Differential equations and integral geometry

A.B.Goncharov April 20, 1997

Contents

1 Introduction 1 2 Examples 10 3 Basic facts about D-modules 13 4 The Green class of M 16 5 General linear maps and natural linear maps between solution

spaces 20 6 Natural linear maps for excellent D-modules. 27 7 Integral geometry on the family of spheres 34 8 Holonomic kernels and their composition: the bicategory of

D-modules 45

1 Introduction

1. An example. Let f(x) be a smooth function in ℝ^{m} and I : f(x) ⟼ If(y; r) := ∫_{|ω|=1} f(y + ω · r)dω

be the operator of mean value over a radius r sphere centered at y ∈ ℝ^{m}. The integral transform I is clearly injective.

Let C be a compact hypersurface in ℝ^{m} isotopic to a sphere. Theorem 1.1 Let f(x) be a smooth function vanishing near C. Then one can recover f from its mean values along the spheres tangent to C, and the inversion is given by an explicit formula.

In fact we will show that this theorem is true for any compact manifold C satisfying a mild condition. The only known before case was the family of all spheres tangent to a plane (horospheres in the hyperbolic geometry, see [GGV]).

The function If(y; r) satisfies the Darboux differential equation

(\frac{∂^{2}}{∂r^{2}} − ∑_{i=1}^{m} \frac{∂^{2}}{∂y_{i}^{2}} + \frac{m − 1}{r} \frac{∂}{∂r} )If(y; r) = 0

Let Sol(Δ_{D}, C^{∞}(ℝ^{m} × R_{+}^{∗})) be a space of smooth solutions of the Darboux equation Δ_{D}. We will construct an inverse operator J as a map

J : Sol(Δ_{D}, C^{∞}(ℝ^{m} × ℝ_{+}^{∗})) ⟶ C^{∞}(ℝ^{m})

Namely, let A^{m}(X) be the space of smooth differential m-forms on a manifold X. We will define a differential operator ν : C^{∞}(ℝ^{m} × ℝ_{+}^{∗}) → A^{m}(ℝ^{m} × ℝ_{+}^{∗}) such that the m-form νφ is closed if (and only if) Δ_{D}φ = 0. For a solution φ(y, r) we define (Jφ)(x) integrating the (closed!) differential m-form νφ(y, r) over a certain m-cycle. In particular restricting this form to the m-dimensional subvariety of all spheres tangent to C and integrating over it we get the theorem, see chapter 7 for details and generalizations.

2. General problem. Let X be a smooth manifold of dimension n and M a system of linear partial differential equations on X. Denote by Sol(M, C^{∞}(X)) the space of smooth solutions to M.

Let N be a linear system of PDE on a manifold Y. Let K(x, y)dx be a (n, 0)-form on X × Y with a compact support along X. Assume that it satisfies the system N along Y. Then the kernel K(x, y)dx defines a linear map I_{K} : C^{∞}(X) ⟶ Sol(N , C^{∞}(X)). Its restriction to Sol(M, C^{∞}(X)) gives an operator Sol(M, C^{∞}(X)) ⟶ Sol(N , C^{∞}(X)). However if M is non trivial the functional dimension of Sol(M, C^{∞}(X)) is less then n, so many kernels represents the same operator.

In this paper I adress the following

Problem. What is the natural description for the linear maps

Sol(M, C^{∞}(X)) ⟶ Sol(N , C^{∞}(Y))  (1)

When Y is a point we come to the question of natural description for linear functionals on the space Sol(M, C^{∞}(X)). On the other hand the composition of a linear map (1) with the evaluation at a point y ∈ Y gives a linear functional on Sol(M, C^{∞}(X)). So these questions are closely related.

In chapters 5-6 we suggest a construction of operators between solution spaces of linear PDE called natural linear maps. Unlike the operators given by the Schwartz kernels K(x, y)dx, the natural linear maps are obtained by integration of closed forms over certain cycles in X. We apply these ideas to solve some old problems in integral geometry.

In chapter 5 a general construction of linear maps between solution spaces is given. The natural linear maps seems to be the most interesting particular case of that construction.

To discuss these questions we need the language of D-modules.

3. Systems of linear PDE and D-modules. Let D_{X} (or D) be the sheaf of rings of differential operators on a smooth manifold X. Suppose we have a linear system M of p differential equations on q functions f_{1}, ..., f_{q}:

M = {∑_{j=1}^{q} P_{ij}f_{j} = 0, i = 1, ..., p}

Then we can assign to M a left coherent D-module M with q generators e_{1}, ..., e_{q} and p relations:

M = \frac{⊕D · e_{j}}{(+D(∑ P_{ij}f_{j}))} = Coker(D^{p} ⟶ D^{q})

On the other hand a coherent D-module M = Coker(D^{p} ⟶ D^{q}) provides a linear system of p differential equations on q functions.

A solution f to the system M in some space of functions F is nothing else then a morphism of D-modules α_{f} : M ⟶ F.

