Elementary pseudoconcavity and fields of CR meromorphic functions

Elementary Pseudoconcavity and fields of CR meromorphic functions.

C. DENSONHILL(*) - MAURONACINOVICH(**)

LetMbe a smoothCRmanifold ofCRdimensionnandCRcodimensionk, which is not compact, but has the local extension propertyE. We introduce the notion of

«elementary pseudoconcavity» forM, which extends toCRmanifolds the concept of a «pseudoconcave» complex manifold. This notion is then used to obtain general- izations, to the noncompact case, of the results of our previous paper about algebraic dependence, transcendence degree and related matters for the fieldK(M) ofCR meromorphic functions onM.

It was A. Andreotti who, in a series of papers ([A], [AG], [ASi], [ASt], [AT]), realized that a number of fundamental theorems about compact complex manifolds (or compact analytic spaces) could be carried over to the non-compact case. This involved the introduction of an «elementary no- tion» of pseudoconcavity: Let X be a connected non-compact complex manifold. Then X is called elementary pseudoconcave if one can find a non-empty open subset Y X with the following properties:

i) Yis relatively compact inX, and@Yis smooth.

ii) The Levi form of@Yrestricted to the analytic tangent space has at least one negative eigenvalue at each point of@Y.

In particular, for any pointz02@Ythere is an analytic disc of complex dimension1 which is tangent atz₀to@Y, and is contained inYexcept for the pointz₀. A compactXmay be considered elementary pseudoconcave, since one may takeYˆXand@Y ˆ ;, so that (i) and (ii) are trivially valid.

With this condition it became possible to carry over toXthe basic results of

(*) Indirizzo dell'A.: Department of Mathematics, Stony Brook University, Stony Brook NY 11794, USA. E-mail address: dhill@math.sunysb.edu

(**) Indirizzo dell'A.: Dipartimento di Matematica - UniversitaÁ di Roma «Tor Vergata» - via della Ricerca Scientifica - 00133 Roma, Italy. E-mail address:

nacinovi@ mat.uniroma2.it

Siegel [Si] about the fieldK(X) of meromorphic functions onX. Thus one was able to discuss the transcendence degree ofK(X) overC, algebraic versus analytic dependence, the fact that K(X) is a simple algebraic ex- tension of the fieldC(f₁;. . .;f_d) of rational functions of a given transcen- dence basisf₁;. . .;f_d, as well as an extension of Chow's theorem.

The present work is a sequel to our previous paper [HN12]. There we managed to replace a compactX b y acompact CRmanifoldM ofCRdi- mension nandCR codimension k, which was assumed to have a certain local extension property E. This propertyE was taken as an axiom. In particular this axiom is satisfied in the important case whereM is pseu- doconcave. [Note that here the word «pseudoconcave» is being used in a completely different context, and has a completely different meaning than the «elementary pseudoconcave» mentioned above. M being pseudo- concave is a local property of M, having to do with an arbitrarily small neighborhood of each point ofM: every complex manifold, compact or not, is pseudoconcave in this sense.] The purpose of the present paper is to find an analogue of the notion of «elementary pseudoconcavity», for a non- compact CRmanifoldM, and to use it to obtain generalizations of most of the results of [HN12]. Thus we study the algebraic dependence, trans- cendence degree and related matters for the field K(M) of CR mer- omorphic functions onM.

In order to find a suitable replacement for the notion of «elementary pseudoconcavity», in the context of aCRmanifoldMsatisfying propertyE, we first discuss the complex Hessiani(@@)_M acting on real transversal 1- jets on M (see [MN]). This enables us to consider weakly and strongly pseudoconcave domainsY Mwith smooth boundaries@Y. We prove two lemmas which serve to replace the analytic disc, tangent toz0 2@Y, and otherwise contained inY.

For more information about CR manifolds, we refer the reader to [HN1], [HN2], . . ., [HN12] and to a number of interesting papers of C. Laurent-ThieÂbaut and J. Leiterer.

The first author would like to express his gratitude for the kind hos- pitality of the Humboldt UniversitaÈt zu Berlin and the UniversitaÁ di Roma

«Tor Vergata».

