Correspondence for D-modules and Penrose transform

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Correspondence for D-modules and Penrose transform

Andrea D’Agnolo Pierre Schapira

1 Introduction

Sheaf theory was created by Jean Leray during the second World War. It is one of the most powerful tools of modern mathematics, and we hope to convince the reader of this fact by applying sheaf theory, together with D- module theory, to the study of the Penrose transform.

The Penrose correspondence is an integral transformation which inter- changes global sections of line bundles on open subsets U of the complex projective space P := P³(C) and holomorphic solutions of some partial differential equations (like the wave equation) on corresponding open subsets U of the compactified complex Minkowski space M (see section 2 below forb a more precise statement). This correspondence has been studied by many authors and generalized in various directions (let us mention in particular the books [14], [2], [20] and the papers [6], [5], [21], [22], [9]).

In this paper (which is an extended and corrected version of [3], and which announces the main results of [4]), we will describe a new approach to the

Appeared in: Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 55–68.

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Penrose transform in the language of sheaves and D-modules, in a general geometrical framework.

In our opinion, this approach has several advantages: first, it allows us to distinguish between two kinds of problems arising in this correspondence, one of a topological nature (sheaves), and the other one of an analytical sort (D-modules). Moreover, working in derived categories clarifies many problems, and we will see for instance how some difficulties encountered by many authors are due to the fact that the Penrose transform of a D-module is, in general, not concentrated in degree 0.

Let us describe the main features of this paper. Let Y

g.&^f

Z X

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be a double morphism of complex manifolds, satisfying suitable hypotheses.

Our principal results are the following:

a) We give a general formula which identifies the (generalized) solutions of aD-module N on Z and the corresponding solutions of the Penrose transform of N . This formula is exemplified in section 5 where we easily recover the results of [22] on hyperfunction solutions.

b) We give a criterion which ensures that the Penrose transform of a D- module is concentrated in degree zero. In the particular case of the twistor correspondence, this shows that the Penrose transform of a line bundleOP(k) is concentrated in degree zero if and only if k is negative.

c) We give (under suitable hypotheses) an equivalence of categories between coherentD-modules on Z (modulo flat connections) and coherent D-modules on X with regular singularities along an involutive submanifold V of ˙T^∗Z (modulo flat connections).

2 The twistor correspondence

Let T be a complex four-dimensional vector space, F = F12(T) its five- dimensional flag manifold of type (1, 2), P = F1(T) the projective three-space, and M = F2(T) the four-dimensional Grassmannian of two-dimensional vector subspaces. The manifold M is identified to a conformal compactification of the complexified Minkowski space M⁴, and it is possible to consider the family of massless field equations on M⁴ as a family of differential operators acting between holomorphic bundles on M. This is a family, noted here

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by h, parameterized by a half-integer h called helicity, including Maxwell’s wave equation, Dirac-Weyl neutrino equations and Einstein linearized zero rest mass equations.

The Penrose transform is an integral transform associated to the double fibration

F

g.&^f

P M

(2) (where f (L₁, L₂) = L₂ and g(L₁, L₂) = L₁ for L₁ ⊂ L2 ⊂ T complex subspaces of dimension 1 and 2) allowing one to find the holomorphic solutions of the equation hφ = 0 on some open subsets U ⊂ M in terms of cohomology classes of line bundles on bU = gf⁻¹(U )⊂ P.

More precisely, recall that for k ∈ Z, the line bundles on P are given by the

−k-th tensor powers O^P(k) of the tautological bundle. Set h(k) = −(1+k/2), and for z ∈ Z, set bz = fg⁻¹(z). We then have the following result of Eastwood, Penrose and Wells [6]:

Theorem 2.1. Let U ⊂ M be an open subset such that

U∩ bz is connected and simply connected for every z ∈ bU . (3) Then, for k < 0, the natural morphism associated to (2) which maps a 1−form on bU to the integral along the fibers of f of its inverse image by g induces an isomorphism:

P : H¹( bU ;O^P(k)) ' ker(U; h(k)).

