Let X be a complex analytic manifold, let M ⊂ X be a real analytic hypersurface, and denote by σ the symplectic form on the cotangent bundle π : T ∗ X → X.

APPLICATION TO CONCENTRATION IN DEGREE FOR MICROFUNCTIONS AT THE BOUNDARY

By Andrea D’Agnolo and Giuseppe Zampieri (Received July 15, 1992)

§0. Introduction.

Let X be a complex analytic manifold, let M ⊂ X be a real analytic hypersurface, and denote by σ the symplectic form on the cotangent bundle π : T ^∗ X → X.

Assume that the conormal bundle T _M ^∗ X is symplectic with respect to the imaginary part of σ.

In this paper, we discuss the link between the (real) symplectic form σ| _T

M

X , and the classical Levi form L _M on the complex tangent plane T ^C M = T M ∩ iT M . In particular, we prove that if M is the boundary of a strictly pseudoconvex (or pseudoconcave) domain, and if S is a submanifold of M such that Σ = S × _M T _M ^∗ X is regular involutive in T _M ^∗ X, then S is generic in X.

Define e Σ to be the union of the complexifications of the bicharacteristic leaves of Σ. For p ∈ Σ, set λ S (p) = T p T _S ^∗ X, λ 0 (p) = T p π ⁻¹ π(p). In §2 we show the relationship between λ _S ∩ iλ _S and the null-space of L _M | _T

S . We then show that:

dim _R (T e Σ ∩ λ ₀ ) = 1 + dim _C (λ _S ∩ iλ _S ).

Denote by s ^± (M ) (resp. s ^± (S)) the numbers of positive and negative eigenvalues of L _M (resp. L _M | _T

_S ), and set γ(S) = dim _C (λ _S ∩ iλ _S ∩ λ ₀ ). Denote by µ _W (O _X ) the Sato microlocalization of the sheaf O X of holomorphic functions along a real submanifold W ⊂ X. In §3 we assume that the dimension of λ _S ∩ iλ _S is constant respect to p. Hence e Σ = T _W ^∗ X for a suitable submanifold W of X, and we show that µ W (O X ) is concentrated in degree d = 1 + dim C (λ S ∩ iλ S ) + s ⁻ (S) − γ(S).

When s ⁻ (M ) − s ⁻ (S) = dim C (λ S ∩ iλ S ) − γ(S), we also show that there is a natu- ral morphism µ W (O X ) → µ M (O X ) which induces an injective morphism between the cohomology groups in degree d. This is, in a generalized sense, a theorem of concentration in degree for microfunctions at the boundary.

§1. Symplectic forms and Levi forms.

Let X be a complex manifold of dimension n, π : T ^∗ X → X the cotangent bundle, ∂ the holomorphic differential on X. Let X ^R denote the real underlying manifold to X and d = ∂+ ¯ ∂ its real differential. We shall always identify T ^∗ (X ^R ) → ^∼ (T ^∗ X) ^R . (If φ is a C ¹ function on X ^R , the identification is given by dφ(x) 7→

∂φ(x).) Let ω be the (complex) canonical 1-form on T ^∗ X, σ = ∂ω the canonical

Appeared in: Japan. J. Math. (N.S.) 19 (1993), no. 2, 299–310.

1

2-form, H: T ^∗ T ^∗ X → T T ^∗ X the corresponding Hamiltonian isomorphism. We shall consider the induced one-forms ω ^R = 2 Re ω, ω ^I = 2 Im ω and two-forms σ ^R = 2 Re σ, σ ^I = 2 Im σ on T ^∗ X ^R .

Let M be a real analytic hypersurface of X defined by the equation φ(z) = 0 at z ₀ ∈ M . Let p = (z ₀ , ∂φ(z ₀ )), and set:

T z

M = {v ∈ T z

X; Rehv, ∂φi = 0}

T _z ^C

M = T z

M ∩ iT z

M = {v ∈ T z

X; hv, ∂φi = 0}

λ M (p) = T p T _M ^∗ X λ 0 (p) = T p π ⁻¹ π(p)

ν(p) = C H(ω(p)).

