Fields of CR meromorphic functions

Fields of CR Meromorphic Functions.

C. DENSON HILL(*) - MAURO NACINOVICH(**)

ABSTRACT- LetMbe a smooth compactCRmanifold ofCRdimensionnandCR codimensionk, which has a certain local extension propertyE. In particular, ifMis pseudoconcave, it has propertyE. Then the fieldK(M) ofCRmero- morphic functions on M has transcendence degree d, with dGn1k. If f₁,f₂,R,f_d is a maximal set of algebraically independentCR meromorphic functions onM, thenK(M) is a simple finite algebraic extension of the field C(f₁,f₂,R, f_d) of rational functions of thef₁,f₂, R,f_d. WhenMhas a pro- jective embedding, there is an analogue of Chow’s theorem, andK(M) is iso- morphic to the fieldR(Y) of rational functions on an irreducible projective al- gebraic varietyY, andMhas aCRembedding in regY. The equivalence be- tween algebraic dependence and analytic dependence fails when conditionE is dropped.

Introduction.

In a beautiful paper Siegel [Si], improving upon an idea of Serre [Se], managed to give simple proofs of the basic theorems concerning algebra- ic dependence and transcendence degree for the field of meromorphic functions on an arbitrary compact complex manifold; thereby generaliz- ing classical results about the field of Abelian functions on a complex n dimensional torus. For a detailed discussion of the now nearly 150 year history of these matters, see the paper of Siegel. His proofs were based on his extension ton dimensions of the classical Schwarz lemma. Later, (*) Indirizzo dell’A.: Department of Mathematics, SUNY at Stony Brook, Stony Brook NY 11794, USA. E-mail: dhillHmath.sunysb.edu

(**) Indirizzo dell’A.: Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica, 00133 - Roma, Italy.

E-mail: nacinoviHmat.uniroma2.it

following almost exactly Siegel’s argument, Andreotti and Grauert [AG]

were able to show that the Siegel modular group, which plays a pivotal role in the study of algebraic fields of automorphic functions, is pseudo- concave. Later Andreotti [A] generalized these kinds of results to gen- eral pseudoconcave complex manifolds and spaces; again following Siegel’s method.

In the present work we replace the compact complex manifold of Siegel by a smooth compact pseudoconcave CR manifold M of general CRdimensionnandCRcodimensionk, and study algebraic dependence, transcendence degree and related matters for the field K^{(M) of CR} meromorphic functions on M. Again we follow the method of Siegel, based on the Schwarz lemma, and we incorporate some ideas used by Andreotti. Actually we are able to obtain results under a condition on the CR manifold that is weaker than pseudoconcavity, which we call condi- tion E. In particular we obtain an analogue of Chow’s theorem [C] for compactCRmanifolds. In the situation whereMhas a projective embed- ding, we are able to identifyK(M) with the field D(Y) of rational func- tions on an irreducible algebraic varietyY, in whichMhas a generic CR embedding that avoids the singularities ofY. We show that the possibility for M to have a projective embedding is equivalent to the existence of a complexCRline bundle overM having certain properties. In this context, it is interesting to note that the general abstract notion of a complexCR line bundleF over a CRmanifold is such thatF may fail to be locallyCR trivializable, even in the case where M is CR embeddable [HN8].

For more information about pseudoconcaveCRmanifolds, we refer the reader to the foundational paper [HN3], to the many examples in [HN8], and to [HN1], [HN2],R, [HN11], as well as [BHN], [DCN], and [L].

1. Preliminaries.

An abstract smooth almostCR manifold of type (n,k) consists of: a connected smooth paracompact manifold M of dimension 2n1k, a smooth subbundle HMof TM of rank 2n, that we call the holomorphic tangent space of M, and a smooth complex structure J on the fibers of HM.

LetT^{0 , 1}Mbe the complex subbundle of the complexificationCHMof HM, which corresponds to the 2k₂1 eigenspace of J:

T^{0 , 1}M4 ]X1k₂1JXNXHM(. (1.1)

We say that M is a CR manifold if, moreover, the formal integrability condition

[C^{Q(M,T0 , 1M),}C^{Q(M,T0 , 1M) ]%}C^{Q(M,T0 , 1M)} (1.2)

holds. When k40 , via the Newlander-Nirenberg theorem, we recover the definition of a complex manifold.