4. Natural functionals on solutions to M. Below X usually will be an algebraic manifold over ℝ of dimension n. Let D′(X) be the space of distributions on X understood as the space of linear continuous maps on the space of smooth differential forms of top degree with compact support on X. Denote by D^{m}(X) the space of m-currents on X, i.e linear continuous functionals on the space of smooth differential (n − m)-forms with compact support on X.

The de Rham complex DR(M)^{•} of a D-module M is defined as follows:

M ⟶^{d} M ⊗_{O} Ω^{1} ⟶^{d} ... ⟶^{d} M ⊗_{O} Ω^{n−1} ⟶^{d} M ⊗_{O} Ω^{n}  (2)

where M ⊗_{O} Ω^{n} is sitting in degree 0, d has degree +1 and given by

d(m ⊗ ω) := m ⊗ dω + ∑ \frac{∂}{∂x_{i}} m ⊗ dx_{i} ∧ ω

(it does not depend on coordinates x_{i}), and O is the structural sheaf of X. Consider the complex D-module M:

DR(M ⊗_{O} D′(X))^{•} = DR(M)^{•} ⊗_{O} D′(X)

Notice that DR(D′(X))^{•} coincides with D^{•}(X)[n], the usual de Rham complex of currents on X shifted by n to the left. Therefore any f ∈ Sol(M, C^{∞}(X)) defines a homomorphism of complexes

m ∘ f : DR(M ⊗_{O} D′(X))^{•} ⟶ D^{•}(X)[n]

given by the composition

DR(M ⊗_{O} D′(X))^{•} ⟶^{f} DR(C^{∞}(X) ⊗_{O} D′(X))^{•} ⟶^{m} D^{•}(X)[n]

Here m is induced by the homomorphism of D-modules C^{∞}(X) ⊗_{O} D′(X) ⟶ D′(X) provided by the multiplication. We get a pairing

H^{m}(DR(M ⊗_{O} D′(X))[−n]) ⊗ Sol(M, C^{∞}(X)) ⟶ H^{m}(X, ℝ)  (3)
(κ, f) ⟶ [(m ∘ f)(κ)]

Evaluation on a homology class [γ] ∈ H_{m}(X, ℝ) leades to a functional

Sol(M, C^{∞}(X)) ⟶ ℝ, f ⟶< [(m ∘ f)(κ)], [γ] >  (4)

Such functionals are called the natural functionals on Sol(M, C^{∞}(X)).

How the integer m depends on M? Let Σ_{M} ⊂ T^{∗}X be the characteristic variety for a D-module M. It is coisotropic, so d_{M} := dimΣ_{M} − dimX ≥ 0. The number d_{M} can often be viewed as the functional dimension of the solution space to M. We will see in chapter 3 that

H^{m}(DR(M ⊗_{O} D′(X))[−n]) = 0 for m > d_{M}

In particular natural functionals for a non zero system of PDE are never given by integration over fundamental class of X.

For a D-module M one may ask whether the described above natural functionals give all the dual to Sol(M, C^{∞}(X)). Integral geometry (including the cohomological Penrose transform) provides a wide class of examples where the answer is positive.

Remark. If X is noncompact, the integration over a (may be noncompact) m-cycle γ_{m} defines a linear functional f ⟶ ∫_{γ_{m}} κ(f) on an appropriately chosen class of functions with certain decreasing conditions at infinity. In a sence a system of PDE ”changes topology of the space”, see examples in chapter 2 and s. 5.6 below. I will not pursue this point further and hope to return to it in future. (If M is holonomic the complex of solutions is a constructible complex of sheaves on X ”changing” topology of X).

5. An elementary description of natural functionals. Assume that a D-module M has q generators. Then an m-chain κ in the de Rham complex DR(M ⊗_{O} D′(X))[−n] may be written as

κ = ∑_{j=1}^{q} P_{j} ⊗ ω_{j}, P_{j} ∈ D_{X}, ω_{j} ∈ D^{m}(X)

So we may think about it as of a differential operator

κ̄ : C^{∞}(X)^{q} ⟶ D^{m}(X);  κ̄(f_{1}, ..., f_{q}) ⟼ ∑_{i=1}^{q} P_{j}(f_{j}) · ω_{j}

Suppose that κ is a cycle in the De Rham complex of M. Then the m-current κ̄(f_{1}, ..., f_{q}) is closed on solutions of the system M, i.e. dκ̄(f_{1}, ..., f_{q}) = 0 whenever the functions (f_{1}, ..., f_{q}) satisfy the system M. In this case we will say that the differential operator κ̄ is M-closed.

Remark. This definition makes sence for any system of partial differential equations, not necessarily linear. It leads to a notion of conservation laws for a system of nonlinear PDE.