3 Correspondence for sheaves and D-modules

Let X be a complex manifold, and denote by dX its complex dimension. We denote by D^b(X) the derived category of the category of bounded complexes of sheaves of C-vector spaces on X, and refer to [13] for a detailed exposition of sheaf theory in the framework of derived categories. We make use of the six classical operations on sheaves, Rf^∗, f⁻¹, Rf!, f^!, ⊗, RHom, and we use the notation D_X⁰ (·) = RHom(·, CX).

From now on, we shall consider a correspondence of complex analytic manifolds

Y

g.&^f

Z X

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(H.1) f is proper and g is smooth,

(H.2) (g, f ) : Y ,→ Z × X is a closed embedding.

Definition 3.1. Let H ∈ Ob(D^b(Z)). We set

PH = Rf!g⁻¹(H), (5)

PH = Rfe ^∗g^!(H), (6)

and we similarly define PF and ePF for F ∈ Ob(D^b(X)).

(This definition is not the same as that of [3].)

Note that assuming (H.1), we get a natural isomorphism for H ∈ Ob(D^b(Z)):

PH ' PH[2de Y − 2dZ].

By classical adjunction formulas, such as the Poincar´e-Verdier duality formula, one gets:

Proposition 3.2. (i) The functor eP is a right adjoint to P, i.e. for F ∈ Ob(D^b(X)) and H ∈ Ob(D^b(Z)) we have an isomorphism:

Hom_D^b_(Z)(PF, H)−→ Hom^∼ _D^b_(X)(F, ePH).

(ii) Assume (H.1). Let F ∈ Ob(D^b(X)), H ∈ Ob(D^b(Z)). Then we have the following commutative diagram whose lines are isomorphisms:

RΓc(Z;PD⁰_ZF ⊗ H) −→^∼ RΓc(X; D⁰_ZF ⊗ PH)

↓ ↓

RΓ(Z; RHom(PF [2dY − 2dZ], H)) −→ RΓ(X; RHom(F, PH)).^∼ Let OX denote the sheaf of holomorphic functions on X, DX the sheaf of rings of differential operators. Denote by D^b(DX) the derived category of the category of bounded complexes of left DX-modules. Following [18] (see also [15]), we say that a coherent DX-module M is good if, in a neighborhood of any compact subset of X, M admits a finite filtration by coherent DX-submodulesMk(k = 1, . . . , l) such that each quotientMk/Mk−1 can be endowed with a good filtration. We denote by Modgood(DX) the full subcategory of Modcoh(DX) consisting of good DX-modules. This definition ensures that Modgood(DX) is the smallest thick subcategory of Mod(DX) containing the modules which can be endowed with good filtrations on a neighborhood of any compact subset of X. Note that in the algebraic case, coherent D- modules are good.

Denote by f

!, f_∗ and f⁻¹ the direct and inverse image functors for D- modules. (Refer to [11], [16] or see [17] for a detailed exposition on D- modules.)

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Definition 3.3. Let N ∈ Ob(D^b(DZ)), M ∈ Ob(D^b(DX)). We define the functors P and eP by:

PN = f_!g⁻¹N ,

PM = ge _∗f⁻¹M[dZ− dX].

We define similarly PM and ePN . We call P(N ) the Penrose transform of N .

Proposition 3.4. Assume (H.1). Then, for N ∈ Ob(D^b_good(DZ)), M ∈ Ob(D^b(DX)), the following adjunction formula holds:

RΓ(X; RHomDX(PN , M)) ' RΓ(Z; RHomDZ(N , ePM)).

By the Cauchy-Kowalevski-Kashiwara theorem applied to g, and the direct image theorem applied to f (cf [11], [19]), we get:

Proposition 3.5. Assume (H.1). Let N ∈ Ob(D^b_good(DZ)). Then there is a canonical isomorphism:

P RHomDZ(N , OZ)' RHomDX(PN , OX)[dX − dY].