1. The complex Hessian onCRmanifolds.

In this paperMwill be a smooth (C¹) paracompact manifold, whose dim_RMˆ2n‡k, which has a smoothCRstructure of type (n;k); i.e.nis

theCR dimCandktheCR codimR. We recall what this means: as an abstractCRmanifoldMis really a tripleMˆ(M;HM;J), whereHMis a smooth real vector subbundle of rank 2nof the real tangent bundleTM, and where J:HM!HM is a smooth fiber preserving isomorphism such that J²ˆ I. It is also required that the formal integrability conditions [G(M;T^0;1M);G(M;T^0;1M)]G(M;T^0;1M) be satisfied (G means smooth sections). Here T^0;1MˆfX‡iJXjX2HMg is the complex subbundle of the complexificationCHMofHMcorresponding to the eigenvalue i of J; we have T^1;0M\T^0;1Mˆ0 and T^1;0M T^0;1MˆCHM, where T^1;0Mˆ T^0;1M. When kˆ0, we recover the abstract definition of acomplexmanifold, via the Newlander-Nirenberg theorem.

We denote by H⁰Mˆ fj2TMj hX;ji ˆ0 8X2H_p(j)Mgthe char- acteristic bundleofM. To eachj2H⁰_xM, we associate the Levi form atj:

L(j;X)ˆj([JX;~ X])~ ˆd~j(X;JX) for X2H_xM (1:1)

which is Hermitian for the complex structure ofHxMdefined byJ. It is of interest to carry over to smoothly bounded domains Y of an abstractCR manifoldMthe notions of pseudoconvexity and pseudoconcavity, expressed in terms of locally defining functions, as one has for complex manifoldsX, such as in the classical work of Andreotti-Grauert [AG]. Classically these notions are expressed in terms of the complex Hessian of a locally defining function; however in the case of an abstract CR manifold the complex Hessian of a real function cannot be defined intrinsically as a Hermitian form onHM. In order to obtain an invariant notion, one must consider real transversal 1-jets onM.

We denote byE^j(M) the space of smooth complex valued exterior forms homogeneous of degree j and by E(M)ˆ ^2n‡k_jˆ0 E^j(M) the algebra of complex valued smooth exterior forms onM. Then we consider the idealJ of E(M) generated by the one-forms vanishing on T^0;1M and by J its conjugate, which is the ideal of E(M) generated by the one-forms van- ishing onT^1;0M.

The notion of transversal 1-jet on M is best explained in terms of a choice of a splitting:

l:E¹(M)![J \ E¹(M)][J \ E¹(M)]

(1:2)

for the exact sequence:

0! J \ J \ E¹(M)![J \ E¹(M)][J \ E¹(M)]! E¹(M)!0: (1:3)

Note that such splittings always exist becauseJ,J,J \ J andE(M) are all locally free gradedE(M)-modules. The splittinglˆ(l1;l2) was called a CRgaugein [MN]. ACRgauge is characterized by

aˆl1(a) l2(a); l1(a)2 J \ E¹(M); l2(a)2 J \ E¹(M) (1:4)

for all a2 E¹(M).

In aCRgaugel areal transversal1-jetc is represented by the pair (c₀;c₁)l, where c₀ is a smooth real valued function on M, and c₁ is a smooth section ofH⁰M. In a differentCRgaugel⁰,cis represented by the pair (c⁰₀;c⁰₁)_l⁰, where

c⁰₀ˆc₀

c⁰₁ˆc₁‡il⁰₁(dc₀) il₁(dc₀)

ˆc₁‡il⁰₂(dc₀) il₂(dc₀): 8>

<

>: :

(1:5)

Thecomplex Hessian i(@@)_Mcof a real transversal one jetconMis then the Hermitian form onHMdefined by

i(@@)_Mc(X;JX)ˆd[c₁ il₁(dc₀)](X;JX)

ˆd[c₁ il₂(dc₀)](X;JX); (1:6)

for allX2HM. By (1.5) the right hand side does not depend on the choice of theCRgaugel; hencei(@@)_Mcis an invariant notion on the abstractCR manifoldM. By our definition of the Levi form onM, we have that

L(c₁;X)ˆL(c₁;JX)ˆdc₁(X;JX): (1:7)

Notice that if M were a complex manifold (kˆ0), then we would have J \ J ˆ0, and (1.2) would be an isomorphism, and so there would be a natural unique choice of theCRgauge; moreoverH⁰Mˆ0, so there is no c₁-term, and hence (1.6) reduces to the usual complex Hessian of a function on the complex manifold. Thus for an open subsetY of aCRmanifoldM with a defining functionc₀, there are many ways to extend c₀ to a real transversal 1-jetc; however the various ways lead to complex Hessians (1.6) which differ by a Levi form (1.7) ofM.

