Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 23, 2006, 183–192

Abstract. In the present paper we construct a complex of sheaves associated to a closed differential form

ω

. We study this complex in case

ω

is a) a closed 1-form vanishing at an embedded submanifold, b) a symplectic structure with Martinet singularities. In particular, we prove that, under additional conditions on

ω

, this complex gives a ﬁne resolution for the sheaf of inﬁnitesimal automorphisms of

ω

1. Introduction

For a tensor ﬁeld

t

deﬁning an integrable

G

-structure on a smooth manifold

M

, the Spencer

P

-complex of the Lie derivative

D V = L_{V} t

gives the ﬁne resolution for the sheaf of inﬁnitesimal automorphisms of this structure. In this way one can obtain, for example, the Dolbeaux cohomology of complex manifold and the Vaisman cohomology of foliated manifold [11] (see also [12]). Now suppose that a tensor ﬁeld

t

M

deﬁnes an integrable structure on an everywhere dense open

U \subset M

, and on

Σ = M ∖ U

the regularity condition for

t

fails. In this case we say that

t

is an integrable structure with singularities. Then we have the problem to construct a ﬁne resolution for the sheaf of inﬁnitesimal automorphisms of

t

In the present paper we construct a complex of sheaves associated to a closed differential form

ω

. We study this complex in case

ω

is a) a closed 1-form vanishing at an embedded submanifold, b) a symplectic structure with Martinet singularities [1]. In particular, we prove that, under additional conditions on

ω

, this complex gives a ﬁne resolution for the sheaf of inﬁnitesimal automorphisms of

ω

. Note that the complex constructed in the present paper can be obtained using a version of Spencer

P

-complex for the Lie derivative (we cannot apply the initial construction of Spencer

P

-complex [11] because, in general situation, the Lie derivative fails to be formally integrable for the integrable structure with singularities), however in the present paper we choose a more direct way based on sheaves of differential ideals in the sheaf of differential forms.

For the detailed exposition on 1-forms with singularities, we refer the reader to [2]. The Martinet singularities ﬁrst appeared in [1], then various properties of symplectic manifolds with Martinet singularities were investigated (see the recent papers [3],[4], [5]). In part, we studied inﬁnitesimal deformations of symplectic structures with Martinet singularities and sheaves naturally associated to these structures [9], [10].

In the present paper we deal with the category of smooth manifolds. Thus all manifolds, maps, bundles, etc. are assumed to be smooth. For a smooth manifold

M

, we denote by

X_{M}

the sheaf of vector ﬁelds on

M

, and by

Ω_{M}^{s p q}

the sheaf of

q

-forms on

M

. For a bundle

ξ

over

M

, we denote by

Γ_{ξ}

the sheaf of sections of

ξ

. For a manifold

M

, a closed subset

A \subset X

, and a sheaf

G

A

, by

G^{s p M}

we denote the sheaf on

M

generated by the presheaf:

U \to 0

U \cap A = \emptyset

, otherwise

U \to G (U \cap A)

(see [8]).

2. Preliminary: differential complex associated to a morphism from a vector bundle to the bundle of exterior forms

We start with the following standard algebraic construction. Let

K

be a ring and

d : K \to K

be a differentiation such that

d^{s p 2} = 0

. Let

I

be an ideal in

K

, then

also is an ideal in

K

such that

d : I^{s p'} \to I^{s p'}

. Then we have the following exact sequence of differential rings:

The same construction can be done for sheaves of rings over a smooth manifold

M

. Then, for a sheaf of rings

K

over

M

endowed with a differential

d

such that

d^{s p 2} = 0

, and a subsheaf

ι : ℐ ↪ K

of ideals, we get the sheaf

ℐ^{s p'}

of ideals and the exact sequence of sheaves

For any vector bundles

ξ : E_{ξ} \to M

and

η : E_{η} \to M

, each vector bundle morphism

Q : ξ \to η

determines the morphism

Q : Γ_{ξ} \to Γ_{η}

of sheaves of vector spaces: for each

s \in Γ_{ξ} (U)