Theorem 3.6. Assume hypothesis (H.1). Let N ∈ Ob(D^b_good(DZ)), and let F ∈ Ob(D^b(X)). Then, setting:

A = RΓc(Z; RHomDZ(N , PD⁰_XF ⊗ OZ)),

B = RΓc(X; RHomDX(PN , D_X⁰ F ⊗ OX))[dX − dY], C = RΓ(Z; RHomDZ(N ⊗ PF [2dY − 2dZ],OZ)), D = RΓ(X; RHomDX(PN ⊗ F, OX))[dX − dY],

we have the following commutative diagram whose lines are isomorphisms:

A −→ B^∼

↓ ↓

C −→ D.^∼

Proof. Apply Proposition 3.2 with H = RHomDZ(N , OZ), then apply Propo- sition 3.5.

Remark 3.7. The formulation of Proposition 2.2 and Theorem 2.5 (i) of [3]

was slightly different from that of Proposition 3.2 and Theorem 3.6, and an hypothesis was missing: in [3] we should have assumed that f is smooth.

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This result allows us to distinguish between two kind of problems arising in the Penrose transform:

• to compute the sheaf theoretical Penrose transform of F ,

• to compute the D-module Penrose transform of N .

The first problem is of a topological nature, and under reasonable hypotheses is not very difficult to solve (cf section 5 below for an example).

The transform PN is much more difficult to compute. For example, PN is a complex of D-modules, and it is not necessarily concentrated in degree zero. This does not affect the formulas as far as we use derived categories, but things may become rather complicated when computing explicitly cohomology groups.

In the next section we will study the transformPN . We begin here with an easy corollary of Theorem 3.6.

For x ∈ X, z ∈ Z and U ⊂ X, set bx = gf⁻¹(x), bz = f g⁻¹(z), bU = gf⁻¹(U ).

Corollary 3.8. Assume (H.1) and (H.2). Let N ∈ Ob(D^b_good(DZ)). Then (i) [Germ formula] For x∈ X one has

RΓ(bx; RHomD_Z(N , OZ))' RHomD_X(PN , OX)x[dX − dY].

(ii) [Holomorphic solutions] Let U ⊂ X be an open set such that

U ∩ bz is contractible for every z ∈ bU . (7) Then

RΓ( bU ; RHomD_Z(N , OZ))' RΓ(U; RHomD_X(PN , OX))[dX − dY].

Proof. (i) By Theorem 3.6, it is enough to check that PCx ' Cx_b which follows from hypothesis (H.2) since g is an isomorphism from f⁻¹(x) to bx.

(ii) By Theorem 3.6, it is enough to check that PCU[2dY − 2dZ] ' C_U_b. By hypothesis (H.2) there is an isomorphism U∩ bz ' f⁻¹(U )∩ g⁻¹(z), and hence (7) implies that the fibers of g are topologically trivial on f⁻¹(U ). It remains to use [13, Remark 3.3.10].

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Let G be a holomorphic bundle on Z, G^∗ =HomO_Z(G, OZ) its dual, and consider the locally free DZ-module:

DG = DZ⊗O_Z G.

It is possible to recover G as the sheaf of holomorphic solutions to DG^∗. In fact:

G ' RHomDZ(DG^∗,OZ).

Let us apply the above results to the particular case of the twistor correspondence described in section 2. Applying Corollary 3.8 to N = DG^∗, G = O^P(k), we get the isomorphisms

RΓ(bx;O^P(k)) ' RHomDX(PD^P(−k), OX)x[1],

RΓ( bU ;O^P(k)) ' RΓ(U; RHomDX(PD^P(−k), OX))[1],

where we set D^P(−k) = DO^P(−k). Theorem 2.1 appears now as a corollary of the following result:

Theorem 3.9. For k < 0, the Penrose transform of the D-module D^P(−k) is the D-module associated to the operator h.

For instance, if k = −2 then PDZ(2) is the DX-module associated to the Maxwell wave equation. The proof of Theorem 3.9 is implicit in [6]. In fact, Eastwood et al. work with complexes of locally free O-modules and D- linear morphisms. This category is equivalent to that of filtered D-modules as proved by Saito [15]. This shows that these authors indeed useD-module theory.

Note that [6] being not interested in computing all cohomology groups, they could weaken the topological hypothesis on U and simply assume that U ∩ bz is connected and simply connected for every z ∈ bU .

