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\author{M.A.Malakhaltsev}
\title{Lectures on deformations of $G$-structures}

\thanks{This is a preliminary version of lectures the author gave in the University
``Los Andes'' in September-October, 2006.}

\address{Kazan State University, Kazan, Russia; 
Universidad de Los Andes, Bogota, Colombia}

\email{mikhail.malakhaltsev@ksu.ru}


\begin{document}

\maketitle

The author is thankful  to Departamento de
  Matem\'aticas, Facultad de Ciencias de Universidad de los Andes (Bogot\'a,
  Colombia), and Grupo Sanford  
  for invitation, hospitality, and
  financial support during his
  visit to Universidad de los Andes in September-October, 2006. 
Also he expresses  his deep gratitude to professor Jose Ricardo Arteaga Bejarano for useful
cooperation and constant support. 
 


\section{Pseudogroups. Sheaves}
\label{sec:1}


\subsection{Pseudogroups}
\label{subsec:1}

Recall that a \emph{transformation group} $G$ is a subgroup in a group $Diff(M)$, so 
each $g \in G$ is a diffeomorphism of $M$. 

A  \emph{local diffeomorphism} is a diffeomorphism $f : U \to V$, where $U$ and $V$ 
are open subsets in $M$. We set $D(f)=U$, $E(f)=V$. Certainly, the set of  local
diffeomorphisms is  not a group of transformations because the composition is not
well-defined. 

A generalization of the notion of transformation group is a  \emph{pseudogroup}.

\begin{df}
A \emph{pseudogroup} $\Gamma$ of transformations of a smooth manifold $M$ is a collection of 
local diffeomorphisms such that 
\begin{itemize}
\item 
For any $f \in \Gamma$ and open $U' \subset D(f)$, we have $\rest{f}{U'} \in \Gamma$;
\item 
Let $U = \cup U_i$, where $U_i$ are open. Let local diffeomorphism $f : U \to V$
be such that $\rest{f}{U_i} \in \Gamma$, then $f \in \Gamma$.
\item 
For any $f \in \Gamma$, we have $f^{-1} \in \Gamma$.
\item 
For $f,g \in \Gamma$ such that $E(f) \subset D(g)$ we have $g \circ f \in \Gamma$;
\end{itemize}
\end{df}

\begin{ex}
All local diffeomorphisms of a manifold.
\end{ex}
\begin{ex}
Pseudogroup of automorphisms of a tensor field: a) pseudogroup of analytical diffeomorphisms
b) pseudogroup of symplectic diffeomorphisms.     
\end{ex}
\begin{ex}
The pseudogroup $\Gamma(G)$ generated by a group $G$ of transformations of a
manifold $M$. The elements of $\Gamma(G)$ are the restrictions of elements of
$G$ to open sets in $M$. 
\end{ex}


We can say informally that a subset $\Gamma$ of local diffeomorphism
of $M$ is a pseudogroup if and only if
(i) the  composition and inverse  elements are well-defined,
(ii) the elements of $\Gamma$ are defined by a local condition (in
fact, by a condition on germs). 



\subsection{Sheaves}
Here we give only main definitions and results of the sheaf theory.
For a detailed exposition see \cite{WF}, \cite{Wells}.



\begin{df}
Let $M$ be a topological space. Suppose that to each open $U$ we assign 
a group $\cF(U)$ and if $V \subset U$ we have group homomorphism 
$r^U_V : \cF(U) \to \cF(V)$ such that 
\begin{eqnarray}
  \label{eq:1_1}
r^U_V r^V_W = r^U_W
\\
r^U_U = \Id    
\end{eqnarray}
Then we get a \emph{presheaf} $\cF$ of groups.
\end{df}
In the same way we can define presheaves of rings, vector spaces, etc.

We will call the elements of $\cA(U)$ \emph{the sections of $\cA$ over $U$}.

\begin{ex}
$\hphantom{x}$
  \begin{itemize}
  \item The presheaf of smooth sections of a smooth bundle $\xi$: 
    to each open
    set $U$ we put in correspondence the set of sections of $\xi$ over $U$, and
    the map $r^U_V$ is the restriction map.  
  \item The presheaf of constant functions on a manifold: to each open
    set $U$ we put in correspondence the set of constant functions on $U$, and
    the map $r^U_V$ is again the restriction map.  
  \end{itemize}
\end{ex}


Let $\cA$, $\cB$ be preasheaves. A presheaf morphism 
$\phi : \cA \to \cB$ is the collection of homomorphisms 
$\phi_U : \cA(U) \to \cB(U)$ given for each open $U$ such that, for every 
$V \subset U$, we have $r^U_V \phi_U = \phi_V r^U_V$. 

If $\cA$, $\cB$ are preasheaves of abelian groups, then we can
define the kernel and the image of $\phi$: the kernel of $\phi$ is the presheaf
given by 
$$
U \to \Ker(\phi : \cA(U) \to \cB(U)),
$$
and the image of $\phi$ is the presheaf given by 
$$
U \to \Im(\phi : \cA(U) \to \cB(U)).  
$$


\begin{df}
A presheaf $\cA$ is called a \emph{sheaf} if 
\begin{enumerate}
\item 
If $U = \cup U_i$ and $s,t \in \cA(U)$ and $r^U_{U_i}(s) =
r^U_{U_i}(t)$, then $s=t$;
\item
Let $U = \cup U_i$. Then for any collection $s_i \in \cA(U_i)$ such that 
$r^{U_i}_{U_i \cap U_j}(s_i) = r^{U_j}_{U_i \cap U_j}(s_j)$, there
exists a (unique) $s \in \cA(U)$ such that $r^U_{U_i}(s) = s_i$.
\end{enumerate}
\end{df}

\begin{ex}
  $\hphantom{x}$
  \begin{enumerate}
  \item The presheaf of local sections of a bundle is a sheaf.
  \item The presheaf of constant functions on a manifold is not a sheaf.
  \end{enumerate}
\end{ex}


\subsubsection{Passing from a presheaf to a sheaf}
Let $\cA$ be a presheaf on a manifold $M$. Then, for each $x \in M$,
we can define a germ of a section $s \in \cA(U)$ in the standard way. Namely, we
consider the set of pairs $(W, s \in \cA(W))$ such that $x \in W$ and take the
equivalence  relation on this set defined as follows: 
$(U,s \in \cA(U)) \sim (V,t \in \cA(V))$ if and only if there exists 
$x \ni W \subset U \cap V$ such that $r^U_W(s) = r^V_W(t)$.
Denote by $\cA_x$ the set of germs of sections of $\cA$ at $x$.


Now we consider the set $E(\cA) = \cup_{x \in M} \cA_x$ and introduce
a topology on $E(\cA)$ in the following way. The idea is to set up a
minimal topology such that the sections of the presheaf are 
continuous maps. The prebase of the topology on $E(\cA)$ consists of
sets
\begin{equation}
  \label{eq:10_11}
\Omega = \{<s>_x \mid s \in \cA(U)\}  
\end{equation}
Note that this topology has a lot of open sets, and, in general,  is not
Hausdorff. 

We have the natural projection $\pi : E(\cA) \to M$, $<s>_x \to x$ and thus we obtain
a covering $(E(\cA), \pi, M)$, in the sense that for each point $y \in E(\cA)$
there exists a neighborhood in $E(\cA)$ homeomorphic to a neighborhood of 
$x = \pi(y)$. Then the sheaf $\widehat\cA$ associated with $\cA$ 
is the sheaf of  continuous sections of $\pi : E(\cA) \to M$. 
We can describe the sections of this sheaf in the following way:
a section $s : U \to E(\cA)$, $x \to <\sigma_x>_x$ is continuous if
for each $x \in U$ there exist a neighborhood $x \ni W \subset U$ and
a section $\sigma \in \cA(U)$ such that $<s_x>_x = <\sigma>_x$. 

\begin{ex}
The sheaf associated with the presheaf of constant functions is the
sheaf of locally constant functions.
\end{ex}


Let $\cA$ and $\cB$ be sheaves of abelian groups.
If $\phi: \cA \to \cB$ is a sheaf morphism, then we have the image presheaf
$U \to \phi(\cA(U))$ which is not  a sheaf in general. However we denote by 
$\Im \phi$ the sheaf associated with the image sheaf.

Also, for sheaves $\cB \subset \cA$ we define the presheaf 
$U \to \cA(U)/\cB(U)$, and we denote by
$\cA/\cB$ the sheaf associated to this presheaf.
 
 


\subsubsection{Sheaf cohomology}
Let $\cG$ be a sheaf of abelian groups on a manifold $M$. 
Take at most countable open covering $\cU =  \{U_i\}$ of $M$. Define the
\emph{nerve} of $\cU$ be $N(\cU) = \sqcup N^m(\cU)$, 
where 
\begin{equation}
  \label{eq:1_2}
  N^m(\cU) = \{ \{i_0 \dots i_m \}  
\mid  
U_{i_0} \cap U_{i_2} \cap \dots U_{i_m}  
\ne \emptyset\} 
\end{equation}

Now an \emph{$m$-cochain} on $M$ is a family of sections 
$\{c_{i_0 \dots i_m} \in
\cG(U_{i_0} \cap U_{i_2} \cap \dots U_{i_m}) \mid \{i_0 \dots i_m \} \in N^m(\cU)\}$. 
The order of indices is unessential, so we
assume  that $c_{i_0i_1 \dots i_m}$ is skew-symmetric with
respect to $i_0$, $i_1$, \ldots, $i_m$.
 