6. Natural linear maps between solution spaces: a naive version. Let M and N be systems of linear PDE on manifolds X and Y. A natural linear map

I : Sol(M, C^{∞}(X)) ⟶ Sol(N , D′(Y))

is defined as follows. Let κ_{y} : C^{∞}(X) ⟶ D^{m}(X) be an M-closed differential operator whose coefficients are distributions on Y satisfying the system N. Then

I : f ⟼ ∫_{γ_{m}} κ_{y}f ∈ Sol(N , D′(Y))

where γ_{m} is an m-cycle in X and by definition ∫_{γ_{m}} κ_{y}f :=< [γ_{m}], κ_{y}f >. The key idea of this paper is the following:

If there is a (continuous) linear functional on solutions to a system of linear partial differential equations M or an operator between solution spaces to M and N, then one should look for its natural realization.

7. Relation with integral geometry. Let B be a manifold of dimension n and a linear operator

I_{K} : C_{0}^{∞}(B) ⟶ C^{∞}(Γ)  f(x) ⟼ ∫_{B} K(x, ξ)dx  (5)

enjoys the following properties:

It is injective, transforms functions f(x) to solutions of a linear system of PDE N on Γ, and I_{K}(C_{0}^{∞}(B)) is dense in Sol(N , C^{∞}(Γ)).

Usually K(x, ξ) satisfies a holonomic system of differential equations.

Such a situation is typical in integral geometry and appears as follows. Let {B_{ξ}} be a family of submanifolds of B parametrized by a manifold Γ. Suppose on {B_{ξ}} densities μ_{ξ} (depending smoothly on ξ) are given. Then there is an integral operator

I : C_{0}^{∞}(B) ⟶ C^{∞}(Γ),  f(x) ⟼ ∫_{B_{ξ}} f(x)μ_{ξ}  (6)

So here K(x, ξ) = μ(x, ξ) · δ(A)db, where db is a volume form on B, and

A := {(x, ξ)|x ∈ B_{ξ}} ⊂ B × Γ

is the incidence subvariety. The integral transform I often satisfies the list of properties above. This was discovered by F.John [J] for the family of all lines in ℝ^{3}, and developed much further by Gelfand, Graev, Shapiro [GGS], (see also [GS2])). Here are some examples.

Example 1. Consider the integral transformation

I : f(x_{1}, ..., x_{n+1}) ⟶ ∫ f(t_{1}, ..., t_{n}, a ∑_{i=1}^{n} t_{i}^{2} + ∑_{i=1}^{n} b_{i}t_{i} + c)d^{n}t  (7)

related to the (n + 2)-parametrical family of paraboloids in ℝ^{n+1}. Let S(ℝ^{n+1}) be the Schwartz space of functions in ℝ^{n+1}.

Lemma 1.2 If f ∈ S(ℝ^{n+1}) then (\frac{∂^{2}}{∂a∂c} − ∑_{i=1}^{n} \frac{∂^{2}}{∂b_{i}^{2}} )If = 0.

The integral transformation I is injective on S(ℝ^{n+1}).

Proof. Applying \frac{∂^{2}}{∂a∂c} to the right-hand side of (7) we get

∫ ∑_{i=1}^{n} t_{i}^{2}ft_{n+1}^{′′}(t_{1}, ..., t_{n}, a ∑_{i=1}^{n} t_{i}^{2} + ∑_{i=1}^{n} b_{i}t_{i} + c)d^{n}t

Applying ∑_{i=1}^{n} \frac{∂^{2}}{∂b_{i}^{2}} we get the same result. Let a = 0. Then I is the Radon transform and so the lemma follows from its standard properties.

Example 2. Consider the integral transformation

I_{k} : f(x_{1}, ..., x_{n}) ⟶ ∫ f(t_{1}, ..., t_{k}, ∑_{j=1}^{k} a_{1}^{j}t_{j} + a_{1}^{0}, ..., ∑_{j=1}^{k} a_{n−k}^{j}t_{j} + a_{n−k}^{0})dt_{1}...dt_{k}

related to the family of k-planes in ℝ^{n}: x_{k+i} = ∑_{j=1}^{k} a_{i}^{j}x_{j} + a_{i}^{0}.

Theorem 1.3 ([GGS]) a) If f(x) ∈ S(ℝ^{n}) then

(\frac{∂^{2}}{∂a_{i_{1}}^{j_{1}}∂a_{i_{2}}^{j_{2}}} − \frac{∂^{2}}{∂a_{i_{1}}^{j_{2}}∂a_{i_{2}}^{j_{1}}} )If = 0

where k + 1 ≤ i_{1}, i_{2} ≤ n, 0 ≤ j_{1}, j_{2} ≤ k.

b) I_{k} is injective on S(ℝ^{n}) and provides an integral formula for solutions the system of PDE above.

Let us return to the integral transform I_{K} (see (5)). Its properties implies that its inverse would provide a continuous linear map

J_{K} : Sol(N , C^{∞}(Γ)) ⟶ C^{∞}(B)

Definition 1.4 An integral transform I_{K} admits a universal inversion formula if the inverce operator J_{K} is given by a natural linear map.

