Summary. Let M be a real analytic hypersurface of a complex manifold X, and let S be an analytic submanifold of M . If the Levi form L

WITH CONSTANT LEVI RANK

Andrea D’Agnolo Corrado Marastoni Giuseppe Zampieri

Summary. Let M be a real analytic hypersurface of a complex manifold X, and let S be an analytic submanifold of M . If the Levi form L

of M has constant rank, by a result of [Z], the conormal bundle Λ = T

X admits a local partial complexification Λ ⊂ T ˜

X which is regular involutive. Assume moreover that L

is semi-definite and that Σ = S ×

T

X is regular involutive in the real symplectic space Λ. We prove here that S is generic in X, and that the bicharacteristic flow of ˜ Λ issued from Σ is locally the conormal bundle to a hypersurface W ⊂ X. We then consider a complex of “microfunctions at the boundary” naturally associated to this geometrical setting.

Concerning this complex, we get vanishing theorems for its cohomology groups, and we prove that it satisfies the principle of analytic continuation along the integral leaves of ˜ Λ.

Notice that the above results extend those obtained in [D’A-Z 2] under the as- sumption of maximal rank for L

.

Microfunzioni lungo sottovariet` a a rango di Levi costante

Riassunto. Sia M una sottovariet` a analitica reale di una variet` a complessa X, e sia S una sottovariet` a analitica di M . Nell’ipotesi che la forma di Levi L

di M abbia rango costante, per un risultato di [Z], il fibrato conormale Λ = T

X ammette una complessificazione parziale locale ˜ Λ ⊂ T

X regolare involutiva. Si assuma inoltre che L

sia semi-definita e che Σ = S ×

T

X sia regolare involutiva nello spazio simplettico reale Λ. In tali ipotesi si mostra che S ` e generica in X, e che il flusso bicaratteristico di ˜ Λ sortito da Σ ` e localmente il fibrato conormale ad un’ipersuperficie W ⊂ X. Si considera quindi un complesso di “microfunzioni al bordo” naturalmente associato a questa situazione geometrica. Riguardo ad esso, si mostrano teoremi di annullamento per i suoi gruppi di coomologia, e si prova che soddisfa il principio del prolungamento analitico lungo le foglie integrali di ˜ Λ.

Si noti che i risultati citati estendono quelli ottenuti in [D’A-Z 2] nell’ipotesi di rango massimo per L

.

§1. Some results on symplectic geometry.

Let X be a complex manifold of dimension n. Denote by T X the tangent bundle to X, by T ^∗ X the cotangent bundle, and let π : T ^∗ X → X be the projection.

Denote by X ^R the real underlying manifold to X. We will identify identify T ^∗ (X ^R ) and (T ^∗ X) ^R by df = ∂f + ∂f 7→ ∂f , for any real function f . Let ω be the canonical one-form, σ = dω be the canonical two-form on T ^∗ X, and decompose it as σ = σ ^R + iσ ^I , for σ ^R and σ ^I real symplectic two-forms. The forms σ and σ ^I induce Hamiltonian isomorphisms that we denote by H and H ^I respectively.

Appeared in: Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 111–121.

Dipartimento di Matematica Pura e Applicata; Universit` a di Padova; via Belzoni 7; I-35131 Padova

1

Let M be a real analytic submanifold of X ^R , and denote by T _M ^∗ X the conormal bundle to M in X. For p ∈ ˙ T ^∗ X = T ^∗ X \ T _X ^∗ X, z = π(p), we set:

T _z ^C M = T z M ∩ iT z M, λ M (p) = T p T _M ^∗ X,

λ ◦ (p) = T p π ⁻¹ π(p), ν(p) = CHω(p),

δ _M (p) = λ _M (p) ∩ iλ _M (p), d M (p) = dim _C (δ M (p)),

γ M (p) = dim _C (δ M (p) ∩ λ _◦ (p)).

If no confusion may arise, we shall drop down the indices p or M in the above notations.