The morphism:

M → T _M ^∗ X z 7→ (z, ∂φ(z)).

induces the morphism

ψ : T z

M → λ M (p)

v 7→ (v, ∂h∂φ, vi + ∂h ¯ ∂φ, ¯ vi)

by which T _z ^C

M is identified to {(v, ∂h ¯ ∂φ, ¯ vi); h∂φ, vi = 0}. Let L M be the Levi form of M at p. Recall that, if (z ₁ , . . . , z _n ) is a local system of coordinates at z ₀ , L M is the Hermitian form on T _z ^C

M represented by the matrix (∂ i ∂ ¯ j φ(z 0 )) i,j . One immediately checks that:

ψ ^∗ (σ| _ψ(T

z0

M ) ) = L M .

Let S be a real analytic submanifold of M with codim X M = 1, codim M S = r.

We shall set:

γ(S, p) = dim C (λ S (p) ∩ iλ S (p) ∩ λ 0 (p)).

One says that S × _M T _M ^∗ X is regular when ω| _S×

_T

M

X (= ₂ ⁱ ω ^I | _S×

_T

M

X ) 6= 0.

Proposition 1.1. The following assertions are equivalent:

(1.1) S × M T _M ^∗ X is regular at p;

(1.2) dim ^R (λ _S (p) ∩ ν(p)) = 1.

Proof. Let S be locally defined by the equations φ i = 0 (i = 1, ..., r + 1), where φ ₁ ∼ = φ is a local equation for M , and recall that p = (z 0 , ∂φ(z ₀ )). We have:

ψ ^∗ (ω ^I ) = − Re(i ∂φ).

Thus:

S × _M T _M ^∗ X is regular ⇔ i ∂φ

∂z (z ₀ ) / ∈ (T _S ^∗ X) _z

⇔ dim _R (λ _S (p) ∩ ν(p)) = 1.



Remark 1.2. Let codim _X S = 2, textcodim _X M = 1. Then S × _M T _M ^∗ X is regular iff γ(S, p) = 0 (i.e. S is “generic”).

We shall assume from now on that

(1.3) T _M ^∗ X is I-symplectic

i.e. that σ ^I | _λ

is non-degenerate. This is equivalent to λ _M (p) ∩ iλ _M (p) = {0} or else to

L _M is non-degenerate,

(cf [S1]). In fact via ψ one may easily identify λ _M (p) ∩ iλ _M (p) with the null-space of L M (cf [D’A-Z1]).

Corresponding to the non-degenerate forms σ ^I and L M we may thus consider two Hamiltonian isomorphisms H ^I and e H ^I :

T ^∗ T _M ^∗ X −−−−→ T T ^HI _M ^∗ X

ψ





y ^ψ

x



 (T ^C M ) ^{∗ H e}

I

−−−−→ T ^C M.

Consider the morphism π ^∗ _M : T _z ^∗

M → T _p ^∗ T _M ^∗ X induced by the projection π M : T _M ^∗ X → M .

Proposition 1.3. For θ ∈ T _z ^∗

0

M , we have

(1.4) H ^I (π _M ^∗ (θ))| _ψ(T

_{M )} = ψ(v)

(in the identification λ _M (p) f →λ ^∗ _M (p) given by w 7→ σ(w, ·)), where v ∈ T _z ^C

0

M is the unique solution of

(1.5) ∂h ¯ ∂φ, ¯ vi = i

2 θ| _T

_M (in the identification T ^C M f →T ^C M ^∗ given by v 7→ hv, ·i).

Proof. The vectors v which satisfy (1.4) must verify for every u ∈ T _z ^C

M : 1

2 [hθ, ui + h¯ θ, ¯ ui] = σ(ψ(v), ψ(u))

= −[ih∂h ¯ ∂φ, ¯ vi, ui − ih∂h ¯ ∂φ, ¯ ui, vi]

= −[ih∂h ¯ ∂φ, ¯ v, i, ui + ih∂h ¯ ∂φ, ¯ vi, ui], and hence:

Reh i

2 θ, ·i| _T

_M = Reh∂h ¯ ∂φ, ¯ vi, ·i| _T

_M Reasoning in the same way for iu we get the conclusion. 

