Deformation of singular foliations, 1: Local deformation cohomology

TL;DR

This paper introduces a new cohomology framework to study local deformations of singular foliations, integrable forms, and Nambu structures, providing tools to analyze their infinitesimal changes near a point.

Contribution

It defines deformation cohomology for singular foliations and related structures, focusing on local behavior, which is a novel approach in the field.

Findings

Established the concept of local deformation cohomology for singular foliations.

Analyzed the properties of this cohomology in the neighborhood of a point.

Provided foundational results for future deformation studies of singular geometric structures.

Abstract

In this paper we introduce the notion of deformation cohomology for singular foliations and related objects (namely integrable differential forms and Nambu structures), and study it in the local case, i.e., in the neighborhood of a point.

Equations

$$1)\omega \wedge i_A\omega =0\text{and}2)d\omega \wedge i_A\omega =0.$$

$$\omega =fd{x}_1\wedge \dots \wedge d{x}_p,$$

$$\omega =i_{\Lambda }\Omega$$

$$\Lambda =f\frac{\partial }{\partial x_{p+1}}\wedge \dots \wedge \frac{\partial }{\partial x_n}.$$

$$\mathrm{codim}(S(\mathcal{F})\setminus S(\Lambda ))\ge 2$$

$$\Pi =X_1\wedge \dots \wedge X_q,$$

$$X\wedge \Lambda =0\text{and}{\mathcal{L}}_X\Lambda =g\Lambda \text{ for some function }g.$$

$$(\omega +\epsilon \eta )\wedge i_A(\omega +\epsilon \eta )\equiv 0(\mathrm{mod}{\epsilon }^2)\text{and}d(\omega +\epsilon \eta )\wedge i_A(\omega +\epsilon \eta )\equiv 0(\mathrm{mod}{\epsilon }^2)$$

$$i_A\omega \wedge \eta +i_A\eta \wedge \omega =0\text{and}{i}_A\omega \wedge d\eta +i_A\eta \wedge d\omega =0,\forall (q-1)\text{-vector fields }A.$$

$$DH(\omega )=\frac{\mathcal{Z}(\omega )}{\mathcal{B}(\omega )}.$$

$$DH(\Lambda )=\frac{\mathcal{Z}(\Lambda )}{\mathcal{B}(\Lambda )}.$$

$$X_{i}=(-1)^{p-i}i_{dx_{1}\wedge\ldots\widehat{dx_{i}}\ldots\wedge dx_{q}}\big{(}\Lambda+\varepsilon\Pi\big{)}\,.$$

$$X_i=\partial /\partial x_i+\epsilon \sum _{k=q+1}^n{f}_k^{(i)}\partial /\partial x_k.$$

$$X_1\wedge \dots \wedge X_q=\Lambda +\epsilon {\mathcal{L}}_X\Lambda (\mathrm{mod}{\epsilon }^2)$$

$$\Pi ={\mathcal{L}}_X\Lambda +f\Lambda$$

$$DH({\mathcal{F}}_{\Lambda })=\frac{\mathcal{Z}(\Lambda )}{\mathcal{B}({\mathcal{F}}_{\Lambda })}.$$

$$DH({\mathcal{F}}_{\omega })=\frac{\mathcal{Z}(\omega )}{\mathcal{B}({\mathcal{F}}_{\omega })}.$$

$$DH({\mathcal{F}}_{\Lambda })\cong DH({\mathcal{F}}_{\omega })\text{and}DH({\mathcal{F}}_{\Lambda })\cong DH({\mathcal{F}}_{u\Lambda }).$$

$$DH(f)$$

$$DH({\mathcal{F}}_f)$$

$$\Lambda =f\frac{\partial }{\partial x_1}\wedge \dots \wedge \frac{\partial }{\partial x_n}$$

$$DH({\mathcal{F}}_{\Lambda })\cong \frac{{\mathcal{O}}_n}{⟨f,\frac{\partial f}{{\partial }_{x_1}},\dots ,\frac{\partial f}{{\partial }_{x_n}}⟩}$$

$$DH(\Lambda )\cong \frac{{\mathcal{O}}_n}{\{X(f)-(divX)f|X\in \mathfrak{X}({\mathbb{K}}^n)\}}.$$

$$\omega =ud{f}_1\wedge \dots \wedge d{f}_p,$$

$$\eta =a_{0}d{f}_1\wedge \dots \wedge d{f}_p+\sum _{i=1}^{p}d{f}_1\wedge \dots \wedge d{f}_{i-1}\wedge d{a}_i\wedge d{f}_{i+1}\wedge \dots \wedge d{f}_p.$$

$$i_A\eta \wedge d{f}_1\wedge \dots \wedge d{f}_p+i_A(d{f}_1\wedge \dots \wedge d{f}_p)\wedge \eta$$

$$i_A(d{f}_1\wedge \dots \wedge d{f}_p)\wedge d\eta$$

$$d{f}_i\wedge d\eta =0(\forall i=1,\dots ,p).$$

$$\eta =d\theta +cd{f}_1\wedge \dots \wedge d{f}_p,$$

$$d{f}_i\wedge d{f}_j\wedge \eta =0\forall i,j=1\dots ,p.$$

$$d{f}_i\wedge d\theta =d{f}_1\wedge \dots \wedge d{f}_p\wedge {\beta }_i$$

Full text

1. Introduction

This is the first in our series of papers on the problem of deformations of singular foliations in the sense of Stefan-Sussmann [13, 14]. In this paper we will concentrate on the local case, i.e., germs of singular foliations (analytic, smooth or formal), and study the deformation cohomology which governs their infinitesimal deformations. In the subsequent papers, we will discuss the global deformation cohomology, the rigidity problem of singular foliations, and so on.