To clarify the meaning of this definition consider the composition J_{b} of the operator J with the δ-functional at a point b ∈ B. Its natural realization is given by an N -closed differential operator ν_{b} : C^{∞}(Γ) ⟶ D^{n}(Γ) and a certain n-dimensional cycle γ_{b} in Γ such that

∫_{γ_{b}} ν_{b}(I_{K}f) = c_{[γ_{b}]} · f(b)

Here c_{[γ_{b}]} is a constant depending linearly on the homology class of γ_{b} and n = dimB. We define the left hand side as < ν_{b}(I_{K}f), [γ_{b}] >. To compute the integral we may use any cycle γ_{b} transversal to the wave front of the distribution κ_{b}(I_{K}f). Then the restriction of this distribution to γ_{b} is defined and we can integrate it over the fundamental class of γ_{b}. So we can find the value f(b) if we know only the values of I_{K}(f) at an infinitesimal neighborhood of any such a cycle. This explanes the name ”universal inversion formula”.

8. Local and nonlocal inversion formulas in integral geometry. Let us discuss in more details the general Radon transform (6).

Definition 1.5 A local universal inversion formula for the Radon transform (6) is given by a differential operator κ_{b} : C^{∞}(Γ) ⟶ A^{k}(Γ_{b}) such that κ_{b}(If) is closed (on Γ_{b}) and

∫_{γ_{k}} κ_{b}(If) = c_{[γ_{k}]} · f(b)

where c_{[γ_{k}]} is a constant (depending linearly on the homology class [γ_{k}]).

In particular the value of any smooth function f on B at any point b can be recovered from its integrals over the submanifolds B_{ξ} passing through an infinitesimal neighborhood of b.

Let Γ_{b} be the variety parametrizing all the subvarieties B_{ξ} passing through a given point b. Set k := dimB_{ξ}. Notice that dimΓ − dimB = dimΓ_{b} − dimB_{ξ}. So if dimΓ > dimB the degree of the form κ_{b}(If) is less then dimΓ_{b}.

A first example of local universal inversion formula was discovered in 1967 by I.M.Gelfand, M.I.Graev and Z.Ya.Shapiro for integral transformation I_{k}^{ℂ} related to the family of all k-planes in ℂ^{n} ([GGS]). Here we treat complex planes as real submanifolds and integrate smooth functions along them. Later more generic examples were studied, including local universal inversion formulas for the families of complex curves, see [GGiG], [BG], [Gi].

However in integral geometry there are many examples where there are no local inversion formulas. This is quite typical in “real” integral geometry (i.e. we integrate over family of real submanifolds). For instance in examples 2 (resp 3) the inversion formula is nonlocal if the dimension of hyperboloids (resp. planes) is odd. It is always non local for integral transformations related to any family of real curves.

A very interesting approach to integral geometry on k-planes in ℝ^{n} was suggested by I.M.Gelfand and S.G.Gindikin [GGi], (see also [GGR]). However it was based on the Fourier transform in ℝ^{n} and so can not be generalized to families of “curved” submanifolds, like in examples 1-2. What is even more important, the differential k-form κ was replaced by a k-density, so a possibility to use the Stokes formula was missed. It seems that this approach to integral geometry was not really understood yet.

As a result the nature of the form κ_{b} and inversion of the general integral transformations, especially if they do not admit a local inversion formula, were the key unsolved problems in integral geometry.

The main idea of this paper is that

Inversion formulas in integral geometry are given by natural linear maps between solution spaces of systems of partial differential equations.

Let me explain how the local universal inversion formulas fit in this concept. The form κ_{b}(If) is a differential k-form on Γ_{b}. Since n − k = dimΓ − dimB, a k-form on Γ_{b} defines an n-current on Γ. The n-current corresponding to κ_{b}(If) leads to a natural liear map given by integration of κ_{b}(If) over an n-cycle K ⊂ Γ. We will demonstrate this for the Radon transform over spheres in ℝ^{m}.

In general our approach leads to a universal inversion formula where the functional J_{b} is represented by a differential n-form on Γ. The fact that this n-form does not concentrated on a subvariety Γ_{b} (or a certain bigger subvariety of Γ) means that we get a nonlocal universal inversion formula. So we treat simultaneously both local and nonlocal inversion formulas.

The form κ_{b} appeared in [GGS] as a construction ”ad hoc” and looks like a very special phenomena. In our approach the universal inversion formula is a very general property of the corresponding system of linear PDE. Its locality, however, is a rather rare phenomena, which generalizes the Huygens principle or, more generally, the notion of lacunas for hyperbolic differential equations.

In particular in these examples our natural functionals describe the whole dual to the space of solutions of a linear system of PDE.