Let us remark now that (when (1.3) holds):

dim _R (λ _S (p) ∩ ν(p)) = 1 ⇔ i H ^I (π ^∗ (p)) / ∈ λ _S (p),

where π ^∗ : T _z ^∗

X → T _p ^∗ T ^∗ X denotes the map associated to π : T ^∗ X → X. We may

then correspondingly rephrase the equivalent conditions of Proposition 1.1. We also

have

Proposition 1.4. Let S ⊂ M ⊂ X with codim _X M = 1 and with L _M being positive or negative definite at p. Then

(1.6) S × _M T _M ^∗ X is regular involutive ⇒ γ(S, p) = 0.

Proof. Let S be locally defined by the equations φ i = 0 (i = 1, ..., r + 1), where φ ₁ ∼ = φ is a local equation for M . We have already seen that:

S × _M T _M ^∗ X is regular ⇔ i ∂φ / ∈ R∂φ ₁ + · · · + R∂φ _r+1

⇔ ∂φ 2 | _T

_M , . . . , ∂φ r+1 | _T

_M are R-independent.

(1.7)

Let v _i solve (1.4) for θ _i = ∂φ _i . Let χ and ψ be functions on M which vanish on S and let u, v ∈ ⊕

i Rv i be such that: ψ(u) = H ^I (π _M ^∗ (dχ)), ψ(v) = H ^I (π ^∗ _M (dψ)).

Owing to Proposition 1.3, we have:

{χ, ψ} _I = −i L _M (¯ u ∧ v), ({·, ·} I denoting the Poisson bracket). Hence:

(1.8) {χ, ψ} I = 0 ∀χ, ψ (i.e. S × M T _M ^∗ X is involutive) ⇒ u 6= i v ∀u, v.

(In fact if v = i u, then σ(H u, H v) = i σ ^I (H u, i H u) = −2i L _M (¯ u ∧ u) 6= 0 since L M definite (> 0 or < 0).) We also remark that

v 7→ h∂h ¯ ∂φ, ¯ vi, ·i| _T

_M is injective,

due to (1.4). By (1.7), (1.8), this implies that ∂φ i | _T

_M , i ≥ 2 are C-independent.

This is in turn equivalent to the fact that ∂φ _i , i ≥ 1 are C-independent.

§2. Degenerate R-Lagrangian submanifolds.

Let X be a complex manifold of dimension n, and let ¯ X be the complex conjugate manifold to X. We shall identify X ^R to the diagonal X × X X. The manifold X × ¯ ¯ X is then a natural complexification of X ^R .

Let M be a C ^ω -hypersurface of X ^R defined at z ₀ ∈ M by the equation φ(z) = 0, and set p = (z 0 , ∂φ(z 0 )). We shall identify C ⊗ R T z

M with {(u, ¯ v) ∈ T z

X × T _z

X; h∂φ, ui + h ¯ ¯ ∂φ, ¯ vi = 0} and consider:

ψ ^C : C ⊗ R T z

M → λ M (p) + i λ M (p)

(u, ¯ v) 7→ (u; ∂h∂φ, ui + ∂h ¯ ∂φ, ¯ vi).

(2.1)

Let S ⊂ M be a submanifold with codim _M S = r. Consider the null space:

NS(L M | _T

z0

S ) = {v ∈ T _z ^C

S; ∂h ¯ ∂φ, ¯ vi ∈ π ^∗ ((T _S ^∗ X) z

)}.

Using T X ,→ T X × T X T ¯ X and ψ, we get

Proposition 2.1. We have an identification (2.2) λ _S (p) ∩ iλ _S (p) ∼ = NS(L M | _T

z0

S ) ⊕ (λ _S (p) ∩ iλ _S (p) ∩ λ ₀ (p)).