Next we defineT*^{1 , 0}Mas the annihilator ofT^{0 , 1}M in the complexi- fied cotangent bundleCT*M. We denote byQ^{0 , 1}Mthe quotient bundle CT*M/T*^{1 , 0}M, with projectionpQ. It is a rankncomplex vector bundle onM, dual toT^{0 , 1}M. The¯_M-operator acting on smooth functions is de- fined by ¯_M4pQid. A local trivialization of the bundle Q^{0 , 1}M on an open setUinMdefinesnsmooth sectionsL1,L2,R,LnofT^{0 , 1}MinU;

hence

¯_Mu4(L₁u,L₂u,R,L_nu) , (1.3)

whereu is a function in U. Solutions u of ¯_Mu40 are called CR func- tions. We denote by C R(U) the space of smooth (C^{Q) functions on an} open subsetUofMthat satisfy¯_Mu40 . Note that¯_Mis a homogeneous first order partial differential operator and hence the spaceC R^{(U) is a} commutative algebra with respect to the multiplication of functions. We denote by C RM,a4lim

UKa

C R(U) the local ring of germs of smooth C R functions at aM.

Let M1,M2be two abstract smooth CRmanifolds, with holomorphic tangent spaces HM1, HM2, and partial complex structures J1, J2, re- spectively. A smooth map f : M₁KM₂ is CR if f

*(HM₁)%HM₂, and f*(J₁v)4J₂f

*(v) for every vHM₁.

ACR embedding(¹)f of an abstractCR manifold M into a complex manifold X, with complex structure J_X, is a CR map which is a smooth embedding satisfying f

*(HaM)4f

*(TaM)OJX(f

*(TaM) ) for every aM. We say that the embedding isgenericif the complex dimension of X is (n1k), where (n,k) is the type of M.

LetM be a smooth abstractCRmanifold of type (n,k). We say that Mislocally embeddableataM, ifahas an open neighborhoodvainM which admits a CR embedding into some complex manifold Xa. In this case we can always take forXaan open subset of C^n1kand assume that the embeddingv_a^%KX_ais generic. The property of being locally embed- (¹) In this case we shall often identify M with the submanifold f(M) of X.

dable atais equivalent to the fact that there exist an open neighborhood va of a and functions f₁,f₂,R,f_n1kC R^(va) such that

df₁(a)Rdf₂(a)RRRdf_n1k(a)c0 . (1.4)

The functionsf₁,f₂,R,f_n1k can be taken to be the restrictions tovaof the coordinate functions z₁,z₂,R,z_n1k of X_a%C^n1k. For this reason one can say that they provide CR coordinates on M near a.

The characteristic bundle H⁰M is defined to be the annihilator of HMinT*M. Its purpose it to parametrize the Levi form: recall that the Levi form of M at x is defined for jH_x⁰M and XH_xM by

L^(j;X)₄^djA(X,JX)₄aj, [J XA,XA]b, (1.5)

wherejAC^{Q(M,H0M) andXA}C^Q(M,HM) are smooth extensions of jand X. For each fixedj it is a Hermitian quadratic form for the com- plex structure J_x on H_xM.

A CR manifold M is said to be q-pseudoconcave if the Levi form L(j;Q) has at leastqnegative andqpositive eigenvalues for everyaM and every nonzero jH_a⁰M.

By the termpseudoconcave CR manifold Mwe mean an abstractCR manifold which is: (i) locally embeddable at each point, and (ii) 1-pseudoconcave.

In this paper we shall be concerned with CR manifolds M of type (n,k) which have a certain propertyE(Eis forextension).M is said to have property E iff there is an E-pair (M,X). By an E-pair we mean that

(i) M is a generic CR submanifold of the complex manifold X, and

(ii) for each aM, the restriction map induces an isomorphism OX,aKC RM,a.

REMARK. If M is a pseudoconcave CR manifold, then M has proper- ty E.

In fact, property (i) for a pseudoconcaveMwas proved in Proposition 3.1 of [HN3]; however, Theorem 1.3 below gives a new simplified proof of this fact. Property (ii) for a pseudoconcaveM was proved in [BP], [NV];

however, a very short proof of this fact is also given by Theorem 13.2 in [HN7]. Thus propertyEis to be regarded as a somewhat weaker hypoth- esis on M than pseudoconcavity.

Whenk40 , soMis of type (n, 0 ), thenM is ann-dimensional com- plex manifold, and we obtain an E pair by choosing X4M. Hence we adopt the convention that any complex manifold has property E.

Whenn40 , soMis of type ( 0 ,k), thenM is a smooth totally realk- dimensional manifold, and we can never obtain an E-pair, (unless M4 4X4a point), because then any smooth function belongs to C R^(M).

THEOREM 1.1. Let(M,X) be an E-pair. Then for any open setv%

%M there is a corresponding open set V%X such that (i) VOM4v, and

(ii) r:O^(V)KC R^(v) is an isomorphism.

