Logarithmic Differential Operators and Prequantization of Logsymplectic Poisson Structure

In this paper, we study the prequantization condition of logsymplectic structure using integrality of such structure on the complement of associated divisor D.


Introduction
The first problem of geometric quantization concerns the kinematic relationship between the classical and quantum domains.At the quantum level, the states of physical system are represented by the rays in Hilbert space H, and the observable by a collection O of symmetric operators on H, while in the limiting classical description, the state space is a symplectic manifold or more generally Poisson manifold (X, ω) and the observable are algebra C ∞ (X) of smooth functions on X.The kinematic problem is: given X and ω, is it possible to reconstruct H and O? According to Dirac's general principles, the canonical transformations of X generated by the classical observable should correspond to the unitary transformations of H generated by the quantum observable, and Poisson brackets of classical observable should correspond to commutators of quantum observable.Each classical observable f : X → R should correspond to an operator φ( f ) ∈ O such that 1. the map f → φ( f ) is linear over R 2. if f is constant, then φ( f ) is the corresponding multiplication operator Therefore, the Poisson bracket is the classical analogue of the quantum commutator.Linear map φ satisfying (1 → 2) is called prequantization formula .When the prequantization formula exists, C ∞ (X) acts faithfully on the Hilbert space H.In the case where classical phase space if represent by symplectic manifold, J. Soureau and B. Kostant; independently presented necessary and sufficient conditions on which symplectic manifold (X, ω) shall be prequantizable.Their proved that (X, ω) is prequantizable if and only if the De Rham cohomolgy class of ω is element of the image of homology map induced by the inclusion of Z in R.More generally, I. Vaisman introduced the notion of contravariant derivative, Poisson Chern class and used it to generalize the Kostant-Soureau integral theorem.They prove that the obstruction to prequantization of symplectic manifold is measured by De Rham cohomology while Poisson cohomolgy measures the obstruction to the prequantization of Poisson manifold.
Between symplectic manifold and Poisson manifold, there exist the logarithmic manifold.The notion of logsymplectic structure taking their origin from particular meromorphic forms having at most simple poles along certain divisor D of a giving complex manifold X.Such forms are amply studied in (Saito, K. (1980)).Using the notion Lie-Rinehart algebra, we give the algebraic generalization of such notion.
The aim of this work is to study integrality of logsymplectic structure.Of course, we first use the notion of Lie-Rinehart to introduce the algebraic versus of logsymplectic structure.Secondly, we use the notion of logarithmic differential operator to define Dirac principle of prequantization of such structures.Since each 2n dimensional logsymplectic manifold has 2n − 2 dimensional leaves, we study the impact of the integrality of such leaves on the all manifold.
The main result of this paper is as follow: Theorem 2.4 If the divisor D is free and irreducible hypersurface of X and the sheaf of logarithmic 1-form is generated by closed forms, then for all logarithmic 2-form ω, the following conditions are equivalent The structure of the paper is as follows: 1. [Section 1] It is devoted to algebraic formalism of the notion of logsymplectic structure.Using the notion of Lie-Rinehart algebra, we define the notion of logsymplectic structures and we prove in Proposition 2.11 that such structures induce two Lie-structure on the base algebra.we also prove that Poisson structure associated to logsymplectic structure are logarithmic Poisson structure.We give two examples of logsymplectic structures and one example of non logsymplectic structure.

[Section 2]
In this section, we recall the notion of logsymplectic manifold and we prove the main theorem of the note.
3. [Section 3] This section is devoted to the interpretation of Dirac principle for logsymplectic manifold.Of cause, we introduce the notion of logarithmic differential operator and we prove in Proposition 4.14 that the module of logarithmic differential operator is a Lie-Rinehart algebra.We also prove that the latter is central extension of logarithmic vector fields along the sheaf of holomorphic functions of complex manifold with free divisor.

[Section 4]
We use extension of sheaf to study integrality condition of logsymplectic algebra.

[Section 5]
In this section, we introduce Lie-algebroide formalism to study integrality of logsymplectic structures.