9. Some general remarks on analytic theory of overdetermined systems. The classical theory of PDE usually study systems of p linear partial differential equations on p unknown functions on X i.e. the characteristic variety of the system has codimension 1. (Of cource there are some exceptions with extremely reach analytic theory, like the Cauchy-Riemann system). It seems that one of the reasons is this. A system P_{1}f = P_{2}f = 0 of two general differentaial equations on one unknown function has no solutions because the corresponding D-module is equal to zero (even if P_{i} are differential operators of order one). This shows that overdetermined systems (i.e. the ones where the codimension of the characteristic subvariety is greater then 1) can not describe a physical process in a way similar to systems of p equations on p unknown functions (like Laplas, Schrodinger, etc. equation): a small perturbation of the experimental data leads to a system without solutions. Therefore one should not expect an analytic theory of general overdetermined systems, i.e. a theory stable under a perturbation of a system

The theory of D-modules is a tool providing nontrivial linear systems of PDE. We think that an interesting overdetermined system of PDE should be a part of a reacher data. For example for a system N on a variety Γ appearing in integral geometry (see s.6) we should also remember the kernel K(x, ξ) on B × Γ. So perturbing such a system we should deform the whole data, not only the system N on Γ.

We may wonder about the goals of analytic theory for some special overdetermined systems. It seems that the problem of natural description of linear maps between solution spaces looks quite promising.

10. The structure of the paper. Chapter 2 contains examples of functionals and natural functionals on solution spaces of systems of PDE. In chapter 3 we recall some general information about D-modules, including the duality on the derived category of D-modules, needed for applications to integral geometry. In chapter 4 our key tool appears: the Green class of a D-module. It generalizes the classical Green formula for a single differential operator. Chapter 5 contains a definition and properties of general linear maps between solution spaces of (complexes of) D-modules. Then we define natural linear maps as a quite special case of general linear maps. The definitions uses the language of derived categories. This is necessary for many reasons including:

1) Even nice systemes like M = {x_{1} · f = ... = x_{k} · f = 0} may have no smooth solutions, so one should consider the spaces of “higher” solutions. (In the example above only Ext_{D}^{k}(M, C^{∞}(ℝ^{n})) ≠ 0; it is isomorphic to C^{∞}(ℝ^{n−k})).

2) The duality may send a D-module to a complex of D-modules.

In applications the dual complex for a D-module M is often concentrated in just one degree. Such M’s will be called excellent D-modules. In chapter 6 we define natural linear maps between solution spaces for excellent D-modules. This allows to eliminate derived categories and makes the story more elementary. I made this chapter independent of chapter 5, so those who interested only in applications to “nice” systems of PDE could go directly to chapter 6.

In chapter 7 we demonstrate how the general method works for the family of all spheres in ℝ^{m} (see section 1.1 above). Our approach leads to universal inversion formulas which are nonlocal when m is odd and local when m is even. The corresponding problem of integral geometry was unsolved even for the family of circles in the plane.

In fact we study in chapter 7 integral operators I_{λ} more general then the Radon transform over the family of spheres. They are intertwiners for the group O(m + 1, 1) acting from the space of sections of a line bundle over S^{m} to the space of sections of a line bundle over the manifold X_{m+1} of oriented hyperplane sections of S^{m}. (The hyperplane sections of S^{m} can be identified with spheres in ℝ^{m} by a stereographic projection). The image of I_{λ} is described by differential equations. So the inverse operators gives examples of intertwiners which are well defined only on a subrepresentation.

See also [GR] where a problem of integral geometry for the family of conics on real or complex plane was solved using the methods of this paper.

The next problem after the definition of natural linear maps would be development of ”calculas of natural linear maps”. In particular there are the following questions:

a) How to compose natural linear maps.

b) How to compute their composition. For instance when the composition of two natural linear maps is equal to a given natural linear map.

A universal inversion formula for the integral transform I_{K} is a natural linear map J_{K} : Sol(N , C^{∞}(Γ)) ⟶ C^{∞}(B) such that the composition J_{K} ∘I_{K} equals to the identity map, so this is a very special case of the problem b).

The development of this program should include a version of the theory of Fourier Integral Operators as a special case when M = D_{X}, N = D_{Y}.

In chapter 8 we study an algebraic version of the problem of composition of natural linear maps. It turns out that one can organize neatly the algebraic structures responsible for that introducing a bicategory of D-modules.

The objects of this bicategory are pairs (X, M), where M is a complex of D-modules on a variety X. A 1-morphism (X, M) → (Y, N) is the algebraic part of the data needed to construct a linear map RHom_{D}(M, C^{∞}(X)) ⟶ RHom_{D}(N , D′(Y)). Composition of 1-morphisms mirrors the composition of linear maps. A 2-morphism between two 1-morphism reflects coincidence of the the corresponding maps on functions.

In the end of chapter 8 we consider the simplest examples of composition of 1-morphisms and 2-morphisms relevant to integral geometry.