PROOF. We fix a Hermitian metric g on X, with associated distance d(x,y). Let av and consider

Fn4

m

g

g

hh _N

g

h

n

(1.6)

Here B

g

h

n in X, centered at a. Note that eachFnis a closed subspace of a Fréchet-Schwartz space, and hence a Fréchet-Schwartz space itself. For each n, the map

p_n:Fn(f,fA)KfC R^(v) (1.7)

is linear and continuous. By our hypothesis,

n

0

pn(Fn)4C R^{(v) .} (1.8)

Hence by the Baire category theorem, somepn₀(Fn0) is of the second cat- egory. It follows from a theorem of Banach thatp_n0:Fn0KC R(v) is sur- jective. Now we denote B

g

h

Next we fix a tubular neighborhoodUofMinX, withp: UKMde- noting the orthogonal projection. By letting rC^{Q(v,R1) vary, we} produce a fundamental system of open neighborhoods

Vr4 ]zUNp(z)v, and d(z,p(z) )Er(p(z) )( (1.9)

of v in X. We choose r0C^Q(v,R1) such that V_r0%_av

0

(1.10)

Since M is a deformation retract of Vr₀, and the local holomorphic ex- tension of CRfunctions from the generic M is unique, the different ex- tensions to each Ba of a given fC R(v) match at points of V_r0. This completes the proof with V4V_r0.

COROLLARY 1.2. In the situation of Theorem 1.1 we have, in addi- tion, that

(iii) If fC R^(v),and f vanishes of infinite order at av, then ff0 in the connected component of a in v.

(iv) (r*f)(V)4f(v).

(v) IfNfNhas a local maximum at a point av,then f is constant on the connected component of a in v.

PROOF. IffC R(v) vanishes of infinite order atav, then alsor*f vanishes of infinite order ataand, by the strong unique continuation of holomorphic functions,r*fvanishes identically in the connected compo- nent of V containing a, and we obtain (iii).

To prove (iv), we assume by contradiction thatr*ftakes some value z0C at some point ofV, but that fdoes not assume that value at any point of v. Then the function

g4 1 f2z₀ (1.11)

belongs toC R(v), and has no holomorphic extension toV, contradicting (ii).

By (ii) and (iv), a local maximum offC R^{(v) ata}v, is also a local maximum ofr*fataV; therebyr*fis constant on the connected com- ponent of a in V and we obtain (v).

COROLLARY 1.3. Let(M,X)and(N,Y)be E-pairs, and let f :MK KN be a smooth CR isomorphism. Then there are E-pairs(M,X8)and (N,Y8),with X8%X and Y8%Y,such that f extends to a biholomorphic diffeomorphism fA: X8 KY8.

PROOF. We first consider the case where X and Y are open sets in C^n1k. By Theorem 1.1, there is an open neighborhoodVofMinXwhere fhas a holomorphic extension fA:VKC^n1k. By shrinking V to V8, we can arrange thatfA(V8)%Y and the Jacobian determinant of fA is differ- ent from zero onV8. Likewise there is an open neighborhoodV9ofNin

Ywheref²¹4gextends togA, withg(V9)%V8and the Jacobian determi- nant ofgAbeing nonzero in V9. By uniqueness of holomorphic extension of CR functions from M to gA(V9), it follows that gA_ifA

4identity on a neighborhood of M in X.

Now we consider the general case. Introducing local holomorphic co- ordinates charts onXandY, we may use the special case above to pro- duce local holomorphic extensions. The local holomorphic extensions patch together, by unique continuation, to give the desired fA.

We may now use Corollary 1.3 to show thatM having property Eis actually a local property of M.

THEOREM 1.4. M has property E if and only if for each aM,there is an open neighborhood va of a in M such that va has property E. PROOF. By hypothesis we have an E-pair (va,Xa), for each aM. We can assume that v_aLM, and that p_a:X_aKv_a is the orthogonal projection from a tubular neighborhood, with a distance function d_a(x,y). By Corollary 1.3, whenevervaOvbc¯, there are open neigh- borhoodsX_abofvaOvbinX_aandX_baofvaOvbinX_b, and a unique bi- holomorphic mapfA

ab:XabKXba, extending the identity map onvaOvb. We may select a locally finite open covering]v_a(ofM, parametrized by aA%M. By shrinking, we refine the]v_a( to an open covering]v_a8 (, with v8^aLva. With eaD0 sufficiently small, we define

X^a84p2a1(v8_a)O]da(x,v8^a)Eea(, (1.12)

so as to have

p2a1(v8^aOv8^b)OX^a8%X_ab, (1.13)

for all bA such that v8^bOv8^ac¯. Set X^ab84fab21(Xab)OX^a8. (1.14)

Then X is obtained by gluing together the Xa8’s, by X^a8&X^ab8K

fA

ab X^ba8%X^b8. (1.15)

This completes the proof.

We now turn to the object of main concern in this paper, which are theCRmeromorphic functions on an M satisfying propertyE. The ring C R(v) of smooth (C^Q)CRfunctions onv%M is an integral domain ifv is connected. Let D(v) be the subset of C R(v) of divisors of zero; i.e.

