The Radon-Penrose correspondence II: Line bundles and simple D-modules

Andrea D’Agnolo  Pierre Schapira

February 14, 1997

Abstract

On a complex manifold X of dimension ≥ 3, we show that coherent D_{X}- modules which are “simple” all over P^{∗}X are classified by Pic(X). As a corollary, using the Penrose transform, we obtain that on the complex Minkowski space 𝕄, simple D_{𝕄}-modules along the characteristic variety of the wave equation are classified by ℤ/2ℤ (the so-called helicity).

Introduction

Consider a complex manifold X of dimension ≥ 3 and a regular involutive submanifold V of P^{∗}X, the projective cotangent bundle to X. A natural problem is to classify all systems of linear PDE (i.e., coherent D_{X}-modules) with simple characteristics along V. In the extreme case where V = P^{∗}X, we solve this problem by showing that such modules are classified —modulo flat connections— by Pic(X). Starting from this result, the theory of integral transformations for D-modules allows us to treat the case of other involutive manifolds, such as the characteristic variety of the wave equation in the conformally compactified Minkowski space or, more generally, the case where X is a Grassmann manifold G_{p}(ℂ^{n}), and V is the image of the conormal bundle to the incidence relation in G_{1}(ℂ^{n}) × G_{p}(ℂ^{n}). We show in particular that on such a space, simple modules along V are determined, up to flat connections, by an integer corresponding to the so-called helicity. In [5] it was shown that the Penrose transform allows one to obtain the whole family of massless field equations. By the results of this paper, one gets that there are no other simple modules than those of this family.

1991 AMS Mathematics Subject Classification. 30E20, 32L25, 58G37.

The Radon-Penrose correspondence II: Line bundles and simple D-modules

1 Simple D-modules

We shall use the classical notations concerning D-modules. References are made to [6], [9], [10].

Let X be a complex manifold. We denote by d_{X} its dimension, by π : T^{∗}X ⟶ X its cotangent bundle, and by O_{X} its structural sheaf. The sheaf D_{X} of linear holomorphic partial differential operators on X is naturally endowed with a structure of filtered ring, the filtration being given by the subsheaves D_{X}(k) of operators of degree at most k. The associated graded ring GD_{X} is identified to ⨁_{k∈ℕ} π_{∗}O_{T^{∗}X}(k), where O_{T^{∗}X}(k) denotes the subsheaf of O_{T^{∗}X} whose sections are homogeneous of degree k in the fiber variables.

Let M be a coherent D_{X}-module. A filtration on M is an increasing sequence {M_{k}}_{k∈ℤ} of O_{X}-submodules of M, such that M = ∪_{k}M_{k}, and D_{X}(l)M_{k} ⊂ M_{k+l}. A filtration {M_{k}}_{k∈ℤ} is called good if the M_{k}’s are O_{X}- coherent and, locally on X, M_{k} = 0 for k ≪ 0, and D_{X}(l)M_{k} = M_{k+l} for any l ≥ 0 and for k ≫ 0. If M is endowed with a good filtration, we set:

\overline{G}M = O_{T^{∗}X} ⊗_{π^{−1}GD_{X}} π^{−1}GM,

where GM = ⊕_{k}M_{k}/M_{k−1} is the associated graded module. This is a coherent O_{T^{∗}X}-module. Recall that any coherent D_{X}-module locally admits a good filtration, that \overline{G}M does not depend on the choice of the good filtration, and that char(M), the characteristic variety of M, is defined as the support of \overline{G}M.

Let us denote by Mod(D_{X}) the abelian category of D_{X}-modules, and by Mod_{coh}(D_{X}) its full abelian subcategory of coherent D_{X}-modules. We denote by D^{b}(D_{X}) the bounded derived category of Mod(D_{X}), and by D_{coh}^{b}(D_{X}) the full triangulated subcategory of D^{b}(D_{X}) whose objects have coherent cohomology groups.

Definition 1.1. Let φ : M ⟶ N be a morphism of coherent D_{X}-modules. We shall say that φ is an isomorphism modulo flat connections (an m-f-c isomorphism for short) if ker φ and coker φ are flat connections (i.e., coherent D_{X}-modules whose characteristic varieties are contained in the zero-section).