The set $\vC^m(\cU;\cG)$ of $m$-cochains is
evidently an abelian group. We have the group homomorphism:
\begin{align}
  \label{eq:1_3}
&  d : \vC^m(\cU;\cG) \to \vC^{m+1}(\cU;\cG)  
\\
&  (dc)_{i_0 \dots i_{m+1}} = \sum^{m+1}_{k=0}
(-1)^k
r^{U_{i_0 \dots \widehat{i_k} \dots i_{m+1}}}_{U_{i_0 \dots i_{m+1}}}
(c_{i_0 \dots \widehat{i_k} \dots i_{m+1}})
\notag
\end{align}
For example, 
\begin{align}
  \label{eq:1_4}
  & d : C^0(\cU;\cG) \to C^1(\cU;\cG), 
&dc_{\beta\alpha} = c_\beta -  c_\alpha
\\
& d : C^1(\cU;\cG) \to C^2(\cU;\cG), 
&dc_{\alpha\beta\gamma} = c_{\beta\gamma} -  c_{\alpha\gamma} + c_{\alpha\beta}   
\end{align}

One can prove that $d \circ d = 0$ by direct calculations, hence we
obtain a complex $(\vC^*(\cU;\cG),d)$ called the Cech complex of the
covering $\cU$ with coefficients in the sheaf $\cG$.
Certainly, the cohomology of this complex depends on the covering.

\begin{ex}
$\vH^0(\cU;\cG) = \cG(M)$.
\end{ex}

Now the set of open coverings is a partially ordered set with respect
to the inscribing relation $\le$, and, if $\cV \le \cU$, we have 
the natural map $\vH(\cU;\cG) \to \vH(\cV;\cG)$. Thus we can define 
the Cech cohomology of $M$ with coefficients in $\cG$:
\begin{equation}
\vH(M;\cG) = \lim_{\longrightarrow} \vH(\cU;\cG)
\label{eq:1_5}
\end{equation}

The following theorems make it possible to calculate the Cech cohomology
without taking the direct limit.


Let $\cG$ be a sheaf of abelian groups on $M$.
An open covering is said to be `fine' if $\vH^q(U_I;\cG) \cong 0$, $q>0$, for
each $I \in N(\cU)$. 

\begin{thm}[Leray]
If $\cU$ is a fine covering on $M$, then $\vH(M;\cG) \cong \vH(\cU;\cG)$.
\end{thm}


An exact sequence of sheaves
\begin{equation}
  \label{eq:1_6}
0 \to \cG \xrightarrow{i} \cF_0 \xrightarrow{d} \cF_1 \xrightarrow{d} \dots 
\end{equation}
is called a \emph{resolution} of $\cG$. If all the sheaves  $\cF$ are
fine, then we say that \fref{eq:1_6} is a fine resolution of $\cG$.
From the fine resolution \ref{eq:1_6} we get the complex $(F_k(M),d)$.
 
\begin{thm}[Abstract de Rham Theorem]
Given a fine resolution $\cF_k$ of a sheaf $\cG$, 
we have  $\vH(M;\cG) \cong H(\cF_*(M),d)$.
\end{thm}


\begin{ex}
The sheaf $\R_M$ of locally constant functions on a manifold $M$ admits 
the fine resolution 
$$
0 \to \R_M \to \Omega^0_M \xrightarrow{d} \Omega^1_M \xrightarrow{d} 
\dots \xrightarrow{d} \Omega^n_M \to 0,
$$
where $\Omega^q_M$ is the sheaf of differential $q$-forms on $M$,
therefore the cohomology of $H(M;\R_M)$ with coefficients in the sheaf of
locally constant functions is isomorphic to the de Rham cohomology.  
\end{ex}



\begin{thm}
If $0 \to \cA  \xrightarrow{i} \cB \xrightarrow{p} \cC \to 0$ is a
exact sequence of sheaves, we have a long exact cohomology sequence 
\begin{equation}
  \label{eq:1_777}
\dots \to \vH^{m-1}(M;\cC)  \xrightarrow{\delta} \vH^m(M;\cA) \xrightarrow{i_*} \vH^m(M;\cB)
\xrightarrow{p_*} \vH^m(M;\cC)  \xrightarrow{\delta} \vH^{m+1}(M;\cA)
\to \dots 
\end{equation}
\end{thm}


 






\section{Deformations of pseudogroup structures associated with
  integrable first order $G$-structures}
\label{sec:2}

We will apply deformation theory to the integrable first order
$G$-structures  (for the general theory of $G$-structures we refer the
reader to e.\,g. \cite{Kob1}, \cite{Al-Molino}, \cite{Sternberg}).


\subsection{Integrable $G$-structures}
\label{subsec:2_11}

Let $M$ be a smooth $n$-dimensional manifold, $L(M)\to M$ the frame bundle
of $M$, and ${\goth X}(M)$ the Lie algebra of vector fields on $M$.
For a Lie subgroup $G \subset GL(n)$, consider the bundle
$E_G(M) = L(M)/G \to M$. 

\begin{rem}
$E_G$ is a natural bundle: this means that $E_G$ is 
the functor $\cM an \to \cB undles$, where $\cM an$ is the category of smooth
manifolds whose morphisms are diffeomorphisms, 
and  $\cB undles$ is the category of fiber bundles. This functor 
sends a manifold $M$ to the fiber bundle $E_G(M)$, 
and a diffeomorphism $f : M \to M'$ to the natural bundle
morphism $f^c : E_G(M) \to E_G(M')$.
\end{rem}


\begin{df}
\emph{A first order $G$-structure} is a section $s\colon M \to E_G(M)$.
\end{df}

  \begin{ex}[\textbf{C}]
    \emph{An almost complex structure} is a section $s\colon M \to E_G(M)$,
    where $G = \GL(n,\C) \subset \GL(2n,\R)$. One can easily prove
    that the sections of $E_G(M)$ are in 1-1-correspondence with the linear
    operator fields $J$ such that $J^2=-I$.
  \end{ex}
  \begin{ex}[\textbf{F}]
   \emph{A distribution} of codimension $q$ 
   can be identified with  a section  $s\colon M \to E_G(M)$,
    where $G = \{
\left(
    \begin{array}{cc}
    A & 0
    \\
    B & C      
    \end{array}
\right)
\} \subset \GL(n,\R)$, where $A \in GL(q,q)$, $B \in Mat(n-q, q)$, 
$C \in GL(n-q,n-q)$.
  \end{ex}
  \begin{ex}[\textbf{S}]
 \emph{An almost symplectic structure} is a section of $s\colon M \to
 E_G(M)$, where $G = \Sp(n) \subset \GL(2n,\R)$. One can easily prove
    that the sections of $E_G(M)$ are in 1-1-correspondence with
    non-degenerate differential 2-forms on $M$.
  \end{ex}

\begin{df}
We say that a \emph{first order $G$-structure $s$ is integrable} iff
$s$ is locally equivalent to a given first order $G$-structure $s_0$ on $\R^n$.   
\end{df}

  \begin{ex}[\textbf{C}]
    \emph{A complex structure} is a section $s\colon M \to E_G(M)$
    locally equivalent to $J \in T^1_1(\R^n)$ such that 
    $J \pd_k = \pd_{n+k}$, $J \pd_{n+k} = - \pd_k$, $k=\ov{1,n}$.
  \end{ex}
  \begin{ex}[\textbf{F}]
   \emph{A foliation} of codimension $q$ is a section  $s\colon M \to E_G(M)$ locally
   equivalent to the distribution $\Delta$ given by equations
   $dx^1=0$, $dx^2=0$, \dots, $dx^q=0$.
  \end{ex}
  \begin{ex}[\textbf{S}]
 \emph{A symplectic structure} is a section  $s\colon M \to E_G(M)$ which is 
locally equivalent to 
$\omega_0 = dx^1 \wedge dx^2 + \dots + dx^{2n-1} \wedge dx^{2n}$. 
  \end{ex}

 
\begin{rem}
For all the classical structures, like the complex structure, the
foliation structure, etc., the section $s_0$ is invariant under the
translation group, so $s_0$ is determined by its value at a point
$0\in\R^n$.  
\end{rem}


\subsection{$\Gamma$-structure associated with integrable $G$-structure}
\label{subsec:2_2}

Let $\Gamma_0$ be a pseudogroup of transformations of $\R^n$. A \emph{$\Gamma_0$-structure
on a manifold $M$} is a maximal atlas $\{(U_\alpha,\phi_\alpha)\}$ on $M$ whose transition
functions $\phi_\beta\phi_\alpha^{-1}$ lie in $\Gamma_0$.


Given an integrable $G$-structure $s : M \to L(M)/G$, and so the model
section $s_0 : \R^n \to L(\R^n)/G$, we obtain
the pseudogroup $\Gamma_0=\Gamma(s_0)$ of automorphisms of $s_0$. For classical
$G$-structures, this pseudogroup is transitive (see the remark above).

It is clear that a \emph{manifold $M$ is endowed with an integrable
$G$-structure if and only if $M$ is endowed with $\Gamma(s_0)$-structure.} 

For a manifold $M$ endowed with a $\Gamma_0$-structure
$\{(U_\alpha,\phi_\alpha)\}$, we have the
pseudogroup $\Gamma$ of diffeomorphisms of $f : M \to M$ 
such that the local representations $\phi_\beta f \phi_\alpha^{-1}$ lie in
$\Gamma_0$. 
In case the $\Gamma_0$-structure corresponds to
an integrable $G$-structure $s : M \to L(M)/G$, the pseudogroup
$\Gamma$ consists of local automorphims of $s$: $f \in \Gamma$ if and only if
$f^*(s)=s$. 


A vector field $X$ is called a \emph{$\Gamma$-vector field} if the flow
$\phi_t$ consists of elements of $\Gamma$. 
On each manifold $M$ endowed with a $\Gamma_0$-structure 
we have the sheaf $\gX_\Gamma$ of $\Gamma$-vector fields. 
In case the $\Gamma_0$-structure corresponds to
an integrable $G$-structure $s : M \to L(M)/G$, 
a vector field $X$ lies in $\gX_\Gamma(U)$ if and only if $\cL_X s = 0$.   