Recall that T _M ^∗ X is said to be I-symplectic when NS σ ^I 

 _λ

_(p)
= {0}, where NS denotes the null-space. It is immediate to see that NS σ ^I 

 _λ

_(p)
= δ _M (p).

Fix p ◦ ∈ ˙ T _M ^∗ X, set z ◦ = π(p ◦ ), and let φ 1 = 0, . . . , φ l = 0 be a set of independent equations for M near z _◦ . The vectors w _i = ∂φ _i (z _◦ ) constitute a basis for (T _M ^∗ X) _z

. Consider the morphism

ψ : R ^l × M → T _M ^∗ X, ((t i ), z) 7→ (z;

l

X

i=1

t i ∂φ i (z)).

Let φ be a real function such that φ| M ≡ 0, (z ◦ ; ∂φ(z ◦ )) = p ◦ . The Levi form of M at p _◦ is defined by:

(1.1) L _M (p _◦ ) = ∂∂φ 

 _T

M .

This is a good definition in the sense that if ˜ φ is any other function with ˜ φ  

 ^M ≡ 0 and ∂ ˜ φ(z ◦ ) = p ◦ , the corresponding form ∂∂ ˜ φ

 ^T

^M has the same signature and rank as (1.1).

Let (t _◦i ) ∈ R ^l be such that ∂φ(z _◦ ) = P l

i=1 t _◦i w _i . Consider the tangent morphism to ψ at ((t ◦i ), z ◦ ):

(1.2) ψ ∗ : R ^l × T z

M → λ M (p ◦ ), ((t i ), v) 7→ v;

l

X

i=1

t i w i + ∂h∂φ, vi + ∂h∂φ, vi
.

We remark that the pull back of σ| λ

by ψ ∗ is described as

(1.3) ψ ^∗ _∗ ( σ| λ

) ((t i ), v), ((˜ t i ), ˜ v)
= dt ∧ ∂φ
((t i ), v) ∧ ((˜ t i ), ˜ v)
+ ∂∂φ(v ∧ ˜ v).

Lemma 1.1. The following relation holds:

dim NS (L M (p _◦ )) = d M (p _◦ ) − γ M (p _◦ ).

Proof. Let ((t _i ), v) be such that in (1.3) we get 0 for all ((˜ t _i ), ˜ v). Then



 

 

 

 

 

 

∂∂φ(v ∧ ˜ v) +

l

X

i=1

ht _i w _i , ˜ vi = 0 ∀˜ v ∈ T _z

M,

l

X

i=1

h˜ t _i w _i , vi = 0 ∀(˜ t _i ) ∈ R ^l

(here, to obtain the first condition we put (˜ t _i ) = 0 in (1.3), and to obtain the second we put ˜ v = 0). By the second equality we get v ∈ T _z ^C

M ; by the first we have

=m h∂h∂φ, vi, ˜ vi +

l

X

i=1

ht i w i , ˜ vi
= 0 ∀˜ v ∈ T z

M,

which is equivalent to

∂h∂φ, vi +

l

X

i=1

t i w i ∈ i(T _M ^∗ X) z

,

or else to



 



 



∂h∂φ, vi ∈ (T _M ^∗ X) z

+ i(T _M ^∗ X) z

,

l

X

i=1

t i w i ∈ <e −∂h∂φ, vi
+ ((T _M ^∗ X) z

∩ i(T _M ^∗ X) z

)

(where <e is the real part, well-defined modulo (T _M ^∗ X) z

∩i(T _M ^∗ X) z

). Summarizing up,

dim NS(σ| _λ

_(p

₎ )
= dim{v ∈ T _z ^C

M : ∂h∂φ, vi ∈ (T _M ^∗ X) z

+ i(T _M ^∗ X) z

} + γ M (p _◦ )

= dim NS(L M (p ◦ )) + γ M (p ◦ ).