D(v) is the set of thoseCRfunctions onvwhich vanish in some connect- ed component of v. Let M(v) be the quotient ring of C R(v) with re- spect toC R(v)0D(v). This means that M(v) is the set of the equiva- lence classes of pairs (p,q) with pC R^{(v) and q}C R^(v)0D(v). The equivalence relation (p,q)A(p8,q8) is defined by pq8 4p8q. If v8%v is an inclusion of open subsets ofM, the restriction map r_v^v8:C R^(v)_K KC R^(v₈^{) sends}C R^(v)0D(v) intoC R^(v₈⁾0D(v8) and thus induces a homomorphism of rings:

r_v^v8:M^(v)_KM^(v₈^{) .} (1.16)

We obtain in this way a presheaf of rings. We shall call the correspond- ing sheaf M ^the sheaf of CR meromorphic functions on M. By a CR meromorphic function on an open setv%M, we mean a continuous sec- tionfofM^{over v. Ifv}is connected, the space of all such sectionsK^(v) forms a field. Since we always assume that M is connected, we have in particular K^(M), the field of CR meromorphic functions on M.

We recall these standard notions: LetF be a field andF₀%F a sub- field. Then f₁,f₂,R,f_l F are said to be algebraically dependent over F0 iff there is a nonzero polynomial PF0[x1,x2,R,x_l] with coeffi- cients in F0 such that

P(f₁,f₂,R,f_l)40 ; (1.17)

otherwise they are calledalgebraically independent. The transcendence degree of F over F0 is the cardinality of a maximal set S%F such that every finite subset ofSis algebraically independent overF0. If the tran- scendence degree ofFoverF0is zero, we say thatFisalgebraicover (or is analgebraic extension of)F₀. The cardinal [F: F₀] denotes the dimen- sion ofFoverF₀, as a vector space. The fieldFis said to be asimple alge- braic extension ofF₀if there exists an element uFsuch that anyfF can be written as a polynomial inu with coefficients inF0. When F0has characteristic zero, the primitive element theorem says that any finite algebraic extension of F0 is simple.

Finally we discuss the general notion of a smooth complex CR line bundle FK^p M, which was introduced in [HN8]. By this we mean thatF is a smooth complex line bundle over M such that:

(i) FandM are smooth abstract CRmanifolds of type (n11 ,k) and (n,k), respectively,

(ii) p:FKM is a smooth CR submersion,

(iii) F5F(j₁,j2)Kj11j2F and C3F(l,j)KlQjF are CR maps.

Note that the Whitney sums F5F and C3F have natural struc- tures of smooth CR manifolds of type (n12 ,k); see [HN8]. There we also introduced the notion of the tangential CR operator ¯^F_M, acting on smooth sections ofF. We may take a smooth (not necessarilyCR) trivial- ization (U_a,s_a) ofF, wheres_ais a smooth non vanishing section ofFon U_a. Then a smooth section s of Fhas a local representation s4s_as_ain U_a, wheres_ais a smooth complex valued function inU_a, ands_a4g_abs_bin U_aOU_b, with g_ab4s_b/s_a. In eachU_a the tangentialCRoperator acting on s has a representation of the form:

¯^F_Ms4(¯_Ms_a1A_as_a)7s_a, (1.18)

where A_aC^Q(Ua,Q^{0 , 1M) and ¯}MA_a40 . On U_aOU_b we have:

A_b2A_a4g_ba¯_Mg_ab, with g_ba4g_ab²¹. (1.19)

If s satisfies ¯^F_Ms40 , it is called a CR section of F.

The

l

¯^F_M^l t4(¯_Mt_a1

l

(1.20)

in U_a, where t is a smooth section of F^l, with t4t_asla in U_a.

If the local trivialization is a CR trivialization, then the ( 0 , 1 ) forms A_a in (1.18) and (1.20) are equal to zero. On the other hand, if the ¯_M- closed forms A_a are locally ¯M-exact, then F and F^l are locally CR trivializable.

LetfK(M), where we now assume thatMhas propertyE. Then we can associate to fa smooth complex line bundleFK^p M, which is locally CR trivializable. By definition, f has local representations

f4p_a

q_a on v_a, (1.21)

withp_a,q_aC R^(va). Moreover we may arrange that their holomorphic extensionspA_aandqA_ato a neighborhoodVaof vainXhave no nontrivial common factor at each point ofVa. Then there are uniquely determined

non vanishing functions g_abC R^(vaOvb) such that q_a4g_abq_b on vaOvb. (1.22)

The]g_ab( are then the transition functions of a smooth complexCRline bundle F over M, and F is therefore locally CR trivializable. The ]p_a( and]q_a( give global smooth sections p andq of F over M, whose quo- tient p/q is the CR meromorphic function f.