In most deformation theories of objects of some given category (e.g., Lie algebras, complex structures, group actions, etc.), one can define a cohomology group which controls infinitesimal deformations, and other higher cohomology groups which may play the role of obstructions to integrating these infinitesimal deformations into true deformations. One wants to do the same thing for singular foliations. In order to do that, one first needs to “algebraize” or “tensorize” them, turn them into objects which can be manipulated with algebraic operations. Our approach to algebraization of singular foliations is via integrable differentiel forms and their dual multi-vector fields, called Nambu structures (see, e.g., [5, Chapter 6] and [9]).

We note that some authors, including Androulidalik, Skandalis and Zambon, consider locally finitely-generated involutive modules of vector fields and Lie algebroids as proxies for singular foliations, and obtain many interesting results with this approach, see, e.g., [1, 2]. Our approach is different from theirs. We believe that our approach is more directly related to the problem of deformations of singular foliations of a given dimension, and that the two approaches are complementary to each other and can be combined for the study of various problems concerning singular foliations.

We refer to [5, Section 1.5 and Chapter 6] for a brief introduction to singular foliations and some basic results, including the Stefan-Sussmann theorem [13, 14], which says that a distribution $\mathcal{D}$ on a manifold $M$ generated by a family $\mathcal{C}$ of (smooth, analytic or formal) vector fields on $M$ (i.e., at every point $x\in M$ the corresponding tangent subspace $\mathcal{D}_{x}$ of $\mathcal{D}$ is spanned by the vectors $\{X(x),X\in\mathcal{C}\}$) is integrable, i.e., is the tangent distribution of a singular foliation à la Stefan-Sussmann, if and only if $\mathcal{D}$ is invariant with respect to $\mathcal{C}$, i.e. the local flow of every element of $\mathcal{C}$ preserves $\mathcal{D}$.

In the case when $\dim\mathcal{D}_{x}$ does not depend on $x$ then $\mathcal{D}$ is called a regular distribution, and in this case its integrability condition (i.e., $\mathcal{D}$ is the tangent distribution of a regular foliation) is equivalent to the Frobenius involutivity condition: the Lie bracket of any two vector fields tangent to $\mathcal{D}$ is again tangent to $\mathcal{D}$. In the singular case (when $\dim\mathcal{D}_{x}$ is not constant but drops on a subset called the singular set), the involutivity condition is still necessary but not sufficient. A simple counter example is the distribution $\mathcal{D}$ on $\mathbb{R}^{2}$ given by $\mathcal{D}_{(x,y)}=Span(\partial/\partial x)$ if $x\leq 0$ and $\mathcal{D}_{(x,y)}=Span(\partial/\partial x,\partial/\partial y)$ if $x>0$, which is involutive but not integrable. However, according to a theorem of Hermann [7], for locally finitely generated disributions (i.e., the family $\mathcal{C}$ of vector fields which generates $\mathcal{D}$ can be chosen to be finite, at least locally) the involutivity condition is sufficient for integrability. To avoid pathologies, we will be mainly interested in singular foliations whose tangent distributions are locally finitely generated.

For regular foliations, the problem of stability (rigidity) was studied by Reeb [12] and Thurston [15], among other authors, and a deformation cohomology (which governs infinitesimal deformations) was defined by Heitsch [6]. We want to extend these theories of stability and infinitesimal deformations of foliations to the case of singular foliations. The motivation is clear: similarly to the fact that most functions in practice admit singular points, most foliations that we encounter (e.g., in geometric control theory, sub-Riemannian geometry, dynamical systems, symplectic and Poisson geometry, algebraic geometry, etc.) are in fact singular, and many interesting things (including global invariants) are localized at singularities, so one should include singularities in the study.

2. Tensorization of singular foliations

2.1. Integrable differential forms and Nambu structures

Let us recall that a differential $p$-form $\omega$ on a manifold $M^{n}$ ($0\leq q\leq n$) is called integrable if it satisfies the following two conditions for any $(p-1)$-vector field $A$:

[TABLE]

In particular, when $p=1$ then the first condition is trivial ($\omega\wedge\omega=0$ for any 1-form $\omega$), and the second condition is the usual integrability condition for a differential 1-form $\omega\wedge d\omega=0$. If $\omega$ is an integrable $p$-form and $z$ is a regular point of $\omega$, i.e. $\omega(z)\neq 0$, then in a neighborhood of $z$ there is a local coordinate system $(x_{1},\ldots,x_{n})$ in which

[TABLE]

where $f$ is some function such that $f(z)\neq 0$ . The kernel of an integrable $p$-form $\omega$ near a point $z$ where $\omega(z)\neq 0$ is an involutive distribution of corank $p$ which gives a codimension $p$ foliation outside singular points.

