, then Xp ∈CTpM.

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The real numbers R may be regarded as the points z ∈ C that are unchanged by conjugation. The following theorem shows that a similar result holds for tangent spaces; the real tangent space TpM is exactly the collection of complexified tangent vectors X_p ∈ CT_pM which satisfy X_p = Xp.

Theorem 3.4. Let M ⊂ Cⁿ be an embedded submanifold and let p ∈ M. Then

T_pM =

X_p ∈CT_pM | X_p =X_p . (46) In other words T_pM is exactly the real subspace of complexified tangent vectors X_p∈CT_pM of the form

Xp =

n

X

j=1

cj

∂

∂zj

p

+cj

∂

∂zj

p

. (47)

Proof. First suppose that X_p = Pn j=1a_j_∂x^∂

j

p +b_j_∂y^∂

j

p ∈ T_pM. Define c_j =a_j+ib_j, then a simple computation shows that

Xp =

n

X

j=1

cj

∂

∂zj

p

+cj

∂

∂zj

p

. (48)

In other wordsXp =Xp. Conversely, ifXp∈CTpM satisfiesXp =Xp, then X_p is of the form

Xp =

n

X

j=1

cj

∂

∂zj

p

+cj

∂

∂zj

p

. (49)

Writingcj =aj+ibj whereaj and bj are real, we see that Xp =

n

X

j=1

aj

∂

∂x_j p

+bj

∂

∂y_j p

. (50)

Which is clearly a real tangent vector, that isXp ∈TpM. 3.2 The complex tangent space

LetM ⊂Cⁿ be an embedded submanifold of codimension 0≤d≤2nand let p∈M. We may regard the tangent space TpM as a subspace of Cⁿ by identifying each tangent vector

Xp =

n

X

j=1

aj

∂

∂x_j p

+bj

∂

∂y_j p

∈TpM (51) with its corresponding geometric tangent vector(a1+ib1, . . . an+ibn)∈Cⁿ. Under this identification we may regardT_pM as a real(2n−d)-dimensional

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subspace of Cⁿ. In general this space does not have the structure of a complex vector subspace ofCⁿ, but we may always decompose

TpM =HTpM⊕XpM, (52) whereHTpM is the maximal complex subspace

HTpM ={z∈TpM | iz ∈TpM}, (53) and whereXpM is the orthogonal complement of HTpM in TpM. In order to make sense of this decomposition whenT_pM is regarded as the usual al- gebraic tangent space we introduce the standard smooth structure onT_pCⁿ. Definition 35. Let p ∈ Cⁿ, we defined the standard smooth structure on T_pCⁿ as the linear map J:T_pCⁿ→T_pCⁿ

J





n

X

j=1

aj

∂

∂xj

p

+ ∂

∂yj

p



=

n

X

j=1

−b_j ∂

∂xj

p

+aj

∂

∂yj

p

. (54) Note that theJ-operator corresponds to multiplication by the imaginary unitiin the geometric tangent spaceTpCⁿ⊂Cⁿ. This suggests the following result.

Theorem 3.5. Let M ⊂ Cⁿ be an embedded submanifold and let p ∈ M. The tangent space TpM decomposes as TpM = HTpM ⊕XpM where HT_pM ⊂T_pM is the complex tangent space

HT_pM ={X_p ∈T_pM | J(X_p)∈T_pM}, (55) and where X_p ⊂T_pM is the real part of T_pM given as the orthogonal complement ofHTpM in TpM.

Similar to the definition of the tangent bundle we introduce the following notation.

Definition 36. Let M ⊂ Cⁿ be an embedded submanifold, we define the subsets HT M ⊂T M andXM ⊂T M by

HT M = [

p∈M

{p} ×HTpM, (56) and

XM = [

p∈M

{p} ×X_pM. (57) We shall later see that the subsetsHT M, XM ⊂T M need not be subbundles. On the other hand the decomposition TpM = HTpM ⊕XpM clearly implies that HT M ⊂T M is a subbundle if and only ifXM ⊂T M is a subbundle.

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Theorem 3.6. Let M ⊂ Cⁿ be an embedded submanifold and let p ∈ M. ThenHTpM has a basis of the form

{X₁, J(X₁), . . . , X_m, J(X_m)}, (58) where X_j ∈T_pM for each 1≤j≤m.

Proof. Let Xp ∈HTpM be any non-zero vector, then Xp and J(Xp) form a linearly independent set. The existence of a basis as above then follows from a simple induction argument.

The next theorem gives an important bound on the dimension of the complex tangent spaceHTpM.

Theorem 3.7. Let M ⊂ Cⁿ be an embedded submanifold of codimension 0 ≤ d ≤ 2n and let p ∈ M. The dimension of the complex tangent space HTpM ⊂TpM is an even number which satisfies the inequality.

2n−2d≤dimHTpM ≤2n−d. (59) Proof. The fact that HTpM is even-dimensional follows immediately from the previous theorem. The upper bound dimHTpM ≤2n−dis an imme- diate consequence of the inclusion HT_pM ⊂ T_pM. In order to obtain the lower bound we write HTpM =TpM ∩J(TpM) and consider the quotient space

TpM⊕J(TpM)

T_pM∩J(T_pM). (60) Noting thatTpM⊕J(TpM)⊂TpCⁿ we see that

dimTpM ⊕J(TpM)

T_pM ∩J(T_pM) ≤dimTpM⊕J(TpM)≤dimTpCⁿ= 2n. (61) It follows that

dimTpM + dimJ(TpM)−dimHTpM = dimTpM ⊕J(TpM)

TpM∩J(TpM) ≤2n. (62) One easily verifies that dimT_pM = dimJ(T_pM) = 2n−d which when combined with the above inequality easily yields the necessary lower bound.