, the section

Q (s) (p) = Q_{p} (s (p))

p \in U

, lies in

Γ_{η} (U)

. The kernel of

Q

is the sheaf

K (U) = {s \in Γ_{ξ} (U) ∣ Q (s) = 0}

of modules over the ﬁne sheaf

C^{s p \infty}

, therefore

K

is also ﬁne. Now let us consider the presheaf

ℱ (U) = {t \in Γ_{η} (U) ∣ t = Q (s)}

Proof. Let $U$ be an open subset in $M$ , and $U_{α}$ be the open covering of $U$ . Assume we are given $t_{α} = Q (s_{α})$ and, for any $α$ , $β$ such that $U_{α} \cap U_{β} \neq \emptyset$ , at each point of $p \in U_{α} \cap U_{β}$ we have $t_{α} (p) = t_{β} (p)$ . Then, $t_{α}$ is the restriction of a section $t \in Γ_{η} (U)$ and $s_{α β} = s_{α} - s_{β}$ is a 1-cocycle on the covering ${U_{α}}$ with coefficients in $K$ . Since $K$ is ﬁne, we can ﬁnd ${\tilde{s}}_{α} \in K (U_{α})$ such that $s_{α β} = {\tilde{s}}_{α} - {\tilde{s}}_{β}$ . Then the sections ${\bar{s}}_{α} = s_{α} - {\tilde{s}}_{α}$ glue to a section $s \in Γ_{ξ} (U)$ such that $t = Q (s)$ , hence $t$ lies in $ℱ (U)$ . □

Let

ξ : E_{ξ} \to M

be a vector bundle, and

A : ξ \to Λ^{s p q} M

be a vector bundle morphism. Denote by

A : Γ_{ξ} \to Ω_{M}

the sheaf morphism corresponding to

A

. Then the subsheaf

A (Γ_{ξ})

generates the subsheaf

ℐ_{M} \subset Ω_{M}

of ideals. Let us denote by

ℱ_{M} = \oplus ℱ_{M}^{s p q} \subset Ω_{M}

the corresponding graded sheaf

ℐ_{M}^{s p'}

of differential ideals.

We take an open

U \subset M

such that

ξ

and

Λ M

are trivial over

U

. Then, on

U

we get

q

-forms

ω_{a}

a = \bar{1, rank ξ}

, which span

A (U)

over the ring

C^{s p \infty} (U)

of functions on

U

. One can easily see that

Thus, to any morphism

A : ξ \to Λ^{s p q} M

we associate the complex

(ℱ^{s p *}, d)

of sheaves, which is a subcomplex of the de Rham complex

(Ω_{M}, d)

considered also as a complex of sheaves. Also, we have the exact sequence of sheaves

Remark 2.1. Let $q = 1$ , and $A : ξ \to Ω^{s p 1}$ be a morphism. Then the sheaf $A (Γ_{ξ})$ is the sheaf of sections of a subbundle (with singularities) in $T^{s p *} M$ . If $rank A$ is constant, then $A$ determines a distribution on $M$ , and if, in addition, this distribution is integrable, $(ℱ^{s p *} (M), d)$ is the complex generated by the basic forms of the corresponding foliation, which is widely used in the foliation theory [6], [7].

Hereafter we assume that

A

is surjective on an open everywhere dense set

M ∖ Σ

, where

ι : Σ ↪ M

is an embedded submanifold. Then, for each open

U

such that

U \cap Σ = \emptyset

we have

ℱ^{s p k} (U) = Ω^{s p k} (U)

, hence

G (U) = 0

. From this follows that the sheaf

G

is supported on

Σ

For a closed

ω \in Ω^{s p q + 1} (M)

, the Lie derivative

D (V) = L_{V} ω = d i_{V} ω

is a ﬁrst order differential operator

T M \to Λ^{s p q + 1} M

. The operator

D

can be included to the complex of sheaves associated to the vector bundle morphism

I_{ω} : T M \to Λ^{s p q}

I (V) = i_{V} ω

where

X_{Γ}

is the sheaf of inﬁnitesimal automorphisms of

ω

. Evidently, all the sheaves in (2), except for

X_{Γ}

are ﬁne. Therefore, if (2) is locally exact, it gives a ﬁne resolution for

X_{Γ}

. However, in general, (2) fails to be locally exact.