Lie-Rinehart-logsymplectic Algebras and Associated Lie Brackets
Throughout this section, k denotes a field of characteristic 0 and A a commutative k-algebra with unity 1.
Let Der A be the A-module of k-derivations of A and Ω A the A-module of formal differentials of A. It is proved in (Huebschmann, J. (2013)) that Ω A is generated by {da, a ∈ A} together with the relations where d is the canonical derivation associated to Ω A .An element D of Der A is said to be logarithmic along an ideal I of A if D(I) ⊂ I.We denote by Der A (log I) the set of derivations of A, logarithmic along I.By definition, Der A (log I) is a sub-Lie algebra of Der A .Let S = {u 1 , ..., u p } be a subset of p nonzero and nonunit elements of A. An element D of Der A is said to be logarithmic principal along the ideal generated by S if for all u i ∈ S , D(u i ) is an element of the ideal u i A generated by u i .We denote by Der A (log I) the set of derivations of A which are logarithmic principals along the ideal I of A generated by S .Der A (log I) is a sub-Lie algebra of Der A (log I).We denote Ω A (log I) the A-module generated by We have the following: Lemma 2.1.Let I be the ideal A generated by S = {u 1 , ...; u p , u i ∈ A; 1 ≤ i ≤ p}.The A-module Der A (log I) of derivations of A logarithmic principal along I is the dual over A of Ω A (log I).
We recall that a Lie-Rinehart algebra over A is a pair (L, ρ); where L is Lie algebra which is also an A-module and ρ : L → Der k (A) is Lie algebras homomorphism which satisfies the following equality.
Remark 2.2.The notion of Lie-Rinehart algebra is particular case of a more general notion called P-Lie-Rinehart algebra; where P is an A-algebra.More explicitly, a P-Lie-Rinehart algebra is a pair (L, ρ) formed by a P-module L and A-linear map ρ : L −→ Diff 1 (P) which is also a Lie-algebras homomorphism satisfies (1) for all a ∈ P and l 1 , l 2 ∈ L.
Where Diff 1 (P) denote the A-module of first differentials operators of P. In the next section, we will recall the definition and give its logarithmic counterpart.
It follows from this definition that Der A endowed with identity is Lie-Rinehart algebra.On the other hand, we can easily prove that the inclusion map of Der A (log I) in Der A is a structure of Lie-Rinehart algebra on Der A (log I).
Let (L, ρ) be a Lie-Rinehart algebra over A. By definition, the associated structure ρ is also a representation of the Amodule L; by derivations of A. For all q ∈ N, a q-linear alternating mapping of L into A is called a q-dimensional cochain.
Definition 2.4.The cohomology of the complex (Lalt q (L, A), The following is easy to prove.Definition 2.7.see (Huebschmann, J. ( 2013) such that for all x 1 , ..., x q−1 ∈ L; and for all f ∈ Lalt q (L, A) we have Remark 2.8.If X is a smooth manifold then (i) Every Poisson structure on X is a Lie-Rinehart-Poisson structure on Ω X with Lie-Rinehart structure the Hamiltonian map.
(ii) Every symplectic structure on X is a Lie-Rinehart-Poisson-symplectic on the O X -module of smooth vector fields on X It follows from the definition that Der A (log I) is a Lie-Rinehart algebra; with inclusion as structure.
In particular, when A is the algebra of holomorphic functions on a 2n dimensional complex manifold X, a logsymplectic structure on A is algebraic analogous logsymplectic form on X (see (Goto, R. (2002))).According to above the definition, logsymplectic structure are 2-forms on Der A (log I) which are closed under the De Rham differential.Since Der A (log I) is sub-module of Der A its algebraic dual over A is bigger than Der A one's; which is not well defined since, in general the bi-dual of Ω A is not Ω A .In (Saito, K. (1980)), the author proves that the sheaf of logarithmic forms and the sheaf of logarithmic vector field on final dimensional complex manifold are dual each other.
Let µ be a logsymplectic structure.Since µ is non degenerated, its contraction by logarithmic derivation induce an isomorphism of A-modules between Der k (log I) and its dual Der k (log I) * which is the module of logarithmic forms (see (Saito, K. (1980))).Therefore, for all a ∈ A, there is an unique δ a ∈ Der k (log I) such that: a is called logarithmic Hamiltonian element and δ a is called logarithmic Hamiltonian field.For all a, b ∈ A, consider: We have the following proposition.
Proposition 2.10.(Dongho, J.(2012) Let µ be a logsymplectic structure on A. The following bracket is a well defined logarithmic Poisson structure on A In the sequel, we suppose that I is generated by S = {u 1 , ..., u q } and µ, is a logsymplectic structure.Then for all u i ∈ S there exist a unique δu i ∈ Der k (log I) such that But since du i ∈ Ω A ⊂ Ω A (log I), there exist δ u i such that i δ u i µ = du i .It is easy to prove that δ u i = u i δu i .We can then consider the following bracket: By direct computation, we have the following proposition.
Proposition 2.11.A Logsymplectic structure µ on A induces two Lie structures {−, −} and {−, −} sing such that for all u, v ∈ I − 0, Example 2.12.(Dongho, J.( 2012) y] the algebra of two variables polynomials and I = xA, J = yA, K = x 2 A the ideal generated respectively by x, y and x 2 .We have: and Where < U > B denotes the B-module generated by U for all U ⊂ L and B ⊂ A. The following 2-forms ) and its determinant is x 2 which is not an inversible element of A. Therefore, ω 3 is a Lie-Rinehart-Poisson structure which is not a logsymplectic structure.