Now we continue considering the above examples and present the
corresponding pseudogroups $\Gamma_0$, as well as the sheaves of
$\Gamma_0$-vector fields.            

  \begin{ex}[\textbf{C}]
    \emph{A complex structure}. $\Gamma_0$  consists of local 
diffeomorphisms $f : \R^{2n} \to \R^{2n}$ which satisfy the
Cauchy-Riemann equations, i.e. $df$ commutes with $J_0$, where 
$$
J_0 = 
\left(
    \begin{array}{cc}
    0 & -I
    \\
    I & 0      
    \end{array}
\right).
$$


The sheaf $\gX_{\Gamma}$ of $\Gamma$-vector fields on $M$ 
consists of holomorphic vectors fields of $(M,J)$. 
\end{ex}

\begin{ex}[\textbf{F}]
   \emph{A foliation} of codimension $q$. 
Let $(x^a,x^\alpha)$, $a=\ov{1,q}$,   $\alpha=\ov{q+1,n}$, be coordinates on $\R^n$.
The pseudogroup $\Gamma_0$ consists of local diffeomorphisms 
$f : \R^n \to \R^n$ whose germs
   are of the form $y^a=f^a(x^b)$,    $y^\alpha=f^\alpha(x^b,x^\beta)$.
In fact, a local diffeomorphism lies in $\Gamma$ if and only if
$\frac{\pd f^a}{\pd x^\alpha}=0$.  And the sheaf $\gX_\Gamma$ consists 
of vector fields $V$ with local expression 
$$
V = V^a(x^b) \pd_a + V^\alpha(x^b,x^\beta)\pd_\alpha  
$$
  \end{ex}
  \begin{ex}[\textbf{S}]
 \emph{A symplectic structure}. A local diffeomorphism lies in
 $\Gamma_0$ if and only if $f^*\omega_0 = \omega_0$. The sheaf $\gX_\Gamma$ consists 
of vector fields $V$ which are locally Hamiltonian vector fields:
$V \in \gX(U)$ if and only if $\cL_V \omega = 0$ if and only if $d i_V \omega =
0$ if and only if
$V$ has local expression  $V^k = \omega^{ks}\pd_k f$.
\end{ex}



\subsection{Deformations of pseudogroup structure.} 

\label{subsec:2_1} 
Let
us recall definitions and results of the theory of deformations of
pseudogroup structures (see \cite{Spen1}--\cite{Spen3}
and also \cite{GuiSten}, \cite{Kodaira}). 

Let $\Gamma$ be a pseudogroup of local diffeomorphisms of a
manifold $M$. 
\begin{df}
  A \emph{ one-parameter family $\phi_s$ of elements of $\Gamma$ is said to
  be smooth} if $U = \{(x,s) \in M \times \R \mid x \in \cD(\phi_s)\}$,
  where $\cD(g)$ is the domain of $g \in \Gamma$, is open in $M \times
  \R$ and $\Phi : U \to \R^n$, $\Phi(x,s) = \phi_s(x)$, is smooth.  \emph{A
  smooth one-parameter family of elements of $\Gamma$ will be called a
  curve in $\Gamma$}.
\end{df}

Let $\phi_s$, $|s| < \varepsilon$, be a curve in $\Gamma$ such
that $\phi_0$ is the identity map of an open set
$U \subset M$.
Then, on $U$ we have the vector field
$V(x) =
\frac{d}{ds}\big|_{s=0}\phi_s(x)$, which is called \emph{the vector
field tangent to $\phi_s$ at $s=0$}. Let us recall that a vector
field $V$ on an open set $U \subset M$ is called a
$\Gamma$-vector field if its flow consists of elements of
$\Gamma$.

\begin{df}
  If any vector field tangent to a curve in $\Gamma$ is a
  $\Gamma$-vector field, then \emph{the pseudogroup is said to have the Lie
  pseudoalgebra} \cite{Spen1}. If a pseudogroup $\Gamma$ has Lie
  pseudoalgebra, then the set $\gX_\Gamma(U)$ of $\Gamma$-vector
  fields on $U$ is a Lie subalgebra in the Lie algebra $\gX(U)$ of
  vector fields on $U$, and the correspondence $U \to \gX_\Gamma(U)$
  determines a sheaf $\gX_\Gamma$ of Lie algebras on $M$.
\end{df}


\begin{rem}
Let $\Gamma$ be the pseudogroup associated with an 
integrable first order structure  $s : M \to L(M)/G$. 
Then $\Gamma$ has the Lie pseudoalgebra, and this pseudoalgebra 
coincides with $\gX_\Gamma$. The reason is that in the definition of
the Lie derivative: $\cL_X s = \frac{d}{dt} |_{t=0} \phi^*_t(s)$ one
can take any smooth curve of diffeomorphisms tangent to $X$.
\end{rem}


A \emph{deformation of a $\Gamma_0$-atlas}
$\cA=\{(U_\alpha, \phi_\alpha)\}$ is a family of
$\Gamma_0$-atlases $\cA(s)=\{(U_\alpha, \phi_\alpha(s))\}$, where
$|s|<\varepsilon$, such that

1) $U_\alpha \times (-\varepsilon,\varepsilon) \to \R^n$,
$(x,s) \to \phi_\alpha(s)(x)$  is smooth for each $\alpha$;

2) $\phi_{\beta\alpha}(s) = \phi_\beta(s)\phi^{-1}_\alpha(s)$ is a
curve in $\Gamma_0$ for every $\alpha$, $\beta$;

3) $\cA(0) = \cA$.

A deformation $\cA(s)$ of a $\Gamma_0$-atlas $\cA$ is said to be
{\em nonessential} if $ \phi_\alpha(s)\phi^{-1}_\alpha$ lie in
$\Gamma_0$, i.\,e. for every $s$, $\cA(s)$ is compatible
with the maximal $\Gamma_0$-atlas determined by $\cA$.


Let $\cA(s)=\{(U_\alpha, \phi_\alpha(s))\}$ be a deformation of a
$\Gamma_0$-atlas $\cA=\{(U_\alpha,\phi_\alpha)\}$.
Then $\gamma_{\beta\alpha}(s) = \phi_\beta^{-1}
\phi_{\beta\alpha}(s)\phi_\alpha$ is a curve in 
$\Gamma$ and $\gamma_{\beta\alpha}(0)=\Id$.


$\frac{d}{ds}\big|_{s=0}\phi_{\beta\alpha}\phi_{\alpha\beta}(s)$ is a
$\Gamma_0$-vector field on
$\phi_{\beta\alpha}(\phi_\alpha(U_\alpha \cap U_\beta)) \subset
\phi_\beta(U_\beta)$.

We define the vector field $V_{\beta\alpha}$ on  $U_\alpha \cap
U_\beta$,
\begin{equation}\label{eq8} V_{\beta\alpha}(p)
= d\phi_\beta^{-1}\bigg(\frac{d}{ds}\Big|_{s=0}
\phi_{\beta\alpha}(s)(\phi_\alpha(p)) \bigg),
\end{equation}
and the vector field $W_\alpha$ on $U_\alpha$,
\begin{equation}\label{eq91}
W_\alpha(p) = d\phi_\alpha^{-1} \bigg(\frac{d}{ds}\Big|_{s=0}
\phi_\alpha(s)(p) \bigg).
\end{equation}

\begin{stat}\label{st2}

$1)$ $V_{\beta\alpha}=\rest{W_\beta}{U_\alpha \cap U_\beta}-
\rest{W_\alpha}{U_\alpha \cap U_\beta};$

$2)$ $\{V_{\beta\alpha}\}$ is a  $1$-cocycle on the covering
$\cU=\{U_\alpha\}$ with coefficients in $\gX_{\Gamma};$

$3)$ if the deformation $\cA(s)$ is nonessential,
the cohomology class $[V_{\beta\alpha}] \in H^1(M; \gX_{\Gamma})$
is trivial.
\end{stat}

The cocycle  $\{V_{\beta\alpha}\}$ is called an {\em infinitesimal
deformation of the $\Gamma_0$-structure}. The class
$[V_{\beta\alpha}] \in H^1(M; \gX_\Gamma)$ is called an
{\em essential infinitesimal deformation} of the
$\Gamma_0$-structure.

In general, not every infinitesimal deformation is generated by a
deformation of $\Gamma_0$-structure. In \cite{Kodaira} it is
proved that if an essential infinitesimal deformation represented
by a cocycle $\{V_{\beta\alpha}\}$ is generated by a deformation
of $\Gamma_0$-structure, then  the cocycle
$\{W_{\alpha\beta\gamma} = [V_{\alpha\beta}, V_{\beta\gamma}]\}$
of $\Gamma$-vector fields, where  $[\ ,\ ]$ is the Lie bracket of
vector fields, represents the trivial cohomology class in
$H^2(M;\gX_\Gamma)$. The cohomology class
$[W_{\alpha\beta\gamma}] \in H^2(M, \gX_\Gamma)$ is called
the {\em obstruction to integrability of the infinitesimal
deformation} $\{V_{\beta\alpha}\}$.


\section{Fine resolution for the sheaf of $\Gamma$-vector fields.}
\label{sec:3}


We know that the space of infinitesimal essential deformations of a
$\Gamma$-structure is $H^1(M;\gX_\Gamma)$, which is defined in terms of
the \v Cech complex. However, for calculations and for establishing
relations with other geometrical structures we often need to have a
differential complex whose cohomology is isomorphic to 
$H^*(M;\gX_\Gamma)$. Using the Spencer $P$-complex, 
we will construct such a complex for integrable first order
$G$-structures.  

\subsection{Lie derivative of a section of a natural bundle}
\label{subsec:3_1}

Let $M$ be a manifold and  $E : \cM an \to \cB undles$ be a natural
bundle. Let us take a section $s : M \to E(M)$ and a vector field  
$X \in \gX(M)$. Denote by $\phi_t$ the flow of $X$, and since the
bundle $E(M)$ is natural, we have the one-parameter group 
$\phi^c_t : E(M) \to E(M)$, and hence the complete lift $X^c \in
\gX(E(M))$. Now denote by $VE(M) \to E(M)$ the vertical bundle of
$E(M)$, then the pullback bundle $s^*(VE(M))$ is a vector bundle over $M$, and
for each $p \in M$ there is defined the isomorphism 
$\Pi_p\colon (VE(M))_{s(p)} \to (s^*(VE(M))_p$.
Then
$$
({\cal L}_X s)_p \overset{def}{=} \Pi({X^c}(s(p)) - ds_p(X(p))),
$$
is the Lie derivative of $s$ with respect to $X$ at a point
$p \in M$.

\par
From the definition it immediately follows that ${\cal L}_X s = 0$ if
and only if $X$ is an infinitesimal automorphism of $s$, i.e., if the flow
$\phi^c_t$ preserves~$s$.
\par
Now let $s\colon M \to E_G(M)$ be an integrable first order structure.
We consider the Lie derivative ${\cal L}_X s$ as a first-order
differential operator ${\cal D_{\cal L}} \colon TM \to s^*(VE)$.
In \cite{Pommaret},  for a differential operator ${\cal D}\colon\Gamma(\xi)\to
\xi'$, where $\xi$, $\xi'$ are vector bundles and $\Gamma(\xi)$ is the sheaf of
sections of $\xi$, a differential complex was constructed (the Spencer
P-complex)
$$
0 \to \Theta \to \Gamma(\xi) \stackrel{\cal D}{\longrightarrow} F^0\to
F^1 \to \dots ,
$$
where $\Theta$ is the kernel of $\cal D$. This construction, applied
to a Lie derivative associated with a general pseudogroup structure,
gives the deformation complex of this structure \cite{Pommaret}.
Then, specializing this complex to the case of the pseudogroup
associated to an integrable $G$-structure, we obtain the fine
resolution for the sheaf $\gX_\Gamma$ (for details, see \cite{MA3}).
In the next subsection we present this complex in terms 
of tensor fields and covariant derivatives.