3. Differential complex associated with closed 1-form with singularities

Let

η

be a closed 1-form on an

n

-dimensional differential manifold

M

, and

Σ = {p \in M ∣ η (p) = 0}

be an embedded submanifold of codimension

k

. Assume that for each

p \in Σ

there exists a coordinate system

(u^{s p a}, u^{s p α})

a = \bar{1, k}

α = \bar{k + 1, n}

, such that

Σ

is given by the equations

u^{s p a} = 0

and

Remark 3.1. Under these assumptions we can write coordinate expression for $η$ . For any $p \in M ∖ Σ$ , one can take a coordinate system ${u^{s p a}}$ in a neighborhood $U$ of $p$ such that $η| U = d u^{s p 1}$ .

Let $p \in Σ$ . Since $η$ is closed, we have $η = d f$ , where $f$ is a function on a neighborhood $U$ of $p$ . From our assumptions it follows that

f = A_{a b} u^{s p a} u^{s p b} + B_{a b α β} (u^{s p c}, u^{s p γ}) u^{s p a} u^{s p b} u^{s p α} u^{s p β},

where $∣ ∣ A_{a b} ∣ ∣$ is a constant symmetric matrix, and $B_{a b α β} (u^{s p c}, u^{s p γ})$ are smooth functions (see [13], Ch. 6). Hence

η = (A_{a b} + B_{a b α β} (u^{s p c}, u^{s p γ}) u^{s p α} u^{s p β}) u^{s p a} d u^{s p b} + u^{s p a} u^{s p b} d (B_{a b α β} (u^{s p i}) u^{s p α} u^{s p β}) .

Let us denote by

Ω_{Σ, 0}^{s p q}

the sheaf over

M

consisting of

q

-forms

θ

such that

ι^{s p *} θ = 0

. This means that, for each open

U \subset M

such that

U \cap Σ \neq \emptyset

, a form

θ \in Ω^{s p q} (U)

lies in

Ω_{Σ, 0}^{s p q} (U)

if and only if

ι^{s p *} θ = 0

. If

U \cap Σ = \emptyset

, then

Ω_{Σ, 0}^{s p q} (U) = Ω^{s p q} (U)

. It is clear that the exterior differential

d

maps

Ω_{Σ, 0}^{s p q}

Ω_{Σ, 0}^{s p q + 1}

The 1-form

η

is a vector bundle morphism from the tangent bundle

T M

to the trivial vector bundle

Λ^{s p 0} M = M \times ℝ

. Denote by

\tilde{η}

the corresponding sheaf morphism

X_{M} \to Ω_{M}

Proof. Take a chart $(U, u^{s p i})$ , and let $η = η_{i} d u^{s p i}$ . Then,

\tilde{η} (X_{M}) (U) = {η_{i} f^{s p i} ∣ f^{s p i} \in Ω^{s p 0} (U)}

(3)

and (1) gives us

ℱ^{s p k} (U) = {φ^{s p i} \land η_{i} + ψ^{s p i} \land d η_{i} ∣ φ^{s p i} \in Ω^{s p k} (U), ψ^{s p i} \in Ω^{s p k - 1} (U)}

If $p \in U$ does not lie in $Σ$ , then at least one $η_{i}$ does not vanish at $p$ . Hence in a neighborhood $V$ of $p$ we have $ℱ^{s p k} (V) = Ω^{s p k} (V)$ . If $p \in U \cap Σ$ , then $η_{i} (p) = 0$ for all $i = \bar{1, n}$ . Moreover, for any $X \in T_{p} Σ$ , we have $d η_{i} (X) = 0$ . Hence $ℱ^{s p k} (U) \subset Ω_{Σ, 0}^{s p k} (U)$ .

