On holomorphic foliations admitting invariant CR manifolds

arXiv:1910.01930v2 [math.CV] 6 Jul 2021

On holomorphic foliations admitting invariant CR manifolds

Judith Brinkschulte¹

Abstract

We study holomorphic foliations of codimensionk≥1 on a complex manifoldX of dimensionn+kfrom the point of view of the exceptional minimal set conjecture. For n ≥2 we show in particular that if the holomorphic normal bundleNF is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection ofksmooth real hypersurfaces inX.

1 Introduction

Let X be a complex manifold and F a (singular) holomorphic foliation of codimensionk≥1 onX. In complex dynamics, one is interested in under- standing the structure of the set of accumulation point of the leaves of F.

A natural question asks under what condition on X and/or F does it hold that every leaf of F accumulates to the singular set Sing(F)?

In this context, Brunella in [Br] stated the following conjecture, which may also be formulated for holomorphic foliations of codimensionk≥1:

LetX be a compact connected complex manifold of dimension ≥3, and let F be a codimension one holomorphic foliation on X whose normal bundle N_F =T X/TF is ample. Then every leaf of F accumulates to Sing(F).

Assume to the contrary that not every leaf ofF accumulates to Sing(F):

Then X contains a nonempty compact subset M which is invariant by F and disjoint from Sing(F) (a so-called exceptional minimal set). The aim of this paper is to prove that such an Mcannot be a smooth intersection of k real hypersurfaces whose normals are complex linearly independent.

More precisely, our main result is the following

1Universit¨at Leipzig, Mathematisches Institut, PF 100920, D-04009 Leipzig, Germany.

E-mail: brinkschulte@math.uni-leipzig.de

Key words: Levi-flat CR manifolds, holomorphic foliation, positive normal bundle, ex- ceptional minimal set

2000 Mathematics Subject Classification: 32V40, 32V25, 37F75

Theorem 1.1

Let X be a complex manifold of dimension n+k, and let M ⊂ X be a compact smooth real submanifold that is a complete intersection of k real hypersurfaces in X. Suppose that on some neighborhood U of M, there exists a codimension k holomorphic foliation F which leaves M invariant.

Then, for n ≥ 2, the holomorphic normal bundle N_F of F in U does not admit any Hermitian metric h such that (N_F, h) is Griffiths positive on U.

By a complete intersection we mean that M is defined by k smooth functionρj ∈ C^∞(X):

M ={z∈X|ρ₁(z) =. . .=ρ_k(z) = 0}, with∂ρ₁∧. . .∧∂ρ_k 6= 0 onM.

Theorem 1.1 generalizes the main result of [Br] in two ways. First of all, we pass from codimension one to codimension k ≥1. Second, X is an arbitrary complex manifold in Theorem 1.1, possibly noncompact, whereas in [Br] it was assumed to be compact K¨ahler.

Another interpretation of Theorem 1.1 is from the point of view of clas- sifying compact Levi-flat CR manifolds. Let us remind that a CR manifold of type (n, k) is given by a smooth real manifold M of dimension 2n+k and a complex subbundle T^1,0M of rank n of C⊗M that is stable un- der the Lie-bracket. It is also required that T^1,0M ∩T^0,1M = {0}, where T^0,1M =T^1,0M. If moreoverT^1,0M+T^0,1M is closed under the Lie bracket, then M is called Levi-flat. It follows from the theorems of Frobenius and Newlander-Nirenberg that Levi-flat CR manifolds are locally foliated by complex n-dimensional submanifolds.

The normal bundle of a Levi-flat CR manifold of type (n, k) is the vector bundle of rankkover M defined by

N_M =C⊗T M/(T^1,0M +T^0,1M).

NM carries the structure of a CR vector bundle overM, that is the restric- tion to each leaf of the foliation ofM is a holomorphic vector bundle.

Remark that due to a theorem of Andreotti-Fredricks [AF], a real-analytic CR manifold always admits a generic CR embedding into a complex mani- fold. IfM is also Levi-flat, then the Levi-foliation ofM can be holomorphi- cally extended to some neighborhood U of M (see [R]). Also a metric on N_M with leafwise positive curvature can always be extended to a metric on the extended bundle with positive curvature. Thus Theorem 1.1 enables us to state the following corollary:

Corollary 1.2

Let X be a complex manifold of dimension n+k, and let M ⊂ X be a real-analytic compact CR manifold of type (n, k). If M is Levi-flat and a complete intersection of k real hypersurfaces in X, then, for n ≥ 2, the holomorphic normal bundle N_M of M in X does not admit any Hermitian metric h such that(N_M, h) is Griffiths positive along the leaves of M.

As a very special case, Corollary 1.2 therefore includes the nonexistence results for Levi-flat real hypersurfaces in CPⁿ, n ≥3, because the normal bundle of a real hypersurface in CPⁿ naturally inherits positivity from the ambient complex manifold (see [LN] and [S]). Fork >1, the corresponding nonexistence result for smooth Levi-flat CR manifolds in complex projective spaces can be found in [B1]. We also refer the interested reader to [FSW], where the authors weaken the assumptions on the Levi-flat CR manifold under a more restrictive assumption on k.

In [B3] Corollary 1.2 was proved for k = 1, also if M is only smooth instead of real-analytic (M is then a compact Levi-flat real hypersurface), thereby generalizing previous results by Brunella [Br] and Ohsawa [O2].

Note that Theorem 1.1 does not hold for Levi-flat CR manifolds of CR- dimensionn= 1. Counterexamples can be found in [Br] and [O3].