\subsection{$P$-complex for the Lie derivative}
Let $s : M \to E_G(M) = L(M)/G$ be an integrable first order
$G$-structure. Let us denote by $E_{\goth g}$ the subbundle of the bundle
$T{}^1_1(M)$ consisting of linear operators whose matrices written
with respect to the corresponding $\Gamma_0$-atlas 
lie in the Lie algebra $\goth g$ of $G$, and let
$F_{\goth g} = T{}^1_1(M)/E_{\goth g}$.


The $P$-complex of the Lie derivative is isomorphic to the complex
$(C^q(P), d)$, where
$$
C^q(P) =
\frac{\Omega^k(M) \otimes TM}{\Alt(\Omega^{k-1}(M)\otimes E_{\goth g})},
$$
where $\Alt(t^j_{i_1 \dots t_{k-1} t_k}) = t^j_{[i_1 \dots t_{k-1} t_k]}$
and the differential $d : C^q(P) \to C^{q+1}(P)$ is induced by the
differential operator $D = \Alt \circ \nabla$, where $\Alt$ is the
alternation and $\nabla$ is the covariant derivative of
a torsion-free connection adapted to the $G$-structure $s$: $\nabla s = 0$.
This means that, with respect to local coordinates adapted to $s$ (the charts of
the $\Gamma_0$-atlas),  we have 
$(D\omega)_{i_1 \dots}{}_{i_{q+1}}^j = \nabla{}_{[i_1}\omega{}_{i_2 \dots}{}_{i_{q+1}]}^j$.

The definition of $d$ does not depend on an adapted  connection choice because, 
if $\nabla (s) = \ov{\nabla} (s) = 0$, we have 
$T^k_{ij} = \Gamma^k_{ij} -  \ov{\Gamma}^k_{ij}$ in 
$E_{\goth g} \otimes T^*M$, so 
$
\Alt(\nabla \omega) - \Alt(\ov\nabla \omega)  
$
lies in $\Alt(\Omega^{k-1}(M)\otimes E_{\goth g})$.
  


\begin{ex}[\textbf{C}]
  For a complex structure, the complex $(C^q, d)$ is the Dolbeaut
  complex of vector-valued forms (see, e.g., \cite{GriffithsHarris}). 
\end{ex}

\begin{ex}[\textbf{F}]
  Let a foliation structure on a smooth manifold $M$ be given by the 
integrable distribution $\Delta$.  
Then $E_{\goth g} = \{ A \in T{}^1_1(M) \ | \ A(\Delta) \subset \Delta
  \}$.  Therefore $C^p = \Omega^p(\Delta) \otimes (TM/\Delta)$. If
  $(x^i, x^\alpha)$ are adapted local coordinates, i.e., if $\Delta$
  is given by the equations $dx^i = 0$, then $d$ can be written
  locally as $(d\omega){}_{\alpha_1 \dots}{}_{\alpha_{q+1}}^i =
  \partial{}_{[\alpha_1}\omega{}_{\alpha_2
    \dots}{}_{\alpha_{q+1}]}^j$.  Thus we arrive at Vaisman's foliated
  cohomology \cite{Vaisman}.
\end{ex}

\begin{ex}[\textbf{S}]
Let us consider a symplectic manifold $(M,\theta)$. 
Then the subbundle $E_{\goth g}$ consists of those
linear operators that are skew-symmetric with respect to $\theta$: 
$\theta(AX,Y) + \theta(X,AY) = 0$, and using
the isomorphism $T{}^1_1(M) \to T^2(M)$ determined by $\theta$, we
obtain that $C^q \cong \Omega^{q+1}(M)$ and . The adapted connection
$\nabla$ satisfies $\nabla \omega = 0$ (a symplectic connection), and
the differential $d \colon C^q \to C^{q+1}$ 
is the exterior differential.  Thus we find that the kernel of~$d$ is
the Lie algebra of (locally) Hamiltonian vector fields.
\end{ex}


\subsection{Fine resolution for $\gX_\Gamma$}

It is clear that the construction of $(C^*,d)$ transforms to the sheaf
level. So we have the sequence of sheaves 
\begin{equation}
0 \to \gX_\Gamma \xrightarrow{i} \gX \xrightarrow{\cL(s)} C^0
\xrightarrow{d} C^1 \xrightarrow{d} C^2 \xrightarrow{d} \dots 
\label{eq:3_1}
\end{equation}

From the construction, it follows that these sheaves are fine, since
these ones are sheaves of modules over the fine sheaf $C^\infty$.

So we arrive at 
\begin{thm}
If the sheaf sequence \fref{eq:3_1} is exact (the ``Poincare lemma''
holds), then the space of essential deformations 
$H^1(M;\gX_\Gamma)$ is isomorphic to $H^1(C^*(M),d)$.
\end{thm}


\section{Deformation of symplectic structure with Martinet singularities}
\label{sec:4}

\subsection{Symplectic structure with singularities}

Let $\omega_0$ be a closed differential $2$-form on
$\R^{2n}$ such that
$\Sigma_0 = \{x \in \R^{2n} \mid \det\omega_0(x)=0\}$
is an embedded manifold, and $\Gamma_0$ be the pseudogroup
consisting of local diffeomorphisms $f$ of $\R^{2n}$ such that
$f^*\omega_0 = \omega_0$.

A symplectic structure with singularities of type $\omega_0$ on a
$2n$-dimensional manifold $M$ can be defined in two \emph{equivalent} ways.

\begin{df} \label{def1}
{\em A symplectic structure with singularities of type
$\omega_0$} on a $2n$-dimensional manifold $M$ is a
$\Gamma_0$-structure on  $M$, i.\,e.
a maximal atlas whose transition functions lie in
$\Gamma_0$.
\end{df}

\begin{df} \label{def2}
A 2-form $\omega$ on a $2n$-dimensional smooth manifold
$M$ is called a
{\em symplectic structure with singularities of type }
$\omega_0$ if for any point $p \in M$ there exist
a neighborhood $U(p)$ and a diffeomorphism
$\phi_p : U(p) \to \cO \subset \R^{2n}$, where
$\cO$ is an open subset in $\R^{2n}$, such that
$\rest{\omega}{U(p)} = \phi^*_p(\omega_0)$.
\end{df}


{\it The singular submanifold of a symplectic structure with
singularities}. Let a 2-form $\omega$ be a symplectic structure
with singularities of type $\omega_0$ on a $2n$-dimensional
manifold $M$.

\begin{lm} \label{lm2}
The set
$\Sigma = \{ p \in M \mid \det\omega(p)=0 \}$
is an embedded closed submanifold in $M$, and
$\dim \Sigma = \dim \Sigma_0$.
\end{lm}


The submanifold $\Sigma$ will be called {\em the singular
submanifold} of a symplectic structure with singularities.

\begin{lm} \label{lm1}
Let $\omega$ be a symplectic structure with singularities of type
$\omega_0$ on a manifold $M$,
and $U \subset M$ be an open set. For  $V,W \in \gX(U)$, from
$i_V\omega = i_W\omega$ it follows that $V=W$.
\end{lm}


\subsection{Infinitesimal deformations of symplectic structure
with singularities}

We have defined the symplectic structure with singularities to be
a pseudogroup structure (Definition \ref{def1}), and in the
equivalent way, to be a closed 2-form (Definition \ref{def2}).
Therefore, there are defined two spaces of essential infinitesimal
deformations of symplectic structure with singularities. The first
one is the space of essential infinitesimal deformations of
pseudogroup structure, which is isomorphic to the \v Cech
cohomology group
$\check H^1(M;\gX_\Gamma)$ with coefficients
in the sheaf of $\Gamma$-vector fields, where
$\Gamma$ is the pseudogroup of automorphims
of symplectic structures with singularities. The second one is the
space $\cD_{\ess}(\omega)$ of essential infinitesimal deformations
of the closed 2-form $\omega$. In this section, for a compact
manifold $M$ we construct a homomorphism
$\phi : \check H^1(M; {\gX}_\Gamma) \to \cD_{\ess}(\omega)$
and study its properties.

{\em In what follows we assume that $M$ is compact}.



\subsubsection{Infinitesimal deformations of a closed differential
form.}
\label{subsubsec:4_2} 

In \ref{subsec:2_1} we have defined the deformations of pseudogroup 
structure and now we define the deformations of a closed differential form. 
Let $\eta$ be a closed differential form on a
manifold $M$. A {\em deformation} of $\eta$ is a smooth family of
closed forms $\eta(s)$, $|s|<\varepsilon$, where $\eta(0)=\eta$.
Let $\phi(s)$ be a flow of local diffeomorphisms of
$M$. Then the family of forms $\eta(s)= \phi(s)^*(\eta)$
is called a {\em nonessential deformation} of
$\eta$. A closed form $\theta$ is called
a (nonessential) infinitesimal deformation of
$\eta$, if there exists a (nonessential) deformation
$\eta(s)$ of $\eta$ such that
$\theta = \frac{d}{ds}\big|_{s=0} \eta(s)$.
Note that a closed form $\theta$ is a nonessential infinitesimal
deformation  of $\eta$ if and only if there exists a vector field
$V \in \gX(M)$ such that
$\theta=L_V \eta = d (i_V \eta)$.

For any closed $p$-form $\theta$, the family
$\eta(s) = \eta + s \theta$ is a family of closed forms,
and $\theta =\frac{d}{ds}\big|_{s=0} \eta(s)$, therefore the set
of infinitesimal deformations $\cD(\eta)$ of
$\eta \in \Omega^p(M)$ coincides with the space of
closed $p$-forms. The set of nonessential infinitesimal
deformations is the subspace $\cD_0(\eta) \subset \cD(\eta)$
consisting of exact $p$-forms $d(i_V \eta)$, where
$V \in \gX(M)$. Then the quotient space
$\cD_{\ess}(\eta)=\cD(\eta)/\cD_0(\eta)$
is called the {\em  space of essential infinitesimal deformations
of the closed form} $\eta$.