We denote by π̇: Ṫ^{∗}X ⟶ X the cotangent bundle with the zero-section removed, by τ: P^{∗}X ⟶ X the projective cotangent bundle, and by γ: Ṫ^{∗}X ⟶ P^{∗}X the natural projection. For V ⊂ T^{∗}X, we set V̇ = V ∩ Ṫ^{∗}X.

Denote by Ê_{X} the sheaf of formal pseudo-differential operators on P^{∗}X. If M is a coherent D_{X}-module, we set:

M̂ = Ê_{X} ⊗_{τ^{−1}D_{X}} τ^{−1}M.

If φ : M ⟶ N is a morphism of coherent D_{X}-modules, we denote by φ̂ : M̂ ⟶ N̂ the associated morphism of Ê_{X}-modules.

Lemma 1.2. (i) A morphism φ : M ⟶ N is an m-f-c isomorphism if and only if φ̂ is an isomorphism.

(ii) The set of m-f-c isomorphisms is a multiplicative system (as defined e.g. in [7, Definition 1.6.1]) in Mod_{coh}(D_{X}).

Proof. Set K = ker φ, L = coker φ. Since Ê_{X} is flat over τ^{−1}D_{X}, K̂ = ker φ̂, L̂ = coker φ̂. Recalling that Ṫ^{∗}X ∩ char(M) = γ^{−1} supp(M̂), it is clear that K and L are flat connections if and only if φ̂ is an isomorphism. This proves (i). To prove (ii), we have to check that properties (S 1)–(S 4) of [7, Definition 1.6.1]) are satisfied. (S 1), asserting that the identity morphisms are m-f-c isomorphisms, is obvious. (S 2) requires that a composition of two m-f-c isomorphisms be again an m-f-c isomorphism, and follows from (i). A simple proof of (S 3) and (S 4) is obtained by working in the derived category D_{coh}^{b}(D_{X}), along the lines of Proposition 1.6.7 of loc. cit. Let us prove for example that any diagram

[figure: P; φ; M; f; N ,]

where φ is an m-f-c isomorphism, can be completed into a commutative diagram

[figure: Q; ψ; P; φ; M; f; N ,]  (1.1)

with ψ an m-f-c isomorphism. (We shall leave the other verifications to the reader.) Embed φ in a distinguished triangle P ⟶_{φ} N ⟶_{g} R ⟶_{+1}, and g ∘ f in

a distinguished triangle Q̃ ⟶ M ⟶_{h} R ⟶_{+1}. The diagram

[figure: M; h; f; R; id; N; g; R]

gives rise to a morphism of distinguished triangles

[figure: Q̃; ψ̃; M; R; +1; P; N; R; +1; .]

Since char(R) ⊂ T_{X}^{∗}X, ψ = H^{0}(ψ̃) is an m-f-c isomorphism. Setting Q = H^{0}(Q̃), we thus get (1.1).

Definition 1.3. We denote by Mod_{coh}(D_{X}; O_{X}) the quotient category of Mod_{coh}(D_{X}) by the multiplicative system of m-f-c isomorphisms.

Recall that an m-f-c isomorphism of Mod_{coh}(D_{X}) becomes an isomorphism in Mod_{coh}(D_{X}; O_{X}).

Definition 1.4. Let V be a closed conic involutive submanifold of Ṫ^{∗}X, and let M be a coherent D_{X}-module. We shall say that M is simple along V if M can be endowed with a good filtration \{M_{k}\} such that \overline{G}M|_{Ṫ^{∗}X} is locally isomorphic to O_{V} as an O_{Ṫ^{∗}X}-module.

If L is a locally free O_{X}-module, we set:

DL = D_{X} ⊗_{O_{X}} L.

The D_{X}-module DL has a natural good filtration DL_{k} = D_{X}(k)⊗_{O_{X}} L, and is thus an example of simple module along Ṫ^{∗}X. We show in the next theorem that, essentially, any simple module along Ṫ^{∗}X is of this form.

For m ∈ ℤ, denote by O_{P^{∗}X}(m) the −m-th tensor power of the canonical line bundle on P^{∗}X.