Proof. We shall often omit the indices z 0 and p in the following. The projection R : Hπ ^∗ (T _S ^∗ X + iT _S ^∗ X) −→ Hπ ^∗ (T _S ^∗ X) is well defined modulo λ S ∩ iλ S ∩ λ 0 . It is easy to check that (cf also [D’A-Z1]):

λ _S ∩ iλ _S λ S ∩ iλ S ∩ λ 0

= {ψ ^C (v) − 2R(∂h ¯ ∂φ, ¯ vi); v ∈ T ^C S, ∂h ¯ ∂φ, ¯ vi ∈ Hπ ^∗ (T _S ^∗ X + iT _S ^∗ X)}.

Then (2.2) follows. 

We suppose all through this section that T _M ^∗ X is I-symplectic and that S × _M T _M ^∗ X is a regular involutive submanifold of T _M ^∗ X. We also set Σ = S × M T _M ^∗ X, and define e Σ to be the union of the complexifications of the bicharacteristic leaves of Σ. This is a germ of R-Lagrangian submanifold of T ^∗ X ^R .

Proposition 2.2. We have an identification

(2.3) T _p Σ ∩ iT e _p Σ ∼ e = ψ({v ∈ T z

S + iT _z

S; ∂h ¯ ∂φ, ¯ vi ∈ Hπ ^∗ (T _S ^∗ X _z

+ iT _S ^∗ X _z

)}).

Proof. We have by definition

(2.4) T e Σ = T Σ + iT Σ ^⊥ ,

(where · ^⊥ denotes the symplectic orthogonal in (T _M ^∗ X, σ ^I ).) But

(2.5) T Σ ^⊥ = ψ({u ∈ T ^C M ∩ T S; ∂h ¯ ∂φ, ¯ ui| _T

_M ∈ iHπ ^∗ (T _S ^∗ X) | _T

_M }).

Then (2.3) follows immediately. 

Proposition 2.3. Under the above assumptions, we have (2.6) dim C (T p Σ ∩ iT e p Σ) = r. e

Proof. We have T Σ ∩ iT Σ = 0 (due to the fact that T _M ^∗ X is I-symplectic). It follows:

T e Σ ∩ iT e Σ = (T Σ + iT Σ ^⊥ ) ∩ (T Σ ^⊥ + iT Σ)

= T Σ ^⊥ + iT Σ ^⊥ = C ⊗ _R T Σ ^⊥ .



For v ∈ T e Σ + iT e Σ (resp v ∈ T Σ + iT Σ, resp v ∈ λ _S + iλ _S ) let us denote by v ^c

e (resp v ^c

, resp v ^c

) the “conjugate” with respect to T e Σ (resp T Σ, resp λ _S ).

This is well defined modulo T e Σ ∩ iT e Σ = (T e Σ + iT e Σ) ^⊥ (resp T Σ ∩ iT Σ = {0}, resp λ S ∩ iλ S = (λ S + iλ S ) ^⊥ ). We remark now that T e Σ + iT e Σ = T Σ + iT Σ and thus (2.7) σ(v, w ^c

e

)| _{v,w∈(T e Σ+iT e Σ)∩λ}

0

∼ σ(v, w ^c

)| v,w∈(T Σ+iT Σ)∩λ

(where “∼” means equivalent in signature and rank). On the other hand:

( ψ ^C (T ^C S) + (λ _S ∩ λ ₀ ) ^C = (λ _S + iλ _S ) ∩ λ ₀ ψ ^C (T ^C S) ∩ (λ _S ∩ λ ₀ ) ^C = λ _S ∩ iλ _S ∩ λ ₀ and:

(T Σ + iT Σ) ∩ λ 0 = ψ ^C (T ^C S) ⊕ (λ S ∩ λ 0 ) ^C

(where · ^C = · + i· and where we make the identification T ^C S ,→ {0} × T ¯ X). It follows:

(2.8) σ(v, w ^c

)| v,w∈(T Σ+iT Σ)∩λ

∼ σ(v, w ^c

)| _v,w∈(λ

_+iλ

_)∩λ

.