From the definition of the space of essential  infinitesimal
deformations it follows that for any closed $p$-form
$\eta$ we have the surjective homomorphism
$\psi : \cD_{\ess}(\eta) \to H^p(M)$ which maps each
essential infinitesimal deformation of
$\eta$ to the corresponding de Rham cohomology class.


\subsection{Infinitesimal deformations of symplectic structure with
singularities.} \label{ss3} 
Let us establish relationship between
infinitesimal deformations of symplectic structure with
singularities of type $\omega_0$ considered as a pseudogroup
structure (see \ref{subsec:2_1}) and infinitesimal deformations of the
corresponding closed 2-form $\omega$ (see \ref{subsubsec:4_2}).


Let a symplectic structure with  singularities of type
$\omega_0$ be given on a manifold $M$ by a $\Gamma_0$-atlas
$\cA = \{(U_\alpha,\phi_\alpha)\}$. Let $\omega
\in \Omega^2(M)$ be the corresponding closed 2-form (see
Definitions  \ref{def1} and \ref{def2}). Let us denote by
$\Gamma$ the pseudogroup consisting of
local diffeomorphisms  $f$ of $M$ such that $f^*\omega = \omega$.
Hence follows that a vector field $V$ is a section of the sheaf
$\gX_\Gamma$ of $\Gamma$-vector fields  if and only if $L_V
\omega=0$.

Each deformation $\cA(s)=\{(U_\alpha,\phi_\alpha(s))\}$ of the
$\Gamma_0$-atlas determines a deformation $\omega(s)$ of
the form $\omega$.  Namely, the family of closed  2-forms
$\omega(s)$ is given by
$\omega(s)|_{U_\alpha} = \phi_\alpha^*(s)\omega_0$.
Let $\eta=\frac{d}{ds}\big|_{s=0}\omega(s)$ be the corresponding
infinitesimal deformation of $\omega$, then
$\eta |_{U_\alpha} = L_{W_\alpha}\omega$, where the vector field
$W_\alpha \in \gX(U_\alpha)$ is defined by
(\ref{eq91}). And $V_{\alpha\beta} = W_\alpha - W_\beta$  is an
infinitesimal deformation of $\cA$ determined by the deformation
of $\cA(s)$ (see Statement~\ref{st2}).

Now we define a map
$\phi$ from the space $\check H^1(M, \gX_\Gamma)$ of
essential infinitesimal deformations of the $\Gamma_0$-atlas to
the space $\cD_{\ess}(\omega)$ of essential infinitesimal
deformations of the closed 2-form determined by this atlas. We
proceed as follows. Let an essential infinitesimal deformation be
given by a cocycle of $\Gamma$-vector fields  $V_{\beta\alpha}$ on
the covering
$\cU=\{U_\alpha\}$ of $M$.
Since the sheaf $\gX$ of vector fields on $M$ is fine, there exist
vector fields $W_\alpha \in \gX(U_\alpha)$ such that
$V_{\beta\alpha} = W_\beta - W_\alpha$. We set
$\eta_\alpha = L_{W_\alpha} \omega$, then, since
$L_{V_{\beta\alpha}} \omega =0$, we get
$\left.\eta_\alpha\right|_{U_{\alpha\beta}} =
\left.\eta_\beta\right|_{U_{\alpha\beta}}$. Therefore, we get the
form $\eta$ such that
$\left.\eta\right|_{U_\alpha}=\eta_\alpha$, and
$d\eta = 0$.

If $V_{\beta\alpha} = W'_\beta - W'_\alpha$, then on  $U_\alpha
\cap U_\beta$ we have
$W'_\beta - W'_\alpha = W_\beta - W_\alpha$.
Therefore, we obtain the vector field
$W$ such that $\rest{W}{U_\alpha}=W'_\alpha-W_\alpha$,
and
$$
\eta' - \eta = L_W\omega = d(i_W\omega).
$$
Thus, $\eta$ and $\eta'$ determine the same class in the space
$\cD_{\ess}(\omega)$.

Thus we obtain the map
$$
\wt\phi : \check Z^1(\cU; \gX_\Gamma) \to \cD_{\ess}(\omega),
\quad \wt\phi(V_{\beta\alpha}) = [\eta].
$$
From our construction follows that, if  cocycles
$V'_{\beta\alpha}$ and $V_{\beta\alpha}$ are cohomologous,
they define the same closed form $\eta$, and, therefore,
$\wt\phi(V'_{\beta\alpha})=\wt\phi(V_{\beta\alpha})$,
i.\,e. we obtain the linear map
\begin{equation}\label{eq1}
\phi : \check H^1(M; {\cal X}_\Gamma) \to \cD_{\ess}(\omega).
\end{equation}



\begin{stat}
Let us consider a symplectic structure $\omega$ with singularities
of type $\omega_0$ given by a $\Gamma_0$-atlas $\cA$. Given a
deformation $\cA(s)$ of $\cA$, we denote by $\omega(s)$ the
corresponding deformation of $\omega$. If $[V_{\beta\alpha}] \in
H^1(M, \gX_\Gamma)$ is an essential infinitesimal deformation
determined by $\cA(s)$, and $\eta =
\frac{d}{ds}\big|_{s=0}\omega(s)$, then
$\phi([V_{\beta\alpha}]) = [\eta]$.
\end{stat}

\emph{Properties of the map $\phi$.}\label{ss4}

\begin{stat} \label{st3}
$\phi$ is a monomorphism.
\end{stat}


Let us define by $\Omega^p$  the sheaf of $p$-forms on
$M$. Let $\iota : \gX \to \Omega^1$ be the sheaf morphism
given by  $\iota_U : \gX(U) \to \Omega^1(U)$,
$\iota(V) = i_V\omega$.

\begin{stat}\label{st4}
$\phi$ is surjective if and only if
$\Omega^1 = \iota(\gX)+d\Omega^0$.
\end{stat}


\subsection{The sheaf of local Hamiltonians and the map $\phi$}

\emph{The sheaf of local Hamiltonians} is  a subsheaf  $\cT$ of the sheaf $\Omega^0$ of smooth
functions on $M$. For any open set $U$ a smooth function
$f$ lies in $\cT(U)$ if and only if for any point
$ p \in U$ there exist a neighborhood $U' \subset U$ of $ p$
and $V \in \gX(U')$ such that $ df = i_V\omega$. 
We will  denote by $i$ the inclusion $\R_M \to \cT$.

Let us construct a sheaf morphism
$\pi : \cT \to \gX_\Gamma$.
For $f \in \cT(U)$ there exists a covering
$\cU =\{U_\alpha\}$ of $U$
such that on each $U_\alpha$ there exists a vector field
$V_\alpha$ with the property that $df = i_{V_\alpha}\omega$,
which is uniquely defined by virtue of Lemma \ref{lm1}.
Then, on
$U_\alpha \cap U_\beta$ we have $ V_\alpha = V_\beta$. Thus, on
$U$ we obtain the vector field $V$ such that $df =i_V\omega$,
which is uniquely defined. Furthermore, $L_V\omega = d(i_V
\omega) = d(df) = 0$, therefore $V \in \gX_\Gamma(U)$.
Thus we have obtained the map
$\pi_U : \cT(U) \to \gX_\Gamma(U)$,
which is evidently linear. All the maps $\pi_U$ determine a sheaf
morphism $\pi :  \cT \to \gX_\Gamma$.

In fact $\cT$ is the sheaf of algebras. 
The \emph{Poisson bracket} $\{\ ,\ \}$ of sections of
$\cT$ is defined in the following way. Let $f,g \in \cT(U)$,  $df=i_V\omega$,
and $dg=i_W\omega$, where $V, W \in \gX_\Gamma(U)$ are uniquely
defined. Set
\begin{equation}\label{eq13}
\{f,g\} = dg(V) - df(W) = 2 \omega(W,V).
\end{equation}


\begin{stat}\label{st20}
$$
0 \to \R_M \overset{i}{\longrightarrow} \cT
\overset{\pi}{\longrightarrow} \gX_\Gamma \to 0
$$
is an exact sequence of sheaves.
\end{stat}


\begin{cor}\label{cor111}
1. The sequence
\begin{multline} \label{eq6}
\dots \to
\check H^0(M;\gX_\Gamma) \overset{\delta}{\to}
\check H^1(M;\R_M) \overset{i_*}{\to}
\check H^1(M;\cT)  \overset{\pi_*}{\to}
\\
\overset{\pi_*}{\to}
\check H^1(M;\gX_\Gamma) \overset{\delta}{\to}
\check H^2(M;\R_M) \overset{i_*}{\to} \dots
\end{multline}
is exact.

2) The  diagram
\begin{equation}\label{eq7}
\begin{array}{ccc}
\check{H}^1(M;\gX_\Gamma) & \overset{\phi}{\longrightarrow}
& \cD_{\ess}(\omega) \\
{\scriptstyle\delta}{\Big\downarrow}\phantom{\delta} & & \phantom{\psi}
{\Big\downarrow}{\scriptstyle\psi} \\
\check{H}^2(M;\R_M) & \underset{\cong}{\overset{\eta}{\longrightarrow}}
& H^2(M),
\end{array}
\end{equation}
is commutative. Here $\psi$ is defined in \ref{subsubsec:4_2},
$\delta$ is the connecting homomorphism of
exact sequence (\ref{eq6}), and $\eta$ is the standard isomorphism
between the \v Cech cohomology with coefficients in
$\R_M$ and the de Rham cohomology.
\end{cor}

\subsubsection{Obstructions to integrability of infinitesimal deformations.}

\begin{stat}
Let $[v=\{V_{\beta\alpha}\}] \in H^1(M;\gX_\Gamma)$ be an
essential infinitesimal deformation of a symplectic structure with
singularities of type $\omega_0$, and
$[w=\{W_{\alpha\beta\gamma}\}] \in H^2(M, \gX_\Gamma)$ be the obstruction
to integrability of this infinitesimal deformation
$($see\,$1.1)$.
Then $[w]$ lies in the image of the map
$\pi_* : H^2(M; \cT) \to H^2(M;\gX_\Gamma)$
(see $(\ref{eq6})$).
\end{stat}

Let a 1-cocycle $v \in Z^1(\cU; \gX_\Gamma)$, where $\cU$ is a
covering of $M$ consisting of contractible open sets, represent an
infinitesimal deformation of a symplectic structure with
singularities of type $\omega_0$. Then the 2-cocycle
$w=\{W_{\alpha\beta\gamma} = [V_{\alpha\beta}, V_{\beta\gamma}]\}
\in Z^2(\cU; \gX_\Gamma)$ represents the obstruction
to integrability of $v$.
Given $v$, we have the cochain
$f = \{f_{\beta\alpha}\} \in C^1(\cU; \cT)$ such that
$\pi(f) = v$ (see Statement \ref{st20}). Take the cochain
$g_{\alpha\beta\gamma} =\{f_{\alpha\beta},f_{\beta\gamma}\}
\in C^2(\cU; \cT)$, where $\{\ ,\ \}$ is the Poisson bracket
(\ref{eq13}). Since $\pi$ is a morphism between sheaves of Lie
algebras, we obtain that $\pi(g)=w$. 


