Maurer-Cartan equation in the DGLA of graded derivations

Let M be a smooth manifold and $\Phi$ a differential 1-form on M with values in the tangent bundle TM. We construct canonical solutions $e_\Phi$ of Maurer-Cartan equation in the DGLA of graded derivations D*(M) of differential forms on M by means of deformations of d depending on $\Phi$. This yields to a classification of the canonical solutions of the Maurer-Cartan equation according to their type: $e_\Phi$ is of finite type r if there exists $r\in N$ such that $\Phi^r[\Phi,\Phi]_{FN} = 0$ and r is minimal with this property, where $[.,.]_{FN}$ is the Fr\"olicher-Nijenhuis bracket. A distribution $\xi\subset TM$ of codimension k>1 is integrable if and only if the canonical solution $e_\Phi$ associated to the endomorphism $\Phi$ of TM which is trivial on $\xi$ and equal to the identity on a complement of $\xi$ in TM is of finite type $\leq 1$, respectively of finite type 0 if k = 1.


Introduction
In [9], one of the last papers of their seminal cycle of works on deformations of differentiable and complex structures, K. Kodaira  They defined a DGLA structure (D * (M ) , , [·, ·]) on the graded algebra of graded derivations introduced by Frölicher and Nijenhuis in [5] and the deformations of the multifoliate structures are related to the solutions of the Maurer-Cartan equation in this algebra. This was done in the spirit of [11], where A. Nijenhuis and R. W. Richardson adapted a theory initiated by M. Gerstenhaber [6] and proved the connection between the deformations of complex analytic structures and the theory of differential graded Lie algebras (DGLA).
In the paper [1], the authors elaborated a theory of deformations of integrable distributions of codimension 1 in smooth manifolds. Our approach was different of K. Kodaira and D. C. Spencer's in [9] (see remark 14 of [1] for a discussion). We considered in [1] only deformations of codimension 1 foliations, the DGLA algebra (Z * (L) , δ, {·, ·}) associated to a codimension 1 foliation on a co-oriented manifold L being a subalgebra of the the algebra (Λ * (L) , δ, {·, ·}) of differential forms on L. Its definition depends on the choice of a DGLA defining couple (γ, X), where γ is a 1-differential form on L and X is a vector field on L such that γ (X) = 1, but the cohomology classes of the underlying differential vector space structure do not depend on its choice. The deformations are given by forms in Z 1 (L) verifying the Maurer-Cartan equation and the moduli space takes in account the diffeomorphic deformations. The infinitesimal deformations along curves are subsets of of the first cohomology group of the DGLA (Z * (L) , δ, {·, ·}).
This theory was adapted to the study of the deformations of Levi-flat hypersurfaces in complex manifolds: we parametrized the Levi-flat hypersurfaces near a Levi-flat hypersurface in a complex manifold and we obtained a second order elliptic partial differential equation for an infinitesimal Levi-flat deformation.
In this paper we consider the graded algebra of graded derivations defined by Frölicher and Nijenhuis in [5] with the DGLA structure defined by K. Kodaira and D. C. Spencer in [9]. We construct canonical solutions of the Maurer-Cartan equation in this algebra by means of deformations of the d-operator depending on a vector valued differential 1-form Φ and we give a classification of these solutions depending on their type. A canonical solution of the Maurer-Cartan equation associated to an endomorphism Φ is of finite type r if there exists r ∈ N such that Φ r [Φ, Φ] F N = 0 and r is minimal with this property, where [·, ·] F N is the Frölicher-Nijenhuis bracket. We show that a distribution ξ of codimension k on a smooth manifold is integrable if and only if the canonical solution of the Maurer-Cartan equation associated to the endomorphism of the tangent space which is the trivial extension of the k-identity on a complement of ξ in T M is of finite type 1. If ξ is a distribution of dimension s such that there exists an integrable distribution ξ * of dimension d generated by ξ, we show that there exists locally an endomorphism Φ associated to ξ such that the canonical solution of the Maurer-Cartan equation associated to Φ is of finite type less than r = min m ∈ N : m d s . In the case of integrable distributions of codimension 1, we study also the infinitesimal deformations of the canonical solutions of the Maurer-Cartan equation in the algebra of graded derivations by means of the theory of deformations developped in [1].