The main results of this paper were annonced in [G1]. A different approach to integral geometry via D-modules was independently developed by A. D’Agnolo and P. Shapira [A], [AS1-2]. Inversion formulas for real quadrics were also considered by S.G. Gindikin [Gi2].

Acknowledgment. I am very grateful to Joseph Bernstein for many stimulating discussions and suggestions.

This work was partially supported by NSF Grant DMS-9500010.

The paper was finished during my stay in the Max-Planck-Institute fur Mathematik in Bonn, whose financial support and hospitality are gratefully acknowledged.

2 Examples

1. Analitycal functionals [GS]. Let X = ℂ and M be the Cauchy - Riemann equation \frac{∂}{∂z̄}f(z, z̄) = 0. Let g(z) be a holomorphic function. Then

f(z, z̄) ⟶ g(z)f(z, z̄)dz

is an M-closed operator of order 0. The corresponding linear functional should be

f(z) ⟶ ∫_{γ_{1}} g(z)f(z)dz  (8)

To make sense out of this consider the space Z_{1} of holomorphic functions f(z) such that |z|^{q} · |f(z)| ≤ C_{q} · e^{a·Imz} for any q > 0 (the constants a and C_{q} may depend on f). Let ℂ̄ = {ℂ ∪ S^{1}} be a compactification of the complex plane by a circle such that each line compactifies by endpoints at + and - infinity and two lines have the same endpoints if and only if they are parallel. Let x_{−} and x_{+} are the endpoints of the x - axis (z = x + iy). Let γ_{1} be a cycle representing the nontrivial homology class in H_{1}(ℂ̄, x_{−} ∪ x_{+}). Then the right side of (8) is convergent for g(z) ∈ Z_{1} and defines a continuous linear functional on Z_{1}.

One can also take g(z) to be a meromorphic function and integrate along compact 1-cycles in ℂ\\{poles of g(z)\}. For example if g(z) = \frac{1}{z−z_{0}} we get the Cauchy formula

f(z_{0}) = \frac{1}{2πi} ∫_{γ} \frac{f(z)}{z − z_{0}}

It can be interpreted as the natural realization for the δ-functional f(z) → f(z_{0}).

Now let M be the Cauchy-Riemann system in ℂ^{n}. Let g(z) be a holomorphic function. Then

f(z, z̄) ⟶ g(z)f(z, z̄)dz_{1} ∧ ... ∧ dz_{n}

is an M-closed operator of order 0. The corresponding natural functional is

f(z) ⟶ ∫_{γ_{n}} g(z)f(z)dz_{1} ∧ ... ∧ dz_{n}

where f(z) belongs to the space Z_{n} of holomorphic functions satisfying some grouth condition ([GS]). So any g(z) ∈ Z_{n} defines an element of DSol(M)_{n}

However for n > 1 there are another M - closed operators. Namely, let us look at the classical Bochner - Martinelly formula

f(z_{0}) = ∫_{s_{2n−1}} f(z)\frac{ω^{∗}(z̄) ∧ ω(z)}{|z − z_{0}|^{2n}}  (9)

where ω(z) = dz_{1} ∧ ... ∧ dz_{n} and ω^{∗}(z) = ∑_{i=1}^{n}(−1)^{i}z_{i}dz_{1} ∧ ...d̂z_{i}... ∧ dz_{n} and [s_{2n−1}] is a generator in H_{2n−1}(ℂ^{n}\z_{0}) The zero order operator

f(z, z̄) ⟶ f(z)\frac{ω^{∗}(z̄) ∧ ω(z)}{|z − z_{0}|^{2n}}

represents a non-zero element of DSol(M)_{2n−1} for X = ℂ^{n}\z_{0}

In fact all “integral formulas” in complex analysis (like the Cauchy formula in a polydisc, the Weil formula ...) are examples of elements in DSol(M)_{m} where n ≤ m ≤ 2n − 1 given by zero order operators.

2. The Green function of a differential operator and a natural realization of the δ-functional. Let P = ∑ a_{I}(x)∂_{x}^{I} be a differential operator and P^{t} = ∑ ∂_{x}^{I}a_{I}(x) the transposed one (I is the multyindex). The classical Green formula is

(P^{t}u · v − u · P v)dx_{1} ∧ ... ∧ dx_{n} = dω_{n−1}(u; P; v)  (10)

where ω_{n−1}(u; P; v) is an (n − 1)-form depending linearly on u and v. For example if Δ = ∑_{i=1}^{n} ∂_{x_{i}}^{2} then

ω_{n−1}(Δ; u, v) = ∑_{i=1}^{n}(u′_{x_{i}}v − uv′_{x_{i}})dx_{1} ∧ ...d̂x_{i}... ∧ dx_{n}

A Green function g(x, y) for P is a generalized function on X × X satisfying

P_{x}^{t}g(x, y) = P_{y}g(x, y) = δ(x − y)

Let us put in (10) u := g(x, y) and suppose P v = 0. Then

v(x) = ∫_{s_{n−1}} ω_{n−1}(g(x, y); P; v)

where s_{n−1} is a small (n − 1)-sphere around x. Therefore ω_{n−1}(g(x, y); P; v) provides a natural realization of the δ-functional f → f(x).