Let τ denote the inertia index for a triple of Lagrangian planes in the sense of [K-S1] and [K-S2]. We recall that ¹ ₂ τ (T e Σ, iT e Σ, λ 0 ) and n − dim R (T e Σ ∩ λ 0 ) + 2 dim _C (T e Σ ∩ iT e Σ ∩ λ ₀ ) − dim _C (T e Σ ∩ iT e Σ) are respectively the signature and the rank of σ(v, w ^c

e

)| _{v,w∈(T e Σ+iT e Σ)∩λ}

0

. In the same way ¹ ₂ τ (λ S , iλ S , λ 0 ) and n − 1 − r + 2γ(S) − dim _C (λ _S ∩ iλ _S ) are signature and rank of σ(v, w ^c

)| _v,w∈λ

_∩iλ

_∩λ

due to [D’A-Z1]. By (2.7), (2.8) the above signatures and ranks have to coincide. In particular

dim R (T e Σ ∩ λ 0 ) + dim C (T e Σ ∩ iT e Σ) − 2 dim C (T e Σ ∩ iT e Σ ∩ λ 0 )

= 1 + r + dim C (λ S ∩ iλ S ) − 2γ(S).

(2.9)

We also notice that

T e Σ ∩ iT e Σ ∩ λ ₀ = Hπ ^∗ (T _S ^∗ X ∩ iT _S ^∗ X) | _T

_M

∼ = Hπ ^∗ (T _S ^∗ X ∩ iT _S ^∗ X)

= λ S ∩ iλ S ∩ λ 0 . (2.10)

Thus by using (2.10) and (2.6) in (2.9) we get at once the following statement.

Theorem 2.4. We have

(2.11) dim _R (T _p Σ ∩ λ e ₀ ) = 1 + dim _C (λ _S (p) ∩ iλ _S (p)).

§3. A vanishing theorem for generalized microfunctions at the bound- ary.

Let X be an open set of C ⁿ , and let M be a C ^ω -hypersurface of X. We de- note by D ^b (X) the derived category of the category of complexes of sheaves with bounded cohomology. We denote by O X the sheaf of holomorphic functions on X and we will consider its microlocalization along M , µ _M (O _X ) (µ _M being the Sato microlocalization functor). For any C ^ω -submanifold W of X ^R , we shall similarly define µ _W (O _X ).

Let M be defined in local coordinates by φ(z) = 0 and let p = (z 0 , ∂φ(z 0 )) with

z ₀ ∈ M . Let s ^± (M, p) denote the numbers of positive and negative eigenvalues for

the Levi form L M . Let S be a C ^ω -submanifold of M with codim M S = r. We

shall denote by s ^± (S, p) the corresponding numbers of eigenvalues for L _M | _T

S (cf

[D’A-Z1]). Set now Λ = T _M ^∗ X, Σ = S × M T _M ^∗ X and assume that Λ is I-symplectic

and Σ is regular involutive in Λ. Let us recall that the latter hypothesis implies that dim R (λ S (p) ∩ ν(p)) = 1 (recall that ν denotes the Euler vector field), and moreover, if M is the boundary of a strictly pseudoconvex or pseudoconcave domain, it implies that γ(S, p) := dim C (λ S (p) ∩ iλ S (p) ∩ λ 0 (p)) = 0.

Define e Σ to be the union of the complexifications of the bicharacteristic leaves of Σ. Let us assume

(3.1) δ(S, p) := dim _C (λ _S (p) ∩ iλ _S (p)) is constant with respect to p.

Let us remark that if (3.1) holds, then by (2.11) there exists W with S ⊂ W ⊂ X and with codim X W = 1 + δ(S, p) such that e Σ = T _W ^∗ X.

We also know from [K-S1, Proposition 11.3.5] that

(3.2) µ M (O X ) is concentrated in degree 1 + s ⁻ (M, p).

(In fact the assumptions of that Proposition are fulfilled since, T _M ^∗ X being I- symplectic, s ⁻ (M, p) is constant.) Let S ⊂ M ⊂ X, codim X M = 1, codim M S = r.

Our result goes as follows.

Theorem 3.1. Let T _M ^∗ X be I-symplectic, let S × M T _M ^∗ X be regular involutive in T _M ^∗ X, and assume (3.1) to hold. Then

(3.3) H ^j µ W (O X ) = 0 for j 6= 1 + δ(S, p) + s ⁻ (S, p) − γ(S, p).