Lemma 1.5. If d_{X} ≥ 2, there is a natural isomorphism:

Pic(X) × ℤ ⟶^{∼} Pic(P^{∗}X)  (1.2)
(F, m) ↦ τ^{−1}F ⊗_{τ^{−1}O_{X}} O_{P^{∗}X}(m).

Proof. We may assume X is connected.

(o) If x ∈ X and L is a line bundle on P^{∗}X, denote by L_{x} its holomorphic restriction to P_{x}^{∗}X. First, notice that if U is an open ball in ℂ^{n}, then Pic(P^{∗}U) ≃ ℤ. Hence, the Chern class of L_{x} is locally constant w.r.t. x.

(i) The map is injective. In fact, if τ^{−1}F ⊗_{τ^{−1}O_{X}} O_{P^{∗}X}(m) is trivial, by restriction to P_{x}^{∗}X we find that m = 0. Next, by taking the direct image on X, we find that F is trivial.

(ii) The map is surjective. In fact, if L is a line bundle on P^{∗}X, then L_{x} ≃ O_{P_{x}^{∗}X}(m), and m does not depend on x ∈ X by (o). Let L′ = L ⊗_{O_{P^{∗}X}} O_{P^{∗}X}(−m). Then L′_{x} is trivial for all x ∈ X, and hence the natural morphism L′ ⟶ τ^{∗}τ_{∗}L′ is an isomorphism. One concludes, since then L is the image of (τ_{∗}L′, m) by (1.2).

Theorem 1.6. Let M be a coherent D_{X}-module. Assume M is simple along Ṫ^{∗}X, and assume d_{X} ≥ 3. Then, there exists a line bundle L on X and an m-f-c isomorphism M ⟶ DL.

Proof. If {M_{k}} is a good filtration of M, M̂ has a natural filtration given by

M̂_{k} = ∑_{l∈ℤ} Ê_{X}(k − l)τ^{−1}M_{l},

where Ê_{X}(k) is the subsheaf of Ê_{X} of operators of degree at most k. Since {M_{k}} is a good filtration, the above sum is finite, and hence the M̂_{k}’s are Ê_{X}(0)-coherent. Moreover,

M̂_{k} = Ê_{X}(k)M̂_{0}

for all k ∈ ℤ. Since M is simple along Ṫ^{∗}X, M̂_{0}/M̂_{−1} is a line bundle on P^{∗}X, and by Lemma 1.5, there exists a line bundle F on X and m ∈ ℤ, such that

M̂_{0}/M̂_{−1} ≃ τ^{−1}F ⊗_{τ^{−1}O_{X}} O_{P^{∗}X}(m).

By shifting the filtration, we may assume m = 0. Let us cover X by Stein open affine charts U. Then, M̂_{0}/M̂_{−1}|_{P^{∗}U} ≃ O_{P^{∗}U}. In particular, this implies:

M̂_{k}/M̂_{k−1}|_{P^{∗}U} ≃ O_{P^{∗}U}(k).  (1.3)

Since U is affine, P^{∗}U ≃ U × P, where P is a d_{X} − 1-dimensional complex projective space. Since d_{X} − 1 > 1 and U is Stein, H^{1}(P^{∗}U; O_{P^{∗}U}(k)) ≃ Γ(U; O_{U})ˆ⊗H^{1}(P; O_{P}) = 0 for k < 0. Applying the functor RΓ(P^{∗}U; ·) to the exact sequence:

0 ⟶ M̂_{k}/M̂_{k−1} ⟶ M̂_{0}/M̂_{k−1} ⟶ M̂_{0}/M̂_{k} ⟶ 0,

we thus get, for k < 0, the surjectivity of the morphism:

Γ(P^{∗}U; M̂_{0}/M̂_{k−1}) ⟶ Γ(P^{∗}U; M̂_{0}/M̂_{k}).