Let us now consider the situation whereMis a genericCRsubmanifold of a complex manifoldX. Iffis a smooth real valued function defined onX, we shall associate to it the real transversal 1-jetconMdefined by

cˆ fj_M;il1(dfj_M) i(@f)j_M j_l; (1:8)

wherelis some fixedCRgauge onM. Then the complex Hessiani(@@)_Mc

of the associated transversal 1-jet is the restriction toHMof the pullback toMof the complex Hessiani@@f onX:

i(@@)_Mcˆi(@@f )j_M on HM: (1:9)

On the other hand, given a smooth transversal 1-jet c on M, by the Whitney extension theorem, we can find a smooth real valued functionfon Xsuch that (1.9) holds.

2. TheE-property.

As in [HN12], also in this paper we shall considerCRmanifoldsM of type (n;k) which have propertyE(Eis forextension). We recall thatM is said to have property Eiff there is anE-pair (M;X). By anE-pair we mean that

i) Mis a genericCRsubmanifold of the complex manifoldX, and ii) for each a2M, the restriction map induces an isomorphism O_X;a! CR_M;a.

We use the notation: O(X) and OX;a are the spaces of holomorphic functions on X and of germs of holomorphic functions defined on a neighborhood of a pointa2X, respectively; likewiseCR(M) andCR_M;aare the spaces of smoothCRfunctions onMand the space of germs of smooth CRfunctions defined on a neighborhood of a pointa2M, respectively.

REMARK. If M is a pseudoconcave CRmanifold, then M has property E.(see [HN12]).

Whenkˆ0, soMis of type (n;0), thenMis ann-dimensional complex manifold, and we obtain anEpair by choosingXˆM. Hence we adopt the convention thatany complex manifold has property E.

When nˆ0, so M is of type (0;k), thenM is a smooth totally real k-dimensional manifold, and we can never obtain an E-pair, (unless MˆXˆa point), because then any smooth function belongs toCR(M).

In [HN12] it was proved:

THEOREM2.1. Let(M;X)be an E-pair. Then for any open setvM there is a corresponding open setVX such that

i) V\Mˆv,and

ii) r:O(V)! CR(v)is an isomorphism.

iii) If f 2 CR(v), and f vanishes of infinite order at a2v, then f 0 in the connected component of a inv.

iv) (rf)(V)ˆf(v).

v) Ifjfjhas a local maximum at a point a2v, then f is constant on the connected component of a inv.

COROLLARY2.2. Let(M; X)and(N; Y)beE-pairs, and letf :M!N be a smoothCRisomorphism. Then there areE-pairs(M; X⁰)and(N; Y⁰), withX⁰X andY⁰Y, such thatf extends to a biholomorphic diffeo- morphismf~:X⁰!Y⁰.

THEOREM2.3. M has property E if and only if for each a2M, there is an open neighborhoodvaof a in M such thatvahas property E.

3. Weakly and strongly pseudoconcave domains.

In this section we consider domains YM with smooth boundaries.

We say that Y isweakly pseudoconcave at a2@Y if there is a smooth transversal 1-jetfˆ(f₀;f₁)_l, defined on an open neighborhoodUofain Msuch that:

1) f₀:U!Ris a smooth locally defining function forY ata: this means thatdf₀(a)6ˆ0 andY\Uˆ fp2Ujf₀(p)<0g;

2) i(@@)_Mf(p)‡L(j;) has at least 2 negative eigenvalues onH_pM for everyp2Uand everyj2H⁰_pM.