Let us return now to the smooth complex CR line bundle FK^p M, which may not be locallyCRtrivializable. In this context, it is natural to consider smooth abstractCRmanifoldsM, which may not have property E, but which areessentially pseudoconcave, as defined (²) in [HN8]. The important consequence of the assumption of essential pseudoconcavity onM is that one has the weak unique continuation property forCRsec- tions ofF. Note that 1-pseudoconcave abstract CRmanifolds are essen- tially pseudoconcave. Under these assumptions we can give a more gen- eral notion of what is aCRmeromorphic function onM: We associate a CRmeromorphic function fto any pair (p,q), wherepandqare smooth globalCRsections of a smooth complexCRline bundleFK^p M, withqg g0 . Another pair (p8,q8), which are smoothCRglobal sections of another suchF8 K^p8M, with q8g0 , define the same fiffpq8 4p8q as sections of F7F8. Note thatf4p/qis a well defined smoothCRfunction whereqc c0 . With this more general definition, we get a new collectionK^{×(M) of ob-} jects called CR meromorphic functions on M. Observe that K^{×(M) is a} field. For an essentially pseudoconcaveM, which has propertyE,K^(M) is a subfield of K^×(M). If in addition M is 2-pseudoconcave, then all smooth complexCRline bundles overM are locallyCRtrivializable, and then K^(M)₄K^×(M).

2. CR meromorphic functions on compact CR manifolds.

Let M be a connected smooth compact CR manifold of type (n,k), having property E. Then:

(²) Misessentially pseudoconcaveiff it isminimal, i.e. does not contain ger- ms ofCRmanifolds with the sameCRdimension and a smallerCRcodimension, and admits a Hermitian metric onHMfor which the traces of the Levi forms are zero at each point.

THEOREM 2.1. The field K^(M) of CR meromorphic functions on M has transcendence degree over C less or equal to n1k.

Setting k40 above, we recover Satz 1 in Siegel [Si].

PROOF. According to the discussion in § 1, the statement means:

Given n1k11 CR meromorphic functions f0,f1,R,f_n1k on M, there exists a non zero polynomial with complex coefficients F(x₀,x₁,R,x_n1k) such that

F(f₀,f₁,R,f_n1k)f0 on M. (2.1)

From the preceding section, we may regardMas a genericCRsubmani- fold of an n1k dimensional complex manifold X.

For each pointaM there is a connected open coordinate neighbor- hood Va, in which the holomorphic coordinate z_a is centered at a. We chooseVa in such a way thatva4VaOM is a connected neighborhood ofain M. Moreover we can arrange that, forj40 , 1 ,R,n1k, each f_j has a representation

fj4 p_ja

q_ja on v_a (2.2)

withp_jaandq_ja being smoothCRfunctions inva. According to Theorem 1.1 we may also assume that the restriction mapO^(Va)KC R^(va) is an isomorphism. For eachCRfunctiong onva, we denote its unique holo- morphic extension toV_abygA. By a careful choice of thep_ja andq_ja, and an additional shrinking of va, Va, we can also arrange that

fA

j4 pA_ja

qA_ja on Va, (2.3)

with the functionspA_jaandqA_ja being holomorphic and having no nontrivial common factor at each point in a neighborhood of Va. For each pair of points a,b on M we have the transition functions

qA_ja4g_jabqA_jb, (2.4)

which are holomorphic and non vanishing on a neighborhood ofVaOVb. Again, for each aM we consider the polydiscs:

Ka4 ]NzaN Gra( and La4 ]NzaNEe²¹ra(, (2.5)

whereNz_aN denotes the max norm in C^n1k, and r_aD0 is chosen so that K_aLVa. By the compactness ofM, we may fix a finite number of points a1,a2,R,amonM, such that theLa₁,L_a2,R,L_amprovide an open cov- ering ofM. Then we choose positive real numbersmandnto provide the bounds:

Ng0abNEe^m and

N

^»

N

(2.6)

on V_aOV_b for a,b4a₁,a₂,R,a_m.

Consider a polynomial with complex coefficients to be determined later, F(x₀,x₁,R,x_n1k) of degree s with respect to x₀and of degree t with respect to eachxifori41 , 2 ,R,n1k. The number of coefficients to be determined is

A4(s11 )Q(t11 )^n1k. (2.7)

Now, lettinga stand for any one of thea₁,a₂,R,a_m, we introduce the functions

Qa4qA^s_0a

»

j41 n1k

qA_ja^t , Pa4QaF(fA

0,fA

1,R,fA

n1k) (2.8)

which are holomorphic on a neighborhood ofVa. For a positive integer h, to be made precise later, we wish to impose the condition, for a4 4a₁,a₂,R,a_m, thatP_avanishes to orderhata. In terms of our local co- ordinatesza, this means that all partial derivatives of orderGh21 must vanish atz_a40 . This imposes a certain number of linear homogeneous conditions on the unknown coefficients of the polynomialF. The number of such conditions is

B4m

g

h

(2.9)

If we can arrange that BEA, then this system of linear homogeneous equations has a non trivial solution.