A Nambu structure of order $q$ on a manifold $M$ is a $p$-vector field $\Lambda$ on $M$ such that its contraction

[TABLE]

with a (local) volume form $\Omega$ is a (local) differential $p$-form (where $p+q=n$ is the dimension of the manifold). An equivalent condition (for smooth or analytic) Nambu structures is as follows: A (smooth or analytic) $q$-vector field $\Lambda$ is a Nambu structure if and only if near every point $x$ such that $\Lambda(x)\neq 0$ there is a local coordinate system $(x_{1},\dots,x_{n})$ such that

[TABLE]

In fact, by a change of coordinates, one can put $f=1$ in Formula (4). This formula cannot be used in the singular formal case, so in order to define a formal singular Nambu structure one has to use Condition (1) (applied to the dual differential form) instead (see, e.g.,[5]).

2.2. From singular foliations to Nambu structures and back

Given a singular foliation $\mathcal{F}$, we will say that a Nambu structure $\Lambda$ is a tangent Nambu structure to a $\mathcal{F}$ if

[TABLE]

and near each point $x\notin S(\Lambda)\cup S(\mathcal{F})$ there is a local coordinate system in which $\Lambda=\partial/\partial x_{1}\wedge\ldots\wedge\partial/\partial x_{p}$ and $\mathcal{F}$ is generated by $\partial/\partial x_{1},\dots,\partial/\partial x_{p}$. Here $S(\Lambda)$ denotes the singular set of $\Lambda$, i.e. the set of points where $\Lambda$ vanishes, and $S(\mathcal{F})$ denotes the set of singular points of $\mathcal{F}$, i.e., the set of points where the dimension of the tangent distribution drops. If, moreover, \mathrm{codim}\big{(}S(\Lambda)\setminus S(\mathcal{F})\big{)}\geq 2, and $\Lambda$ is without multiplicity in the sense that $\Lambda$ can’t be written as $\Lambda=f^{2}\Lambda^{\prime},$ where $f$ is a function which vanishes at the origin, then we say that $\Lambda$ is an associated Nambu structure to $\mathcal{F}.$

The above definition works well in the analytic and formal categories, and also in the smooth category under some finiteness conditions: the local associated Nambu structure exists and is unique up to multiplication by a non-vanishing function. (See [9] for the details). It can be constructed as follows. Take $p$ local vector fields $X_{1},\ldots,X_{p}$ which are tangent to $\mathcal{F}$ and which are linearly independent almost everywhere, where $p$ is the dimension of $\mathcal{F}$. Put

[TABLE]

then factorize $\Pi$ as $\Pi=h\Lambda$, where $\mathrm{codim}S(\Lambda)\geq 2.$ If $\mathrm{codim}S(\mathcal{F})\geq 2$ then $\Lambda$ is an associated Nambu structure of $\mathcal{F}$. If $\mathrm{codim}S(\mathcal{F})=1$, then we find a reduced function $s$ such that $S(\mathcal{F})=\{s=0\}$, and $s\Lambda$ is an associated Nambu structure of $\mathcal{F}$.

For example, let $\mathcal{F}$ be the codimension-1 quadric foliation on $\mathbb{R}^{3}$ or $\mathbb{C}^{3}$ with leaves $\{x^{2}+y^{2}+z^{2}=const\}$. Take two tangent vector fields $X=y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}$, $Y=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$, and put $\Pi=X\wedge Y=z\left(x\frac{\partial}{\partial y}\wedge\frac{\partial}{\partial z}+y\frac{\partial}{\partial z}\wedge\frac{\partial}{\partial x}+z\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}\right).$ Then $\Lambda=\dfrac{\Pi}{z}$ is an associated Nambu structure of $\mathcal{F}$.

Conversely, given a Nambu structure $\Lambda$, consider the set (or the sheaf), denoted by $CIT(\Lambda)$, of all (local) conformally invariant tangent (CIT) vector fields of $\Lambda$, i.e. vector fields $X$ satisfying

[TABLE]

Then one checks easily that $CIT(\Lambda)$ generates an integrable distribution. The corresponding foliation is denoted by $\mathcal{F}_{\Lambda}$ and called the associated foliation of $\Lambda$. If $f$ is a non-vanishing function then $\mathcal{F}_{f\Lambda}=\mathcal{F}_{\Lambda}$.

The above forward and backward functors give an “almost one to one” correspondence between local singular foliations and local Nambu structures (up to multiplication by non-vanishing functions) under some mild conditions on the singularities. (See [9] for precise statements in the holomorphic case). This justifies our use of Nambu structures as a proxy for singular foliations.

Nambu structures will allow us to study deformations of singular foliations. They also allow us to talk about (quasi)homogeneous singular foliations (i.e., foliations associated to linear and/or (quasi)homogeneous Nambu structures in some coordinate system), and study the local normalization problem near a singular point. See [16] and references therein for recent results on the problem of local linearization of singular foliations and of Nambu structures. Many operations with singular foliations, e.g., pull-back and reduction, can also be done naturally via associated Nambu structures and integrable differential forms.

Globally, on a manifold, we have a sheaf of local tangent Nambu structures, which is a locally free module of rank one over the ring of functions. In other words, this sheaf is a line bundle, which is nothing but the anti-canonical line bundle of the foliation. Since this line bundle may be twisted and does not necessarily admit a global section, we do not necessarily have a global Nambu structure associated to a singular foliation, only local ones. This will be discussed in more detail in our subsequent paper.