Later we shall use the above lower bound in order to define the concept of a genericCR-submanifold. It is therefore of some interest to investigate whether this is the best possible bound for a given n ∈ N and a given 0≤d≤2n. Suppose first that d≤n, thenM =C^n−d×R^d⊂Cⁿ is clearly an embedded submanifold of codimensiond. Furthermore, if p∈M then

HTpM = span ( ∂

∂x₁ p

, ∂

∂y₁ p

, . . . ∂

∂xn−d

p

, ∂

∂yn−d

p

)

, (63)

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so thatdimHT_pM = 2n−2d. If insteadn < d, thenM =R^2n−d× {0}^d−n⊂ Cⁿ is an embedded submanifold of codimensiond. Now if p∈M then one easily shows thatHTpM is trivial, so thatdimHTpM = 0. This shows that the given lower bound is indeed the best possible bound.

We now turn our attention towards the complexified complex tangent spaceCHT_pM. We first introduce the following useful notation.

Definition 37. Let M ⊂ Cⁿ be an embedded submanifold, we define the subsets CHT M ⊂CT M and CXM ⊂CT M by

CHT M = [

p∈M

{p} ×CHTpM, (64) and

CXM = [

p∈M

{p} ×CXM. (65)

We now introduce the complex vector subspaces Tp^(1,0)M ⊂CTpM and Tp^(0,1)M ⊂CTpM. We then show that these spaces lead to a nice decomposition ofCHTpM.

Definition 38. Letp∈Cⁿ, we define the complex vector subspacesTp^(1,0)Cⁿ⊂ CTpCⁿ and Tp^(0,1)Cⁿ⊂CTpCⁿ by

T_p^(1,0)Cⁿ= span_C ( ∂

∂z₁ p

, . . . , ∂

∂z_n p

)

, (66)

and

T_p^(0,1)Cⁿ= span_C ( ∂

∂z₁ p

, . . . , ∂

∂z_n p

)

. (67)

Definition 39. Let M ⊂ Cⁿ be an embedded submanifold of codimension 0 ≤ d ≤ 2n and let p ∈ M. We define the complex vector subspaces T_p^(1,0)M ⊂CT_pM and T_p^(0,1)M ⊂CT_pM by

T_p^(1,0)M =CT_pM ∩T_p^(1,0)Cⁿ, (68) and

T_p^(0,1)M =CT_pM ∩T_p^(0,1)Cⁿ. (69) The space Tp^(1,0)M is often referred to as the holomorphic tangent space atM atp, while the spaceTp^(0,1)M is often referred to as the antiholomor- phic tangent space ofM atp.

The following theorem gives a simple way of computing the spacesTp^(1,0)M and Tp^(0,1)M. The proof is a trivial consequence of Theorem3.2.

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Theorem 3.8. Let M ⊂ Cⁿ be an embedded submanifold of codimension 0 ≤ d ≤ 2n and let p ∈ M. If r:U → R^d is a locally defined defining function forM atp, then

(i) Tp^(1,0)M = n

Xp ∈Tp^(1,0)Cⁿ | Xprj = 0 for 1≤j ≤d o

, (ii) Tp^(0,1)M =n

X_p ∈Tp^(0,1)Cⁿ | X_pr_j = 0 for 1≤j ≤do .

Theorem 3.9. Let M ⊂ Cⁿ be an embedded submanifold and let p ∈ M. Then the following holds

(i) Tp^(1,0)M ∩Tp^(0,1)M ={0}, (ii) Tp^(0,1)M =Tp^(1,0)M.

Proof. The first property is clear from the fact thatTp^(1,0)Cⁿ∩Tp^(0,1)Cⁿ. In order to prove the second property, let0≤d≤2nbe the codimension ofM and let r:U → R^d be a locally defined defining function forM atp. Note from Theorem 3.8 that X_p = P_n

j=1a_j_∂z^∂

j

p ∈CT_pM lies inTp^(0,1)M if and only ifX_pr_j = 0for all 1≤j ≤d. Equivalently, using that r is real-valued we see that Xp lies inTp^(0,1)M if and only if Xprj = 0 for all 1≤j≤d. By Theorem3.8this is equivalent toXp ∈Tp^(1,0)M.

Definition 40. Let M ⊂ Cⁿ be an embedded submanifold. We define the sets T^(1,0)M ⊂CT M and T^(0,1)M ⊂CT M by letting

T^(1,0)M = [

p∈M

{p} ×T_p^(1,0)M, (70) and letting

T^(0,1)M = [

p∈M

{p} ×T_p^(0,1)M. (71) We shall later see that the subsets T^(1,0)M, T^(0,1)M ⊂ CT M need not be subbundles. Note however that from Theorem 3.9 the set T^(1,0)M will be a subbundle if and only if T^(0,1)M is a subbundle.

Definition 41. Let M ⊂ Cⁿ be an embedded submanifold. A C^s-smooth (0 ≤ s ≤ ∞) (1,0)-vector field over M is a C^s-smooth complex vector field of the form X:M → T^(1,0)M. Similarly a C^s-smooth (0 ≤ s ≤ ∞) (0,1)-vector field over M is a C^s-smooth complex vector field of the form X:M →T^(0,1)M.

Let V be a real vector space. A linear map J:V → V is called a complex structure ifJ²(x) =−x for eachx∈V. A basis result on complex structures says that any such map extends to aC-linear mapJ:CV →CV. This immediately implies the following.

CR extensions with a classical Several Complex Variables point of view