For other results related to the exceptional minimal set conjecture for holomorphic foliations of arbitrary codimension, we refer the reader to [CF].

The organization of this paper is as follows: We argue by contradic- tion and assume that we are given a holomorphic foliation of codimension k with N_F Griffiths positive that leaves invariant a compact complete in- tersection M of k real hypersurfaces as in the statement of Theorem 1.1.

In section 3 we prove that M admits a neighborhood basis having certain q-convexity/concavity properties. This allows us to holomorphically extend CR sections of (detN_F)^⊗ℓ in section 4. Combining a version of Kodaira’s embedding theorem for Levi-flat CR manifolds proved by Ohsawa with the holomorphic extension of CR sections of (detNF)^⊗ℓ, we then show that a tubular neighborhood of M can be generically embedded into a projective K¨ahler space ˆX (section 5). In section 6, we prove some L²-Hodge-type symmetry results on the regular part of ˆX. The final argument is then given in section 7: we first extend the Chern curvature form of detN_F to a d-exakt (1,1)-form on the regular part of ˆX. Using theL²-Hodge-type sym- metry results from section 6, we may then find a smooth potential for the Chern curvature form of detN_F on an open neighborhood of M. This will contradict the maximum principle on the leaves of M and therefore prove our main result.

Remark: As a referee pointed out, in Theorem 1.1 and Corollary 1.2, it is possible to relax the assumption on M of being a complete intersection ofk hypersurfaces to the CR dimension ofM being equal tonand the line bundle det(N_M) being topologically trivial. In this situation, in the proof of Proposition 3.1, one has to replace the functionρwith the distance function to M with respect to the metricω_o.

2 Preliminaries

Let Y be a complex manifold of dimension n endowed with a Hermitian metric ω, and let E be a holomorphic vector bundle on Y with a Hermi- tian metric h. We recall that (E, h) is said to be Griffiths positive if for all y∈Y and all non-zero decomposable tensorsv⊗e∈T Y ×Y_y, the curvature term hiΘ_h(E)(v, v)e, ei is positive, where iΘ_h(E) ∈ C_1,1^∞(Y,Herm(E, E)) is the curvature of the Chern connection of (E, h).

For integers 0≤p, q≤n, we use the following notations:

D^p,q(Y, E) denotes the space of smooth, compactly supported E-valued (p, q)-forms on Y.

L²_p,q(Y, E, ω, h) denotes the Hilbert space obtained by completingD^p,q(Y, E) with respect to theL²-norm k · k_ω,h induced by ω and h.

As usual, the differential operator ∂ is extended as a densely defined closed linear operator onL²_p,q(Y, E, ω, h), whose domain of definition is

Dom∂={f ∈L²_p,q(Y, E, ω, h)|∂f ∈L²_p,q+1(Y, E, ω, h)},

where∂f is computed in the sense of distributions. The Hilbert space ad- joint of∂ will be denoted by ∂^∗ (=∂^∗_ω,h).

We also define the space of harmonic forms,

H^p,q(Y, E, ω, h) =L²_p,q(Y, E, ω, h)∩Ker∂∩Ker∂^∗_ω,h, and theL²-Dolbeault cohomology groups ofY,

H_L^p,q₂(Y, E, ω, h) =L²_p,q(Y, E, ω, h)∩Ker∂/L²_p,q(Y, E, ω, h)∩Im∂.

Whenever we feel that it is clear from the context, we will omit the de- pendency of theL²-spaces, norms, operators etc. on the hermitian metrich of the vector bundles under considerations.

In section 4, we shall also use the following variant of the ∂-operator:

by∂_c we denote the strong minimal realization of∂onL²_p,q(Y, E, ω, h). This

means thatu∈Dom∂c ⊂L²_p,q(Y, E, ω, h) if there existsf ∈L²_p,q+1(Y, E, ω, h) and a sequence (u_ν)_ν∈N ⊂ D^p,q(Y, E) such that u_ν −→ u and ∂u_ν −→ f =

∂_cu inL²_p,q+1(Y, E, ω, h).

The Hilbert space adjoint of ∂c will be denoted by ϑ; it is the weak maximal realization of the formal adjoint of∂ on L²_p,q(Y, E, ω, h).

3 Convexity properties of tubular neighborhoods

From now on, M will always denote a smooth submanifold of real codi- mension k in a complex manifold X of dimensionn+kthat is a complete intersection of k smooth real hypersurfaces Σj ={z ∈ U |ρj(z) = 0} and that is invariant by a holomorphic foliation F on some neighborhood of M.

The aim of this section is to find a real-valued function onU\M, with a certain growth order nearM, whose Levi-form has n+ 1 positive andk−1 negative eigenvalues.

For that purpose, we fix a Hermitian metric ω_o on X and define ρ = (ρ²₁+. . .+ρ²_k)¹².

Note the analogy with the following model case: IfX is a complex man- ifold of dimension n+k and Y ⊂X is a compact complex submanifold of dimensionn inX such that the normal bundle N_{Y /X} of Y inX is positive in the sense of Griffiths, then Y admits a tubular neighborhood basis con- sisting of domainsU such that the Levi form of∂U hasnnegative andk−1 positive eigenvalues (see [G]).

For k= 1, the following Proposition was proved in [Br].