LEMMA3.1. Assume that M has property E. Let Y be an open subset of M with a smooth boundary and let U be a connected relatively compact open subset of M. If Y is weakly pseudoconcave at every point of@Y\U, then for every function u, which is defined and CRon a neighborhood of U, we have:

@Y\Usup juj sup

@(Y\U)n@Y\Ujuj: (3:1)

PROOF. Using a partition of unity, we may assume thatfˆ(f₀;f₁)_lis globally defined on an open neighborhoodU⁰ofU, thatdf₀6ˆ0 on an open neighborhoodWof@Y\U, and that condition (2) of the definition is valid at each pointp2W.

We argue by contradiction. If the statement where false, one could find a CR function u, that we can assume to be defined on U⁰, and a point

p02@Y\Usuch that

mˆ ju(p0)j ˆmax

Y\Ujuj> sup

@(Y\U)n@Y\Ujuj ˆm⁰: (3:2)

Set Fdˆ fp2Ujf₀(p) dg. Then for small d>0 the function m(d)ˆmaxp2Fdju(p)jis continuous and decreasing. Choose a smalld0 such thatmm(d)>m⁰,@F_dn(@(Y\U)n@Y\U)W, andm²(d) is not a critical value for the smooth real valued function juj², which we may do by Sard's theorem. Then m(d)ˆ ju(p_d)j for some p_d2@Y_d\U, b e- cause ju(p)j m⁰<m(d) on @U\Ydn(U\@Yd)@(U\Y)n(U\@Yd).

Then we have djuj²(pd)ˆk df₀(pd) with some k>0, that we can ar- range to be 1.

LetXbe a complex manifold such that (M;X) is anEpair and consider the open U~ X corresponding toU given by Theorem 2.1. We denote by~fa smooth real valued function onU~ defining onUthe transversal 1-jetf. Let r₁;. . .;r_kbe smooth real valued functions on a neighborhoodVofpdinXsuch that @r₁^ ^@r_k6ˆ0 in V and M\Vˆ fz2Vjr₁ˆ0; ;r_k ˆ0g.

Then, denoting byd_X the differential onX, for the holomorphic extensionu~ of u to U~ we have d_Xj~uj²(p_d)ˆd_X~f(p_d)‡P_k

iˆ1j_id_Xr_i(p_d), for suitable j₁;. . .;j_k2R. Then from the inclusion:

F_d fz2U~ j j~uj²m²(d)g (3:3)

it follows that we can find a large constantC>0 such that j~uj² m²(d)~f d‡X^k

iˆ1

j_ir_i‡CX^k

iˆ1

r²_i (3:4)

on a neighborhood ofpd.

By the E.E. Levi theorem [L], all holomorphic functions defined in V ˆ fz2U~ j~f d‡P_k

iˆ1j_ir_i‡CP_k

iˆ1r²_i <0g have a holomorphic ex- tension to an open neighborhood of the pointp_d2@V\U. However, the~ functionz!…u(z)~ u(pd)† ¹is holomorphic inVand cannot be holomor- phically extended to an open neighborhood of pd in U. This gives a con-~ tradiction, completing the proof of the lemma.

Next we fix a Hermitian metric h on the complex vector bundle HM!M. Iffˆ(f₀;f₁)_lis a transversal 1-jet defined on an open subset U ofM, we can consider for eachp2U andj2H⁰_pMthe eigenvalues

l1(f;j)l2(f;j) ln(f;j) (3:5)

of the Hermitian formi(@@)_Mf(p)‡L(j;) with respect toh.

LetY be an open subset ofMwith smooth boundary. We say thatYis strongly pseudoconcave ata2@Y if there is a smooth transversal 1-jet fˆ(f₀;f₁)_l, defined on an open neighborhoodUofainM such that:

1) f₀:U!Ris a smooth locally defining function forYata;

2) there exists a positive constantc0>0 and an open neighborhoodv ofainUsuch that

l2(f;j) c0<0 8p2v; 8j2H⁰_pM: We obtain the following

LEMMA3.2. Assume that M has property E. Let Y be an open subset of M with a smooth boundary. If Y is strongly pseudoconcave at a point

a2@Y, then for every open neighborhood U of a in M there is an open

neighborhoodvof a in U such that supv juj sup

Y\Ujuj 8u2 CR(U): (3:6)

PROOF. We can assume that the open setUis so chosen that there is a 1-jet fˆ(f₀;f₁)_l, defined on a neighborhood of U, with f₀ being a locally defining function for Y at all points of @Y\U, and l₂(f;j) c₀<0 for allp2U and j2H⁰_pM. Letc₀ be a non negative smooth real valued function, with compact support in U, and with c₀(a)>0. Set f_tˆ(f₀ tc;f₁)_l. Then there exists d>0 such that l2(f_t;j) c0=2<0 for allt2Rwithjtj d. Using the previous lemma we obtain (3.6) withvˆ fp2Ujf₀(p) dc(p)<0g.