Let \overline{s}_{U} be a free generator of M̂_{0}/M̂_{−1} on P^{∗}U. By induction on k, using the above surjection we get a section

s_{U} ∈ \underleftarrow{lim}_{k} Γ(P^{∗}U; M̂_{0}/M̂_{k}) ≃ Γ(P^{∗}U; \underleftarrow{lim}_{k} M̂_{0}/M̂_{k})
≃ Γ(P^{∗}U; M̂_{0})

whose class modulo M̂_{−1} is \overline{s}_{U} (here, the last isomorphism follows from [9, Proposition II 3.2.5]). Consider the morphism of Ê_{U}(0)-modules:

Ê_{U}(0) ⟶_{φ_{U}} M̂_{0}|_{P^{∗}U},

given by φ_{U}(P) = P s_{U}, and set K = ker φ_{U}, L = coker φ_{U}. By construction, φ_{U} induces an isomorphism Ê_{U}(0)/Ê_{U}(−1) ⟶^{∼} M̂_{0}/M̂_{−1}|_{P^{∗}U}. It follows that K/Ê_{U}(−1)K ≃ L/Ê_{U}(−1)L ≃ 0, and hence, again by loc. cit., K = L = 0.

Summarizing up, we have shown that M̂_{0}|_{P^{∗}U} is a free Ê_{U}(0)-module of rank one. This implies that M̂|_{P^{∗}U} is a free Ê_{U}-module of rank one. Hence, τ_{∗}M̂ is a locally free D_{X}-module of rank one. Since the only invertible differential operators are of degree zero, τ_{∗}M̂ ≃ DL for a line bundle L on X.

Let φ: M ⟶ τ_{∗}M̂ be the natural adjunction morphism. To conclude, by Lemma 1.2 (i), it is enough to check that φ̂ is an isomorphism. Since M̂ is free of rank one on P^{∗}U, we are reduced to prove that the natural morphism:

E_{U} ⟶ E_{U} ⊗_{τ^{−1}D_{X}} τ^{−1}τ_{∗}E_{U}

is an isomorphism, which is obvious.

2 Integral transforms

Let X and Y be complex analytic manifolds of dimension d_{X} and d_{Y}, respectively, let Λ ⊂ Ṫ^{∗}(X × Y) be a closed smooth Lagrangian submanifold, and consider the natural projections:

X ⟵_{q_{1}} X × Y ⟶_{q_{2}} Y,  Ṫ^{∗}X ⟵_{p_{1}} Λ ⟶_{p_{2}^{a}} Ṫ^{∗}Y,  (2.1)

where we denote by p_{2}^{a} the composition of p_{2} with the antipodal map. Here, we will make the following assumptions:

\{ \begin{matrix}(i) & q_{1} and q_{2} are proper on π̇(Λ), \\ (ii) & Λ ∩ (Ṫ^{∗}X × T_{Y}^{∗} Y) = Λ ∩ (T_{X}^{∗}X × Ṫ^{∗}Y) = ∅, \\ (iii) & p_{1} is smooth and surjective on Ṫ^{∗}X, and has \\ & connected and simply connected fibers, \\ (iv) & p_{2}^{a} is a closed embedding identifying Λ to a \\ & closed regular involutive submanifold V of Ṫ^{∗}Y. \end{matrix}  (2.2)

(Recall that a conic involutive submanifold V of T^{∗}X is called regular if the restriction to V of the canonical one form never vanishes.)

If f: S ⟶ X is a morphism, we denote by \underline{f}_{!} and \underline{f}^{−1} the proper direct image and inverse image for D-modules, and we denote by \underline{⊠} the exterior tensor product. To M ∈ D^{b}(D_{X}) we associate its dual

\underline{D}′M = RHom_{D_{X}} (M, D_{X} ⊗_{O_{X}} Ω_{X}^{⊗−1}),

where Ω_{X} is the sheaf of holomorphic forms of maximal degree. We also set \underline{D}M = \underline{D}′M[d_{X}]. Thus, \underline{D}′M and \underline{D}M belong to D^{b}(D_{X}).