Now let $p \in Σ$ . By assumption, $det ∣ ∣ \partial_{a} η_{b} ∣ ∣ \neq 0$ , hence we can take a coordinate system such that $u^{s p a} = η_{a}$ . From this follows that (3) is just the ideal of functions vanishing on $Σ \cap U$ . Then, for each $θ \in Σ_{Σ, 0}^{s p k} (U)$ , we have $θ = d u^{s p a} \land ψ_{a} + τ$ , where the coordinate expression of $τ$ does not involve any $d u^{s p a}$ . Since $i^{s p *} θ = 0$ , all the coordinates of $τ$ vanish at $Σ$ , hence $τ = u^{s p a} φ_{a}$ . Therefore, we have $θ = d u^{s p a} \land ψ_{a} + u^{s p a} φ_{a}$ , hence $Ω_{Σ, 0}^{s p k} = ℱ^{s p k} (U)$ . □

Let

Ω_{Σ}^{s p q}

be the sheaf of

q

-forms on

Σ

, and

{(Ω_{Σ}^{s p q})}^{s p M}

be the corresponding sheaf over

M

. Let

ι : Σ \to M

be the embedding. For each open

U \subset M

, we have the following exact sequence of complexes of differential forms:

Note that if

U \cap Σ = \emptyset

, then

Ω_{M}^{s p *} (U) = 0

and

Ω_{Σ, 0}^{s p *} (U) \overset{i}{\to} Ω_{M}^{s p *} (U)

is the identity mapping. Thus we obtain the exact sequence of complexes of sheaves:

where the sheaf morphism

ι^{s p *} : Ω^{s p q} \to Ω_{Σ}^{s p q}

is induced by the embedding

ι

Now let

X_{Γ}

be the sheaf of inﬁnitesimal automorphisms of

η

, this means that

X_{Γ} (U) = {V \in X (U) ∣ L_{V} η = 0}

. By assumption,

η = 0

Σ

, hence the function

i_{V} η

vanishes on

Σ

. Since

L_{V} η = d i_{V} η

, we have

L_{V} η (W) = 0

for each

p \in Σ

and

W \in T_{p} Σ

. Thus, we get a sheaf morphism

D : X \to Ω_{Σ, 0}^{s p 1}

D (V) = L_{V} η

, and the sheaf

X_{Γ}

is the kernel of

D

Statement 3.2. Let $η$ be a closed 1-form on a differential manifold $M$ , and $Σ = {p \in M ∣ η (p) = 0}$ . Assume that for each $p \in Σ$ there exists a coordinate system $(u^{s p a}, u^{s p α})$ , $a = \bar{1, k}$ , $α = \bar{k + 1, n}$ such that $Σ$ is given by the equations $u^{s p a} = 0$ and

η = A_{a b} u^{s p a} d u^{s p b},

(5)

where $A_{a b}$ is a constant symmetric matrix of rank $k$ . Then the sequence of sheaves and their morphisms

0 \to X_{Γ} \overset{i}{\to} X \overset{D}{\to} Ω_{Σ, 0}^{s p 1} \overset{d}{\to} Ω_{Σ, 0}^{s p 2} \overset{d}{\to} \dots Ω_{Σ, 0}^{s p n}

(6)

is a ﬁne resolution of the sheaf $X_{Γ}$ .

Proof. It is clear that the sheaves $X$ and $Ω_{Σ, 0}^{s p q}$ are ﬁne. Let us prove the exactness of (6).