EXAMPLE 1. Let q3 and consider in CP^{2q 1} homogeneous co- ordinates (w;z) withz;w2C^q. LetKdenote the union of the two disjoint projectiveq 1-planes:

Kˆ fwˆizg [ fwˆ izg:

Let M be theCR submanifold of type (2q 3;2) of points inCP^{2q 1}nK satisfying the homogeneous equations

X^q

jˆ1

jw_jj²ˆX^q

jˆ1

jz_jj² X^q

jˆ1

w_jz_j ‡z_jw_jˆ0: 8>

>>

>>

<

>>

>>

>:

Then, for every 1=2<e<1 the relatively compact open subset Y_eˆ fminfjz‡iwj²;jz iwj²g>e(jzj²‡ jwj²)g \M

has a smooth boundary and is strongly pseudoconcave. To see this we observe that the boundary consists of two disjoint pieces and that we can take the functions fˆe(jzj²‡ jwj²) jziwj² as locally defining func- tions for@Y_e. Note also thatMis (q 2)-pseudoconcave at each point and hence has propertyE.

EXAMPLE 2. Denote by w0;w1;w2;z0;z1;z2 the homogeneous coordi- nates ofCP⁵. LetQbe the ruled quadric described by the homogeneous equations

w₀ˆ0; z₀ˆ0; w₁z₂ˆw₂z₁:

Then consider the noncompactCRmanifoldMconsisting of the points of CP⁵nQwhose homogeneous coordinates satisfy:

w₀w₁‡w₁w₀ˆz₀z₁‡z₁z₀ w₀w₂‡w₂w₀ˆz₀z₂‡z₂z₀

ThenMis 1-pseudoconcave at all points which do not belong to the 3-plane Sˆ fw₀ˆ0; z₀ˆ0g. The Levi form is identically zero at the points of S\M, because SnQM. Let d be the distance in the Fubini-Study metric ofCP⁵and denote byYethe set of points ofMhaving a distance>e fromQ. SinceQis smooth, for smalle>0 the boundary ofYeis smooth and one can verify, by using the functionf(p)ˆe d(p;Q), that for smalle>0 the domainY_eis strongly pseudoconcave at all pointsa2@Y_e. Note thatM does not have the propertyE, because it is not minimal at the points of SnQ.

4. CRmeromorphic functions on elementary pseudoconcaveCRmani- folds.

LetMbe a connected smoothCRmanifold of type (n;k). We say thatM iselementary pseudoconcaveif it contains a relatively compact non empty open subsetY, with a smooth boundary which is strongly pseudoconcave at every pointa2@Y.

Note that for a compactMwe can takeYˆM, so that@Yˆ ;and the condition above is trivially satisfied: hence compactCRmanifolds are tri-

vially elementary pseudoconcave. The CRmanifolds M described in the Examples 1,2 at the end of the previous section provide examples of non- compact elementary pseudoconcaveCRmanifolds, the first having and the second not having propertyE.

For the notion ofCRmeromorphic functions we refer to our previous article [HN12].

THEOREM4.1. Let M be a connected smooth CRmanifold of type(n;k).

If M has property E and is elementary pseudoconcave, then the fieldK(M) of CRmeromorphic functions on M has transcendence degree overCless or equal to n‡k.

Settingkˆ0 above, we recover Andreotti's generalization [A] of Satz 1 of Siegel [Si].

PROOF. The statement means: Given n‡k‡1 CR meromorphic functions

f0; f1; . . .; f_n‡k onM,

there exists a non zero polynomial with complex coefficients F(x0;x1;. . .;x_n‡k) such that

F(f₀;f₁;. . .;f_n‡k)0 on M: (4:1)

Because of property (E) we may regardMas a genericCRsubmanifold of ann‡kdimensional complex manifoldX.