The DGLA of graded derivations
In this paragraph we recall some definitions and properties of the DGLA of graded derivations from [5], [9] (see also [10]). Notation 1. Let M be a smooth manifold. We denote by Λ * M the algebra of differential forms on M , by X (M ) the Lie algebra of vector fields on M and by Λ * M ⊗ T M the algebra of T M -valued differential form on M , where T M is the tangent bundle to M . In the sequel, we will identify Λ 1 M ⊗ T M with the algebra End (T M ) of endomorphisms of T M by their canonical isomorphism: for σ ∈ i∈N is a family of C-vector spaces and d : V * → V * is a graded homomorphism such that d 2 = 0. An element a ∈ V k is said to be homogeneous of degree k = deg a.
2) [·, ·] : V * × V * → V * defines a structure of graded Lie algebra i.e. for homogeneous elements we have and 3) d is compatible with the graded Lie algebra structure i.e.
Definition 2. Let (V * , d, [·, ·]) be a DGLA and a ∈ V 1 . We say that a verifies the Definition 6. Let α ∈ Λ * M and X ∈ X (M ). We define L α⊗X , I α⊗X by where L X is the Lie derivative and ι X the contraction by X.

By linearity, for every
In [5] the graded derivations of L (M ) (respectively of I (M )) are called of type d * (respectively of type ι * ).

Lemma 3.
( We denote L Φ = L (D) and    By Lemma 1 and the Jacobi identity, for every This gives the following In particular In particular

Canonical solutions of Maurer-Cartan equation
Then Proof. It is sufficient to prove the assertion for Ψ = α ⊗ X, Since The following Theorem is a refinement of results from [3] and [4] : Proof. Since both terms of (3.1) are derivations of degree 1, it is enough to prove (3.1) on Λ 0 (M ) and Λ 1 (M ). Let f ∈ Λ 0 (M ) and X ∈ X (M ). Then and by linearity we obtain Since I Φ is of type i * , I Φ f = 0 and therefore (3.1) is verified for every f ∈ Λ 0 (M ).
We will prove firstly that By using Remark 3, we have By Remark 1 it follows that , by comparing (3.5), (3.6) and (3.7) it follows that (3.1) is verified for each form in Λ 1 M and the Lemma is proved.
It follows that Since L is injective, this implies By Lemma 6 we obtain which is equivalent to .
called the canonical solution of Maurer-Cartan equation associated to Φ.

Canonical solutions of finite type of Maurer-Cartan equation
Proof. We remark that for r 2, γ r ∈ I (M ), so by 2.5 it follows that γ p , γ q ∈ I (M ) for p, q 2. Since ℵ I(M) = 0 we have ℵ γ p , γ q = 0 for p, q 2 and so We will show by induction that for every r 2 Suppose that for every r 3 and by 2.6 we have (4.4)