However, in order to apply the Schwarz lemma later, we need also to arrange that s, t and h satisfy

ms1ntEh. (2.10)

To this end we fixsto be an integer withsDmn^n1k. Thus, for each posi- tive h, we denote by t_h the largest positive integer satisfying st_h^n1kE

Emh^n1k. In this way we obtain that

BGmh^n1kGs(th11 )^n1kE(s11 )(th11 )^n1k4A. (2.11)

On the other hand, since thK Q as hK Q, by choosing h sufficiently large we have

m

g

h

(2.12)

which implies (2.10) for t4t_h. Set Y4 max

1GiGm max

K_ai NP_aiN. (2.13)

This maximum is obtained at some pointz* belonging to some K_a*, for a* equal to some one ofa₁,a₂,R,a_m. Since z*K_a*%Va*, because of our choices of theva,Va, according to (iv) in Corollary 1.2, there is an- other point z**va* such that

P_a*(z* )4P_a*(z** ) . (2.14)

But the pointz** belongs to someL_a**%K_a**, wherea** is one of thea₁, a2,R,am. Hence by the Schwartz lemma of Siegel [Si] we obtain

NP_a**(z** )N GYe^2h. (2.15)

However

P_a*(z** )4P_a**(z** )

k

^»

g_ja^t_*a**(z** )

l

(2.16)

Hence from (2.6), (2.14), (2.15) we obtain

Y4 NP_a*(z** )N GYe^ms1nt2h. (2.17)

By (2.10) this implies thatY40 . Hence eachPa_jf0 , which in turn yields F(fA

0,fA

1,R,fA

n1k)f0 . Therefore restricting to M we get (2.1). This completes the proof.

3. Analytic and algebraic dependence of CR meromorphic functions.

Let f₀,f₁,R,f_lK(M). We say that they are analytically depen- dent if

df0Rdf1RRRdf_l 40 where it is defined . (3.1)

THEOREM 3.1. Let M be a connected smooth compact CR manifold of type (n,k), having property E. Let f₀,f₁,R,f_l K^{(M). Then they} are algebraically dependent overC if and only if they are analytically dependent.

PROOF. First we observe that algebraic dependence implies analytic dependence. Assume that there is a nontrivial polynomialF, with com- plex coefficients, of minimal total degree, such thatF(f0,f1,R,f_l )f0 . Then

j

!

l ¯F

¯x_j(f₀,f₁,R,f_l )df_j40 (3.2)

where it is defined. It follows that some coefficient in (3.2) is a nonzero CRmeromorphic function onM. This implies (3.1) on an open dense sub- set of M, and hence whenever it is defined.

For the proof in the other direction, we can assume thatf₁,R,f_l are analytically independent. Our task is to show that there exists a nonzero polynomial with complex coefficients F(x₀,x₁,R,x_l ) such that

F(f0,f1,R,f_l )f0 on M. (3.3)

To this end we repeat the proof of Theorem 2.1, withn1kreplaced by

l

down to the line below (2.8). We shall replace s,t,n,A,B by new s8,t8,n8,A8,B8. After that we choose additional pointsa₁8,a₂8,R,a_m8 with a^j8va_j and a^j8 sufficiently close to aj, for j41 , 2 ,R,m. These points are chosen so that

K_aj84 ]Nz_aj2z_aj(a_j8)N Gr_aj(LVa_j, (3.4)

theL_aj84 ]Nz_aj2z_aj(a_j8)N Ee²¹r_aj( still give an open covering ofM, the fA

0,fA

1,R,fA

l are holomorphic at eachaj8, andfA

1,R,fA

l can be completed to a system of holomorphic coordinates in a neighborhood of each aj8. This is possible because the set of points onM, where thefA

1,R,fA

l are

holomorphic, and thed fA

1,R,d fA

l are linearly independent, is open and dense. Our assumption (3.1) that thef0,f1,R,f_l are analytically depen- dent implies that, near each point a^j8, fA

0 is a holomorphic function of fA

1,R,fA

l. We modify the proof of Theorem 2.1 by requiring that the holomorphic functionsP_ajvanish to orderhata^j8, forj41 ,R,m. To ac- complish this, we require thatF(fA

0,fA

1,R,fA

l ) vanish to orderhat each a^j8. This amount to requiring that all partial derivatives of orderG(h2

21 ) ofF(fA

0,fA

1,R,fA

l ) with respect tofA

1,R,fA

l should vanish ata^j8. The number of homogeneous linear equations is now

B8 4m

g ^l

_l

h

(3.5)

Now we fix an integers8withs8 Dm(n8)^l . Just as in the proof of Theo- rem 2.1, we choosehsufficiently large, and taket8to be the largest posi- tive integer satisfying s8(t8)^lEmh^l , so as to obtain