If $p \notin Σ$ , we take a contractible neighborhood $U ∋ p$ such that $U \cap Σ = \emptyset$ . Then, $Ω_{Σ, 0}^{s p q} (U) = Ω^{s p q} (U)$ , and, by the Poincare lemma, we obtain that (6) is exact at $Ω_{Σ, 0}^{s p q} (U)$ for each $q > 1$ . If $θ$ lies in $Ω_{Σ, 0}^{s p 1} (U) = Ω^{s p 1} (U)$ and $d θ = 0$ , then $θ = d f$ , where $f \in Ω^{s p 0} (U)$ , and since $η$ does not vanish on $U$ , we can ﬁnd $V \in X (U)$ such that $f = η (V)$ . Then $D (V) = L_{V} η = d i_{V} η = d f = θ$ .

Now take $p \in Σ$ , and a contractible $U ∋ p$ such that $U \cap Σ$ is also contractible. Hence $H^{s p q} (Ω (U), d) ≅ H^{s p q} (Ω_{Σ} (U \cap Σ), d) ≅ 0$ for $q > 0$ . Then, from the exact cohomology sequence associated with the exact sequence (4) of complexes we get that $H^{s p q} (Ω_{Σ, 0} (U), d) ≅ 0$ for $q > 1$ . From this follows that (6) is exact at $Ω_{Σ, 0}^{s p q} (U)$ for each $q > 1$ .

Since $ι^{s p *} : H^{s p 0} (Ω^{s p 0} (U)) \to H^{s p 0} (Ω_{Σ}^{s p 0} (U \cap Σ))$ is an isomorphism, we have that $H^{s p 1} (Ω_{Σ, 0} (U), d) ≅ 0$ . Hence, if $θ \in Ω_{Σ}^{s p 1} (U)$ and $d θ = 0$ , then we can ﬁnd $f \in Ω_{Σ} (U)$ such that $d f = θ$ . As above, to prove the exactness at $Ω_{Σ}^{s p 1}$ , we need to ﬁnd $V$ such that $f = η (V)$ .

Let $(u^{s p i}, u^{s p α})$ be the coordinate system with respect to which $η$ has canonical form (5). Then, $f (0, u^{s p α}) = 0$ , hence $f (u^{s p a}, u^{s p α}) = u^{s p a} g_{a} (u^{s p i}, u^{s p α})$ . Therefore, we set $V = A^{s p a b} g_{a} \partial_{b}$ , where $A^{s p a b}$ is the matrix inverse to $A_{a b}$ , and $V$ is the required vector ﬁeld. □

4. Differential complex associated to symplectic form with Martinet singularities

Let

ω

be a closed 2-form on a smooth manifold

M

such that

det ω = 0

on a closed submanifold

i : Σ ↪ M

and

rank ω = 2 m

is constant on

Σ

. Then

ω

determines the vector bundle morphism

I_{ω} : T M \to Λ^{s p 1} M

I_{ω} (V) = i_{V} ω

, which is a vector bundle isomorphism over

M ∖ Σ

. The kernel of

I_{ω}

is a vector bundle over

Σ

, call it

ε : E \to Σ

. Denote by

ℐ_{ω}

the corresponding sheaf morphism

X_{M} \to Ω_{M}^{s p 1}

. From Lemma proved in [10] it follows that

τ \in ℐ_{ω} (X_{M})

if and only if

τ| E = 0

. We will consider the symplectic structures with Martinet singularities [1].

Let

ω

be a closed 2-form on a

2 n

-dimensional manifold. Assume that for each point

p \in M

one can take a chart

(U, u^{s p i})

such that

Proof. With respect to the canonical coordinates (7), the submanifold $Σ$ is given by the equation $u^{s p 1} = 0$ , and the subbundle $E$ is spanned by the vector ﬁelds $\partial_{1}$ , $\partial_{2}$ along $Σ$ . Therefore, $η = η_{a} d u^{s p a}$ lies in $ℐ (X) (U)$ if and only if $η_{1}$ , $η_{2}$ vanish on $U \cap Σ$ . Note that

