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INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1995

FOLIATIONS WITH COMPLEX LEAVES

G I U S E P P E T O M A S S I N I

Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

1. Preliminaries

1. In the following new results on foliations with complex leaves are announced.

Complete proofs will appear elsewhere.

A foliation with complex leaves is a (smooth) foliation X of dimension 2n + k whose local models are domains U = V × B of Cn× Rk, V ⊂ Cn, B ⊂ Rk and whose local transformations are of the form

(∗)  z0= f (z, t),

t0= h(t),

where f is holomorphic with respect to z. A domain U as above is said to be a distinguished coordinate domain of X and z = (z1, . . . , zn), t = (t1, . . . , tk) are said to be distinguished local coordinates. k is called the real codimension of X.

As an example of such foliations we have the Levi flat hypersurfaces of Cn ([13], [4], [11]).

If X is a smooth foliation as above, then the leaves are complex manifolds of dimension n. Let D be the sheaf of germs of smooth functions, holomorphic along the leaves (namely the germs of CR-functions on X). D is a Fr´echet sheaf and we denote by D(X) the Fr´echet algebra Γ (X, D).

It is natural to study foliations with complex leaves in the spirit of the theory of complex spaces, in particular, the convexity with respect to the algebra D(X) and the cohomology of X with values in D. In this talk I will discuss some recent results obtained in a joint paper with G. Gigante.

2. Let X be a smooth foliation with complex leaves. X is said to be a q-complete foliation if there is an exhaustive, smooth function Φ : X → R which is strictly q-pseudoconvex along the leaves. X is a Stein foliation if

(a) DX separates points of X,

1991 Mathematics Subject Classification: 32F, 53C.

The paper is in final form and no version of it will be published elsewhere.

[367]

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(b) X is D-convex,

(c) for every x ∈ X there exist f1, . . . , fn, h1, . . . , hk ∈ D(X) such that rank∂(f1, . . . , fn, h1, . . . , hk)

∂(z1, . . . , zn, t1, . . . , tk) = n + k (z1, . . . , zn, t1, . . . , tk distinguished local coordinates at x).

One can prove that a Stein foliation is 1-complete.

R e m a r k. If we replace Rk by Ck and in (∗) we assume t ∈ Ck and that f , h are holomorphic with respect to z, t then we obtain the notion of complex foliation of (complex) codimension k.

3. Every real analytic foliation can be complexified. Precisely, we have the following

Theorem 1. Let X be a real analytic foliation with complex leaves, of codi- mension k. Then there exists a complex foliation eX of codimension k such that : (1) X ,→ eX by a closed real analytic embedding which is holomorphic along the leaves;

(2) every real analytic CR-function f : X → R extends holomorphically to a neighbourhood of X;

(3) if X is a q-complete foliation with exhaustive function Φ then for every c ∈ R, Xc = {Φ ≤ c} has a fundamental system of neighbourhoods which are q-complete manifolds.

R e m a r k. eX with the properties (1)–(3) is essentially unique.

As a corollary, using the approximation theorem of M. Freeman ([5]) we prove the following

Theorem 2. Under the assumptions of Theorem 1, if X is 1-complete, a smooth CR-function on a neighbourhood of Xc can be approximated by smooth global CR-functions.

R e m a r k. A similar argument can be applied to prove that in the previous statement Xccan be replaced by an arbitrary D-convex compact K (i.e. bK = K).

2. Applications

1. The approximation theorem allows us to prove an embedding theorem for real analytic Stein foliations ([7]).

Let X be a smooth foliation with complex leaves of dimension n and of codi- mension k. Let us denote by A(X; CN) the set of smooth CR-maps X → CN. Then A(X; CN) is Fr´echet. We have the following

Theorem 3. Assume X is a real analytic Stein foliation. Then there exists a smooth CR-map X → CN, N = 2n + k + 1, which is one-to-one, proper and regular.

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2. We apply the above theorem to obtain information about the topology of X.

Theorem 4. Let X be a real analytic Stein foliation. Then Hj(X, Z) = 0 for j ≥ n + k + 1 and Hn+k(X, Z) has no torsion.

Sketch of proof. Embed X in CN and consider on X the distance function

% from a point z ∈ CN \ X. z can be chosen in such a way that % is a Morse function. Next we show that % has no critical point of index j ≥ n + k + 1 ([14]).

Corollary 5. Let X ⊂ PN(C) be a closed oriented real analytic foliation and let W be a smooth algebraic hypersurface which does not contain X. Then the homomorphism

Hj(X, Z) → Hj(X ∩ W, Z)

induced by X ∩ W → X is bijective for j < n − 1 and injective for j = n − 1.