Since NS(σ| _λ

_(p

₎ ) = δ M (p _◦ ), we get the conclusion.  For p ∈ ˙ T _M ^∗ X, consider the numbers s ^+,−,0 _M = s ^+,−,0 _M (p) given by:



 

 

 

 

s ⁰ _M = d _M − γ _M ,

s ⁺ _M + s ⁻ _M + s ⁰ _M = n − codim _{T X} T ^C M, s ⁺ _M − s ⁻ _M = 1

2 τ (λ M , iλ M , λ ◦ ),

where τ is the Maslov index in the sense of [K-S]. In [D’A-Z 1] it is proved that s ^+,−,0 _M are equal to the numbers of respectively positive, negative, and null eigenvalues for the Levi form L M . The latter are moreover equal to the corresponding numbers of eigenvalues for σ(v, v ^c )| _δ

_∩λ

, where v ^c is the conjugate with respect to λ _M in the quotient λ M + iλ M /λ M ∩ iλ M . (Note that these results in case of codim X M = 1 were obtained in [S 1] and [K-S].)

In the following we will write Λ = ˙ T _M ^∗ X. Let Σ be a conic submanifold of Λ, and recall that Σ ⊂ Λ is called regular if ω ^I 

 _Σ is everywhere different from zero.

Proposition 1.2. (cf [Z]) Let p _◦ ∈ Σ, and assume that for a conic neighborhood V of p ◦ in T ^∗ X we have:



 

 

 



 

 

 



d M (p) = d ∀p ∈ Λ ∩ V, dim ν(p _◦ ) ∩ λ _M (p _◦ )
= 1, T p Σ ⊃ δ M (p) ∀p ∈ Σ ∩ V, Σ is regular in V ,

T _p Σ/δ(p) is involutive in δ(p) ^⊥ /δ(p) ∀p ∈ V ∩ Σ.

Then, there exist complex manifolds X ⁰ , X ⁰⁰ of dimension n − d and d respectively, real analytic submanifolds S ⁰ ⊂ M ⁰ ⊂ X ⁰ with codim X

M ⁰ = 1, codim M

S ⁰ = codim Λ Σ, and a germ of symplectic transformation at p _◦

χ : T ^∗ X → T ^{∼ ∗} (X ⁰ × X ⁰⁰ ),

such that, identifying X ⁰⁰ with T _X ^∗

X ⁰⁰ , the following isomorphisms hold locally at p ◦ :

χ(Λ) = Λ ⁰ × X ⁰⁰ , Λ ⁰ = T _M ^∗

X ⁰ I-symplectic, (i)

χ(Σ) = Σ ⁰ × X ⁰⁰ , Σ ⁰ = S ⁰ × _M

T _M ^∗

X ⁰ regular involutive in T _M ^∗

X ⁰ . (ii)

Proof. (i) We give here a new and simpler proof with respect to [Z].

Performing a first complex contact transformation, it is not restrictive to assume that Λ is the conormal bundle to a hypersurface M such that s ⁻ _M (p ◦ ) = 0 (cf. [K-S]).

By Lemma 1.1, the form L _M has constant rank (in particular, M is the boundary of a pseudoconvex domain). By [R], [F] we may find an analytic foliation of M by the (complex) integral leaves of NS(L _M ). By a suitable choice of an equation φ = φ 1 for M , we may assume that the associated morphism

ψ : R × M → T M ^∗ X

is such that for every complex submanifolds Γ of M in X, and for every t ∈ R, ψ(t, Γ) is a complex submanifold of T _M ^∗ X in T ^∗ X. Thus we obtain a foliation of Λ by the complex integral leaves of λ _M ∩ iλ _M . This is a real analytic submersion Λ → Λ ⁰ with complex fibers. Moreover, using [F, th. 2.14] we may find a complex analytic manifold X ⁰⁰ of dimension d and a germ of real analytic isomorphism at p ◦ :