\subsection{Infinitesimal deformations of symplectic structure with
  Martinet singularities}

\subsubsection{Martinet singularities}

In  \cite{Mart1}--\cite{GolTish} it was proved that on $\R^4$
five generic types of germs of closed 2-forms exist, among
them  the following four types  are stable:


Type 0
$$
\omega_0 = dx^1\wedge dx^2 + dx^3\wedge dx^4,
$$

Type I
$$
\omega_0 = x^1 dx^1\wedge dx^2 + dx^3\wedge dx^4,
$$

Type II-e (the elliptic type)
\begin{equation}
\omega_0 = dx^1 \wedge dx^2 + x^3 dx^1 \wedge dx^4 + x^3 dx^2 \wedge dx^3
+ x^4 dx^2 \wedge dx^4 + (x^1- (x^3)^2) dx^3 \wedge dx^4,
\label{eq3}
\end{equation}

Type II-h (the hyperbolic type)
\begin{equation}
\omega_0 = dx^1 \wedge dx^2 + x^3 dx^1 \wedge dx^4 + x^3 dx^2 \wedge dx^3
- x^4 dx^2 \wedge dx^4 + (x^1- (x^3)^2) dx^3 \wedge dx^4.
\label{eq4}
\end{equation}

One can characterize these types in the following way (see
\cite{Mart1} for details). 
Let  $\omega$ be a closed 2-form on a four-dimensional manifold  $M$. 
Let us denote by $\Sigma$  the set of points, where $\omega$ is
degenerate.  In what follows we assume that $\Sigma$ is a
three-dimensional submanifold of $M$.

If a point $p$ does not lie in  $\Sigma$, then, by the Darboux theorem,
$\omega$ has type 0 at  $p$.

Now let  $p \in \Sigma$ and $\omega(p) \ne 0$. If the kernel $E_p$ of 
 $\omega(p)$ is transversal to  $\Sigma$, then $\omega$ has type I at
 $p$. The set $V = U \cap \Sigma$ is open in  $\Sigma$ and consists of
 points of type I, and $U \setminus V$ is everywhere dense in  $U$
 and consists of points of type 0.

Let  $\Sigma' \subset \Sigma$~be the set of points  $p$ such that 
$E_p \subset T_p \Sigma$. If $\Sigma'$ is a submanifold in a
neighborhood of $p$, and  $E_p$ is transversal to  $\Sigma'$,
then  $\omega$ has type II-e, or II-h, at  $p$. 
Let  $V = U \cap \Sigma$, $W = U \cap \Sigma'$.
Then $U \setminus \Sigma$ is everywhere dense in  $U$ and consists of
points of type  0, $V \setminus W$ is everywhere dense in  $V$ and
consists of points of type  I, and $W$ consists of points of type  II.

Let us consider a closed 2-form $\omega$  on a four-dimensional manifold
$M$. If all points of  $M$ has type  0, then  
$(M,\omega)$ is a symplectic manifold. If any point of $M$ has type  0
or I, then  $(M,\omega)$ will be called a {\em symplectic manifold with
Martinet singularities of type  $\mathrm I$}.
If any point of  $M$ has type  0, I, or II, then 
$(M,\omega)$ will be called a {\em symplectic manifold with Martinet
  singularities  of type  $\mathrm{II}$}.

Note that a symplectic manifold  $(M,\omega)$ with Martinet
singularities is a symplectic manifold with singularities of type 
$\omega_0$, where  $\omega_0$ is given by  (\ref{eq3}) or (\ref{eq4}).
Indeed, let $\phi(p) = (x^1_0, x^2_0, x^3_0, x^4_0)$. Then  $p$ has
type  0 if and only if $x^1_0\ne 0$; $p$ has type I  if and only if 
$x^1_0=0$ and $(x^3_0)^2 + (x^4_0)^2 \ne 0$; and $p$ has type II if and only if  
$x^1_0=0$, $x^3_0=0$, and $x^4_0=0$. Thus the symplectic structure is
a partial case of the symplectic structure with Martinet singularities
of type I, which in turn is a partial case of the symplectic structure
with Martinet singularities of type II.

{\it Example of compact manifold with symplectic structure having Martinet
singularity.} Let us take the four-dimensional torus $\T^4$, and let
$\pi : \R^4 \to \T^4$ be the standard covering. Let us denote by
$\theta^a$, $a=\overline{1,4}$, the 1-forms on $\T^4$ such that
$\pi^*(\theta^a) = dx^a$. Set  $\omega = f \theta^1 \wedge \theta^2 + \theta^3
\wedge \theta^4$, where $\pi^* f = \cos(x^1)$. It is clear that
$d\omega=0$. Let us denote by $p_1, p_2 \in \SS^1$ the zeroes of $f$.
Then $\Sigma = p_1 \times \T^3 \cup p_2 \times \T^3$, and the restriction of
$\omega$ to each connected component of $\Sigma$ is
$\theta^3 \wedge \theta^4$, i.\,e. it has rank 2. Then 
$\omega$ determines a symplectic structure with Martinet singularity on
$\T^4$.


\textbf{Type 0}

If $\omega$ is a symplectic structure without singularities, then the sheaf
$\cT$ coincides with the sheaf of smooth functions $\Omega^0$,
therefore $H^k(M; \cT)=0$ for $k>0$. Furthermore,
$\iota : \gX \to \Omega^1$ is a sheaf isomorphism (see 1.4).
Hence, Corollary  \ref{cor111} and Statement \ref{st4} imply the well-known fact
that for a symplectic structure
$H^1(M; \gX_\Gamma) \cong \cD_{\ess}(\omega) \cong H^2(M)$.

\textbf{Type I}

\begin{stat}
For a symplectic structure $\omega$ with Martinet singularities,
$\phi: H^1(M;\gX_\Gamma) \to \cD_{\ess}(\omega)$ is an isomorphism.
\end{stat}

\textbf{Type II}

In this case the  homomorphism $\phi$ is not surjective. Let
$\omega$ be a symplectic structure with singularities of  type
$$
%\begin{equation}\label{eq10}
\omega_0 = dx^1 \wedge dx^2 + x^3 dx^1 \wedge dx^4 +
x^3 dx^2 \wedge dx^3 + x^4 dx^2 \wedge dx^4 + (x^1- (x^3)^2) dx^3
\wedge dx^4
%\end{equation}
$$
Then the form $\alpha = (x^3)^2 dx^4$ cannot be represented as $i_V\omega
+df$, therefore, by Statement \ref{st4}, $\phi$ is not surjective.


\subsubsection{Calculation of $H^*(M; \cT)$ for Martinet
  singularities}

On the submanifold  $\Sigma$ of singular points we have the foliation  $\cF$ with
singularities whose regular leaves are integral curves of the
one-dimensional distribution on $\Sigma \setminus \Sigma'$ obtained by
the intersection of the kernel of $\omega$ with the spaces tangent to
$\Sigma$, and whose singular leaf is  $\Sigma'$. 
Let us denote by $\cF_b$ the sheaf of basic functions of the foliation  
$\cF$, i.\,e. the sheaf of functions which are locally constant along
the leaves of  $\cF$.

Let  $\mu$ be a volume form on  $M$. Then 
$\omega \wedge \omega = \Pf_\mu(\omega) \mu$, where $\Pf_\mu(\omega)$
is called the {\em Pfaffian of  $\omega$ with respect to  $\mu$}.
It is evident that, if  $\mu' = \lambda \mu$ is another volume form,
where $\lambda$ is nonvanishing function, then
$\Pf_\mu(\omega) = \lambda \Pf_{\mu'}(\omega)$. Hence, in the ring sheaf 
$C^\infty$ of smooth functions we have the  {\em subsheaf $\cI$ of
  principal ideals generated by  $\Pf_\mu(\omega)$}:
$$
 \cI(U) = \{\rest{\Pf_\mu(\omega)}{U} \cdot f \mid f \in C^\infty(U)\},
$$
which does not depend on the choice of $\mu$.

Note that  {\em $\cI$ is the ideal sheaf whose sections are the
  functions vanishing on  $\Sigma$, i.\,e.
$C^\infty/\cI$ is the sheaf of smooth functions on  $\Sigma$}.


Let  $X$ be a topological space, $A \subset X$ be a closed subset, and
$\cG$ be  a ring sheaf  on $A$. Then we have 
the sheaf  $\cG^X$ on $X$ generated by the presheaf: 
$U \to {0}$ if $U \cap A = \emptyset$, otherwise $U \to \cG(U \cap A)$ 
 (see \cite{Bre}).