Proof. (We shall often omit the indices p or z ₀ = π(p) in the following.) We perform a contact transformation χ in T ^∗ X such that

(3.4)



 

 

T _M ^∗ X → T _M ^∗

X codim X M ⁰ = 1, s ⁻ (M ⁰ ) = 0 Σ → S ⁰ × M

T _M ^∗

X codim M

S ⁰ = r, γ(S ⁰ ) = 0 T _W ^∗ X(= e Σ) → T _W ^∗

X codim _X W ⁰ = 1.

(The fact that γ(S ⁰ ) = 0 in the second line follows from Proposition 1.4. This, together with NS(L M

)| _T

_S

= {0} (cf §2), gives δ(S ⁰ ) = 0 due to Proposition 2.1. This implies codim W ⁰ = 1 in the third line of (3.4), due to Theorem 2.4.) Let s ⁰ (S) := δ(S) − γ(S); thus s ⁺ (S) + s ⁻ (S) + s ⁰ (S) = n − 1 − r − γ(S) = codim ^C _T

M T ^C S. We have

(3.5) s ⁺ (W ⁰ ) ≡ n − 1 − r, s ⁰ (W ⁰ ) ≡ r

(since δ(W ⁰ ) = dim _C (e Σ ∩ ie Σ) = r). In particular s ⁻ (W ⁰ ) = 0 and hence T _W ^∗

X is the (exterior) conormal to the boundary of a (weakly) pseudoconvex domain and therefore by the results of [H1]:

(3.6) µ W

(O X ) is concentrated in degree 1.

By (3.5) we have 1

2 τ (λ W

, iλ W

, λ 0 ) = n − 1 − r. We recall from §2:

σ(v, w ^c

)| _v,w∈(λ

_+iλ

_)∩λ

∼ σ(v, w ^c

)| (T Σ+iT Σ)∩λ

(3.7)

∼ σ(v, w ^c

)| _(λ

_+iλ

_)∩λ

, dim C (T e Σ ∩ iT e Σ) = r,

(3.8)

dim C (T e Σ ∩ iT e Σ ∩ λ 0 ) = γ(S).

(3.9)

In particular s ^± (W ) = s ^± (S). It follows 1

2 τ (λ _W , iλ _W , λ ₀ ) = s ⁺ (W ) + s ⁻ (W ) − 2s ⁻ (W )

= n − 1 − δ(S) − r + 2γ(S) − 2s ⁻ (S).

(3.10)

Let

d _W,W

= 1

2 [codim W ⁰ − codimW − 1

2 τ (λ _W

, iλ _W

, λ ₀ ) + 1

2 τ (λ W , iλ W , λ 0 )].

(3.11)

We know from [K-S1, Proposition 11.2.8] that the contact transformation χ can be quantized to an isomorphism

(3.12) χ _∗ (µ W (O X )) ∼ = µ W

(O X )[d W,W

].

But

d W,W

= 1

2 [1 − (1 + δ(S)) − (n − 1 − r) + (n − 1 − δ(S) + 2γ(S) − 2s ⁻ (W )]

= −δ(S) + γ(S) − s ⁻ (S)



Lemma 3.2. The following are equivalent

s ⁻ (M, p) − s ⁻ (S, p) = δ(S) − γ(S, p) (3.13)

s ⁺ (M, p) − s ⁺ (S, p) = r − γ(S, p).

(3.14)

Proof. We have (s ⁺ (M ) + s ⁻ (M )) − (s ⁺ (S) + s ⁻ (S)) = r + δ(S) − 2γ(S).  Let S ⊂ M ⊂ X, codim X M = 1, codim M S = r.

Proposition 3.3. Let T _M ^∗ X be I-symplectic, let S × _M T _M ^∗ X be regular involutive in T _M ^∗ X, let (3.1) hold, and let

(3.15) s ⁻ (M, p) − s ⁻ (S, p) = δ(S, p) − γ(S, p).

We may then define a natural morphism:

(3.16) µ W (O X ) → µ M (O X ).