4 Equivalence of D-modules

Let ˙T^∗X be the cotangent bundle to X with the zero-section removed. As- suming (H.2), let Λ = T_Y^∗(Z× X) ∩ ( ˙T^∗Z× ˙T^∗X) be the conormal bundle to Y ,→ Z × X, and consider the diagram:

Λ

p^a1.&^p² T˙^∗Z T˙^∗X

,→

T˙^∗Z× ˙T^∗X

p^a1.&^p² T˙^∗Z T˙^∗X

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where p1, p2 are the projections, and p^a₁ is the composite of p1 and the antipodal map. The manifold Λ being Lagrangian, p1 is smooth on Λ if and only if p^a₂ is immersive. We will assume:

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(H.3) p2|Λis a closed embedding and identifies Λ to a closed regular involutive submanifold V of ˙T^∗X and p^a₁|Λ is smooth and surjective on ˙T^∗Z.

LetEX be the sheaf of microdifferential operators of finite order on T^∗X (cf [16], [10], or [17] for a detailed exposition). IfM is a coherent DX-module, we set EM = EX ⊗_π−1DX π⁻¹M, where π denotes the projection from T^∗X to X. We say that M has regular singularities on V , if so has EM. Let us recall that the notion of regular singularities was introduced by [12]. De- note by ModRS(V )(DX) the thick subcategory of Modgood(DX) whose objects have regular singularities on V , and by D^b_{RS(V )}(DX) the full triangulated subcategory of D^b_good(DX) whose objects have cohomology groups belonging to ModRS(V )(DX).

Theorem 4.1. Assume (H.1)–(H.3).

(i) Let N ∈ Ob(D^b_good(DZ)). Then P(N ) belongs to D^b_{RS(V )}(DX).

(ii) If N is concentrated in degree zero (i.e. is a DZ-module), then for j 6= 0, H^j(PN ) has its characteristic variety contained in the zero- section (in other words it is a flat holomorphic connection).

(iii) Let G be a holomorphic vector bundle, and recall that DG = DZ⊗O_ZG.

Then, X being connected, PDG is concentrated in degree zero if and only if there exists x∈ X such that H^j(bx;G^∗) = 0 for every j < dY−dX. Proof. One first shows that P “commutes to microlocalization”. More precisely, denoting by f_!, f_∗ and f⁻¹ the direct and inverse image functors on E-modules, one extends P to the category D^b(EZ) using the same formula as in Definition 3.3, and using (H.3) one proves that

P(EM) = EP(M).

To prove (ii), one notices that in view of hypothesis (H.3), the Penrose transform acting onEZ-modules is an exact functor outside the zero-section. As- sertion (iii) is proved using (ii) and the germ formula of Corollary 3.8 (i).

Notation 4.2. Denote by D^b_good(DZ;OZ) the localization of D^b_good(DZ) by the null system of objects whose characteristic variety is contained in the zero-section. This is a triangulated category which inherits of the natural t- structure of D^b_good(DZ), and we denote by Modgood(DZ;OZ) its heart for this t-structure. We similarly define D^b_{RS(V )}(DX;OX) and ModRS(V )(DX;OX).

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Since P sends objects of D^b_good(DZ) whose characteristic variety is contained in the zero-section to objects of D^b_{RS(V )}(DX) with the same property, this functor acts on the localized categories. By composition with H⁰(∗), we get the functorP⁰ acting from Modgood(DZ;OZ) to ModRS(V )(DX;OX).

Theorem 4.3. Assume (H.1)–(H.3) and also:

(H.4) f is smooth, g is proper,

(H.5) g has connected and simply connected fibers.

Then the functor:

P⁰ : Modgood(DZ;OZ)−→ ModRS(V )(DX;OX) is an equivalence of categories.

In other words, modulo flat connections, every good DX-module with regular singularities along V is the Penrose transform of a unique goodDZ- module. Notice that on a simply connected space, the sheaf of holomorphic solutions of a flat connection is a constant sheaf of finite rank. In this sense one can say that P⁰ is almost an equivalence of categories between Modgood(DZ) and ModRS(V )(DX).

Remark 4.4. See Brylinski [1] for a related result to Theorem 4.3 in the framework of perverse sheaves.

5 Back to the twistor case

Let us now return to the twistor correspondence (2) F

g.&^f

P M,

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where F = F12(T), P = F1(T), and M = F2(T). Hypotheses (H.1), (H.2), (H.4), and (H.5) are clearly satisfied.