Proposition 3.1

Let X be a complex manifold of dimension n+k, and let M ⊂ X be a smooth compact real submanifold that is a complete intersection of k real hypersurfaces inX. Suppose that on some neighborhoodU ofM, there exists a codimension k holomorphic foliation F which leaves M invariant. We moreover assume that the holomorphic normal bundle NF of F is positive in the sense of Griffiths on U. Then, after possibly shrinking U, there exist a smooth nonnegative real-valued function v on U and constants c, c^′ > 0 such that if λ1 ≤ . . . ≤ λ_n+k denote the eigenvalues of i∂∂(−logv) with respect to ω_o, then

(i) ρ²λ_j ∈[−c^′,−c], j= 1, . . . , k−1, (ii) λ_j ∈[c, c^′], j=k, . . . , n+k−1,

(iii) ρ²λ_n+k ∈[c, c^′]

Moreover we havev=O(ρ²).

Proof. The proof is an adaptation of the argument given in [O2].

We may choose a finite covering of M by open sets U_α, such that F is defined byk holomorphic 1-forms̟_α¹, . . . , ̟_α^k on U_α (i.e. Ker̟^j_α⊂T^1,0F).

Then there exist holomorphick×k-matrices G_αβ onU_α∩U_β satisfying

̟_α=G_αβ̟_β, (3.1)

with̟_α = (̟¹_α, . . . , ̟^k_α)^T; in fact the matricesG_αβ are the transition matri- ces for the holomorphic normal bundleN_F with respect to the dual frames (̟_α)^∗ of ̟_α.

The fiber metric ofN_F may then be identified with a system of positive definite Hermitian matricesH_α on U_α such that

H_α= h(̟^j_α)^∗,(̟_α^ℓ)^∗ik j,ℓ=1. We then have

H_α=G^T_βαH_βG^∗_βα on U_α∩U_β. (3.2) Since M is a smooth Levi-flat CR manifold in X, we may assume that M is locally defined by M ∩U_α ={z ∈U_α |(Imfα^j)(z) = 0, j = 1, . . . , k}, wheref_α^j is a function such that ∂f_α^j vanishes to infinite order on M∩U_α and df_α¹∧. . .∧df_α^k 6= 0 on M∩U_α (this can be seen using the parametric equations forM, see e.g. [HT]).

On Uα∩M we have ̟α =Aαdfα for some smooth matrix Aα which is invertible on U_α and holomorphic along the leaves of M. From (3.1) and (3.2) it follows that we have

(A_αdf_α)^TH_αA_αdf_α= (A_βdf_β)^TH_αA_βdf_β on U_α∩U_β∩M But then

(A_αImf_α)^TH_αA_αImf_α−(A_βImf_β)^TH_βA_βImf_β =O(ρ³) along U_α∩U_β∩M Therefore, invoking Whitney’s extension theorem, there exists a non- negative real-valuedC² functionv defined in a tubular neighborhood ofM, smooth away fromM, such thatv= (A_αImf_α)^TH_αA_αImf_α+O(ρ³) onU_α. It now remains to estimate −i∂∂logv = −i^∂∂v_v +i∂logv∧∂logv on U\M.

It suffices to estimate the Levi-form of−log (AαImfα)^THαAαImfα on Uα \M. Setting H_α^′ = A^T_αHαAα, we have to estimate the Levi-form of

−log Imf_α^TH^′_αImf_α

on U_α\M.

Therefore we let V ∈T^1,0U be a unitary vector that we decompose or- thogonally intoV =Vt+Vn, withVt∈Tk

j=1Ker(∂Imfα^j). A straightforward computation then gives that up to terms vanishing on M

−i∂∂logv(V_t, V_t) =iImf_α^T∂∂H^′_αImf_α

v +Imf_α^T∂H^′_αImf_α∧Imf_α^T∂H^′_αImf_α v²

(V_t, V_t)

=iImf_α^TH^′_α∂ (H^′_α)⁻¹∂H^′_α Imf_α v

(V_t, V_t)+

iImf_α^T∂H^′_α∧(H^′_α)⁻¹∂H^′_αImfα

v

(Vt, Vt) +i|Imf_α^T∂H^′_αImfα|²(Vt)

v² .

Both terms in the last line of the above equation are nonnegative since

Imf_α^T∂H^′_α∧(H^′_α)⁻¹∂H^′_αImf_α

(V_t, V_t) =|(H_α^′)^−1/2∂H_α^′Imf_α(V_t)|² Thus we get

−i∂∂logv(V_t, V_t)≥iImf_α^TH^′_α∂ (H^′_α)⁻¹∂H^′_α Imf_α v

(V_t, V_t).

SinceH_αrepresents the metric ofN_Fin a local holomorphic trivialization over U_α we have

iΘ(N_F) =i∂ (H_α)⁻¹∂H_α on U_α

But as the matrix A_α is holomorphic along the leaves ofM, we may as- sume that∂A_α vanishes to finite order alongM. But theni∂ (H^′_α)⁻¹∂H^′_α is the Chern curvature tensor ofN_F in the almost holomorphic trivialization given by the change of basisA_α. We may therefore findc >0 such that

−i∂∂logv(V_t, V_t)≥(c+ǫ)ω_o(V_t, V_t), (3.3) whereǫcan be made as small as we wish by shrinking U.

Moreover, since ∂fα^j vanishes to infinite order on U_α ∩M and V_t ∈ T_k

j=1Ker(∂Imfα^j), a careful computation shows that

|i∂∂logv(Vt, Vn)|.|Vt| · |Vn| Therefore we get

|i∂∂logv(V_t, V_n)| ≤ǫω_o(V_t, V_t) +C

ǫω_o(V_n, V_n) (3.4)

for some constant C.