Let K be a simple D_{X×Y} -module along Λ. In particular, K is regular holonomic, and hence \underline{D}K is concentrated in degree zero. For M ∈ D^{b}(D_{X}), N ∈ D^{b}(D_{Y}), we set:

\begin{matrix}\underline{Φ}_{K}M = \underline{q_{2}}_{!}(K ⊗_{O_{X×Y}}^{L} \underline{q_{1}}^{−1}M), & \underline{Φ}_{K}^{j}M = H^{j}\underline{Φ}_{K}M, \\ \underline{Ψ}_{K}N = \underline{q_{1}}_{!}(\underline{D}K ⊗_{O_{X×Y}}^{L} \underline{q_{2}}^{−1}N)[d_{X} − d_{Y}], & \underline{Ψ}_{K}^{j}N = H^{j}\underline{Ψ}_{K}N . \end{matrix}

Theorem 2.1. Assume (2.2). Let M be a simple D_{X}-module along Ṫ^{∗}X, and let N be a simple D_{Y} -module along V. Then:

(o) \underline{Φ}_{K}^{0} and \underline{Ψ}_{K}^{0} send m-f-c isomorphisms to m-f-c isomorphisms.

(i) \underline{Φ}_{K}^{0}M is simple along V, and \underline{Ψ}_{K}^{0}N is simple along Ṫ^{∗}X. Moreover, \underline{Φ}_{K}^{j}M and \underline{Ψ}_{K}^{j}N are flat connections for j ≠ 0.

(ii) The natural adjunction morphisms M ⟶ \underline{Ψ}_{K}^{0}\underline{Φ}_{K}^{0}M and \underline{Φ}_{K}^{0}\underline{Ψ}_{K}^{0}N ⟶ N are m-f-c isomorphisms.

Proof. The above theorem has been proved in [3] in the case where Λ = Ṫ_{S}^{∗}(X × Y), for a smooth submanifold S ⊂ X × Y, and K = B_{S}, the sheaf of holomorphic hyperfunctions along S. There, we also show that this statement is of a microlocal nature, i.e., local on Λ. The general case follows since, in a neighborhood of any point of Λ, one may find a quantized contact transformation interchanging the pair (Λ, K) with the pair (Ṫ_{S}^{∗}(X × Y), B_{S}).

Theorem 2.2. Assume (2.2), and assume d_{X} ≥ 3. Let N be a simple D_{Y} - module along V. Then N is isomorphic to \underline{Φ}_{K}^{0}DL in Mod_{coh}(D_{Y}; O_{Y}) for a line bundle L on X, and such an L is unique up to O_{X}-linear isomorphisms. In particular, simple D_{Y} -modules along V are classified, up to flat connections, by Pic(X).

Proof. By Theorem 2.1 (i), \underline{Ψ}_{K}^{0}(N) is simple along Ṫ^{∗}X. Hence, by Theorem 1.6, there exists a line bundle L on X and an m-f-c isomorphism DL ⟶ \underline{Ψ}_{K}^{0}(N). By Theorem 2.1 (o), we get an m-f-c isomorphism \underline{Φ}_{K}^{0}(\underline{Ψ}_{K}^{0}(N)) ⟶ \underline{Φ}_{K}^{0}DL. We conclude by recalling that Theorem 2.1 (ii) gives an m-f-c isomorphism \underline{Φ}_{K}^{0}(\underline{Ψ}_{K}^{0}(N)) ⟶ N.

Theorem 2.1 gives an equivalence of categories between simple D_{X}-mod-ules on Ṫ^{∗}X, and simple D_{Y} -modules on V, modulo flat connections. However, if one is interested in calculating explicitly the image of a D_{X}-module associated to a line bundle, one way to do it consists in “quantizing” this equivalence. This is the purpose of the next result.

With the same notations as in Theorem 2.2, let M be a simple D_{X}-module along Ṫ^{∗}X, and let N be a simple D_{Y} -module along V. Then \underline{D}′M \underline{⊠} N is a simple D_{X×Y} -module along Ṫ^{∗}X × V. Let p ∈ Λ, and let u be a generator of (\underline{D}′M \underline{⊠} N)^{∧} (the Ê_{X×Y} -module associated to \underline{D}′M \underline{⊠} N) in a neighborhood of p. We say that a section

s ∈ Hom_{D_{X×Y}} (\underline{D}′M \underline{⊠} N , K)  (2.3)

is non degenerate at p if s(u) is a non degenerate section of K in the sense of [9]. (One checks immediately that this definition does not depend on u.) We say that s is non degenerate on Λ if s is non degenerate at any p ∈ Λ. There is a natural isomorphism (see e.g., [4, Lemma 3.1]):

α: Hom_{D_{X×Y}} (\underline{D}′M \underline{⊠} N , K) ⟶^{∼} Hom_{D_{Y}} (N , \underline{Φ}_{K}(M)).