(1.4) f : Λ → Λ ^{∼ 0} × X ⁰⁰ ,

with the submersion mentioned above corresponding to the first projection. Note

that, for p = (p ⁰ , z ⁰⁰ ), T _p f ⁻¹ (p ⁰ ×X ⁰⁰ ) = δ(p), and hence σ ^I induces a non-degenerate

real symplectic form on Λ ⁰ . A complexification (Λ ⁰ ) ^C of Λ ⁰ is then endowed with a

non-degenerate complex symplectic two-form, and we may locally identify it with

the cotangent bundle to a complex manifold X ⁰ . Finally, since dim _R (T Λ ⁰ ∩ ν ^δ ) =

1, by applying a contact transformation in T ^∗ X ⁰ it is not restrictive to assume

Λ ⁰ = T _M ^∗

X ⁰ , for a hypersurface M ⁰ of X ⁰ which is the boundary of a pseudoconvex domain.

(ii) Due to (i), and the identifications T Σ ^δ ' T Σ ⁰ , T Λ ⁰ ' δ ^⊥ /δ, we have χ(Σ) = Σ ⁰ × X ⁰⁰

with Σ ⁰ regular involutive in Λ ⁰ = T _M ^∗

X ⁰ . Finally, since Σ ⁰ ⊂ T _M ^∗

X ⁰ and M ⁰ is a hypersurface, we may find S ⁰ ⊂ M ⁰ with codim _M

S ⁰ = codim _Λ Σ such that

Σ ⁰ = S ⁰ × M

T _M ^∗

X ⁰ . 

§2. Genericity and involutivity.

Let M be an analytic hypersurface of X, let S ⊂ M be a submanifold, and set Λ = ˙ T _M ^∗ X, Σ = S × M T ˙ _M ^∗ X. Assuming that L M has constant rank, we shall discuss here the link between “genericity” of S and “regular involutivity” of Σ in the real symplectic space (Λ, σ ^I ). (These results were obtained in [D’A-Z 2] under the hypothesis of L _M being non-degenerate.)

By the results from the previous section, keeping the above notations it is not restrictive to assume (after a contact transformation) that Λ = Λ ⁰ × X ⁰⁰ and Σ = Σ ⁰ × X ⁰⁰ at p ◦ ∈ Σ. For z ◦ = π(p ◦ ) ∈ S, write z ◦ = (z ◦ 0 , z ◦ 00 ) with z ◦ 0 ∈ S ⁰ and z _◦ ⁰⁰ ∈ X ⁰⁰ . Let φ = 0 be a local equation for M ⁰ in X ⁰ at z _◦ ⁰ (hence an equation of M ' M ⁰ × X ⁰⁰ in X at z ◦ ), and set p ◦ 0 = (z ◦ 0 ; ∂φ(z ◦ 0 )) ∈ Σ ⁰ , so that p ◦ = (p ◦ 0 , z ◦ 00 ).

We consider the diagram

T z

M −−−−→ λ ^ψ

M (p _◦ )



 y



 y T _z

M ⁰ −−−−→ λ ^ψ

_M

(p _◦ ⁰ )

where the horizontal arrows are induced from (1.2) with (t i ) = 0 and the vertical arrows are the projections v = (v ⁰ , v ⁰⁰ ) 7→ v ⁰ . One has

ψ ^∗ σ(v, w) = h∂h∂φ, vi, wi − h∂h∂φ, wi, vi

= h∂h∂φ, v ⁰ i, w ⁰ i − h∂h∂φ, w ⁰ i, v ⁰ i

= ψ ^∗ σ((0, v ⁰ ), (0, w ⁰ )).

In [D’A-Z 2] it is proved that ∀θ ⁰ ∈ T _z ^∗

M ⁰ H ^I (π ^∗ θ ⁰ ) 

 _ψ

∗

T

z◦0

M

= ψ _∗ (v ⁰ ), where v ⁰ is the unique solution of

∂h∂φ, v ⁰ i = i θ ⁰ | _T

M

in the identification T _z ^C

◦0

M ^{0 ∼} → (T _z ^C

◦0

M ⁰ ) ^∗ , given by v 7→ hv, ·i.