B8 Gmh^l E(s8 11 )(t8 11 )^l4A8, (3.6)

ms8 1n8t8 Eh. (3.7)

By (3.6) we can choose a nontrivialF, of degrees8inx₀and of degreet8 in each x₁,R,x_l, such that all P_aj vanish to order h at a^j8. Set

Y4 max

1GjGmmax

K_a8j NPa_jN. (3.8)

This maximum is attained at some pointz8belonging to some K_aj8

0. Since z8K_aj8

0%Vaj0, as before by (iv) in Corollary 1.2 there is another point z9va_j0 such that

P_aj

0(z8)4P_aj

0(z9) . (3.9)

This point z9 belongs to some L_aj8

1%K_aj8

1. So by the Schwarz lemma we obtain

NP_aj1(z9)N GYe^{ms8 1nt8 2h}. (3.10)

Thus as before we obtain Y4 NP_aj

1(z9)N GYe^{ms8 1nt8 2h}, (3.11)

showing that Y40 . This implies (3.3), completing the proof.

In order to make the exposition more clear, we have divided the dis- cussion into two parts; however Theorem 2.1 is a direct consequence of Theorem 3.1.

Algebraic dependence always implies analytic dependence. However, in the absence of propertyE, the converse may be false. We give a gen- eral counterexample:

PROPOSITION 3.2. Let M be a connected smooth compact CR mani- fold of type (n,k). Assume that M has a smooth CR immersion into some Stein manifold. Then

(1) condition E is violated, and

(2) there exists an infinite sequence of smooth CR functions on M, any two of which are analytically dependent, and which are alge- braically independent over C.

PROOF. By the embedding theorem for Stein manifolds, we can as- sume thatMhas a smoothCRimmersion in someC^N. Then by a result in [HN1], there exists a point x₀ in M and a jH_x⁰₀M such that the Levi form Lx0(j,Q) is positive definite on H_x0M. This implies that a small neighborhood ofx₀inMis contained in the smooth boundary of a strictly pseudoconvex domainV inC^N. It is well known that there are holomor- phic functions inV, which are smooth on V, and cannot be holomorphi- cally extended beyond x0. Thus condition E is violated at x0.

To prove (2) we first observe that some coordinate z₁ must be non constant onM. Consider the sequence of holomorphic functions]f_l(z)4 4e^z1lNlN(%O^(CN), and denote their pull-backs toMby]f_l*(. Clearly any two of them are analytically dependent. Assume, by contradiction, that some finite collection f_l*₁,f_l*₂,R,f_l*_m, are algebraically dependent.

Then there is a nontrivial polynomialP, with complex coefficients, such that

P(f_l*₁,f_l*₂,R,f_l*_m)40 on M. (3.12)

Since e^z1l1,e^z1l2,R,e^z1lm are algebraically independent in O(C), the en- tire functionz₁KP(e^z1l1,e^z1l2,R,e^z1lm) is not constant and has isolated zeroes. BecauseM is connected, it follows that the pullback onM of the functionz1 is constant, contradicting our assumption that the coordinate z₁ was not constant on M.

4. The field of CR meromorphic functions.

SupposeMis a connected compactCRmanifold of type (n,k), having property E. We have:

THEOREM 4.1. Let d be the transcendence degree of K^{(M) over C,} and let f₁,f₂,R,f_d be a maximal set of algebraically independent CR meromorphic functions inK^(M).ThenK^(M)is a simple finite algebra-

ic extension of the field C(f₁,f₂,R,f_d) of rational functions of f₁,f₂,R,f_d.

Setting k40 above, and taking the special case whered4n, we re- cover Satz 2 of Siegel [Si].

The theorem is a consequence of the following:

PROPOSITION 4.2. Let f1,f2,R,f_l be CR meromorphic functions in K^(M).Then there exists a positive integerk4k(f₁,f₂,R,f_l)such that every f₀K^(M),which is algebraically dependent on f₁,f₂,R,f_l ,satis- fies a nontrivial polynomial equation of degree Gk whose coefficients are rational functions of f1,f2,R,f_l .

PROOF. Without any loss of generality we may assume that f1,f2,R,f_l are algebraically independent. By Theorem 3.1 they are also analytically independent. This puts us in the situation of the second half of Theorem 3.1. The difference, however, is that we useonly the func- tionsf₁,f₂,R,f_l in (2.2), (2.3) and (2.4) to determine theva,Va. In this way the numbersmandn8depend only onf₁,f₂,R,f_l. We fix the inte- gers8 Dm(n8)^l as before. The proof of Theorem 3.1 shows that any CR meromorphic function fK^(M), which can be represented on eachva_i, fori41 , 2 ,R,m, as the quotientp_i/q_iof twoCRfunctions globally de- fined onva_i, satisfies a nontrivial polynomial equation of degree less or equal to k4s8, with coefficients in C(f₁,f₂,R,f_l ). This reduces our task to showing that f0 has such a representation.