Hence, a section s as in (2.3) defines a D_{Y} -linear morphism α(s): N ⟶ \underline{Φ}_{K}(M).

Theorem 2.3. With the above notations, if s is non degenerate on Λ, then α(s) is an m-f-c isomorphism.

Proof. Let α̂(s): N̂ ⟶ \underline{Φ}_{K}(M)^{∧} denote the associated Ê_{X×Y} -linear morphism. It is enough to check that α̂(s) is an isomorphism at each p ∈ V. This can be done as in [4, Theorem 3.3], using [3, Lemma 4.7].

3 Application 1: projective duality

By the methods above, we will recall here some results of [4], [8] on the complex projective Radon transform.

Let us begin with an obvious remark: let S ⊂ X × Y be a smooth hypersurface, and set Ω = (X × Y) \ S. Let B_{S} = T Hom (ℂ_{S}[−1], O_{X×Y}), B_{Ω} = T Hom (ℂ_{Ω}, O_{X×Y}), B_{Ω}^{∗} = \underline{D}B_{Ω}. The short exact sequence:

0 ⟶ ℂ_{Ω} ⟶ ℂ_{X×Y} ⟶ ℂ_{S} ⟶ 0

induces the distinguished triangles:

O_{X×Y} ⟶ B_{Ω} ⟶ B_{S} ⟶_{+1}, B_{S} ⟶ \underline{D}B_{Ω}^{∗} ⟶ O_{X×Y} ⟶_{+1} .

In particular, this implies that for any coherent D_{X}-module M, the modules \underline{Φ}_{B_{S}}^{0}(M), \underline{Φ}_{B_{Ω}}^{0}(M) and \underline{Φ}_{B_{Ω}^{∗}}^{0}(M) are isomorphic in Mod_{coh}(D_{Y}; O_{Y}).

Let ℙ be a complex n-dimensional projective space, ℙ^{∗} the dual projective space, 𝔸 = {(z, ζ); ⟨z, ζ⟩ = 0} the incidence relation, and set Ω = (ℙ × ℙ^{∗}) \ 𝔸. Note that Λ = Ṫ_{𝔸}^{∗}(ℙ × ℙ^{∗}) is the graph of a globally defined contact transformation (the Legendre transform):

Ṫ^{∗}ℙ ⟵_{p_{1}}^{∼} Λ ⟶_{p_{2}^{a}}^{∼} Ṫ^{∗}ℙ^{∗}.

Consider the kernel:

K = T Hom (ℂ_{Ω}, O_{ℙ×ℙ^{∗}}) ≃ O_{ℙ×ℙ^{∗}}(∗𝔸),

the sheaf of meromorphic functions with poles on 𝔸. Denote by O_{ℙ}(k) the −kth tensor power of the tautological line bundle, and set D_{ℙ}(k) = D_{ℙ}⊗_{O_{ℙ}}O_{ℙ}(k),

K^{(n,0)}(k, l) = q_{1}^{−1}(Ω_{ℙ} ⊗_{O_{ℙ}} O_{ℙ}(k)) ⊗_{q_{1}^{−1}O_{ℙ}} K ⊗_{q_{2}^{−1}O_{ℙ^{∗}}} q_{2}^{−1}O_{ℙ^{∗}}(l).

Let

ω(z) = ∑_{j=0}^{n}(−1)^{j}z_{j}dz_{0} ∧ ⋯ ∧ d̂z_{j} ∧ ⋯ ∧ dz_{n}

be the Leray form, and set:

s_{k}(z, ζ) = \frac{ω(z)}{⟨z, ζ⟩^{n+1+k}} , k^{∗} = −n − 1 − k.

Then:

s_{k}(z, ζ) ∈ Γ(ℙ × ℙ^{∗}; K^{(n,0)}(−k, k^{∗}))
≃ Hom_{D_{X×Y}} (D_{ℙ}(k) \underline{⊠} D_{ℙ^{∗}}(−k^{∗}), K).