From now on we will consider the following geometrical situation. Let S and M be submanifolds of X, S ⊂ M , codim X M = 1. Let Σ = S × M T ˙ _M ^∗ X and consider it as a submanifold of Λ = ˙ T _M ^∗ X, endowed with the (degenerate) two-form σ ^I . Let p _◦ be a point of Σ, V a conic neighborhood of p _◦ and set z _◦ = π(p _◦ ).

Recall that S is called generic at z _◦ if (T _S ^∗ X) z

∩ i(T _S ^∗ X) z

= {0}.

Proposition 2.1. (a) Assume that (i) d M (p) = d ∀p ∈ Λ ∩ V , (ii) T _p Σ ⊃ δ _M (p) ∀p ∈ Σ ∩ V . Then

Σ is regular at p _◦ ⇔ dim(ν(p _◦ ) ∩ λ _S (p _◦ )) = 1.

(b) Let us also suppose that L _M is semi-definite (i.e., M is the boundary of a (weakly) pseudoconvex or pseudoconcave domain). Then

( Σ is regular at p _◦

T _p Σ ^δ is involutive in δ(p) ^⊥ /δ(p) ∀p ∈ Σ ∩ V implies

S is generic at z ◦ . (Here ∗ ^δ = (∗ ∩ δ ^⊥ ) + δ
/δ.)

Proof. Let us choose coordinates such that Λ = ˙ T _M ^∗

X ⁰ × X ⁰⁰ and Σ = (S ⁰ × _M

T ˙ _M ^∗

X ⁰ ) × X ⁰⁰ . Then:

T Σ ^δ ' T (S ⁰ × _M

T _M ^∗

X ⁰ ), and

(λ S ∩ λ ◦ ) ^δ ' λ S

∩ λ ◦ .

The conclusion follows by the results of [D’A-Z 2].  Let us denote by ˜ Σ the bicharacteristic flow of ˜ Λ issued from Σ. Note that ˜ Σ is the union of the partial complexifications of the integral leaves of T Σ ^⊥ .

Theorem 2.2. Assume that (i) d M (p) = d ∀p ∈ Λ ∩ V ; (ii) T p Σ ⊃ δ M (p) ∀p ∈ Σ ∩ V ;

(iii) Σ is regular and T p Σ ^δ is involutive in δ(p) ^⊥ /δ(p) ∀p ∈ Σ ∩ V . Then

dim _R (T p Σ ∩ λ ˜ _◦ (p)) = 1 + d S (p) − d M (p), ∀p ∈ Σ ∩ V.

Proof. (Cf. [D’A-Z 2] for a different proof in the case d M (p) ≡ 0.)

Let Λ = ˙ T _M ^∗

X ⁰ × X ⁰⁰ , Σ = (S ⁰ × _M

T ˙ _M ^∗

X ⁰ ) × X ⁰⁰ . Since T ˜ Σ ∩ λ _◦ ' T ˜ Σ ⁰ ∩ λ _◦ , dim(λ S ∩ iλ S ) = dim(λ M ∩ iλ M ) + dim(λ S

∩ iλ S

), then it is not restrictive to suppose d M = 0 from the beginning.

For p ∈ Σ, one has:

T ˜ Σ ' T Σ + iH ^I (T _Σ ^∗ T _M ^∗ X)

' {(u + iv; s + ∂h∂φ, u + ivi + ∂h∂φ, u − ivi) : u ∈ T S, s ∈ (T _M ^∗ X) _z

, v ∈ T ^C M ∩ T S, ∂h∂φ, vi 

 _T

M ∈ iT _S ^∗ X | _T

M } where the first equality is due to T Σ ^⊥ ⊂ T Σ. Thus T ˜ Σ ∩ λ _◦ is described by