The remaining arguments do not involve the Levi-flatness ofM, therefore we may replacevwithρfor the computations involving−i∂∂logv. We have

∂ρ= ¹_ρ(ρ₁∂ρ₁+. . .+ρ_k∂ρ_k) and

−i∂∂ρ=− i ρ(

k

X

j=1

ρ_j∂∂ρ_j+

k

X

j=1

∂ρ_j∧∂ρ_j) + i

ρ³(ρ1∂ρ1+. . .+ρ_k∂ρ_k)∧(ρ1∂ρ1+. . .+ρ_k∂ρ_k) (3.5) The first sum in (3.5) gives only a small contribution with respect to the other two terms as the point approaches M, and it can therefore be neglected.

We now choose a (1,0) vector field ξ /∈ Tk

j=1Ker(∂ρ_j) on U \M such that∂ρ(ξ) = 1 on U \M as follows:

ξ = 1 ρ

k

X

j=1

ρjξj,

whereξj are (1,0) vector fields onU satisfying ∂ρ_ℓ(ξj) =δ_jℓ. A direct com- putation shows that the sum of the second and third term in the right hand side of (3.5) applied to (ξ, ξ) is zero, hence the leading term ofi∂∂logρ(ξ, ξ) isi^{∂ρ∧∂ρ}_ρ2 (ξ, ξ) = _ρ¹₂.

We may thus conclude that

−i∂∂logv(ξ, ξ)≥a1

v (3.6)

for some constant a >0 (shrinking U if necessary).

Combining (3.3), (3.4) and (3.6), the minimum-maximum theorem shows the existence ofc >0 in (ii) and (iii). The existence of c^′ >0 follows from the reversed inequalities in (3.3) (as a consequence of the Levi-flatness of M) and (3.6) (as follows from the definition of the function v).

The (k−1) negative eigenvalues in (i) come from the second term on the right hand side of (3.5).

4 Holomorphic extension of CR sections of det N

M The key result of this section is Proposition 4.3, the extension of CR sections of the normal bundle to holomorphic sections of detN_F over U. This will

enable us to holomorphically embed a tubular neighborhood ofM into some complex projective space in the next section. The proof of this holomorphic extension property needs several steps; it is a modification of the arguments in section 7 of [B3].

From now on, we will exploit the properties of the function ϕ=−logv, wherevas in Proposition 3.1. ShrinkingU if necessary, we may assume that vis actually defined in an open neighborhood ofU and satisfies (i), (ii) and (iii) of Proposition 3.1 on that open neighborhood.

We start with the following lemma that can be proved as Lemma 4.2 in [B2]:

Lemma 4.1

There exists a complete hermitian metric ω_M on U \M with the following properties:

(i) Let γ₁ ≤ . . . ≤ γ_n+k be the eigenvalues of i∂∂ϕ with respect to ω_M. There exists σ >0 such that γ₁+. . .+γ_k> σ on U \M.

(ii) There are constants a, b >0 such that a ω_o≤ω_M ≤b ρ⁻²ω_o. (iii) There is a constant C >0 such that|∂ω_M|_ωM ≤C.

From Lemma 4.1 we obtain the following L²-vanishing result:

Proposition 4.2

Let E −→ U be a holomorphic line bundle. Then there exist N ∈ N such that the following holds: Assume 0 ≤ q ≤ n−1, and let f ∈ L²_0,q(U \ M, E, ω_M,−N ϕ)∩Ker∂. Then there exists a (0, q−1)-form g∈L²_0,q−1(U\ M, E, ωM,−N ϕ) satisfying ∂g =f andkgkωM,−N ϕ≤ kfkωM,−N ϕ.

Proof. We fix 0 ≤ q ≤ n−1 It follows from the generalized Bochner- Kodaira-Nakano inequality (see [D1,Chapter VII, section 1−3]) and stan- dard computations that foru∈ D^0,q(U \M, E) we have

3

2(k∂uk²_{ωM,−N ϕ}+k∂^∗uk²_{ωM,−N ϕ})

≥N ≪(γ₁+. . .+γ_n+k−q)u, u≫_{ωM,−N ϕ}

− ≪(c_q+1+. . .+c_n+k)u, u≫_{ωM,−N ϕ}

−1

2(kτ uk²_{ωM,−N ϕ}+kτ uk²_{ωM,−N ϕ}+kτ^∗uk²_{ωM,−N ϕ}+kτ^∗uk²_{ωM,−N ϕ}), where c₁ ≤ . . . ≤ c_n+k are the curvature eigenvalues of a fixed hermitian metric of E with respect to ω_M, Λ is the adjoint of multiplication by ω_M

andτ = [Λ, ∂ωM].

For N sufficiently big, it therefore follows from the properties of ω_M described in Lemma 4.1 that

kuk²_{ωM,−N ϕ}≤ k∂uk²_{ωM,−N ϕ}+k∂^∗uk²_{ωM,−N ϕ} (4.1) foru∈ D^0,q(U \M, E). By the completeness of ω_M, the estimate (4.1) ex- tends tou∈L²_0,q(U\M, ωM,−N ϕ)∩Dom∂∩Dom∂^∗ with compact support inU.

Note that (4.1) holds also forq =n; it is only now that we have to restrict toq≤n−1: ShrinkingU if necessary, we may assume that the boundary of U is smooth and that its Levi form hasnnegative andk−1 positive eigen- values. From [G,Theorem VI and Theorem 7.4] we may therefore deduce that ifN is sufficiently big, then

kuk²_{ωM,−N ϕ}≤ k∂uk²_{ωM,−N ϕ}+k∂^∗uk²_{ωM,−N ϕ} (4.2) foru ∈L²_0,q(U \M, ω_M,−N ϕ)∩Dom∂∩Dom∂^∗ with compact support in U\K, whereK is a compact containing an open neighborhood ofM inU. Using two cut-off functionsχ₁, χ₂, where χ₁ has compact support in U and equals one in an open neighborhood ofM andχ₂ = 1−χ₁, we may use (4.1) and (4.2) to conclude that ifN is sufficiently big, then

kuk²_{ωM,−N ϕ}≤ k∂uk²_{ωM,−N ϕ}+k∂^∗uk²_{ωM,−N ϕ}

foru∈L²_0,q(U\M, ωM,−N ϕ)∩Dom∂∩Dom∂^∗, 0≤q ≤n−1. From this a priori estimate, the assertion of the Proposition follows in a standard way.