Let  $r: C^\infty_M \to (C^\infty_\Sigma)^M$ be the sheaf morphism
determined by the restriction of functions to  $\Sigma$. This means
that, if 
$U \cap \Sigma = \emptyset$, then $(C^\infty_\Sigma)^M(U) = 0$ and
$r_U : C^\infty_M(U) \to (C^\infty_\Sigma)^M(U)$ is the zero homomorphism;
if $U \cap \Sigma = U' \ne \emptyset$, then 
$r_U : C^\infty_M(U) \to C^\infty_\Sigma(U')$ is the restriction of 
$f \in C^\infty(U)$ to $U'$.
Now, let us denote by $i$ the inclusion  $\cI^2 \to C^\infty_M$, and set 
$\wt\cF_b = \cF_b^M$.

\begin{stat} \label{st1}
\begin{equation} \label{eq9}
0 \to \cI^2 \xrightarrow{i} \cT \xrightarrow{r} \wt\cF_b \to 0.   
\end{equation}
is  the exact  sequence of sheaves on  $M$.
\end{stat}

\begin{cor} \label{cor2}
For a symplectic manifold $(M,\omega)$ with Martinet singularities,
$$
H^q(M;\cT) \cong H^q(\Sigma;\cF_b), \quad q> 0,
$$
where   $\cT$ is the sheaf of local Hamiltonians on  $M$, and $\cF_b$
is the sheaf of basic functions of the foliation with singularities
induced by the kernel of $\omega$ on the singular submanifold  $\Sigma$.
\end{cor}


\begin{cor} For a symplectic manifold $(M,\omega)$ with Martinet singularities such
  that  $\Sigma'=\emptyset$, 
$$
H^q(M;\gX_h) \cong H^{q+1}_{DR}(M), \quad q>2,
$$
where  $\gX_h$ is the sheaf of Hamiltonian vector fields on  $M$,
$H^k_{DR}(M)$ is the de Rham cohomology of $M$.
\end{cor}


\begin{cor} \label{cor1}
If the foliation  $\cF$ is a fiber bundle, then 
$H^q(M;\gX_\Gamma) \cong H^{q+1}_{DR}(M)$ for $q>0$. In particular, 
if $H^2_{DR}(M) \cong 0$, then  $\omega$ is infinitesimally rigid.
\end{cor}



\begin{ex}
\rm
Let  $(\wh M,\wh\omega)$ be a symplectic $2n$-dimensional manifold,
$M$ be a $2n$-dimensional manifold, and $\pi: M \to \wh M$ be a smooth
map. 
Let  $\omega = \pi^*\omega$.
Then  $\omega$ is a closed  2-form on  $M$, and the following diagram
%$$
\begin{equation*}
\begin{CD}
TM @>{\alpha_\omega}>> T^*M
\\
@V{d\pi}VV @AA{d\pi^*}A
\\
T\wh M @>{\alpha_{\Hat \omega}}>> T^*\wh M.
\end{CD}
\end{equation*}
%$$
is commutative.
Then
\begin{equation}
\label{eq16}
\Ann(\omega) = \ker\alpha_\omega =
d\pi^{-1}(\alpha^{-1}_{\Hat\omega}\ker d\pi^*).
\end{equation}

Now let  $\wh M$ be $\R^4$ (with coordinates  $y^\alpha$,
$\alpha=\ov{1,4}$) endowed with the standard symplectic structure 
$$
\wh\omega = dy^1\wedge dy^2 + dy^3 \wedge dy^4,
$$
$M = \SS^4 \subset \R^5$ be the standard sphere given by 
$\sum\limits_{i=1}^5 (x^i)^2 = 1$, and  $\pi : \SS^4 \to \R^4$ be the
restriction of the projection  $\R^5 \to \R^4$,
$y^\alpha = x^\alpha$, $\alpha=\ov{1,4}$.

Let  $\Pi$ be the hyperplane in  $\R^5$ given by  $x^5=0$, and 
$\SS^3_0 = \SS^4 \cap \Pi$ be the equator of  $\SS^4$.
It is clear that  $d\pi_p : T_p\SS^4 \to T_p\R^4$ is an isomorphism
for any $p \in \SS^4 \setminus \SS^3_0$, hence $\omega$ is
nondegenerate on  $\SS^4 \setminus \SS^3_0$. 
Further, by a coordinate calculation with the use of (\ref{eq16}), we
get that at points of $\SS^3_0$ the kernel of $\omega$ is spanned by  
$\pd_5$ and $-x^2\pd_1 + x^1\pd_2 - x^4\pd_3 + x^3\pd_4$. Hence 
$\Ann(\omega) \pitchfork \SS^3_0$, and the one-dimensional foliation
$\cF$
corresponding to the distribution  $\ker(\omega) \cap T\SS^3_0$ on
$\SS^3_0$ is the Hopf bundle. Then  $\omega$ is a symplectic structure
with Martinet singularities of type  I,  which satisfies the
assumptions of 
Corollary~\ref{cor1}. Hence  $\omega$ is rigid.
\end{ex}


\subsubsection{Differential complex associated to symplectic form with Martinet singularities}
$\hphantom{xx}$


Here we expose results of \cite{MA4}.


We start with the following standard algebraic construction. Let $K$
be a  ring and $d : K \to K$ be a differentiation such that $d^2=0$. Let
$I$ be an ideal in $K$, then 
\begin{equation*}
I'=\{a + k_i\, db_i \mid a,b_i \in I\}
\end{equation*}
also is an ideal in $K$ such that $d : I' \to I'$. Then we have the following exact
sequence of differential rings:
\begin{equation*}
0 \to I' \to K \to K/I' \to 0
\end{equation*}
 
The same construction can be done for sheaves of rings over a smooth
manifold $M$. Then, for a sheaf of rings $\cK$ over $M$ endowed with a
differential $d$ such that $d^2=0$, and a subsheaf 
$\iota: \cI \hookrightarrow  \cK$ of ideals, 
we get the sheaf $\cI'$ of ideals and the exact sequence
of sheaves 
\begin{equation*}
0 \to \cI' \to \cK \to \cK/\cI' \to 0
\end{equation*}
and the corresponding  cohomology exact sequence
\begin{equation*}
\dots \to H^k(M;\cI') \to H^k(M;\cK) \to H^k(M;\cK/\cI') \to
H^{k+1}(M;\cI') \to \dots  
\end{equation*}
   
For any vector bundles $\xi : E_\xi \to M$ and $\eta : E_\eta \to M$,
each vector bundle morphism $Q : \xi \to \eta$ determines the morphism   
$\cQ : \Gamma_\xi \to \Gamma_\eta$ of sheaves of vector spaces: 
for each $s \in \Gamma_\xi(U)$, the section 
$\cQ(s)(p) = Q_p(s(p))$, $p \in U$, lies in $\Gamma_\eta(U)$. 
The kernel of $\cQ$ is the sheaf 
$\cK(U) = \{s \in \Gamma_\xi(U) \mid \cQ(s)=0\}$ 
of modules over the fine sheaf $C^\infty$, therefore $\cK$ is also
fine. Now let us consider the presheaf 
$\cF(U) = \{t \in \Gamma_\eta(U) \mid  t = \cQ(s)\}$.
\begin{lm}
The presheaf $\cF$ is a sheaf.
\end{lm}

Let $\xi : E_\xi \to M$ be a vector bundle, and 
$A : \xi \to \Lambda^qM$ be a vector bundle morphism.
Denote by  
$\cA : \Gamma_\xi \to \Omega_M$ the  sheaf morphism corresponding
to $A$. Then the subsheaf $\cA(\Gamma_\xi)$
generates the subsheaf $\cI_M \subset \Omega_M$ of ideals. Let us
denote by $\cF_M = \oplus \cF^q_M \subset \Omega_M$ 
the corresponding graded sheaf $\cI'_M$ of differential ideals.


We take an open $U \subset M$ such that $\xi$ and $\Lambda M$ are
trivial over $U$. Then, on $U$ we get $q$-forms $\omega_a$,
$a=\overline{1, \rank \xi}$, which span 
$\cA(U)$ over the ring $C^\infty(U)$ of functions on $U$. 
One can easily see that  
\begin{equation}
  \label{eq:1_555}
  \cF^k(U) = \{\phi^a \wedge \omega_a + \psi^a \wedge d\omega_a \mid
  \phi^a \in \Omega^k(U), \psi^a  \in \Omega^{k-1}(U)\}
\end{equation}

Thus, to any morphism $A : \xi \to \Lambda^qM$ we associate the
complex  $(\cF^*,d)$ of sheaves, which is a subcomplex  
of the de Rham complex $(\Omega_M,d)$ considered also as a complex of
sheaves. 
Also, we have the exact
sequence of sheaves
\begin{equation*}
  0 \to \cF \to \Omega_M \to \cG=\Omega_M/\cF \to 0.  
\end{equation*}





\begin{rem}
Let $q=1$, and $A : \xi \to \Omega^1$ be a morphism. Then the sheaf 
$\cA(\Gamma_\xi)$ is the sheaf of sections of a subbundle (with
singularities) in $T^*M$. If $\rank A$ is constant, then $A$
determines a distribution on $M$, and if, in addition, this
distribution is integrable,  $(\cF^*(M),d)$ is the 
complex generated by the basic forms of the corresponding foliation, 
which is widely used  in the foliation theory
\cite{Molino1}, \cite{Vaisman}.
\end{rem}

\begin{rem}
If $A : \xi \to \Omega^q$ is surjective, then the associated complex $(\cF^*(M),d)$
is the de Rham complex of $M$.    
\end{rem}


Hereafter we assume that $A$ 
\emph{is surjective on an open everywhere dense set} 
$M \setminus \Sigma$, where $\iota : \Sigma \hookrightarrow M$ is an embedded submanifold.
Then, for each open $U$ such that $U \cap \Sigma = \emptyset$ we have
$\cF^k(U)=\Omega^k(U)$, hence $\cG(U)=0$. 
From this follows that the sheaf $\cG$ is supported on $\Sigma$.


For a closed $\omega \in \Omega^{q+1}(M)$, the Lie derivative
$D(V)=L_V\omega = d i_V \omega$ is a first order differential operator 
$TM \to \Lambda^{q+1}M$. The operator $D$ can be included to the 
complex of sheaves associated to the vector bundle morphism $I_\omega : TM \to \Lambda^q$,
$I(V)=i_V\omega$:
\begin{equation}
  \label{eq:1_7}
0 \to \gX_\Gamma \to \gX_M \overset{D}{\longrightarrow} \cF^1 
\overset{d}{\longrightarrow} \cF^2 
\overset{d}{\longrightarrow}
\dots,    
\end{equation}
where $\gX_\Gamma$ is the sheaf of infinitesimal automorphisms of $\omega$.
Evidently, all the sheaves in (\ref{eq:1_7}), except for $\gX_\Gamma$ are
fine. Therefore, if  (\ref{eq:1_7})  is locally exact, it gives a fine
resolution for $\gX_\Gamma$. However, in general, (\ref{eq:1_7}) fails to
be locally exact.   