(2.1) ( u = −iv ∈ T ^C S

∂h∂φ, u − ivi 

 _T

M ∈ T _S ^∗ X | _T

M ,

Let us set Z = {v ∈ T _z ^C

S; ∂h∂φ, vi ∈ (T _S ^∗ X) _z

| _T

M }. Then dim T ˜ Σ ∩ λ _◦

= 1 + dim Z. One has:

Z + iZ = {v ∈ T _z ^C

S; ∂h∂φ, vi ∈ ((T _S ^∗ X) _z

+ i(T _S ^∗ X) _z

)| _T

M }

= NS(L _M ),

Z ∩ iZ = {v ∈ T _z ^C

S; ∂h∂φ, vi ∈ ((T _S ^∗ X) z

∩ i(T _S ^∗ X) z

)| _T

M }

= {v ∈ T _z ^C

S; ∂h∂φ, vi ∈ (T _S ^∗ X) z

∩ i(T _S ^∗ X) z

}, where the last equality follows from the implication:

dim(λ _S ∩ ν) = 1 ⇒ T _S ^∗ X ∩ iT _S ^∗ X → T ^∼ _S ^∗ X | _T

M ∩ iT _S ^∗ X | _T

M . We then have:

dim _R Z = dim _C (Z + iZ) + dim _C (Z ∩ iZ)

= dim NS(L _S ) + dim(λ _S ∩ iλ _S ∩ λ _◦ )

= dim(λ _S ∩ iλ _S ).

 Corollary 2.3. Assume that

(i) d M (p) = d ∀p ∈ Λ ∩ V ; (ii) T p Σ ⊃ δ M (p) ∀p ∈ Σ ∩ V ;

(iii) Σ is regular and T _p Σ ^δ is involutive in δ(p) ^⊥ /δ(p) ∀p ∈ Σ ∩ V ; (iv) L M is semi-definite.

Then ˜ Σ = T _W ^∗ X for a hypersurface W ⊂ W .

Proof. By Proposition 2.1, S is generic (i.e., γ _S = 0) and NS(L _S ) = NS(L _M ) in Σ. It follows that δ S (p) = δ M (p) ∀p ∈ Σ. In particular, by Theorem 2.2, dim(T ˜ Σ ∩ λ ◦ ) = 1 and thus we may find a hypersurface W = W ⁰ × X ⁰⁰ such that

Σ = T ˜ _W ^∗ X = T _W ^∗

X ⁰ × X ⁰⁰ .

§3. Microlocalization along submanifolds.

Let S, M be submanifolds of a complex manifold X of dimension n with S ⊂ M , codim X M = 1, codim M S = r. Let Σ = S × M T ˙ _M ^∗ X, Λ = ˙ T _M ^∗ X, fix p _◦ ∈ Σ and denote by V a conic neighborhood of p _◦ . Let us suppose

(3.1)



 

 



 

 



d M (p) = d ∀p ∈ Λ ∩ V, T p Σ ⊃ δ M (p) ∀p ∈ Σ ∩ V, Σ regular in V ,

T p Σ ^δ(p) involutive in δ(p) ^⊥ /δ(p) ∀p ∈ Σ ∩ V.

We recall from the preceding section that if L _M ≥ 0 (or L _M ≤ 0), then there exists

a germ of contact transformation χ at p _◦ such that χ( ˜ Σ) = T _W ^∗ X ' T _W ^∗

X × X ⁰⁰

with codim _X W = 1. In this case, we shall consider the numbers s ^+,−,0 _W . For p ∈ Σ, we have the following equalities in T p T ^∗ X:

(T p Σ + iT ˜ p Σ) ∩ λ ˜ ◦ + (λ ◦ ∩ λ S ) ^C = (T p Σ + iT p Σ) ∩ λ ◦ + (λ ◦ ∩ λ S ) ^C

= (λ _S + iλ _S ) ∩ λ _◦ ,

where the first equality holds since T _p Σ = T ˜ _p Σ + iT _p Σ ^⊥ ⊂ T _p Σ + iT _p Σ, and the second follows from the arguments between formulas (2.7) and (2.8) in [D’A-Z 1].