In the next section, we want to holomorphically extend CR sections over M of some high tensor power of the line bundle detN_F, which is a positive line bundle by assumption. On the other hand, we can multiply the metric of detNF bye^{N ϕ}. This adds −N i∂∂ϕ to the curvature, so the curvature of detN_F can be made partly negative near onU\M by taking N sufficiently large. This modification, however, would require the CR sections that we wish to extend to be sufficiently regular.

Since this is false in general, we have to use some approximation argu- ments, reducing the involved∂-equation to compactly supported forms. As a result we can prove

Proposition 4.3

Letℓ∈Nbe sufficiently large, and assume thatsis a CR-section of(detN_F)^ℓ over M of class at least C^k+4. Then there exists a holomorphic section ˜sof (detN_F)^ℓ on U such that s˜_|M =s.

Proof. First we choose aC^k+4-extensionso ofsto X such that ∂so van- ishes to the orderk+ 3 alongM, i.e. |∂s_o|²_ωo =O(|ρ|^2k+6).

Now we consider an exhaustion of U \M by domains Vε = {z ∈ U | v(z) > ε²}; we recall that v = O(ρ²). Moreover, we define the annular domains D_j =V1

j \V2 j.

Then we choose a sequence of smooth cut-off functionsχ_j with compact support in V1

j such that χ_j ≡ 1 on V2

j and |dχ_j|²_ωM ≤ 1 (this is possible sinceωM is complete onU \M). Then

f_j :=∂(χ_j∂s_o)∈L²_0,2(U \M,(detN_F)^ℓ, ω_M)∩Ker∂ (4.3) is compactly supported in D_j.

Applying Lemma 4.4 yields u_j ∈ L²_0,1(U \M,(detN_F)^ℓ, ω_M) supported in D_j satisfying ∂u_j =f_j and

ku_jk²_ωM ≤C²j⁴kf_jk²_ωM .j⁴k∂χ_j∧∂s_ok²_ωM

≤j⁴ Z

Dj

|∂s_o|²_ωMdV_ωM ≤j⁴ Z

Dj

ρ^−2k|∂s_o|²_ωodV_ωo .1 Now gj =χj∂so−uj is∂-closed and supported in V1

j, hence compactly supported in U \M. By Proposition 4.2, there existsN ∈N such that we can find solutionsh_j ∈L²_0,0(U\M,(detN_F)^ℓ, ω_M,−N ϕ) satisfying∂h_j =g_j. Hence g_j ∈ L²_0,1(U \M,(detN_F)^ℓ, ω_M,−kϕ) ∩Im∂. By Lemma 4.5, we can therefore have hj ∈ L²_0,0(U \M,(detNF)^ℓ, ωM,−ϕ) and khjkωM,−ϕ ≤ C_okg_jk_ωM,−ϕ. But

kg_jk²_ωM,−ϕ.kχ_j∂s_ok²_ωM,−ϕ+ku_jk²_ωM,−ϕ .

Z

V1 j

|∂s_o|²_ωoρ^−2k−2dV_ωo+ku_jk²_ωM,−ϕ .

Z

V₁

j

ρ^2k+6ρ^−2k−2dV_ωo +j⁴ Z

Dj

ρ^−2k−2|∂s_o|²_ωodV_ωo .1 Therefore the sequence (h_j) is bounded inL²_0,0(U\M,(detN_F)^ℓ, ω_M,−ϕ), hence has a subsequence that weakly converges to

ho ∈ L²_0,0(U \M,(detNF)^ℓ, ωM,−ϕ). Since ∂hj = ∂so on V2

j, we must therefore have∂h_o =∂s_o inU\M.

Moreover, sinceh_o∈L²_0,0(U \M,(detN_F)^ℓ, ω_M,−ϕ), we have Z

U\M

|h_o|²ρ⁻²dV_ωM <+∞.

This clearly implies that the trivial extension ofho toU satisfies∂ho=∂so

as distributions onU (not only onU\M). Henceh_o is of class at leastC^k+3 by the hypoellipticity of∂, and must therefore vanish onM.

Thus ˜s=s_o−h_o is a holomorphic section of (detN_F)^ℓ over U extending

s.

Lemma 4.4

Letℓ∈N be sufficiently large andf_j be defined by (4.3). For some constant C >0, independent of j ∈N, there exists u_j ∈ L²_0,1(U \M,(detN_F)^ℓ, ω_M), supported in D_j, such that ∂u_j =f_j and

ku_jk_ωM ≤Cj²kf_jk_ωM.

Proof. To abbreviate notations, we set F = (detNF)^ℓ.

Note that the boundary of D_j consists of two parts: the part ∂V1

j whose Levi-form hasnpositive andk−1 negative eigenvalues and the part−∂V2

j

whose Levi-form has n negative and k−1 positive eigenvalues. Since the casek= 1 was treated in [B3] we may assumek≥2. But then D_j satisfies condition Z(n+k−1) (see [FK]), hence the ∂-Neumann problem satisfies subelliptic estimates in degree (p, n+k−1) for all 0 ≤ p ≤ n+k. In particular this implies (see also [FK]) that

∂:L²_n+k,n+k−2(D_j, F^∗, ω_o)−→L²_n+k,n+k−1(D_j, F^∗, ω_o)

has closed range (subelliptic estimates are proved using local computations, therefore they are valid independent of the curvature ofF, sinceF is defined over D_j).