Proof. We may find a contact transformation on T ^∗ X

 T _M ^∗ X → T _R ^∗

C ⁿ ,

Σ → W ⁰ × R

T _R ^∗

C ⁿ , W ⁰ ⊂ R ⁿ , . and hence T _W ^∗ X → T _W ^∗

C ⁿ . Thus the natural morphism

µ W

(O X ) → µ R

(O X )

induces (3.16) (via quantization of χ as in (3.11), (3.12)) if and only if d _W

_,W = d R

,M i. e.

1 + δ(S) − (n + r) − 1

2 τ (λ W , iλ W , λ 0 ) = 1 − n − 1

2 τ (λ M , iλ M , λ 0 ) i. e.

δ(S) − r − (n − 1 − δ(S) − r + 2γ(S)) + 2s ⁻ (S) = −(n − 1) + 2s ⁻ (M ).

The latter is in turn equivalent to (3.15) 

We define now µ _{Λ\e Σ} (O _X ) in D ^b (X) to be the third term of a distinguished triangle:

(3.17) µ W (O X ) → µ M (O X ) → µ _{Λ\e Σ} (O X ) ⁺¹ → . Let S ⊂ M ⊂ X, codim X M = 1, codim M S = r.

Theorem 3.4. Let T _M ^∗ X be I-symplectic, let S × _M T _M ^∗ X be regular involutive, let (3.1) hold, and assume:

(3.18) s ⁻ (M, p) − s ⁻ (S, p) = δ(S, p) − γ(S, p).

Then

(3.19) H ^j µ _{Λ\e Σ} (O X ) = 0 for j 6= 1 + s ⁻ (M, p).

Proof. We have 1 + δ(S) − γ(S) + s ⁻ (S) = 1 + s ⁻ (M ). Hence by Theorem 3.1, µ W (O X ) and µ M (O X ) are both concentrated in degree 1 + s ⁻ (M ). Moreover owing to Proposition 3.3 we have a morphism

H ^1+s

^{(M )} µ _W (O _X ) → H ^1+s

^{(M )} µ _M (O _X ), which is injective due to [S2].

Remark 3.5. We will give in [D’A-M-Z] the following extension of the previous results. Let µ(p) := λ _M (p) ∩ iλ _M (p) and δ(M, p) := dim _C (µ(p)). The preceding results can be extended to the following situation:

(3.20)



 

 

 

 

M is generic

δ(M, p) is constant for p ∈ T _M ^∗ X T p Σ ⊃ µ(p)

T _p Σ ^µ(p) is regular involutive in λ ^µ(p) _M (p).

.

In this frame, the results on involutivity and “genericity” of §1 still hold, and we also have

dim _C (T e Σ ∩ iT e Σ) = r + δ(M ),

dim C (T e Σ ∩ iT e Σ ∩ λ 0 ) = dim C (λ S ∩ iλ S ∩ λ 0 ),

dim R (T e Σ ∩ λ 0 ) = 1 + δ(S) − δ(M ).

In particular if δ(S) is constant in p, then there exists W ⊂ X with e Σ = T _W ^∗ X. We still have

s ^± (W ) = s ^± (S),

s ⁰ (W ) = r + δ(M ) − γ(S)

= s ⁰ (M ) + codim _W S.

Thus repeating step by step the proof of Theorem 3.1 we get the following state- ment:

Assume that (3.20) is fulfilled and that δ(S) is constant in p (and hence e Σ = T _W ^∗ X). Then

(i) H ^j µ W (O X ) = 0 for j 6= 1 + s ⁻ (S) + δ(S) − δ(M ) − γ(S),

(ii) If moreover s ⁻ (M ) − s ⁻ (S) = δ(S) − δ(M ) − γ(S) then there is a natural morphism

µ _W (O _X ) → µ _M (O _X )

and if one defines µ _{Λ\e Σ} (O _X ) as in (3.17), one has:

H ^j µ _{Λ\e Σ} (O X ) = 0 for j 6= 1 + s ⁻ (M ).

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Andrea D’Agnolo D´ ep. de Math´ ematiques

Universit´ e Paris-Nord 93430 Villetaneuse, france

Giuseppe Zampieri Dip. di Matematica Universit` a di Padova

via Belzoni 7, 35131 Padova, Italy