It follows that:

σ(v, v ^c

)| _{(T ˜ Σ+iT ˜ Σ)∩λ}

◦

∼ σ(v, v ^c

)| (T Σ+iT Σ)∩λ

∼ σ(v, v ^c

)| _(λ

_+iλ

_)∩λ

where the symbol ∼ means that the above forms have the same signature and rank. In particular, if L M ≥ 0 or L M ≤ 0 one has L W ∼ L S in Σ and thus s ^± _W = s ^± _S . One may also immediately see that s ⁰ _W = r + d M . It follows that s ⁰ _W = s ⁰ _S + codim _T

W T ^C S = s ⁰ _S + codim M S.

Consider the microlocalization of the sheaf O _X of the holomorphic functions in X along W (cf. [K-S]):

µ W (O X ) = µhom(Z ^W , O X ).

Choosing a quantization of χ by a kernel, we define the complex of microfunctions along ˜ Σ by:

(3.2) C Σ ˜ ' χ ⁻¹ µ W (O X )[1].

This is a good definition (i.e. independent of the choice of χ and its quantization) by the results of [K-S, ch.11]. Let:

C _Λ = µ _M (O _X )[1 + s ⁻ _M ].

Proposition 3.1. Let us assume (3.1). Then:

(i) C Σ ˜ and C Λ are concentrated in degree zero;

(ii) there exists a natural “restriction” morphism

(3.3) C Σ ˜ → C _Λ ,

(iii) the third term C _{Λ\ ˜ Σ} of a distinguished triangle associated to (3.3) is concen- trated in degree zero;

(iv) C Σ ˜ satisfies the principle of the analytic continuation along the integral leaves of T ˜ Σ ∩ iT ˜ Σ.

Proof. (Cf. [S 2].) Let p _◦ = (p _◦ ⁰ , z _◦ ⁰⁰ ) ∈ Σ ⁰ × X ⁰⁰ By a contact transformation χ ⁰ at p ◦ 0 (cf. [K-S]) we may interchange Λ ⁰ (resp. Σ ⁰ ) with the conormal bundle to a strictly (resp. weakly) pseudoconvex domain U (resp. U ⁰ ) with U ⁰ ⊃ U . Setting U = U × X e ⁰⁰ , e U ⁰ = U ⁰ × X ⁰⁰ , and choosing a quantization of χ ⁰ , we get:

H ^j C Λ

p '



 

  lim −→

B

O _X ( e U ∩ B)

O X (B) j = 0

0 j 6= 0,

for p ∈ Λ, (1)

H ^j C Σ ˜

p '



 

  lim −→

B

O X ( e U ⁰ ∩ B)

O _X (B) j = 0

0 j 6= 0,

for p ∈ ˜ Σ,

(2)

where B ranges through the family of open neighborhoods of (π(χ ⁰ (p _◦ ⁰ )), z _◦ ⁰⁰ ). (i) follows. The morphism in (ii) is induced by the restriction O X (U ⁰ ) → O X (U ). (iii) follows. The propagation of (iv) (for microfunctions with holomorphic parameters)

can be proved by the aid of Bochner’s theorem. 

Example 3.2. Let X = C ⁿ , and let χ : T ^∗ X → T ^∗ X be a germ of contact trans- formation with χ(T _R ^∗

X) = T _M ^∗ X, where codim X M = 1 and M is the boundary of a strictly pseudoconvex domain. Let N be an analytic hypersurface of R ⁿ , and write Σ = χ(N × _R

T _R ^∗

X). If R ⁿ \N = Ω ⁺ ∪Ω ⁻ , denoting by C _Ω

_|X = µ _Ω

(O X )[n]

the complexes of [S 1], one has

χ _∗ (C _Ω

_|X ⊕ C _Ω

_|X ) = C _{Λ\ ˜ Σ} .

References

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[S 2] , Conditions de positivit´ e dans une vari´ et´ e symplectique complexe. Application

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