For each pointa2Mthere is a connected open coordinate neighbor- hood Va, in which the holomorphic coordinate za is centered at a. We chooseV_ain such a way thatv_aˆV_a\Mis a connected neighborhood of ainM. Moreover we can arrange that, forjˆ0;1;. . .;n‡k, eachf_jhas a representation

f_jˆp_ja

qja on v_a (4:2)

with p_jaandq_jabeing smoothCRfunctions inv_a. According to Theorem 2.1 we may also assume that the restriction mapO(V_a)! CR(v_a) is an iso- morphism. For eachCRfunctiongonv_a, we denote its unique holomorphic extension toVaby~g. By a careful choice of thepjaandqja, and an additional shrinking ofva,Va, we can also arrange that

~f_jˆ~pja

~q_ja on V_a; (4:3)

with the functions~pjaand~qja being holomorphic and having no nontrivial common factor at each point in a neighborhood ofVa. For each pair of points a;bonMwe have the transition functions

~q_jaˆg_jab~q_jb; (4:4)

which are holomorphic and non vanishing on a neighborhood ofV_a\V_b. Let Y be a relatively compact open subset of Mwith @Y smooth and strongly pseudoconcave at each point. For eacha2Y we can choose the polydiscs:

Kaˆ fjzaj rag and Laˆ fjzaj<e ¹rag; (4:5)

wherejzajdenotes the max norm inC^n‡k, andra>0, so thatKaVaand for allu2 CR(v_a) we have

supKa

j~uj sup

Y\va

juj: (4:6)

This is trivial whena2Y, as we can takeK_aYg\v_ain this case. When a2@Y, we apply Lemma 3.2 to find an open neighborhoodvofainvaso that (3.6) is valid withUˆva, and next we chooseKaV, whereVis the open set inXof Theorem 2.1.

By the compactness of Y, we may fix a finite number of points a₁;a₂;. . .;a_m onM, such that the L_a1;L_a2;. . .;L_am provide an open cov- ering ofY. Then we choose positive real numbersm andyto provide the bounds:

jg_0abj<e^m and ^n‡kY

jˆ1

g_jab

<e^y (4:7)

onV_a\V_b fora;bˆa₁;a₂;. . .;a_m.

Consider a polynomial with complex coefficients to be determined la- ter,F(x₀;x₁;. . .;x_n‡k) of degreeswith respect tox₀and of degreetwith respect to eachxiforiˆ1;2;. . .;n‡k. The number of coefficients to be determined is

Aˆ(s‡1)(t‡1)^n‡k: (4:8)

Now, letting a stand for any one of thea1;a2;. . .;am, we introduce the functions

Q_aˆ~q^s_0a^n‡kY

jˆ1

~q^t_ja; P_aˆQ_aF(~f₀;~f₁;. . .;~f_n‡k) (4:9)

which are holomorphic on a neighborhood ofVa. For a positive integerh, to be made precise later, we wish to impose the condition, for aˆa₁;a₂;. . .;am, thatPavanishes to orderhata. In terms of our local coordinates z_a, this means that all partial derivatives of order h 1 must vanish at z_aˆ0. This imposes a certain number of linear homo- geneous conditions on the unknown coefficients of the polynomialF. The number of such conditions is

Bˆm n‡k‡h 1 n‡k

m h^n‡k: (4:10)

If we can arrange that B<A, 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 thats,tandhsatisfy

ms ‡ yt < h: (4:11)

To this end we fixsto be an integer withs>my^n‡k. Thus, for each positive h, we denote byt_hthe largest positive integer satisfyingst^n‡k_h <mh^n‡k. In this way we obtain that

Bmh^n‡ks t…h‡1†^n‡k<(s‡1)…th‡1†^n‡kˆA: (4:12)

On the other hand, sincet_h! 1ash! 1, by choosinghsufficiently large we have

m ms t_h‡y

_n‡k

<s; (4:13)

which implies (4.11) fortˆt_h. Set Yˆ max

1im max

K_ai jP_aij: (4:14)