By hypothesis our given f0 satisfies an equation G0f0l

1G1f0l21

1R1G_l40 (4.1)

whereG₀,G₁,R,G_lare polynomials inf₁,f₂,R,f_l, and G₀is not iden- tically 0 in M. Let s be an upper bound for the degrees of the G₀,G₁,R,G_l, with respect to each of the f₁,f₂,R,f_l. We set

QA

a4

»

j41 l

qA_ja^s, (4.2)

HA

aa4QA

a aGA

0 a21GA

a (a41 , 2 ,R,l) , (4.3)

SA

a4QA

aGA

0fA

0. (4.4)

Multiplying (4.1) by QA

a lGA

0l21, we obtain that SA^l

1HA

a1SA^l21

1R1HA

al40 on Va. (4.5)

Note thatQA

aGA

b (b40 , 1 ,R,l) and the HA

aa (a41 , 2 ,R,l) are holo- morphic functions on V_a, and SA

a is meromorphic on an open neighbor- hood ofvainVa. SinceSA

asatisfies (4.5), it is locally bounded, and hence actually holomorphic. Then the restrictions

p_0a4SA

aN^va and q_0a4QA

aGA

0N^va. (4.6)

are CR functions on va, and f₀4 p_0a

q0a

on va. (4.7)

The proof is complete.

Now we explain what is the point of Theorem 4.1. Consider a maximal set f1,f2,R,fd of algebraically independent CRmeromorphic functions on M, where d is the transcendence degree of K(M). Consider an f K^{(M). Thenf}is algebraically dependent onf₁,f₂,R,f_d; i.e. it satisfies an equation

f^l1g1f^l211R1g_l40 , (4.8)

where g₁,g₂,R,g_lC(f₁,f₂,R,f_d). The minimal l for which such an equation holds is called thedegreeoffoverC(f₁,f₂,R,f_d). The content of Proposition 4.2 is that this degree is bounded from above by k4 4k(f1,f2,R,f_d). Now choose an elementUK(M) so that its degreeais maximal. For any fK^(M) consider the algebraic extension field C(f₁,f₂,R,f_d,U,f). By the primitive element theorem this extension is simple; i.e. there exists an element hC(f₁,f₂,R,f_d,U,f) such that C(f1,f2,R,fd,U,f)4C(f1,f2,R,fd,h). Then

(4.9) aF[C(f₁,f₂,R,f_d,h) :C(f₁,f₂,R,f_d) ]4

4[C(f₁,f₂,R,f_d,U,f) :C( f₁,f₂,R,f_d,U) ]3 3[C(f1,f2,R,fd,U) :C(f1,f2,R,fd) ]Fa. Hence the first factor on the right must be one; therefore f C(f₁,f₂,R,f_d,U). The conclusion is that

K^(M)4C(f1,f2,R,f_d,U)4C(f1,f2,R,f_d)[U] , (4.10)

andany fK^(M)can be written as a polynomial of degree Eahaving coefficients that are rational functions of f₁,f₂,R,f_d.

From the above remark we derive the

PROPOSITION 4.3. There is an open neighborhood U of M in X such that the restriction map

K^(U)_K^K(M) (4.11)

is an isomorphism. Here K(U) denotes the field of meromorphic func- tions on U.

Let M be a connected smooth abstract CR manifold of type (n,k).

Consider a complexCRline bundleFK^p MoverM. Introduce the graded ring

A^(M,F)₄

0

l40

Q

C R^{(M,Fl ) ,} (4.12)

whereC R^(M,Fl) are the smooth globalCRsections of the

l

power of F. Note that if s1C R^{(M,Fl1) and} s2C R^{(M,Fl2), then} s1s2C R(M,F^l11l2).

Assume that we are in a situation where smooth sections of F have the weak unique continuation property; e.g. we could takeMto be essen- tially pseudoconcave (see [HN8]). ThenA^(M,F) is an integral domain because M is connected. Let

(4.13) Q^(M,F)₄

m

N

^l

n

denote the field of quotients.

Then Q^(M,F)%K^{×(M), and} C R^(M)4A^(M, trivial bundle).

PROPOSITION 4.4. Assume that M is compact and has property E. (1) If F is locally CR trivializable, then Q^(M,F)is an algebraically closed subfield of K^(M).

(2) There exists a choice of a locally CR trivializable F such that Q^(M,F)₄K^(M).

Assume that M is compact and essentially pseudoconcave. Then (3) Q^(M,F) is algebraically closed in K^×(M).

In case M is compact and satisfies both hypothesis, then (4) K^(M) is algebraically closed in K^×(M).