3. Infinitesimal deformations and deformation cohomologies

Let $\omega$ be an integrable differential $q$-form on a $n$ dimensional manifold $M$. By an infinitesimal deformation of $\omega$ we mean a $q$-form $\eta$ such that $\omega+\varepsilon\eta$ is integrable modulo $\varepsilon^{2}$, where $\varepsilon$ is a formal infinitesimal parameter. In other words,

[TABLE]

for any $(q-1)$-vector field $A$. Since $\omega$ is integrable, the above conditions are equivalent to the following family of linear equations on $\eta$ and $d\eta$:

[TABLE]

If $\eta=\mathcal{L}_{X}\omega=i_{X}d\omega+di_{X}\omega$, where $X$ is a vector field, then $\eta$ is called a trivial deformation of $\omega$ (because it is obtained by the pull-back of $\omega$ with respect to the infinitesimal flow of $X$, i.e. $\omega$ is sent to $\omega+\varepsilon\eta$ modulo $\varepsilon^{2}$ by such a flow). Denote by $\mathcal{Z}(\omega)$ the set of infinitesimal deformations of $\omega$, and by $\mathcal{B}(\omega)$ the set of trivial deformations of $\omega$. It is clear that $\mathcal{B}(\omega)$ is a vector subspace of $\mathcal{Z}(\omega)$, and we can define the following quotient vector space, which we denote by $DH(\omega)$ and call the deformation cohomology of $\omega$:

[TABLE]

Suppose that $\Omega$ is a volume form and $\Lambda$ is a Nambu structure of degree $q$ on $M$. The set of infinitesimal deformations $\mathcal{Z}(\Lambda)$ of $\Lambda$ consists of all $q$-vector fields $\Pi$ such that $i_{\Pi}\Omega$ is a infinitesimal deformation of $i_{\Lambda}\Omega$. In other words, $\Pi\in\mathcal{Z}(\Lambda)$ means that $\Lambda+\varepsilon\Pi$ is a Nambu structure modulo $\varepsilon^{2}$. If $\Pi=\mathcal{L}_{X}\Lambda$ for some vector field $X$, then $\Pi$ is called a trivial deformation of $\Lambda$. We denote by $\mathcal{B}(\Lambda)$ the set of trivial deformations of $\Lambda$. The deformation cohomology $DH(\Lambda)$ of $\Lambda$ is defined as follows:

[TABLE]

The definition of $\mathcal{Z}(\Lambda)$ does not depend on the volume form $\Omega$. Usually, $DH(\omega)$ is an infinite dimensional vector space even when $\omega$ is regular. For example, if $\omega=dx_{1}\wedge\ldots\wedge dx_{p}$, with $1\leq p\leq n-1$, then $\dim DH(\omega)=+\infty$ because when a multiform is disturbed it can lose some properties (e.g. the closeness). Here, for any function $f$, the $p$-form $f\omega$ is integrable but not necessarily closed so, does not belong to $\mathcal{B}(\omega)$. Nevertheless, we have the following proposition.

Proposition 1**.**

If $\displaystyle\Lambda=\partial/\partial x_{1}\wedge\ldots\wedge\partial/\partial x_{q}$ in $\mathbb{K}^{n}$, then $DH(\Lambda)=\{0\}$.

Proof.

Consider an infinitesimal deformation $\Lambda+\varepsilon\Pi$ of $\Lambda$. Let us first remark that if $\Pi=f\partial/\partial x_{1}\wedge\ldots\wedge\partial/\partial x_{q}$ for some function $f$ then, we have $\Pi=\mathcal{L}_{X}\Lambda$ where X=\big{(}\int fdx_{1}\big{)}\partial/\partial x_{1}.

Therefore, we can assume that in the deformation $\Lambda+\varepsilon\Pi$, the tensor $\Pi$ does not contain a term of type $f\Lambda$. Consider for $i=1,\ldots,q$ the Hamiltonian vector fields

[TABLE]

We have

[TABLE]

Since $\Lambda+\varepsilon\Pi$ is a Nambu tensor modulo $\varepsilon^{2}$ we have $X_{1}\wedge\ldots\wedge X_{q}=\Lambda+\varepsilon\Pi\,(\mathrm{mod}\,\varepsilon^{2})$ and $\displaystyle[X_{i},X_{j}]=0\,(\mathrm{mod}\,\varepsilon^{2})$.

This last relation then gives $\displaystyle\frac{\partial f_{k}^{(i)}}{\partial x_{j}}=\frac{\partial f_{k}^{(j)}}{\partial x_{i}}$ for any $i\neq j$ and $k>q$. Therefore, by Poincaré’s Lemma, there exist functions $F_{k}$ (for any $k>q$) such that $\displaystyle f_{k}^{(i)}=\frac{\partial F_{k}}{\partial x_{i}}$ for every $i=1,\ldots,q$ and $k>q$.

Now, we put $\displaystyle X=-\sum_{k=q+1}^{n}F_{k}\partial/\partial x_{k}$. We then have

[TABLE]

and the flow $\phi_{X}^{\varepsilon}$ sends $\Lambda$ to $\Lambda+\varepsilon\Pi\,(\mathrm{mod}\,\varepsilon^{2})$.