So, from (4.3) and (4.4) we obtain
But by Lemma 6 It follows that By Theorem 3.1 γ k and the Proposition is proved.    Proof. Let Y, Z ∈ ξ. Since Φ k = Φ for every k 1 and ΦY = ΦZ = 0, Suppose that the canonical solution associated to Φ is of finite type 1. Then Φ [Φ, Φ] F N (Y, Z) = 0 for every Y, Z ∈ ξ and by (4.6 it follows that [Y, Z] ∈ ξ. Therefore ξ is integrable by the theorem of Frobenius. Suppose now that ξ is not integrable. There exist Y, Z ∈ ξ such that [Y, Z] / ∈ ξ. By (4.6) we obtain for every k 1.
Conversely, suppose that ξ is integrable. Then for every V, W ∈ ξ, we have and it follows that the canonical solution associated to Φ is of finite type 1. If so the canonical solution associated to Φ is of finite type 0.
If ξ is integrable and ζ is not integrable, there exists Y, Z ∈ ζ such that [Y, Z] / ∈ ζ, so  Proof. We apply Proposition 4 for η = R [X] which is obviously integrable.
Theorem 5. Let M be a smooth manifold of dimension n and ξ, τ ⊂ T M distributions such that ξ τ . We consider η, ζ ⊂ T M distributions such that τ = ξ ⊕ η and T M = τ ⊕ ζ and let A : η → ξ , B : η → η such that ξ = ker K, where K : τ → τ is defined by K = 0 on ξ and K = A + B on η. We suppose that there exists a natural number m 1 such that K m = 0. Let Φ ∈ End (T M ) defined by Φ = K on τ and Φ = Id on ζ. The following are equivalent: (1) τ is integrable.
(2) The canonical solution associated to Φ is of finite type m.
In order to compute the type of the canonical solution of Theorem 5 we need the following elementary lemma: Then r = min m ∈ N : m d s . Proof. Since K is nilpotent of maximal rank we may suppose that By induction it follows that if d − js > 0, we have and K j = 0 for each j ∈ N * such that d − js 0.
x is independent of x, ξ * is a distribution, but in general dim ξ * x depends on x. If ξ * is a distribution, then ξ * is the smallest integrable distribution containing ξ [12]. Suppose that ξ is not integrable, i. e. d > s. For each x ∈ M there exists a neighborhood U of x and a basis (X 1 , · · ·, X n ) of T M on U such that (X 1 , · · ·, X s ) is a basis of ξ and (X 1 , · · ·, X d ) is a basis of ξ * on U .

Remark 6. In [2]
it is proved that the deformation theory in the DGLA (D * (M ) , , [·, ·]) is not obstructed but it is level-wise obstructed.
Remark 7. An integrable distribution ξ of codimension 1 in a smooth manifold L is called co-orientable if the normal space to the foliation defined by ξ is orientable. We recall that ξ is co-orientable if and only if there exists a 1-form γ on L such that ξ = ker γ (see for ex. [7]). A couple (γ, X) where γ ∈ ∧ 1 (L) and X is a vector field on L such that ker γ = ξ and γ (X) = 1 was called a DGLA defining couple in [1].
If (ξ t ) t∈I is a differentiable family of deformations of an integrable co-orientable distribution ξ, then the distribution ξ t is co-orientable for t small enough. So, if ξ is an integrable co-orientable distribution of codimension 1 in L and (ξ t ) t∈I is a differentiable family of deformations of ξ we may consider a DGLA defining couple (γ t , X t ) for every t small enough such that t → (γ t , X t ) is differentiable on a neighborhood of the origin.
Lemma 9. Let L be a C ∞ manifold and ξ ⊂ T (L) a co-orientable distribution of codimension 1. Let (γ, X) be a DGLA defining couple and denote Φ ∈ End (T M ) the endomorphism corresponding to γ ⊗ X ∈ Λ 1 M ⊗ T M . Then Φ is defined on Proof. Let Y = Y ξ + λX vector fields on L, V ξ ∈ ξ, λ ∈ R. Then Lemma 10. Let L be a C ∞ manifold and ξ ⊂ T (L) a co-orientable distribution of codimension 1. Let (γ, X) be a DGLA defining couple. Then the following are equivalent: Proof. i) ⇐⇒ ii) is a variant of the theorem of Frobenius and it was proved in [1]. ii) ⇐⇒ iii). We have We recall the following lemma from [1]: Lemma 11. Let L be a C ∞ manifold and X a vector field on L. For α, β ∈ Λ * (L), set where L X is the Lie derivative. Then (Λ * (L) , d, {·, ·}) is a DGLA.
In particular δα (V, W ) = 0 for every vector fields V, W tangent to ξ.

Remark 8.
A smooth hypersurface in a complex manifold is Levi flat if it admits a foliation of codimension 1 by complex manifolds. In [1] the authors studied the deformations of Levi flat hypersurfaces and obtained a second order elliptic differential equation for the infinitesimal deformations, which was used to prove the non existence of of transversally parallelizable Levi flat hypersurfaces in the complex projective plane. In [8] it is proved that the results of this paragraph lead to the same second order elliptic differential equation for the infinitesimal deformations of Levi flat hypersurfaces.