This maximum is obtained at some pointz belonging to someK_a, fora equal to some one of a₁;a₂;. . .;a_m. Sincez2K_aV_a, because of our choices of theva,Va, according to (iv) in Theorem 2.1 and (4.6), there is another pointz 2va\Ysuch that

P_a(z)ˆP_a(z): (4:15)

But the pointzbelongs to someL_a K_a, whereais one of thea₁,a₂, . . .,am. Hence by the Schwartz lemma of Siegel [Si] we obtain

P_a(z)

j j Ye ^h: (4:16)

However

Pa(z)ˆPa(z) g^s_0aa(z)^n‡kY

jˆ1

g^t_jaa(z)

" #

: (4:17)

Hence from (4.7), (4.15), (4.16) we obtain

YˆjPa(z)j Ye^{ms‡yt h}: (4:18)

By (4.11) this implies thatYˆ0. Hence eachP_aj 0, which in turn yields F(~f0;~f1;. . .;~fn‡k)0. Therefore restricting toMwe get (4.1). This com- pletes the proof.

We note that our proof follows closely that of the corresponding result (Theorem 2.1) of [HN12], the only change consisting in the restriction of the covering by polycylinders to the points of the closure of the subdomain Y of M, and the use of Lemma 3.2 to reduce again the discussion to polycylinders centered at points ofY.

In a completely similar way we can extend also the other results of [HN12] to the present situation. We shall therefore refer, for the proofs of the following results, to [HN12], as only small changes, similar to those explained in detail in the proof of Theorem 4.1 are needed.

Let f0;f1;. . .;f`2 K(M). We recall that they areanalytically depen- dentif

df₀^df₁^ ^df_`ˆ0 where it is defined:

(4:19)

THEOREM 4.2. Let M be a connected smooth elementary pseudo- concave CRmanifold of type (n;k), having property E. Let f₀;f₁;. . .;f_`2 K(M). Then they are algebraically dependent overCif and only if they are analytically dependent.

THEOREM 4.3. Let M be a connected smooth elementary pseudo- concave CRmanifold of type (n;k), having property E. Let d be the transcendence degree ofK(M)overC, and let f₁;f₂;. . .;f_dbe a maximal set of algebraically independent CRmeromorphic functions in K(M). Then K(M)is a simple finite algebraic extension of the fieldC(f₁;f₂;. . .;f_d)of rational functions of f1;f2;. . .;fd.

Settingkˆ0 above, and taking the special case where dˆn, we re- cover Andreotti's generalization [A] of Satz 2 of Siegel [Si].

As in [HN12], this theorem can be derived from the following:

PROPOSITION 4.4. Let f1;f2;. . .;f` be CRmeromorphic functions in K(M). Then there exists a positive integer kˆk(f1;f2;. . .;f`) such that every f₀2 K(M), which is algebraically dependent on f₁;f₂;. . .;f`, satis- fies a nontrivial polynomial equation of degreekwhose coefficients are rational functions of f<sub>1</sub>;f<sub>2</sub>;. . .;f<sub>`</sub>.

In particular, fix a maximal setf₁;f₂;. . .;f_dof algebraically independent CRmeromorphic functions onM, wheredis the transcendence degree of K(M). Consider an f 2 K(M). Then f is algebraically dependent on f1; f2;. . .;f_d; i.e. it satisfies an equation

f^l‡g1f^{l 1}‡ ‡glˆ0; (4:20)

where g1;g2;. . .;g_l2C(f1;f2;. . .;f_d). The minimall for which such an equation holds is called thedegreeof f over C(f₁;f₂;. . .;f_d). By Propo- sition 4.4 this degree is bounded from above bykˆk(f₁;f₂;. . .;f_d). Now choose an element U2 K(M) so that its degree a is maximal. For any f 2 K(M) consider the algebraic extension fieldC(f1;f2;. . .; fd;U;f). By the primitive element theorem this extension is simple; i.e. there exists an element h2C(f₁;f₂;. . .;f_d;U;f) such that C(f₁;f₂;. . .;f_d;U;f)ˆ C(f₁;f₂;. . .;f_d;h). Then

a[C(f₁;f₂;. . .;f_d;h):C(f₁;f₂;. . .;f_d)]

ˆ[C(f1;f2;. . .;fd;U;f):C(f1;f2;. . .;fd;U)]

[C(f₁;f₂;. . .;f_d;U):C(f₁;f₂;. . .;f_d)]

a: (4:21)

Hence the first factor on the right must be one; therefore f 2C(f1;f2;. . .;f_d;U). The conclusion is that

K(M)ˆC(f1;f2;. . .;f_d;U)ˆC(f1;f2;. . .;f_d)[U]; (4:22)

andany f 2 K(M)can be written as a polynomial of degree <ahaving coefficients that are rational functions of f₁;f₂;. . .;f_d.