\begin{matrix} \begin{aligned} d (u^{s p 1} d u^{s p k}) = d u^{s p 1} \land d u^{s p k} \forall k = \bar{2, 2 n} \\ d (u^{s p i} d u^{s p k}) = d u^{s p i} \land d u^{s p j} \forall i < j, i, j = \bar{2, 2 n} \end{aligned} \end{matrix}

and $u^{s p 1} d u^{s p k}$ , $k = \bar{2, 2 n}$ , and $u^{s p i} d u^{s p j}$ , $i = \bar{2, 2 n}$ , $j = \bar{3, 2 n}$ , vanish at $\partial_{1}$ , $\partial_{2}$ along $Σ$ . This implies that for all $i, j = \bar{2, 2 n}$ we have $d u^{s p i} \land d u^{s p j} = d τ^{s p i j}$ , where $τ^{s p i j} \in ℐ (X) (U)$ . This proves that for each $φ \in Ω^{s p k} (U)$ , $k \geq 2$ , we can write $φ = ψ^{s p a} d ω_{a}$ , where $ω_{a} \in ℐ (X) (U)$ , hence, by (1), $ℱ^{s p k} (U) = Ω^{s p k} (U)$ .

Finally, the fact that $ℱ^{s p 1} (U)$ consists of 1-forms vanishing on $E$ follows from Lemma in [10]. □

Proof. In ([9], Lemma 3) we have proved that, for each germ of 1-form ${〈 ξ 〉}_{p} \in {(Ω_{M}^{s p 1})}_{p}$ , there exist germs ${〈 W 〉}_{p} \in X_{p}$ , ${〈 f 〉}_{p} \in {(Ω_{M})}_{p}^{s p 0}$ such that ${〈 ξ 〉}_{p} = {〈 i_{W} ω + d f 〉}_{p}$ .

Let us take $η \in Ω^{s p 2} (U)$ such that $d η = 0$ . Then $η = d ξ$ , $ξ \in Ω^{s p 1} (U)$ . From the above statement it follows that $η = d i_{W} ω$ for a vector ﬁeld $W \in X (U)$ . Thus (8) is exact in the term $Ω_{M}^{s p 2}$ . The Poincare lemma implies that (8) is exact in the other terms. □

Remark 4.2. The complex associated to $I_{ω} : T M \to Λ^{s p M}$ can be extended in the following way. In [10] we have considered the sheaf of local Hamiltonians for $ω$ :

T (U) = {f \in Ω^{s p 0} (U) ∣ d f \in ℐ (X) (U) \Leftrightarrow d f| E = 0}

Evidently we have complex of sheaves:

0 \to ℝ_{M} \overset{d}{\to} T \overset{d}{\to} ℱ^{s p 1} \overset{d}{\to} ℱ^{s p 2} = Ω_{M}^{s p 2} \overset{d}{\to} \dots

Let us consider another type of Martinet singularities. Let

ω

be a closed 2-form on a four-dimensional manifold, and assume that for each point

p \in M

one can take a chart

(U, u^{s p i})

such that

and the vector bundle

E \to Σ

is spanned by the vector ﬁelds

W_{3}

W_{4}

. Hence a form

ξ

lies in

ℐ_{ω} (X) (U)

if and only if

Thus, in this case we also get the complex of sheaves (8). However, in this case the complex (8) is not locally exact. Let us take, e.g.,

η = d ξ

, where

ξ = {(u^{s p 3})}^{s p 2} θ^{s p 4} \in ℱ^{s p 2} (U)

. Let us prove that

η

cannot be represented as

η = d i_{V} ω

with

V \in X (U)

. Suppose not, then

ξ - i_{V} ω = d f

, where

f

is a function on

U

. Then, the restriction of the last equality to

Σ

gives us the equation system

We differentiate the second equality with respect to

u^{s p 3}

and get

\partial_{2} f = - 2 u^{s p 3}

. Hence,

\partial_{23} = - 2

, this contradicts

\partial_{23} f = 0

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