Now we use a duality argument from [Ch-Sh]: Let∂_c be the strong min- imal realization of∂ on L²_0,1(D_j, F, ω_o). Then by Theorem 3 of [Ch-Sh] the range of∂_cis closed inL²_0,2(D_j, F, ω_o), and∂_c-exact formsf ∈L²_0,2(D_j, F, ω_o) are characterized by the usual orthogonality condition:

Z

Dj

f∧η= 0 ∀η∈L²_n+k,n+k−2(D_j, F^∗, ω_o)∩Ker∂

But, using Stokes’ theorem, we get forη∈ C_n+k,n+k−2^∞ (D_j, F^∗)∩Ker∂

Z

Dj

∂(χ_j∂s_o)∧η= Z

∂Dj

(χ_j∂s_o)∧η=− Z

∂V2 j

∂s_o∧η=− Z

∂V2 j

∂(s_o∧η) = 0,

and this also holds forη ∈L²_n+k,n+k−2(D_j, F^∗, ω_o)∩Ker∂ using a standard approximation argument and the subelliptic estimates in degree (n+k, n+ k−1).

Hence fj = ∂(χj∂so) belongs to the image of ∂c, i.e. there exists u_j ∈ L²_0,1(D_j, F, ω_o) satisfying ∂_cu_j = f_j. As usual, we assume that u_j is the minimal L²-solution i.e. u_j ∈ L²_0,1(D_j, F, ω_o)∩(Ker∂_c)^⊥ ⊂ Kerϑ.

In particular, u_j is smooth on D_j, and the trivial extension of u_j by zero outsideD_j (which we still denote by u_j), satisfies ∂u_j =f_j as distributions onU\M (by definition of the strong minimal realization∂_c). It remains to estimateku_jk_ωM.

First we note that it follows from the subelliptic estimates in degree (n+k, n+k−1) that u_j is sufficiently smooth on D_j: ∗u_j is of bidegree (n+k, n+k−1) and satisfies the elliptic system∂^∗(∗u_j) =∗f_j,∂(∗u_j) = 0.

Sincefj is of classC³ and vanishes outside a compact ofDj,uj is at least in the Sobolev space W³ and smooth up to the boundary outside the support off_j.

We will now estimate u_j by using a priori estimates on the domains W_j = U \V2

j. The Levi-form of ∂W_j has n negative and k−1 positive eigenvalues. Since we may assume k ≥ 2, it follows in particular that the Levi-form of∂W_j has at least one positive eigenvalue everywhere.

We now modify the metric in detN_F^∗ by a bounded factor exp (−mρ²).

This adds to the curvature a term which is mi∂∂ρ²=

k

X

j=1

2mρ_ji∂∂ρ_j + 2mi∂ρ_j∧∂ρ_j .

Taking m sufficiently large, and shrinking U if necessary, we may there- fore assume that the curvature form of detN_F^∗ has at least k ≥ 2 positive eigenvalues. But then the curvature term in the Bochner-Kodaira-Nakano identity of detN_F^∗ is positive in degree (n+k, n+k−1).

It then follows from the Bochner-Kodaira-Nakano formula with bound- ary term (see [G,Theorem 7.2]) that there exists λ >0 such that

kvk²_ωo,Wj ≤ λ

ℓm(k∂vk²_ωo,Wj+k∂^∗vk²_ωo,Wj)

for allv∈L²_n+k,n+k−1(W_j, F^∗, ω_o)∩Dom∂∩Dom∂^∗. From this we infer by Serre duality as in [Ch-Sh] that

kvk²_ωo,Wj ≤ λ

ℓm(k∂_cvk²_ωo,Wj+kϑvk²_ωo,Wj) (4.4) for all v∈L²_0,1(W_j, F, ω_o)∩Dom∂_c∩Domϑ.

We now choose an extension ˜uj of uj to ˜Vj = V1 2j \V2

j such that ˜uj ∈ Dom∂c∩Domϑ(on ˜Vj!) and such that

kϑ˜u_jk²_ω

o,V˜j+k∂u˜_jk²_ω

o,V˜j ≤b(kϑu_jk²_ω

o,D˜j+k∂u_jk²_ωo,Dj+ku_jk²_ωo,Dj) for some constantbnot depending onu_j nor onj. This is possible forj suf- ficiently large by general Sobolev extension methods (locally we flatten the boundary∂Dj and extend the sufficiently smoothuj componentwise across

∂D_j by first order reflection, then we use a partition of unity).

Next we choose a smooth cut-off function ˜χj with compact support in V1

2j \V2

j such that ˜χ_j ≡ 1 on D_j. This is possible with an estimate

|dχ˜_j|² .j².

Applying (4.4) withχ_ju˜_j yields ku_jk²_ωo,Dj ≤ kχ˜_ju˜_jk²_ωo,Wj ≤ λ

ℓm(k∂_c( ˜χ_ju˜_j)k²_ωo,Wj+kϑ( ˜χ_ju˜_j)k²_ωo,Wj) . λ

ℓm(k∂_cu˜_jk²_ωo,Wj+kϑ˜u_jk²_ωo,Wj+j²k˜u_jk²_ωo,Wj)

≤ λ

ℓmb(kϑu_jk²_ωo,Dj+k∂u_jk²_ωo,Dj +j²ku_jk²_ωo,Dj)

= λ

ℓmb(k∂_cu_jk²_ωo,Dj+j²ku_jk²_ωo,Dj)

Form=m(j)∼j², the additional weight factor exp(−mρ²) is bounded on D_j. Hence we may choose firstm and thenℓ sufficiently large so that

ku_jk_ωo ≤ kf_jk_ωo.