Logsymplectic Manifold
In this section we propose a geometric application of the concepts introduced in the above section.As we can see, the geometric analogous of Lie-Rinehart-logsymplectic structure correspond to logsymplectic structure.Throughout this section, X a a final dimensional compact complex manifold with reduced divisor D. To define what is logsymplectic manifold, we need the notion of logarithmic forms which are extensively studied in algebraic geometry; see (Saito, K. (1980)).In addition to the assumptions made in (Saito, K. (1980)), we assume that D is square free.
Let ω be a meromorphic q-form on X with poles only in D. We suppose that D := {z ∈ X, h(z) = 0} where h is some holomorphic map.
ω is said to be logarithmic along D if hω and dh∧ω are holomorphic forms.As in (Saito, K. ( 1980)), we denote Ω q X (log D) the sheaf of logarithmic q-forms on X. dx ∧ dy x 2 .Therefore, according to Theorem 1.8 of (Saito, K. (1980)), the system { dx x 2 , dy} shall be a bases of Ω X (log D).But on the other hand, { dx x , dy} is free bases of Ω X (log D).Then there exist two holomorphic functions a and b such that dx x 2 = a dx x + bdy; but this implies that ax = 1 and b = 0. Then a = 1 x .Which is contradictory to our assumptions.So dx x 2 is not logarithmic 1-form.
It follows from this remark that the assumption that D shall be square free is necessary.
One important notion related to logarithmic forms is associated residue form.According to Theorem 1.1 of (Saito, K. (1980)), if its is a n−dimensional complex manifold and ω is a logarithmic q-form, then there exist an holomorphic function g such that Where ψ and η are holomorphic form.
Thus, the logarithmic forms may have poles outside of D. One form in such a way that residues of all q-form is holomorphic on X if h is irreducible and the residues of all element of Ω X (log D) is holomorphic.
In such cases, global sections of Ω 2 X (log D) are in the form dh h ∧ ψ + η where ψ and η are holomorphic forms.
Let us denote H * (X, C) the De Rham cohomology of X.According to De Rham Theorem, it is isomorphic to the cohomology of the complex of holomorphic forms of X.Since the latter is sub-complex of the complex of logarithmic forms, there is an homomorphism p : is the cohomology group of the logarithmic De Rham complex.So we have the following sequences With those tools, we can prove the following result.
Theorem 3.4.If the divisor D is free and irreducible hypersurface of X and the sheaf of logarithmic 1-form is generated by closed forms, then for all logarithmic 2-form ω, the following conditions are equivalent conversely, if ω 0 + dλ = η and ψ = dβ with ω 0 integral, then Let us denote by D sing the singular part of D and D red the smooth part.The proof of the following is essentially the same as the proof of Darboux Theorem in symplectic geometry.
Lemma 3.5.(Goto, R. ( 2002)) (Log Darboux Theorem).Let (X, D) be a log symplectic manifold with a logarithmic symplectic form ω, where D is reduced divisor.There exist holomorphic coordinates (z 0 , z 1 , ..., z 2n−1 ) of a neighborhood of each smooth point of D red such that ω is given by where {z 0 = 0} = D. we refer to these coordinates as log Darboux coordinates.
It follow from this result that the residue 1-form of a logarithmic 2-form ω if dz 1 in the log Darboux coordinates.We have then an integrable distribution by setting {δ ∈ T D red , Re(ω)(δ) = 0}.Therefore, we have 2n − 2-dimensional leaves on D red .
More explicitly, we have the following result.
It follows from this lemma that in a complement of 2-dimensional sub-manifold of X, logsymplectic forms are symplectic.

Logarithmic Lie-Rinehart Differential Operators
In this paragraph, (X, ω, D) is logsymplectic manifold and E a locally free O X -module of rank 1 and D = {h = 0} a divisor of X.