Let us show that hypothesis (H.3) is also verified. Choose local coordinates (x1, x2, x3, x4), (z1, z2, z3) on affine charts of M and P respectively and denote by (x; ξ), (z; ζ) the associated coordinates on T^∗M and T^∗P respectively. Here (x1, x2, x3, x4) corresponds to the two-plane of T generated by the vectors (x1, x3, 1, 0) and (x2, x4, 0, 1) and (z1, z2, z3) corresponds to the

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line generated by (1, z1, z2, z3)∈ T. The submanifold F of P × M is given by the system of equations





 x1

x₃ 1 0





 ∧





 x2

x₄ 0 1





 ∧





 1 z₁ z2

z3





 = 0.

On the open set x1 6= 0 we find the independent equations

z1− z2x3− z3x4 = 0, 1− z2x1− z3x2 = 0.

The fiber of Λ = TF^∗(P× M) at (z, x) is given by

λdz1+(−λx3−µx1)dz2+(−λx4−µx2)dz3−µz2dx1−µz3dx2−λz2dx3−λz3dx4

for λ, µ ∈ C. Then one checks that p2|Λ is an embedding and V = p₂(Λ) is given by the equation

V ∩ π⁻¹({x; x1 6= 0}) = {(x; ξ); ξ1ξ4 = ξ2ξ3}.

In particular one notices that V is indeed the characteristic variety of the wave equation. Applying Theorem 4.3, we obtain that any coherentD^M- module M with regular singularities on ˙V is (up to a flat connection) the Penrose transform of a unique coherent D^P-module.

In order to apply Theorem 4.1, note that for x ∈ M, the set bx is identified to a projective space P¹ linearly embedded in P.

Lemma 5.1. For x∈ M one has:

H⁰(bx;O^P(k)) =

0 for k < 0,

6= 0 and finite dimensional for k ≥ 0, H¹(bx;O^P(k)) is infinite dimensional for every k,

H^j(bx;O^P(k)) = 0 for j 6= 0, 1 and for every k.

Although this result should be well-known from the specialists, we give here a proof.

Proof. Choose homogeneous coordinates [t0, t1, t2, t3] ∈ P so that bx ⊂ P is given by the equations t2 = t3 = 0. For 0 < a < b let U0 ={[t]; |t1/t0| ≥ b}

and U1 ={[t]; |t0/t1| ≥ a} with coordinates (x1, x2, x3) = (t1/t0, t2/t0, t3/t0)

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and (y1, y2, y3) = (t0/t1, t2/t1, t3/t1) respectively. This is a Leray covering for O^P(k)|_bx formed by two sets, and hence the third assertion follows.

Taking (x1, x2, x3) as coordinates on U0∩ U1, the restriction maps Γ(U0∩ bx; O^P(k))−→

r0

Γ(U0 ∩ U1 ∩ bx; O^P(k))←

r1

Γ(U1∩ bx; O^P(k)) are given by

r₀(f (x₁, x₂, x₃)) = f (x₁, x₂, x₃)

r1(g(y1, y2, y3)) = x^k₁g(x⁻₁¹, x2x⁻₁¹, x3x⁻₁¹).

Take sections

Γ(U₀∩ bx; O^P(k)) 3 f (x1, x₂, x₃) = P

α∈N³

aαx^α₁¹x^α₂²x^α₃³, Γ(U₁∩ bx; O^P(k)) 3 g(y1, y₂, y₃) = P

β∈N³

bβy₁^β¹y^β₂²y^β₃³. (10)

The pair (f, g) represents an element of Γ(U₀∩ U1∩ bx; O^P(k)) if and only if r0(f ) = r1(g), i.e. if and only if

X

α∈N³

aαx^α₁¹x^α₂²x^α₃³ = X

β∈N³

bβx^k−β₁ ¹^−β²^−β³x^β₂²x^β₃³.

In particular we get k− β1− β2− β3 ≥ 0 and hence the first assertion follows.