Let $\omega$ be a closed 2-form on a smooth manifold $M$ such that
$\det \omega = 0$ on a closed submanifold 
$i : \Sigma \hookrightarrow M$ and $\rank \omega = 2m$ is constant on $\Sigma$.  
Then $\omega$ determines the vector bundle morphism 
$I_\omega: TM \to \Lambda^1M$, $I_\omega(V)=i_V\omega$, which is a vector bundle isomorphism over 
$M \setminus \Sigma$. 
The kernel of $I_\omega$ is a vector bundle over $\Sigma$, call it
$\epsilon : E \to \Sigma$.
Denote by $\cI_\omega$ the corresponding sheaf morphism $\gX_M \to \Omega^1_M$. 
From Lemma proved in \cite{MA2} it follows that  
$\tau \in \cI_\omega(\gX_M)$ if and only if $\rest{\tau}{E} = 0$.


We will consider the symplectic structures with Martinet singularities.

Let $\omega$ be a closed 2-form on a $2n$-dimensional manifold.
Assume that  for each point $p \in M$ one can take a chart $(U,u^i)$
such that 
\begin{equation}
  \label{eq:3_2}
  \omega = u^1 du^1 \wedge du^2 + du^3 \wedge du^4 \dots + 
du^{2n-1} \wedge du^{2n}. 
\end{equation}

\begin{stat}
\label{stat:3_1}
Let $(\cF^*,d)$  be the complex of sheaves associated to the symplectic form
$\omega$ with Martinet singularities locally given by \emph{(\ref{eq:3_2})}. 
Then $\cF^1$ is the subsheaf of $\Omega^1_M$ consisting of forms which 
vanish on the subbundle $E \subset \rest{TM}{\Sigma}$, and 
$\cF^k = \Omega^k_M$ for $k \geq 2$.    
\end{stat}



\begin{stat}
\label{stat:3_2}
For $\cD : \gX_M \to \Omega^2_M$, $D(V)=L_V\omega$, 
the sequence of sheaves $($see \emph{(\ref{eq:1_7})}$)$ 
\begin{equation}
  \label{eq:3_5}
0 \to \gX_\Gamma \overset{i}{\longrightarrow} 
\gX_M \overset{D}{\longrightarrow} \Omega^2 
\overset{d}{\longrightarrow} \Omega^3 \overset{d}{\longrightarrow} \dots 
\end{equation}
is a fine resolution for the sheaf $\gX_\Gamma$.   
\end{stat}

\begin{cor}
For $\omega$ with Martinet singularities locally given by
\emph{(\ref{eq:3_2})}, $H^q(M;\gX_\Gamma) \cong H^{q+1}_{DR}(M)$, 
$q \ge 1$, where
$H_{DR}(M)$ is the de Rham cohomology.
\end{cor}



Let us  consider another type of Martinet singularities. 
Let $\omega$ be a closed 2-form on a four-dimensional manifold, and 
assume that  for each point $p \in M$ one can take a chart $(U,u^i)$
such that 
\begin{equation}
\label{eq:3_13}
\begin{split}
\omega_0 = &du^1 \wedge du^2 + u^3 du^1 \wedge du^4 
\\
& + u^3 du^2 \wedge
du^3 + u^4 du^2 \wedge du^4 + (u^1- (u^3)^2) du^3 \wedge du^4,
\end{split}
\end{equation}

\begin{stat}
Let $(\cF^*,d)$  be the complex of sheaves associated to the symplectic form
$\omega$ with Martinet singularities locally given by
\emph{(\ref{eq:3_13})}.
Then $\cF^1$ is the subsheaf of $\Omega^1_M$ consisting of forms
vanishing on the subbundle $E \subset \rest{TM}{\Sigma}$, and 
$\cF^k = \Omega^k_M$ for $k \geq 2$.    
\end{stat}


Thus, in this case  we also get the complex of sheaves
(\ref{eq:3_5}). However, in this case the complex (\ref{eq:3_5}) is not locally
exact. 










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{9}

\bibitem{Pommaret}
Pommaret, J.F.
{\it Systems of partial differential equations and
Lie pseudogroups}, Math. and Appl., {\bf 14} (1978).

\bibitem{Vaisman}
Vaisman I. {\it $d_f$ -cohomologies of Lagrangian
foliations}, Monatshefte f\"ur Math., {\bf 106} (1988), pp.~221--244.



\bibitem{Mart}
Martinet J. {\em Sur les singularit\'es des formes
diff\'eretielles} // Ann. Inst. Fourier. Grenoble. -- 1970. --
V.\,20. -- \No\,1. -- P.\,95--178.

\bibitem{Arnold}
Arnol'd, V.I. {\em Singularities of caustics and wave fronts.} (Osobennosti
kaustik i volnovykh frontov). (Russian) Biblioteka Matematika. 1. Moskva:
Fazis. {\rm x+} 334 p. (1996). [ISBN 5-7036-0021-9]



\bibitem{Spen1}
Spencer D.C. {\em Deformations of structures on manifolds
defined by transitive continuous pseudogroups}. I.
{\em Infinitesimal deformations of structure} // Ann. Math. -- 1962.
-- V.\,76. -- \No\,2. -- P.\,306--398.

\bibitem{Spen2}
Spencer D.C. {\em Deformations of structures on manifolds
defined by transitive continuous pseudogroups}. II.
{\em Deformations of structure} // Ann. of Math. -- 1962. --
V.\,76. -- \No\,3. -- P.\,399--445.

\bibitem{Spen3}
Spencer D.C. {\em Deformations of structures on manifolds
defined by transitive continuous pseudogroups}. III.
{\em Structures defined by elliptic pseudogroup} //
Ann. Math. -- 1965. -- V.\,81. -- \No\,2. -- P.\,389--450.

\bibitem{GuiSten}
Guillemin V., Sternberg S. {\em Deformation theory of
pseudogroup structures} // Mem. Am. Math. Soc. -- 1968. -- V.\,64. --
80~p.

\bibitem{Kodaira}
Kodaira~K. {\it Complex manifolds and deformations
of complex structures}. -- Springer, 1986. -- 400 p.

\bibitem{Fomenko}
Fomenko, A.T. {\em Symplectic geometry. Methods and applications.}
(Simplekticheskaya geometriya. Metody i prilozheniya). (Russian) Moskva:
Izdatel'stvo Moskovskogo Universiteta. 414 p. (1988). [ISBN 5-211-00083-8]





\bibitem{WF} Warner, Frank W.
Foundations of differentiable manifolds and Lie groups. 
(1987). 


\bibitem{Wells}, Wells R.O.jun.
Differential analysis on complex manifolds. 2nd ed. (English)
Graduate Texts in Mathematics. 65. New York - Heidelberg - Berlin:
Springer-Verlag. X, 260 p. 
(1980). 

\bibitem{Sternberg}
Sternberg, Shlomo
Lectures on differential geometry. 2nd ed. (English)
New York: Chelsea Publishing Company. XV, 442 p. (1983).

\bibitem{Kob1}
Kobayashi, Shoshichi
Transformation groups in differential geometry. Reprint of the 1972 ed. (English)
Classics in Mathematics. Berlin: Springer-Verlag. viii, 182 p.


\bibitem{Al-Molino}
Albert, C.; Molino, P.
Pseudogroupes de Lie transitifs. II: Th'eor`ems d'int'egrabilit\`e. 
(Transitive Lie pseudogroups. II: Integrability theorems). (English)
Travaux en Cours, 19. Paris: Hermann. VIII, 131 p.

\bibitem{Mart1} %2
Martinet J. {\em Sur les singularit\'es des formes diff\'erentielles} //
Ann. Inst. Fourier (Grenoble). -- 1970. -- V.\,20. -- \No\,1. -- P.\,95--178.

\bibitem{Mart2} %3
Martinet J. {\em Singularities of smooth functions and maps} // London
Math. Soc. Lecture Note Ser. --1982. -- \No\,58. -- 256 p.

\bibitem{Rous} %4
Roussarie R. {\em Mod\`eles locaux de champs et de formes} // Ast\'erisque.
-- 1975. -- \No\,30. -- 181 p.

\bibitem{GolTish} %5
Golubitsky M., Tischler D. {\em An example of moduli for singular
symplectic forms} // Invent. Math. -- 1977. -- V.\,38. -- \No\,3. -- P.\,219--225.

\bibitem{SGW} %6
Cannas da Silva A., Guillemin V., Woodwart C. {\em On the unfolding of
folded symplectic structures} // Math. Res. Lett. -- 2000. -- V.\,7.
-- \No\,1. -- P.\,35--53.

\bibitem{S} %7
Cannas da Silva A. {\em Fold-forms for four-folds} // Preprint. -- 2003.
-- 16 p.

\bibitem{D1} %8
Domitrz W. {\em Non-local invariants of Martinet's singular symplectic
structure} //
Banach Center Publ. -- 2002. -- V.\,60. -- P.\,122--143.

\bibitem{Fom} %9
Fomenko, A.T. {\em Symplectic geometry. Methods and applications.}
(Simplekticheskaya geometriya. Metody i prilozheniya). (Russian) Moskva:
Izdatel'stvo Moskovskogo Universiteta. 414 p. (1988). [ISBN 5-211-00083-8]



\bibitem{Bre} %10
Bredon, G. E.  {\em Sheaf theory}. 
McGraw-Hill New York 1967

\bibitem{GriffithsHarris}
Griffiths, Phillip; Harris, Joseph
\emph{Principles of algebraic geometry}. 2nd ed. (English)
Wiley Classics Library. New York, NY: John Wiley \& Sons Ltd. xii, 813 p. 


\bibitem{Molino1} 
Molino, Pierre
\emph{Riemannian foliations}. 
Progress in Mathematics, Vol. 73. Boston-Basel: Birkh\"auser. XII, 339 p.




\bibitem{MA1}
Malakhaltsev, M.A.
\emph{Infinitesimal deformations of a symplectic structure with singularities.} 
Russ. Math. 47, No.11, 38-46 (2003); translation from
Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No.11, 42-50 (2003).

\bibitem{MA2}
Malakhaltsev, M.A.
\emph{Sheaf of local Hamiltonians of symplectic manifolds with
Martinet singularities.}
Russ. Math. No.11, 45-52 (2004)



\bibitem{MA3}
Malakhaltsev, M.A.
\emph{The Lie derivative and cohomology of
G-structures}. Lobachevskii Journal of Mathematics, 1999,
Vol.3, pp.215-220 (http://ljm.ksu.ru).


\bibitem{MA4}
Malakhaltsev, M.A.
\emph{Differential complex associated to closed differential forms of nonconstant rank}
Lobachevskii Journal of Mathematics, 1999,
Vol.23, pp.183-192 (http://ljm.ksu.ru).


\end{thebibliography}



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