Logarithmic Connection
The notion of logarithmic connection is original in the work of P. Deligne when he formulated and proved the theorem establishing a Riemann-Hilbert correspondence between monodromy groups and Fuchsian systems of integrable partial equations or flat connections on complex manifolds.He also gives a treatment of the theorem of Griffiths which states that the Gauss-Manin or Picard-Fuchs systems of of differential equations are regular and singular.
where d is the exterior derivative over O X .
This is equivalent to a linear map △ : Der X (log D) → End(E) satisfy the following If ∇ is logarithmic connection K ∇ will denote its curvature and the pair (M, ∇) will refer to logarithmic connection on a locally free O X -module of rank 1 M.
for a giving nowhere vanish section s Let p be a point of D and (z i λ ) a logarithmic coordinate system along D at p.
where a i ∈ H 0 (X, O X ).Therefore,we deduce that: Lemma 4.3.Let D be a normal crossing divisor and α ∈ H 0 (X, Ω X (log D)).If dα = 0 then the residue of α is constant on any component of singular locus of D. Any such form with at least one nonzero residue admits representation Definition 4.6.Let (M, ∇) be a connection on X * = X − D. A meromorphic prolongation of (M, ∇) is a meromorphic connection ( M, ∇) on X such that the restriction is an isomorphism.

Module of Logarithmic Differential Operator
Let A be a commutative ring.For any pair of A-modules M, N we define module Di f f k A (M, N) inductively by putting M).Replacing M by E, the above definition becomes; Definition 4.7.An r-order differential operator on E is a C-linear map φ : E → E such that s → φ( f s) − f φ(s) is an (r-1)-order differential operator on E); for all f ∈ O X In the previous paragraph, we see that each logarithmic connection induces a morphism △ on Der X (log D) such that for all f ∈ O X and X ∈ Der X (log D), we have ] is zero order operator.This motivate the following definition.
Definition 4.8.An r-order differential operator φ is logarithmic along D if s → [φ(hs) − hφ(s)]h −1 is an (r-1)-order differential operator on E. Notation 4.9.We denote Di f f r (E) the set of r-order differential operators and Di f f r log (E) is the subset of r-order differential operators logarithmic along D According to what precedes, △ X ∈ Di f f 1 log (E); for all X ∈ Der X (log D).Lemma 4.10.Let φ be a first order differential operator logarithmic along D, for all sections f of O X , There exists unique Proof.For all s ∈ E, φ(hs) − hφ(s) = hs and there exist g ∈ O X such that φ(hs) − hφ(s) = hgs.Therefore, ( h − hg)s = 0 for all s.
For all f ∈ O X , s ∈ E, we have: From above results, we deduce the following exact sequence of Lie-Rinehart algebras The morphism φ is called quantization formula and it satisfies: a ∈ A, ∇ is a section of g; and v(a) = {a, −}.
In general, ∇ is only an A-module homomorphism.Obstruction to become Lie-morphism is measured by cohomology class of an 2-cocycle K ∇ ; it is usually called curvature of ∇ on H.
When (H, φ) exists and φ satisfies (13) the triplet (H, ∇, K ∇ ) is called prequantum representation of A. The following paragraph is devoted to the construction of H when µ := ω is a logsymplectic structure on a even dimensional complex manifold X with reduced divisor D.
Suppose that the logsymplectic manifold (X, ω, D) admit a prequantum representation (Di Changing the role of f and g, we obtain: It follows that φ is prequantum map of (X, ω, D) if and only if Since ω ∈ H 0 (X, Ω 2 X (log D)), relation ( 14) implies that K ∇ and then ∇ are logarithmic forms.Definition 5.1.We refer to prequantum sheaf on (X, ω, D) a rank 1 connection (M, ∇) satisfy ( 14)