∎

3.1. Deformation cohomology of singular foliations

Let $\Lambda$ be a (germ of a) local (smooth, analytic, or formal) Nambu structure of degree $q$ near the origin in $\mathbb{K}^{n}$ , and $\mathcal{F}_{\Lambda}$ be the foliation generated by $\Lambda$. The set $\mathcal{B}(\mathcal{F}_{\Lambda})$ of trivial deformations of $\mathcal{F}_{\Lambda}$ consists of all (smooth, analyti, or formal) germs of $q$-vector fields $\Pi$ which can be written as

[TABLE]

where $X$ is local a vector field and $f$ is a local function neqr the origin in $\mathbb{K}^{n}$. The deformation cohomology $DH(\mathcal{F}_{\Lambda})$ of the foliation $\mathcal{F}_{\Lambda}$ is defined as follows:

[TABLE]

Similarly, if $\mathcal{F}_{\omega}$ is generated by a local integrable $p$-form $\omega$, then the set of trivial deformations $\mathcal{B}(\mathcal{F}_{\omega})$ consists of all $p$-forms of the type $\mathcal{L}_{X}\omega+f\omega$, where $X$ denotes a local vector field and $f$ denotes a local function. The deformation cohomology of $\mathcal{F}_{\omega}$ can then be defined as follows:

[TABLE]

The following lemma, whose proof is straightforward, says that the cohomology of singular foliations doesn’t depend on the choice of its associated Nambu structures or integrable forms:

Lemma 2**.**

Let $\Lambda$ be a (local or global) Nambu structure on a manifold. Suppose that $\Omega$ is a (local or global) volume form and $u$ is an invertible function on the manifold. If $\omega=i_{\Lambda}\Omega$ then

[TABLE]

3.2. Nambu structures of order 0 (functions)

Suppose that $\Lambda=f$ is a (smooth or analytic) function (i.e. a [math]-vector field) in a neighborhood of 0 in $\mathbb{K}^{n}$ ($\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$).

We denote by $\mathcal{O}_{n}$ the vector space of germs at 0 of (smooth or analytic) functions on $\mathbb{K}^{n}$ and $\mathfrak{X}(\mathbb{K}^{n})$ the vector space of germs at 0 of (smooth, analytic) vector fields.

Theorem 3**.**

With the notations above, we have

[TABLE]

*In particular, $\mathrm{dim}DH(f)=\mu(f)$ (the so-called Milnor number) and $\mathrm{dim}DH(\mathcal{F}_{f})=\tau(f)$ (the so-called Tjurina number).

Proof.

It is obvious. The set of infinitesimal deformations of $f$ is just $\mathcal{O}_{n}$. ∎

Note that in this case, if $\omega=fdx_{1}\wedge\ldots\wedge dx_{n}$ then $DH(\omega)$ is the quotient of $\Omega^{n}(\mathbb{K}^{n})$ by $\{d(f\theta)\,|\,\theta\in\Omega^{n-1}(\mathbb{K}^{n})\}$ (denoting by $\Omega^{k}(\mathbb{K}^{n})$ the vector space of $k$-differential forms) which is isomorphic to the quotient of $\mathcal{O}_{n}$ by $\{X(f)+(divX)f\,|\,X\in\mathfrak{X}(\mathbb{K}^{n})\}$. This deformation space has been computed in [10] (Theorem 3.14) when $f$ is a quasihomogeneous polynomial with an isolated singularity at 0 (its dimension is the Milnor number of $f$).

3.3. Top order multi-vector fields

Assume that

[TABLE]

where $f$ is either a smooth real function or a real or complex analytic function such that $f(0)=0$ and moreover [math] is a singular point of $f$, i.e. $df(0)=0$.

Theorem 4**.**

With the same notations as above, we have

[TABLE]

and $\displaystyle\dim H_{def}(\mathcal{F}_{\Lambda})=\tau(f)$ is the Tjurina number of $f$ at 0. Moreover,

[TABLE]

Proof.

The vector space of infinitesimal deformations of $\Lambda$ is the space of (germs of) $n$-vector fields of type $f\frac{\partial}{\partial x_{1}}\wedge\ldots\wedge\frac{\partial}{\partial x_{n}}$ with $g\in\mathcal{O}_{n}$ which is isomorphic to $\mathcal{O}_{n}$. If $X$ is a vector field, we have \displaystyle\mathcal{L}_{X}\Lambda=\big{(}X(f)-(divX)f\big{)}\frac{\partial}{\partial x_{1}}\wedge\ldots\wedge\frac{\partial}{\partial x_{n}} which gives the expression of $\mathcal{B}(\Lambda)$ and $\mathcal{B}(\mathcal{F}_{\Lambda})$. Finally, one easily checks that \displaystyle\big{\{}X(f)-(divX)f+gf\,|\,X\in\mathfrak{X}(\mathbb{K}^{n}),\,g\in\mathcal{O}_{n}\big{\}} is $\left\langle f,\frac{\partial f}{\partial_{x_{1}}},\ldots,\frac{\partial f}{\partial_{x_{n}}}\right\rangle$.