One has

H¹(bx;O^P(k))' Γ(U₀∩ U1∩ bx; O^P(k))

r0Γ(U0∩ bx; O^P(k)) + r1Γ(U1∩ bx; O^P(k)). Take

Γ(U₀∩ U1∩ bx; O^P(k)) 3 h(x1, x₂, x₃) = X

γ∈Z×N²

cγx^γ₁¹x^γ₂²x^γ₃³,

then [h] = 0 in H¹(bx;O^P(k)) if and only if there exist f and g as in (10) such that h = f + g, i.e. if and only if

X

γ∈Z×N²

cγx^γ₁¹x^γ₂²x^γ₃³ = X

α∈N³

aαx^α₁¹x^α₂²x^α₃³ + X

β∈N³

bβx^k−β₁ ¹^−β²^−β³x^β₂²x^β₃³.

In particular, we notice that the elements of H¹(bx;O^P(k)) whose representa- tives are given by x^γ are different from zero (and different from each other) for k− γ2− γ3 < γ1 < 0. The second assertion follows.

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It is then possible to characterize those line bundles of P whose Penrose transform is concentrated in degree zero.

Proposition 5.2. The complex PD^P(−k) is concentrated in degree zero if and only if k < 0.

Proof. In view of the previous Lemma, this is a consequence of Theorem 4.1 (iii).

This last result explains why many authors restrict their study to the case of positive helicity (i.e. k < 0).

It would be interesting to prove that if M has simple characteristics on V (in the sense of [17, Ch. 1, Def. 6.2.2]) then it is the image of a locally free˙ D^P-module of rank one, i.e. of D^PG, for G a line bundle. This would better explain Penrose’s result.

To conclude, we will show how Theorem 3.6 allows us to easily recover the results of [22] on hyperfunction solutions.

Let φ be a Hermitian form on T of signature (+, +,−, −). Let us choose a basis for T such that

φ =

0 iI2

−iI2 0

where I2 ∈ M2(C) denotes the identity matrix. For A ∈ M2(C), we have:

(A^∗, I2) φ

A I₂

= 0 iff A is Hermitian.

In other words, the local chart

C⁴ −→ M

(x1, x2, x3, x4) 7→







x3− x4 x1+ ix2

x1− ix2 x3+ x4

1 0

0 1







identifies the Minkowski space M⁴ = (R⁴, φ) to an open subset of the com- pletely real compact submanifold M of M defined by:

M ={L2 ∈ M; φ(v) = 0 ∀v ∈ L2}.

Note that M is a conformal compactification of the Minkowski space M⁴. Let us consider

F = {(L1, L₂)∈ F; φ(v) = 0 ∀v ∈ L2}, P = {L1 ∈ P; φ(v) = 0 ∀v ∈ L1},

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and the induced double fibration F

e g.&^f^e

P M

,→

F

g.&^f

P M.

Recall that M is a complexification of M , that P is a real hypersurface of P topologically isomorphic to S²× S³, and that eg is locally isomorphic to a projection P × S¹ −→ P (cf [21]).

The sheaves AM and BM of analytic functions and Sato hyperfunctions respectively are given by

AM = CM ⊗ O^M,

BM = RHom(DM⁰ C_M,O^M).

In order to apply Theorem 3.6 to F = D⁰MC_M, let us calculate PCM. Since f⁻¹(M ) = N , we have f⁻¹C_M = CN. Moreover, since eg is locally isomorphic to P × S¹ −→ P , we find that PCM = Reg^∗C_N is concentrated in degree 0, 1 and H^j(PCM) is locally free of rank one for j = 0 or 1.

Finally, since P ' S² × S³ is connected and simply connected, we have H⁰(PCM)' CP, H¹(PCM)' CP.

Now we assume

k < 0 (11)

so thatPD^P(−k) is a coherent D^M-module (concentrated in degree zero) by Proposition 5.2.

Then, as an easy application of Theorem 3.6, we find the commutative diagram whose lines are isomorphisms

H¹(P ;O^P(k)) −→ Hom^∼ DM(PD^P(−k), AM)

↓ ↓

H_P²(P;O^P(k)) −→ Hom^∼ DM(PD^P(−k), BM) This is Theorem 6.1 of [22].

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Correspondence for D-modules and Penrose transform