If k^{∗} < 0, s_{k} is a non degenerate section on Λ. Similarly, denote by Y (t) the canonical generator of \underline{D}(T Hom (ℂ_{ℂ\\{0\}}, O_{ℂ})) ≃ D_{ℂ}/D_{ℂ}(t∂_{t} − 1), and consider

s̃_{k}(z, ζ) = ⟨z, ζ⟩^{k^{∗}}Y (⟨z, ζ⟩)ω(z).

Then

s̃_{k}(z, ζ) ∈ Γ(ℙ × ℙ^{∗}; \underline{D}K^{(n,0)}(−k, k^{∗}))
≃ Hom_{D_{X×Y}} (D_{ℙ}(k) \underline{⊠} D_{ℙ^{∗}}(−k^{∗}), \underline{D}K).

If k^{∗} ≥ 0, s_{k} is a non degenerate section on Λ.

By Theorem 2.3, we thus get:

Theorem 3.1. With the above notations, one has

(i) If k^{∗} < 0, then α(s_{k}): D_{ℙ^{∗}}(−k^{∗}) ⟶ \underline{Φ}_{K}^{0}D_{ℙ}(−k) is an m-f-c isomorphism.

(ii) If k^{∗} ≥ 0, then α(s̃_{k}): D_{ℙ^{∗}}(−k^{∗}) ⟶ \underline{Φ}_{\underline{D}K}^{0}D_{ℙ}(−k) is an m-f-c isomorphism.

Remark 3.2. Theorem 3.1 was obtained in [4] for −n − 1 < k < 0, in which case it was also shown that \underline{Φ}_{B_{𝔸}}^{j}(D_{ℙ}(−k)) = 0 for j ≠ 0. The kernels K and \underline{D}K were introduced in [8].

The results of section 1 imply that simple D_{ℙ}-modules along Ṫ^{∗}ℙ are classified, up to flat connections, by P ic(ℙ) ≃ ℤ. If M is simple along Ṫ^{∗}ℙ, we denote by ch(M) the Chern class of the line bundle L such that M is m-f-c isomorphic to DL.

Corollary 3.3. Let M be simple along Ṫ^{∗}ℙ. Then:

ch(\underline{Φ}_{B_{𝔸}}^{0}(M)) = −n − 1 − ch(M)

4 Application 2: twistor correspondence

For 1 ≤ q ≤ p ≤ n, denote by F (q, p) the flag manifold of type (q, p) in an (n + 1) dimensional complex vector space V, and denote by G(p) = F (p, p) the Grassmannian manifold of p-planes in V. Let ℙ = G(1), a complex ndimensional projective space, 𝕄 = G(p), and denote by 𝔽 = F (1, p) the incidence relation. To Λ = Ṫ_{𝔽}^{∗}(ℙ × 𝕄) is associated a diagram:

Ṫ^{∗}ℙ ⟵_{p_{1}} Λ ⟶_{p_{2}^{a}} V ⊂ Ṫ^{∗}𝕄,

which satisfies hypotheses (2.2). Consider the kernel K = B_{𝔽}. Theorem 2.2 thus implies:

Theorem 4.1. Simple D_{𝕄}-modules along V ⊂ Ṫ^{∗}𝕄 are classified, up to flat connections, by ℤ.

Let N be a simple D_{𝕄}-modules along V, and let k be the unique integer such that N is isomorphic to \underline{Φ}_{B_{𝔸}}^{0}(D_{ℙ}(−k)) (up to flat connections). Following Penrose and the physics literature, one sets:

h(N) = −(1 + k/2),

and calls it the “helicity” of N.

Remark 4.2. Consider the case n = 4, p = 2, as in [5]. Theorem 4.1 shows in particular that there are no other simple D_{𝕄}-modules along V than those corresponding to the massless field equations in the conformally compactified Minkowski space 𝕄 (see [5] and [1] for related results).

5 Comments

The classification of locally free D-modules (see [2]) or, more generally, simple Ê-modules globally defined on an involutive manifold V, would be, in our opinion, a very interesting task. This paper should be considered as a first step in this direction.

References

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Institut de Mathématiques; Analyse Algébrique; Université Pierre et Marie Curie; Case 247; 4, place Jussieu; F-75252 Paris Cedex 05

email: dagnolo@math.jussieu.fr, schapira@math.jussieu.fr