∎

One can find some computations of this cohomology space in the case where $f$ is a quasihomogeneous polynomial with an isolated singularity at 0 in [10] and [11]. More precisely, if $n=2$, it is the (germified) Poisson cohomology of the Poisson structure $\Lambda$ (see Theorems 4.9 and 4.11 in [11]). If $n\geq 3$, it is related to a Nambu cohomology space associated to $\Lambda$, denoted by $H_{f,n-2}^{n}(\mathbb{K}^{n})$ or $H_{\Lambda}^{2}(\mathbb{K}^{n})$ in [10] (Corollary 3.19). In these two cases, the dimension of the deformation cohomology space is finite and depends on the Milnor number of $f$.

3.4. Decomposable integrable forms with small singularities

In this section, we work on $\mathbb{C}^{n}$, in the complex analytic category. Suppose that $\Lambda$ is an analytic Nambu structure in a neighborhood of 0 in $\mathbb{C}^{n}$ and $\omega=i_{\Lambda}\Omega$, $\Omega$ is a volume form. If $\omega$ is decomposable (i.e. $\omega=\omega_{1}\wedge\ldots\wedge\omega_{p}$) and $\mathrm{codim}(\omega)\geq 3$ then by Malgrange (see [8]):

[TABLE]

where $u$ is a function with $u(0)\neq 0$. According to Lemma 14 we can assume that $u=1$.

Proposition 5**.**

Let $\omega=d{f_{1}}\wedge\ldots\wedge d{f_{p}}$ be a complex analytic integrable $p$-form and $\eta$ is an infinitesimal deformation $\omega$. If $\mathrm{codim}S(\omega)\geq p+2$ then

[TABLE]

It means that $\omega+\epsilon\eta$ is also decomposable and admits first integrals modulo $\epsilon^{2}$.

Proof.

By definition, $\eta$ satisfies for all $(p-1)$-vector field $A$ :

[TABLE]

We first claim that (22) is equivalent to

[TABLE]

Indeed, if $x\notin S(\omega)$ then $df_{1}(x),\ldots,df_{p}(x)$ are independent and if $E_{x}$ is the subspace of $(\mathbb{C}^{n})^{\ast}$ generated by the linear forms $df_{1}(x),\ldots,df_{p}(x)$, we consider constant vector fields $X_{1},\ldots,X_{p}$ such that $\langle df_{i}(x),X_{j}(x)\rangle=\delta_{ij}$ (Kronecker symbol) for all $i$ and $j$. We put $A_{i}=X_{1}\wedge\ldots\widehat{X_{i}}\ldots\wedge X_{p}$ and if (22) is satisfied, it gives $df_{i}(x)\wedge d\eta(x)=0$ for all $i=1,\ldots,p$. The converse is obvious.

Now, using successively the vanishing of the relative de Rham cohomology spaces $H^{p}(\Omega^{\ast}_{f_{1},\ldots,f_{k}})$ for $k=1,\ldots,p$ (see [8]) we get

[TABLE]

for some $(p-1)$-form $\theta$ and function $c$.

In the same way as above, we can show that (21) implies

[TABLE]

Consequently, in the decomposition (24), $d\theta$ satisfies this condition too. For every $i$, using successively the division theorem (see for instance Proposition 1.1 in [8]), one can show easily that

[TABLE]

where $\beta_{i}$ is a 1-form. Now, we get $\displaystyle df_{1}\wedge\ldots\wedge df_{p}\wedge d\beta_{i}=0$ so $\beta_{i}$ is a 1-cocyle in the relative de Rham cohomology $H^{1}(\Omega^{\ast}_{f_{1},\ldots,f_{p}})$, which gives $\beta_{i}=da_{i}+\sum_{j=1}^{p}b_{ij}df_{j}$ for some functions $a_{i}$ and $b_{ij}$. Therefore,

[TABLE]

It gives

[TABLE]

for every $i=1,\ldots,p$, which implies, by the division theorem,

[TABLE]

for some function $b$. The proposition follows. ∎

If, $\omega=df_{1}\wedge\ldots\wedge df_{p}$, we consider $F$ the analytic map from $\mathbb{C}^{n}$ to $\mathbb{C}^{p}$ defined by $F=(f_{1},\ldots,f_{p})$. If $X\in\mathfrak{X}(\mathbb{C}^{n})$ and $H=(H_{1},\ldots,H_{p})$ is an analytic map from $\mathbb{C}^{p}$ to $\mathbb{C}^{p}$, we denote $X.F=(X.f_{1},\ldots,X.f_{p})$ and $H(F)=(H_{1}(f_{1},\ldots,f_{p}),\ldots,H_{p}(f_{1},\ldots,f_{p}))$. Now, we put

[TABLE]

Recall that $\mathcal{Q}_{RL}(F)$ measures the stability of the germ $F$ and the versal deformations of $F$ with respect to the Right-Left-equivalence (see for instance [3]). More precisely, another germ of analytic map $G$ is RL-equivalent to $F$ if there exists a germ of analytic diffeomorphism $\phi$ of $(\mathbb{C}^{n},0)$ and a germ of analytic diffeomorphism $\psi$ of $(\mathbb{C}^{p},0)$ such that $G=\psi\circ F\circ\phi$.

Theorem 6**.**

If $\omega=df_{1}\wedge\ldots\wedge df_{p}$ with $\mathrm{codim}S(\omega)\geq p+2$, then, with the notations above, we have

[TABLE]

Proof.