Extension of Prequantum Sheaf
Our main objective being to determine the existence condition of prequantum sheaf (M, ∇) on (X, D, ω) satisfy ( 14), we intend in a first time to determine in which case integral condition of ω on X − D could be extended to entire X.Of cause we shall know how and when it is possible to extend connection on X − D to logarithmic connection on D. First about, we recall the following proposition.
Proposition 5.2.(Iitaka, S. (1981)) Let F be a closed subset of a nonsingular variety X with F X If ω 1 and ω 2 are rational q−forms such that ω The existence of τ follow from lemma 4.3.
Lemma 5.5.Let (N, ∇ 0 , K ∇ 0 ) be a sheaf of locally free O X * -module of rank 1.If (M, ∇, K ∇ ) is a sheaf of locally free O X -module of rank 1 such that Proof.The prequantum map (14) becomes And we have by simple calculation Definition 6.18.
2. The divisor D ⊂ X is locally quasi-homogeneous if for all x ∈ D there are local coordinates on X, centered at x, with respect to which D has a weighted homogeneous defining equation.
Proposition 6.19. ( F., Narváez-Macarro, L., & Mond, D. (1996)) Let D be a strongly quasihomogeneous free divisor in the complex manifold X, let U be the complement of D in X, and let j : U → X be inclusion.Then the natural morphism from the complex Ω * X (log D) of differential forms with logarithmic poles along D to R j * C is quasiisomorphism.
It follows from 6.19 and Grothendieck's Comparison Theorem that, Cohomology of X−D compute the one of the complex (Ω * X (log D), d).We denote H * dR log (X) the cohomology of the complex (Ω * X (log D), d).Let X be a complex analytic space, F a coherent sheaf on X.
Denote by G k (F ) := {m ∈ X; pro f m F ≤ k}.
We saying that the sheaf F satisfies the condition (s k ) if dim G k (F ) ≤ k − 2 Theorem 6.20.If D is zero dimensional locally homogeneous free divisor of X and if the De Rham cohomology class of ω on X − D live in i * (H 2 (X − D), Z) then (X, ω) have prequantum bundle if the associated prequantum bundle of X − D satisfies the condition (s 2 ) Proof.Since the De Rham cohomology class of ω on X − D live in i * (H 2 (X − D), it follow from B. Kostant in (Kostant, B. (1970)) that there exist a rank one locally free O X−D -module F such that the curvature satisfies the equation 24 with α = −2πi.If F satisfies the condition (s 2 ) and D is zero dimensional analytic divisor of X, then according to Trautmann Theorem, there exist an unique analytic coherent sheaf F on X extending F .Since the curvature of F coincides on X − D with curvature of F it follows from Proposition 5.2 and to the Logarithmic Comparison Theorem that F is prequantum sheaf of X Proposition 2.5.If Der A and Der A (log I) are respectively the Lie-Rinehart algebra of derivations of A and of logarithmic derivations of A, then (i) The Lie-Rinehart cohomology of Der A is the De Rham cohomology.(ii) The Lie-Rinehart cohomology of Der A (log I) is the logarithmic De Rham cohomology Follows J. Huebschmann in (Huebschmann, J. (2013)), we can now introduce the following notion.Definition 2.6.A structure of Lie-Rinehart-Poisson on a Lie-Rinehart algebra (L, ρ) is an alternating 2-form µ µ : L × L → A such that d ρ µ = 0.If µ is a structure of Lie-Rinehart-Poisson on (L, ρ),then (L, ρ, µ) is called a Lie-Rinehart-Poisson algebra.The notion of Lie-Rinehart-Poisson algebra is a part of more general notion defined by I. Krasil'shchik in (Krasil'shchik, I. (1988)) He called it canonical algebra.We denote by L * the algebraic dual over A of a Lie-Rinehart algebra L.
Remark 3.1.(Dongho, J.(2012)) Let X = C 2 and D = {0} × C. In canonical coordinate system (x, y) of X, the divisor D is the set of zero of the ideal I generated by x.But the only useful element of I which represent completely D is h = x.Other element of ideal I are square free.It follows from our definition of logarithmic forms that dx x holomorphic 1-form.But according to K. Saito definition (See Definition 1.2 of(Saito, K. (1980)), dx x 2 and dy are logarithmic forms.In addition, dx x 2 ∧ dy = 1.
10) Proof.Let p be a point of D and U λ an open coordinate neighborhood of p.We have: α = Res(α) dh p h p + α reg where Res(α) is the residue of α.But dα = 0 imply d(Resα) = O; since Res commute with d.However, from Theorem 2.9 in (K.Saito, 1980), Res(Ω 1 X (log D)) = O X , therefore dRes(α) = 0 imply Resα ∈ C. Proposition 4.4.K ∇ = 0 if and only if σ = with a i ∈ C. Proof.K ∇ = dσ; where σ is the connection one form of ∇.The result is then a consequence of Lemma 4.3 Definition 4.5.Let (M, ∇) and (N, δ) be two connections.An homomorphism from (M, ∇) to (N, δ) is a sheaf homomorphism φ : M → N such that the following diagram commute.
dy Are Lie-Rinehart-Poisson structure on Der A (log I), Der A (log J) and Der A (log K) respectively; since dω 0 = dω 1 = dω 2 = 0. We recall that in this case, the Lie-Rinehart structure is just their inclusion in Der A .The matrix of ω 1 relative to the bases (x∂ x , ∂ y ) of Der A (log I) and ( )It is also the matrix of ω 2 relative to the bases (∂ x , y∂ y ) of Der A (log J) and (dx, dy y ) of Ω A (log J).Since de determinant of M ω is inversible element of the ring A, we conclude that ω 1 and ω 2 are logsymplectic structures.On the other hand, the matrix of ω 3 relative to the bases (x∂ x , ∂ y ) of Der A (log I) and ( dx x , dy) of Ω A (log I) is