There are differential operators that do not have a Green function, for example the H. Lewy operator 1/2(∂_{x_{1}} + i∂_{x_{2}}) − (x_{1} + ix_{2})∂_{x_{3}} or the operator ∂_{x} − ix∂_{y}. A natural realization of a functional not always exist.

3. A Green form for an arbitrary system M.

Definition 2.1 A Green form for a system M is an element g_{y} ∈ Sol(M)_{n−1} such that

dg_{y} =^{M} δ(x − y)dx_{1} ∧ ... ∧ dx_{n}

Here =^{M} means M-equivalence, i.e.

dg_{y}(f) = f(y)dx_{1} ∧ ... ∧ dx_{n} f ∈ Sol(M, C^{∞}(X))

If g_{y} is a Green form for M then for any f ∈ Sol(M, C^{∞}(X)) one has

f(y) = ∫_{s_{n−1}} g_{y}(f)

Here s_{n−1} is a small sphere around y. This follows from the Stokes formula.

Example 2.2 g_{y} : f → ω_{n−1}(P; g(x, y), f) (see s 2.2 above) is the Green form for a differential equation P f = 0.

Example 2.3 The Bochner - Martinelly form (9) provides a Green form

f ⟶ f \frac{ω^{∗}(z̄) ∧ ω(z)}{|z − z_{0}|^{2n}}  (11)

for the Cauchy-Riemann system in ℂ^{n}.

4. A universal solution of a boundary value problem. Let M be a system of PDE on X and m := d_{M}. Let Id : Sol(M, C^{∞}(X)) ⟶ Sol(M, C^{∞}(X)). be the identity map. Its natural realization should be given by an M-closed operator G_{x} : C^{∞}(X) ⟶ D^{m}(X) depending on a parameter x ∈ X whose coefficients considered as functions on x are also solutions to M. For a given solution ϕ ∈ Sol(M, C^{∞}(X)) one has G_{x}(ϕ) is an m-form on X such that for any closed m-dimensional manifold Y one has

∫_{Y} G_{x}(ϕ) = c_{[Y]}ϕ(x)  (12)

where c_{[Y]} is a constant depending linearly on the homology class of Y.

According to the definition to compute G_{x}(ϕ) at a point y ∈ Y one has to know the restriction and a finite number of transversal derivatives of ϕ at y. So formula (12) is a universal solution to the Cauchy problem for M on Y. The fact that d_{M} can be often viewed as a “functional dimension” of the space of solutions to M) looks very natural from this point of view. G_{x} will be referred to as the boundary value problem Green form.

Remark 2.5. There are two different realizations for the identity map given in s. 2.3 and s. 2.4. I would like to emphasize the following differences between them. The realization given in s. 2.3 is not a natural one because the form is not M-closed. However one may interpret it as a natural realization for a modification of M at x. Further, in general the cycles for g_{y} and G_{y} are of different dimension and in fact of different nature. Namely, for a Green form g_{y} the cycle always exist and represents a class in H_{n−1}(X\x), while for the boundary value problem Green form G_{y} the cycle in (12) represents a homology class of X of dimension m and its existence is a nontrivial problem.

3 Basic facts about D-modules

For conviniense of the reader I will recall some material about D-modules (see [Be], [Bo]).

1. The bimodule D_{X}^{Ω}. I will assume that X is an algebraic manifold, D = D_{X} is the sheaf of regular differential operators on X and Ω_{X} the O_{X} - sheaf of regular differential forms of highest degree on X. Ω_{X} has a right D_{X}-module structure given by

ω · f = fω, ω · ξ := −L_{ξ}ω

where f ∈ O_{X} and ξ is a vector field. Here L_{ξ}ω := di_{ξ}ω. Set

D_{X}^{Ω} := D_{X} ⊗_{O} Ω_{X}^{−1} = Hom_{O_{X}} (Ω_{X}, D_{X}^{r})  (13)

where D_{X}^{r} is D_{X} viwed as a right D-module via right multiplication. Then (13) carries 2 commuting left D_{X}-modules structures. The first is provided by the left multiplication on D_{X}, and the second, ′′∘′′, is given by

ξ ∘ (λ)(ω) = λ(ω · ξ) − λ(ω) · ξ  (14)

where ξ is a vector field and λ ∈ Hom_{D_{X}} (Ω_{X}, D_{X}^{r}).

The two natural commuting left D_{X}-module structures on D_{X}^{Ω} let us to consider D_{X}^{Ω} as a D-module on X × X. It is canonically isomorphic to the D_{X×X}-modules δ_{Δ} of δ-functions on the diagonal Δ ⊂ X × X. There exists a canonical involution on D_{X}^{Ω} interchanging the two left D_{X}-module structures. For δ_{Δ} it is induced by the switch of the factors of X × X.