We denote

[TABLE]

By Proposition 5, we have

[TABLE]

Clearly, $C$ is included in $A$. If $bdf_{1}\wedge\ldots\wedge df_{p}\in B$ is in $A$, i.e. of the form $\sum_{i=1}^{p}df_{1}\wedge\ldots\wedge df_{i-1}\wedge da_{i}\wedge df_{i+1}\wedge\ldots\wedge df_{p}$ then, we have $db\wedge df_{1}\wedge\ldots\wedge df_{p}=0$ which gives $b=H(f_{1},\ldots,f_{p})$ where $H\in\mathcal{O}_{p}$ (see [8], Theorem 2.1.1). Conversely, if $b=H(f_{1},\ldots,f_{p})$ with $H\in\mathcal{O}_{p}$ then $bdf_{1}\wedge\ldots\wedge df_{p}$ is in $A\cap B$.

Therefore, we have

[TABLE]

where D=\big{\{}H(f_{1},\ldots,f_{p})df_{1}\wedge\ldots\wedge df_{p}\,|\,H\in\mathcal{O}_{p}\big{\}}.

Now, we define $\Phi:(\mathcal{O}_{n})^{p}\longrightarrow DH(\mathcal{F}_{\omega})$ such that if $G=(g_{1},\ldots,g_{p})$ we have

[TABLE]

It is a surjective linear map. It is clear that $\mathcal{I}_{RL}(F)$ is included in the kernel of $\Phi$. Moreover, if $\Phi(G)=0$ then there exist a vector field $X$ on $\mathbb{C}^{n}$ and an analytic map $K$ from $\mathbb{C}^{p}$ to $\mathbb{C}^{p}$ such that

[TABLE]

It implies that for every $i=1,\ldots,p$, we have $df_{1}\wedge\ldots\wedge df_{p}\wedge(dg_{i}-X.f_{i})=0$ which gives (see [8]) that $g_{i}=X.f_{i}+H_{i}(f_{1},\ldots,f_{p})$ for some $H_{i}\in\mathcal{O}_{p}$. Therefore, $G\in\mathcal{I}_{RL}(F)$. ∎

Corollary 7**.**

If $\omega=df$ (Nambu structure of order $n-1$) and $\mathrm{codim}S(df)\geq 3$ then

[TABLE]

In particular, $\mu(f)\geq\mathrm{dim}DH(\mathcal{F}_{df})\geq\tau(f)-1$.

Remark 1**.**

We consider the two ideals of $\mathcal{O}_{n}$, $I_{f}=\left\langle\frac{\partial f}{\partial_{x_{1}}},\ldots,\frac{\partial f}{\partial_{x_{n}}}\right\rangle$ and $J_{f}=\left\langle f,\frac{\partial f}{\partial_{x_{1}}},\ldots,\frac{\partial f}{\partial_{x_{n}}}\right\rangle$, If we assume that $df(0)=0$ then $X.f$ is not a constant for any vector field $X$ so, we have $I_{f}\oplus\mathbb{C}\subset\mathcal{I}_{RL}(f)\subset J_{f}\oplus\mathbb{C}$ which gives $\mu(f)-1\geq\mathrm{dim}DH(\mathcal{F}_{df})\geq\tau(f)-1$. Consequently, if 0 is an isolated singularity of $f$, i.e. $\mu(f)<\infty$, then $\mathrm{dim}DH(\mathcal{F}_{df})<\infty$. If moreover, $f$ is a quasihomogeneous polynomial, then $I_{f}=J_{f}$ which gives $\mathrm{dim}DH(\mathcal{F}_{df})=\mu(f)-1=\tau(f)-1$.

Corollary 8**.**

If [math] is an isolated singularity of $\omega=udf_{1}\wedge\ldots\wedge df_{p}$ then $\mathrm{dim}DH(\mathcal{F}_{\omega})<\infty$

Proof.

We prove that, denoting $F=(f_{1},\ldots,f_{p})$, the quotient \displaystyle\frac{(\mathcal{O}_{n})^{p}}{\big{\{}X.F\,|\,X\in\mathfrak{X}(\mathbb{C}^{n})\big{\}}} has a finite dimension. If we denote by $\mathfrak{M}$ the ideal of $\mathcal{O}_{n}$ formed by the functions vanishing at 0, we prove that there is a positive integer $N$ such that if $g_{i}\in\mathfrak{M}^{N}$, $i=1,\ldots,p$, then there exists a vector field $X$ such that $X.f_{i}=g_{i}$, $i=1,\ldots,p$. The corollary will follow naturally.

We prove it by induction on $p$. The case $p=1$ is a direct consequence of Hilbert’s Nullstellensatz Theorem. If the statement is true for $p-1$, we prove the existence of an integer $N$ such that for all $g\in\mathfrak{M}$, there is a vector field $X$ which satisfies $X.f_{p}=g$ and $X.f_{i}=0$, $i=1,\ldots,p-1$.

We consider the ideal $\mathcal{I}$ of $\mathcal{O}_{n}$ formed by functions $g\in\mathfrak{M}$ such that there exists a vector field $X$ satisfying $X.f_{p}=g$ and $X.f_{i}=0$, $i=1,\ldots,p-1$.