It remains to compare the norms kujkωo and kujkωM. From the con- struction ofω_M (see Lemma 4.1 ), we have dV_ωM ∼ρ^−2kdV_ωo. We also have

|f_j|²_ωo . ρ⁻⁴|f_j|²_ωM since f_j is a (0,2)-form. On the other hand, we have ω_M & ω_o, which implies |u_j|²_ωo & |u_j|²_ωM. Since u_j is supported in D_j, we thus have

ku_jk²_ωM .j^2kku_jk²_ωo ≤j^2kkf_jk²_ωo .j⁴kf_jk²_ωM,

which proves the desired estimate.

The point of the following lemma is that even though ℓ∈N can be ar- bitrary big, the weight function−ϕstays the same (it does not have to be multiplied by a large integer asℓincreases!).

Lemma 4.5

Let ℓ∈N be arbitrary. Then there exists a constant C_o such that for every u∈L²_0,0(U\M,(detN_F)^ℓ, ω_M,−ϕ)with∂u∈L²_0,1(U\M,(detN_F)^ℓ, ω_M,−ϕ) satisfies kuk_ωM,−ϕ ≤C_okfk_ωM,−ϕ.

Proof. We start by working with the metric ωo on U. Recall that in degree (0,0), the curvature term in the Bochner-Kodaira-Nakano identity is given by minus the trace of the curvature form with respect to the metric under consideration. Therefore it is convenient to have

Trace_ωo(i∂∂ϕ)&ρ⁻². (4.5) If the original metric ω_o does not satisfy this condition, then we replace it by ωo +APn+k−1

j=1 ηj ∧η_j, where A > 0 is a real constant and ηj, j = 1, . . . , n+k−1 are (1,0)-forms that are smooth on U\M and bounded on U:

Let ζ1, . . . , ζ_n+k be an orthonormal basis of (1,0)-vector fields on U with respect to the original metric, smooth on U \M and bounded on U, with ζ_n+k=ξ, whereξ is as in the proof of Proposition 3.1. Takingη₁, . . . , η_n+k to be the dual basis, (4.5) is satisfied if A is sufficiently big.

We now modify the metric in (detN_F)^ℓ by a bounded factor exp (mρ²).

This adds to the curvature a term which is

−mi∂∂ρ²=−

k

X

j=1

2mρi∂∂ρ_j + 2mi∂ρ_j∧∂ρ_j .

Taking m sufficiently large, and shrinkingU if necessary, we may therefore assume that Trace_ωo(iΘ((detN_F)^ℓ) negative onU. Thus

−Trace_ωo iΘ((detN_F)^ℓ)−ik∂∂ϕ

&ρ⁻²

onU. Increasingm if necessary, we may even assume that the torsion ofω_o can be absorbed by the right-hand side of the above inequality. But then the above estimate and the Bochner-Kodaira-Nakano inequality implies that for u∈ D^0,0(U\M,(detN_F)^ℓ) we have

Z

U\M

1

ρ²|u|²e^kϕdV_ωo . Z

U\M

|∂u|²_ωoe^kϕdV_ωo.

On the one hand, we have ω_o . ω_M by Lemma 4.1. We even have dV_ωM ∼ρ^−2kdV_ωo by construction of the metricω_M. Therefore the preceed- ing inequality implies that

Z

U\M

|u|²e^ϕdV_ωM . Z

U\M

|∂u|²_ωMe^ϕdV_ωM

foru∈ D^0,0(U\M,(detN_F)^ℓ), which extends tou∈L²_0,0(U\M,(detN_F)^ℓ, ω_M,−ϕ) with compact support in U by the completeness of ω_M on U\M. In addi- tion, the Levi-form of∂U has at leastn≥2 negative eigenvalues. Therefore we may invoke [G,Theorem 7.4] to conclude that

kuk_ωM,−ϕ.k∂uk_ωM−ϕ for allu∈L²_0,0(U \M,(detN_F)^ℓ, ω_M,−ϕ) such that

∂u∈L²_0,1(U \M,(detNF)^ℓ, ωM,−ϕ).

5 Projective embeddings of tubular neighborhoods

In [O1] (see also [H-M] for k = 1), Kodaira’s embedding theorem was generalized to the setting of compact Levi-flat CR manifolds, and it was shown that sufficiently high tensor powers of a positive CR line bundle over a smooth, compact Levi-flat CR manifold M admit enough CR sections s₀, . . . , s_m, so that the CR map [s₀ :. . .:s_m] provides a CR-embedding of M into CP^m. This applies to our situation, since detN_M is assumed to be positive.

In particular, it was proved in [O1] that if ℓis big enough, then theC⁴- smooth CR-sections of (detNM)^ℓ separate the points on M and give local coordinates onM. Using Proposition 4.3, the CR-sections of (detN_M)^ℓ can be extended to holomorphic sections of (detN_F)^ℓ over a tubular neighbor- hood ofM inX.

Arguing by continuity, it is not difficult to see that if ℓ is big enough, then, after possibly shrinkingX, the holomorphic sections of detN_M sepa- rate points and give local coordinates onX. Hence we have a holomorphic embedding Ψ :X ֒→CP^m.