Moreover , the quotient group Hn−1(X ∩ W, Z)/Hn−1(X, Z) has no torsion.

3. Cohomology

1. Given a q-complete smooth foliation X, according to the Andreotti and Grauert theory for complex spaces it is natural to expect that the cohomology groups Hj(X, D) vanish for j ≥ q. This is actually true for domains in Cn× Rk ([1]). More generally, we prove the following:

Theorem 6. Let X be a 1-complete real analytic foliation. Then Hj(X, D) = 0 for j ≥ 1.

Sketch of proof. Assume k = 1 and let Φ be an exhaustive function for X.

Then the vanishing theorem for domains in Cn× Rk, the bumps lemma and the Mayer–Vietoris sequence ([1]) yield the following: for every c > 0 there is ε > 0 such that

(1) Hj(Xc+ε, D) → Hj(Xc, D)

is onto for j ≥ 1 (and this holds true for j ≥ q whenever X is a q-complete smooth foliation).

Now let eX be the complexification of X and consider the compact Xc= {Φ ≤ c}. In view of Theorem 1, Xc has a fundamental system of Stein neighbourhoods U in eX. X is oriented around Xc and consequently U \ X has two connected components U+, U (U is connected).

Denote by O+ (resp. O) the sheaf of germs of holomorphic functions on U+

(resp. U) that are smooth on U+∪ (U+∩ X) (resp. U∪ (U∩ X). Then we have the exact sequence

(2) 0 → O → O+⊕ O → D → 0re

([2]) (here O+ (resp. O) is a sheaf on U+ (resp. U) extended by 0 on all U and re (f ⊕ g) = f|x− g|x). Since U is Stein we derive from (2) that

(3) Hj(U+, O+) ⊕ Hj(U, O)→ H j(U ∩ X, D)

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for j ≥ 1 (and this holds true for j ≥ q whenever X is a q-complete real-analytic foliation of codimension 1).

Let be a j-cocycle of D on a neighbourhood of Xc. In view of (2) we have ξ = ξ+− ξ where ξ+ and ξ are represented by two (0, j)-forms ω+, ω on U+, U respectively which are smooth up to X.

Moreover, according to [6] it is possible to construct pseudoconvex domains U+0 and U0 satisfying the following conditions: U+0 ⊂ U+, U0 ⊂ U0, ∂U+0 , ∂U0 are smooth and ∂U+0 ∩ X, ∂U0 ∩ X contain a neighbourhood of Xc.

Then Kohn’s theorem ([10]) implies that on U+0 and U0 respectively we have ω+ = ∂v+, ω = ∂v where v+ ∈ C(U0+), v ∈ C(U0). It follows that Hj(Xc, D) = 0 for j ≥ 1 and from (1) we deduce that Hj(xc, D) = 0 for every c ∈ R and j ≥ 1.

At this point, in order to conclude our proof we can repeat step by step the proof of the Andreotti–Grauert vanishing theorem for q-complex spaces ([1]).

If k ≥ 2 the situation is much more involed. Using the Nirenberg Exten- sion Lemma ([10]) it is possible to reduce the cohomology H(x, D) to the ∂- cohomology of eX with respect to the differential forms on eX which are flat on X and to conclude invoking a theorem of existence proved by J. Chaumat and A. M. Chollet ([3]).

Assume that X is real analytic and let O0be the sheaf of germs of real analytic CR-functions. Then an analogous statement for O0 is not true. Andreotti and Nacinovich ([2]) showed that H1(X, O0) is never zero. However by Theorem 1 we have for arbitrary k, Hj(Xc, O0) = 0 for j > 0 whenever X is q-complete.

2. Using the same method of proof, under the hypothesis of Theorem 6, we have the following

Theorem 7. Let A = {xν} be a discrete subset of X and let {cν} be a sequence of complex numbers. Then there exists f ∈ D(X) such that f (xν) = cν, ν = 1, 2, . . . In particular , X is D-convex and D(X) separates points of X.

R e m a r k. A vanishing theorem can be also proved for the sheaf of germs of

“CR-sections” of E → X where E is a fibre vector bundle with fibre Cm× Rh. 4. The Kobayashi metric

1. Let X be a foliation with complex leaves of codimension k, and let T (X)π X be the tangent bundle of X. The collection of all tangent spaces to the leaves of X forms a complex subbundle TH(X) of T (X). Let D be the unit disc in C and denote by CR(D, X) the set of all CR-maps D → X.