From the above remark we derive the

PROPOSITION4.5. There is an open neighborhood U of M in X such that the restriction map

K(U)! K(M) (4:23)

is an isomorphism.

HereK(U) denotes the field of meromorphic functions onU.

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

Consider a complexCRline bundleF!^p MoverM. Introduce the graded ring

A(M;F)ˆ[¹

`ˆ0

CR(M;F^`); (4:24)

where CR(M;F<sup>`</sup>) are the smooth global CR sections of the `-th tensor power of F. Note that if s₁2 CR(M;F^{`1</sup>) and s<sub>2</sub>2 CR(M;F<sup>`2}), then s₁s₂2 CR(M;F^`1‡`2).

Assume that we are in a situation where smooth sections ofFhave the weak unique continuation property; e.g. we could takeMto be essentially pseudoconcave (see [HN8]). ThenA(M;F) is an integral domain because Mis connected. Let

Q(M;F)ˆ s1

s₂ s1;s22 CR(M;F<sup>`</sup>) for some` ; ands060

(4:25)

denote the field of quotients.

Then (¹)Q(M;F)K(M), and^ CR(M)ˆ A(M;trivial bundle).

PROPOSITION4.6. Assume that M is elementary pseudoconcave and has property E.

1) If F is locally CRtrivializable, thenQ(M;F)is an algebraically closed subfield ofK(M).

2) There exists a choice of a locally CRtrivializable F such that Q(M;F)ˆ K(M).

Assume moreover that M is essentially pseudoconcave(²).Then

(¹) See [HN12]: We can associate a CRmeromorphic function f to any pair (p;q), wherepandqare smooth globalCRsections of a smooth complexCRline bundle F!^p M, with q60. Another pair (p⁰;q⁰), which are smooth CR global sections of another such F⁰!^p0 M, withq⁰60, define the samef iffpq⁰ˆp⁰qas sections ofFF⁰. Note thatf ˆp=qis a well defined smoothCRfunction where q6ˆ0. With this more general definition, we get a new collectionK(M) of objects^ 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 ofK(M). If^ in additionMis 2-pseudoconcave, then all smooth complexCRline bundles overM are locallyCRtrivializable, and thenK(M)ˆK(M).^

(²) See [HN8]:Misessentially pseudoconcaveiff it isminimal, i.e. does not contain germs of CRmanifolds with the sameCRdimension and a smaller CR codimension, and admits a Hermitian metric onHMfor which the traces of the Levi forms are zero at each point.

3) Q(M;F)is algebraically closed inK(M).^

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

LetMbe a connected smooth elementary pseudoconcaveCRmanifold of type (n;k), having propertyE. Then

THEOREM4.7. Let F!^p M be a locallyCRtrivializable smooth com- plex CRline bundle over M. Then

dimCCR(M;F)<1: (4:26)

THEOREM4.8. Let M be a paracompact connected elementary pseu- doconcave CRmanifold of type(n;k)having property E. Lett:M!CP^N be a smooth CRmap. Suppose thatthas maximal rank2n‡k at one point of M. Then t(M) is contained in an irreducible algebraic subvariety of complex dimension n‡k, and the transcendence degree ofK(M)overCis n‡k.

THEOREM4.9. The following are equivalent:

1) M has a smooth CRembedding as a locally closed CRsubmani- fold of someCP^N.

2) There exists over M a smooth complex CRline bundle F such that the graded ring A(M;F)ˆS₁

`ˆ0CR(M;F<sup>`</sup>) separates points and gives

«local coordinates» at each point of M.

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Manoscritto pervenuto in redazione il 10 luglio 2004