Using Proposition 3.1, we may assume thatV = Ψ(X) is pseudoconcave in the sense of [A]. Therefore we may invoke [A,Th´eor`eme 6] to conclude that the projective closure ofV inCP^m is a projective variety of the same dimension. By denoting ˆX the projective normalization ofV (which exists by a theorem of Zariski), we may moreover assume that ˆX is normal. ˆX inherits the structure of a compact K¨ahler space from the K¨ahler metric of CP^m. This justifies the following proposition.

Proposition 5.1

A sufficiently small tubular neighborhood of M can be holomorphically em- bedded into a compact (normal) K¨ahler space Xˆ of dimension n+k.

6 Hodge symmetry on a compact K¨ ahler space

By the results of the preceeding sections, we may assume that the Levi-flat CR manifold M is embedded into a normal compact K¨ahler space ( ˆX,ω) ofˆ dimensionn+k. We define Ω = Reg ˆX = ˆX\Sing ˆX. By the normality of ˆX, we have codim Sing ˆX ≥2. As observed e.g. in [O3], Ω admits a complete K¨ahler metric of a particular nice form.

Proposition 6.1

There exists a complete K¨ahler metricωΩ on Ω and an exhaustion function ψ: Ω→[0,+∞) such that

1. |∂ψ|²_ωΩ < _2(n+k)¹

2. The eigenvalues λ1 ≤. . .≤λ_n+k of i∂∂ψ with respect to ωΩ satisfy (a) −1− _2(n+k)¹ < λ_j <−1 +_2(n+k)¹ outside a compact K_Ω of Ω for

1≤j≤codimSing ˆX

(b) λ_j < _2(n+k)¹ for j >codimSing ˆX

Proof. The details of the construction can be found in [O3], we only sketch the idea for the convenience of the reader.

Letx_obe a singular point of ˆX, and letf₁, . . . , f_mbe holomorphic functions on a neighborhoodV ofx_o that generate the ideal of holomorphic functions vanishing on Sing ˆX∩V in the ring of holomorphic functions onV. Shrinking V if necessary so that P_m

j=1|f_j|² < e^−e onV, we set ψ_V =εlog(−log(

m

X

j=1

|f_j|²))

and ω_V = Aωˆ −εi∂∂ψ_V for some small ε > 0 and some large A > 0. A simple computation shows that

−εi∂∂ψV = εi∂∂log(Pm

j=1|f_j|²)

−log(P_m

j=1|f_j|²) + i

ε∂ψV ∧∂ψV ≥ i

ε∂ψV ∧∂ψV. It is therefore easy to see thatψV andωV have the required properties onV.

A detailed computation in [O3] shows that ˆω−i∂∂ψ_Vαand ˆω−i∂∂ψ_Vβ are quasi-isometrically equivalent. Therefore the functions ψ_α can be patched together by a partition of unity to define a complete K¨ahler metric on Ω:

LetV ={V_α}be a finite open cover of ˆX by suchV, and letη_α be a smooth partition of unity associated toV. We set

ψ=X

α

η_αψ_Vα

and

ω_Ω =Aˆω−i∂∂ψ.

As in [O3], the function and the metric thus defined have the required prop-

erties.

Proposition 6.2

TheL²-Dolbeault-cohomology groups H_L^0,1₂(Ω, ω_Ω) andH_L^1,0₂(Ω, ω_Ω) are finite dimensional.

Proof. The idea is to use the twisting trick of Berndtsson and Siu, since the function ψ satisfies the Donnelly-Fefferman condition. On the trivial bundle E = Ω×Cwe introduce the auxiliary metric e^−ψ. Let λ₁ ≤. . . ≤ λ_n+k be the eigenvalues of i∂∂ψ with respect to ωΩ as in Proposition 6.1.

The curvature term [i∂∂ψ, ω_Ω] acting on (0,1)-forms is given by −(λ₂ + . . .+λ_n+k) (see e.g. [D1]). Since ˆX is normal, we have codimSing ˆX ≥ 2.

Therefore

−(λ₂+. . .+λ_n+k)≥1− n+k−1

2(n+k) (6.1)

outsideK_Ω from Proposition 6.1. Hence for everyu∈ C_c^0,1(Ω\K_Ω) we have the estimate

1−n+k−1 2(n+k)

Z

Ω

|u|²_ωΩe^−ψdV_ωΩ≤ Z

Ω

(|∂u|²_ωΩ+|∂^∗_ψu|²_ωΩ)e^−ψdV_ωΩ (6.2) Now we substitutev=ue^−ψ/2. It is not difficult to see that

|∂u|²_ωΩe^−ψ ≤2|∂v|²_ωΩ+1

2|∂ψ|²_ωΩ|v|²_ωΩ ≤2|∂v|²_ωΩ+ 1

4(n+k)|v|²_ωΩ. Since ∂^∗_ψ =e^ψ∂^∗e^−ψ, we likewise get

|∂^∗_ψu|²_ωΩe^−ψ ≤2|∂^∗v|²_ωΩ+1

2|∂ψ|²_ωΩ|v|²_ωΩ ≤2|∂^∗v|²_ωΩ+ 1

4(n+k)|v|²_ωΩ. Together with (6.2), these two inequalities imply

kvk²_ωΩ ≤4(k∂vk²_ωΩ+k∂^∗vk²_ωΩ) (6.3) for allv∈ C^0,1c (Ω\K_Ω). It is now standard to conclude that for any compact K containing K_Ω in its interior, there exists a constant C_K such that

kvk²_ωΩ ≤C_K(k∂vk²_ωΩ+k∂^∗vk²_ωΩ+ Z

K

|v|²_ωΩdV_ωΩ)