Given ζ ∈ TH(X) with x = π(ζ) we define the function F = FXon X ×TH(X) by

F (x, ζ) = inf{s ∈ R : s ≥ 0, sϕ0(0) = ζ}

where ϕ ∈ CR(D, X) and ϕ(0) = x.

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When k = 0, F reduces to the Kobayashi “infinitesimal metric” of the complex manifold X ([8]). In particular, if X = Cn× Rk, then F = 0.

If X0 is another foliation as above and φ : X → X0 is a CR-map then dφ : TH(X) → TH(X0) and

FX(φ(x), dφζ) ≤ FX(x, ζ).

Teorem 7. FX is upper semicontinuous.

According to the complex case [8], X is said to be hyperbolic if F (x, ζ) > 0 for every x ∈ X and ζ ∈ TH(X), ζ 6= 0.

R e m a r k s. 1) The fact that all the leaves are hyperbolic does not imply that X itself is hyperbolic.

2) Every bounded domain in Cn× Rk is hyperbolic.

3) Following [12] it can be proved that if X admits a continuous bounded function u, p.s.h. along the leaves and strictly p.s.h. in a neighbourhood of x, then X is hyperbolic at x.

2. Now consider a riemannian metric on X and let V be a smooth distribution of transversal tangent k-spaces. Then every ζ ∈ T (X) splits into ζ0+ ζc where ζ0∈ V , ζc∈ TH(X) and we denote by τ (ζ0) the length of ζ0.

Let F be the infinitesimal Kobayashi metric on X and for ζ ∈ Tx(X) set g(x, ζ) = F (x, ζc) + τ (x, ζ0). Then g is an upper semicontinuous pseudometric.

If γ = γ(s), 0 ≤ s ≤ 1, is a smooth curve joining x, y ∈ X the pseudo-length of γ with respect to g is

L(γ) =

1

R

0

g(γ(s), ˙γ) ds and the pseudo-distance between x, y is

d(x, y) = inf

γ L(γ).

d is a real distance on X inducing the topology of X if X is hyperbolic. X is said to be complete if a field V can be chosen making X complete with respect to d.

For example, the unit ball in C × R is complete for the choice V = λ(t)

 x

∂x+ y

∂y



+ (1 + t2)−1

∂t where λ(t) = 2 arctan t[(1 + t2)−1(1 − arctan2t)−3/2].

The interest of this construction is due to the following

Theorem 8. Let Ω ⊂ Cn× Rk be with the riemannian structure induced by Cn× Rk. If Ω is hyperbolic and complete then Ω is D-convex.

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References

[1] A. A n d r e o t t i et H. G r a u e r t, Th´eor`emes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259.

[2] A. A n d r e o t t i and M. N a c i n o v i c h, Analytic convexity , Ann. Scuola Norm. Sup. Pisa 7 (1980), 287–372.

[3] J. C h a u m a t et A. M. C h o l l e t, Noyaux pour r´esoudre l’´equation ∂ dans des classes ultradiff´erentiables sur des compacts irr´eguliers de Cn, preprint.

[4] M. F r e e m a n, Local complex foliations of real submanifolds, Math. Ann. 209 (1970), 1–30.

[5] —, Tangential Cauchy–Riemann equations and uniform approximation, Pacific J. Math.

33 (1970), 101–108.

[6] R. G a y et A. S e b b a r, Division et extension dans l’alg`ebre A(Ω) d’un ouvert pseudo- convexe `a bord lisse de Cn, Math. Z. 189 (1985), 421–447.

[7] L. H ¨o r m a n d e r, An Introduction to Complex Analysis in Several Variables, North- Holland, 1973.

[8] S. K o b a y a s h i, Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York, 1970.

[9] J. J. K o h n, Global regularity for ∂ on weakly pseudoconvex manifolds, Trans. Amer.

Math. Soc. 181 (1973), 273–292.

[10] L. N i r e n b e r g, A proof of the Malgrange preparation theorem, in: Liverpool Singularities 1, 97–105.

[11] C. R e a, Levi flat submanifolds and biholomorphic extension of foliations, Ann. Scuola Norm. Sup. Pisa 26 (1972), 664–681.

[12] N. S i b o n y, A class of hyperbolic manifolds, in: Recent Developments in Several Com- plex Variables, Ann. of Math. Stud. 100, Princeton Univ. Press, 1981.

[13] F. S o m m e r, Komplexe analytische Bl¨atterung reeler Mannigfaltigkeiten in Cn, Math.

Ann. 136 (1958), 111–133.

[14] G. T o m a s s i n i, Extension d’objets CR, Math. Z. 194 (1987), 471–486.

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