For $1\leq i_{1}<\ldots<i_{p}\leq n$, denoting

[TABLE]

we have $X.f_{i}=0$, $i=1,\ldots,p-1$ and X.f_{p}=det\Big{(}\frac{\partial(f_{1},\ldots,f_{p})}{\partial(x_{i_{1}},\ldots,x_{i_{p}})}\Big{)}. Therefore, the function \displaystyle det\Big{(}\frac{\partial(f_{1},\ldots,f_{p})}{\partial(x_{i_{1}},\ldots,x_{i_{p}})}\Big{)} is in $\mathcal{I}$.

By the hypothesis, the zero locus of $\mathcal{I}$ is $\{0\}$ and it finishes the proof, using Hilbert’s Nullstellensatz Theorem. ∎

3.5. Vector fields and linear Nambu structures

If $\Lambda=X$ is a vector field, the leaves of the associated foliation are integral curves of $X$. The normalization of this foliation is the same as the orbital normalization of $X$.

In the case $\Lambda=X$ is a formal vector field in $(\mathbb{K}^{n},O)$ whose linear part $X^{(1)}$ is non-trivial, we denote by $DH_{lin}(X^{(1)})$ the quotient of the vector space of linear infinitesimal deformations of $X^{(1)}$ by the vector space of linear trivial deformations of $X^{(1)}$.

Then by the classical Poncaré-Dulac formal normalization theory we have the following proposition.

Proposition 9**.**

Suppose that $X^{(1)}$ is non-resonant, then we have :

• (i)

$DH(X)=DH_{lin}(X^{(1)})$* and $\dim DH(\mathcal{F}_{X})=\dim DH_{lin}(X^{(1)})-1$.*

• (ii)

$\dim DH_{lin}(X^{(1)})=n^{2}-d\geq n$* where $d$ is the dimension of the adjoint orbit of $X^{(1)}$ in $\mathfrak{gl}_{n}(\mathbb{K})$.*

• (iii)

In the generic case (i.e. the eigenvalues of $X^{(1)}$ are distincts) we have $\dim DH_{lin}(X^{(1)})=\leavevmode\nobreak\ n$.

Proof.

By hypothesis, the vector field $X$ is formally linearizable, we then can assume that it coincides with its linear part $X^{(1)}$. Moreover, if $X^{(1)}+\varepsilon Y$ is an infinitesimal deformation of $X^{(1)}$ then $Y$ may be written as $Y=Y^{(1)}+\widetilde{Y}$ where $\widetilde{Y}$ contains only terms of degree larger or equal to 2. Since $X^{(1)}$ is non-resonant, we have $\widetilde{Y}=[X^{(1)},\widetilde{Z}]$ for some formal vector field $\widetilde{Z}$ which contains only terms of degree larger or equal to 2. Finally, $DH(X)$ is the quotient of the vector space of linear vector fields by the vector space of vector fields of type $[X^{(1)},Z]$ where $Z$ is a linear vector field (whose dimension is the dimension of the adjoint orbit of $X^{(1)}$). In the same way, $DH(\mathcal{F}_{X})$ is the quotient of the vector space of linear vector fields by the vector space of vector fields of type $[X^{(1)},Z]+\lambda X^{(1)}$ where $Z$ is a linear vector field and $\lambda\in\mathbb{K}$.

Finally, recall that the dimension of the adjoint orbit of $X^{(1)}$ is less than $n(n-1)$ and if the eigenvalues of $X^{(1)}$ are distincts, it is exactly $n(n-1)$. ∎

Note that the point (iii) of this proposition can be true even if $X^{(1)}$ has eigenvalues of multiplicity strictly larger than 1. It is the case if in the Jordan decomposition of $X^{(1)}$, there is only one Jordan block corresponding to each eigenvalue.

In the resonant case, the formal deformation cohomology can be infinite-dimensional.

Let us now recall that there are two types of linear Nambu structures:

Type 1: $\Lambda$ is dual to a decomposable linear integrable differential form $\omega=dx_{1}\wedge\dots\wedge dx_{p-1}\wedge dQ,$ where $Q$ is a quadratic function.

Type 2: $\Lambda$ is decomposable: $\Lambda=\partial/\partial x_{1}\wedge...\wedge\partial/\partial x_{q-1}\wedge(\sum_{i,j=q}^{n}b^{i}_{j}x_{i}\partial/\partial x_{j}).$

It has been shown in [4, 16] that linear Nambu structures of Type 1 with a nondegenerate quadratic function $Q$ in its formula are formally and analytically rigid (and they are also smoothly rigid if $Q$ satisfies a natural condition on its signature). In fact, the proofs in these papers also dealt with deformation cohomology, so we can conclude that the formal and analytic deformation cohomology of a linear Nambu structure of Type 1 is trivial if the quadratic function $Q$ in its formula is nondegenerate. If, moreover, the signature of $Q$ is different from $(1,*)$ then the local smooth deformation cohomology is also trivial.

As regards linear Nambu structure of Type 2, the situation is similar to that of linear vector fields $X=\sum_{i,j=q}^{n}b^{i}_{j}x_{i}\partial/\partial x_{j}$ in the formula. In particular, if $X$ is non-resonant then $\Lambda=\partial/\partial x_{1}\wedge...\wedge\partial/\partial x_{q-1}\wedge X$ has trivial formal deformation cohomology. (See [4, 16] for the details.)