2. The duality functor. Let D_{coh}^{b}(D_{X}) be the derived category of bounded complexes of D_{X}-modules whose cohomology groups are coherent D_{X}-modules. Let us define a duality ⋆ : D_{coh}^{b}(D_{X})^{0} ⟶ D_{coh}^{b}(D_{X}) by

⋆M := RHom_{D_{X}} (M, D_{X}^{Ω})[dimX]

The second D_{X}-structure on D_{X}^{Ω} provides a left D_{X}-module structure on the sheaves Ext_{D_{X}}^{i} (M, D_{X}^{Ω}).

To compute ⋆M we should find a bounded complex P = {⟶ P^{−1} ⟶ P^{0} ⟶ P^{1} ⟶ ...} of locally projective coherent D-modules quasiisomorphic to M and set ⋆M = ⋆P where (⋆P)^{i} = ⋆(P^{−dimX−i}) := Hom_{D_{X}} (P^{−dimX−i}, D_{X}^{Ω}). It is easy to see that ⋆ ⋆ P is isomorphic to P. Therefore ⋆⋆ = Id.

The object ⋆M represents the functor

N ⟶ RHom_{D_{X×X}} (N ⊠ M, δ_{Δ})[dimX]

i.e. one has

RHom_{D_{X}} (N , ⋆M) = RHom_{D_{X×X}} (N ⊠ M, δ_{Δ})[dimX]  (15)

Indeed, there is canonical morphism

RHom_{D_{X×X}} (N ⊠ M, D_{X}^{Ω}) ⟶ RHom_{D_{X}} (N , RHom_{D_{X}} (M, D_{X}^{Ω}))

It is obviously an isomorphism when N = M = D_{X}, and so using locally free resolutions we see that it is an isomorphism in general.

Let SSM be the singular support of a D-module M. The following important result was proved by Roose.

Theorem 3.1 a) M has a finite resolution by locally projective D_{X}-modules. b) codimSS(Ext_{D_{X}}^{i} (M, D_{X}^{Ω}) ≥ i

c) If codimSS(M) = k, then Ext_{D_{X}}^{i} (M, D_{X}^{Ω}) = 0 for i < k.

Notice that H^{i}(⋆M) = Ext_{D_{X}}^{dimX+i}(M, D_{X}^{Ω}) The Roose theorem implies that ⋆M is concentrated in degrees [−dimX, −codimSSM].

Lemma 3.2 DR(M) = Ω_{X} ⊗^{L}_{D_{X}} M.

Proof. Using the Koszule complex we see that DR(D_{X}) is a locally free resolution for the right D_{X}-module Ω_{X}. One has DR(M) = DR(D) ⊗_{O} M. Let D^{b}(Sh_{X}) be the bounded derived category of sheaves on X.

Theorem 3.3 Let M ∈ D_{coh}^{b}(D_{X}) and N ∈ D^{b}(D_{X}). Then there is an isomorphism in D^{b}(Sh_{X}) functorial with respect to M and N

DR(⋆M ⊗_{O} N)[−dimX] = RHom_{D_{X}} (M, N)  (16)

This nature of this isomorphism and the fact that N may not be coherent plays a crucial role, so we will scetch its proof following [Bo], ch. 6. Let us replace M and N by bounded locally projective resolutions P_{M}^{•} and P_{N}^{•}. One can suppose P_{M}^{•} to be locally free from certain low degree on. Therefore according to lemma 3.2 to prove the theorem it is sufficient to construct for a given coherent D_{X}-modules M and N a natural morphism (functorial with respect to M and N)

α : Ω_{X} ⊗_{D_{X}} (Hom_{D_{X}} (M, D_{X}^{r} ⊗ Ω_{X}^{−1}) ⊗_{O_{X}} N ) ⟶ Hom_{D_{X}} (M, N) (17)

which will be an isomorphism if M = D_{X}.

The functors Ω_{X}⊗_{D_{X}} and Hom_{D_{X}} in the left-hand side of (17) are defined using different commuting left D_{X}-structures on D_{X} ⊗ Ω_{X}^{−1}. So we can interchange them and get the canonical morphism from the left-hand side of (17) to

Hom_{D_{X}} (M, Ω_{X} ⊗_{D_{X}} (D_{X} ⊗ Ω_{X}^{−1}) ⊗_{O_{X}} N ))  (18)

There is canonical isomorphism of D_{X}-modules

Ω_{X} ⊗_{D_{X}} (D_{X} ⊗ Ω_{X}^{−1}) ⊗_{O_{X}} N ) = N  (19)

Indeed, the left structure on (D_{X} ⊗ Ω_{X}^{−1}) we used to define Ω_{X} ⊗_{D_{X}} (D_{X} ⊗ Ω_{X}^{−1}) is provided by the left multiplication in D_{X}, therefore Ω_{X} ⊗_{D_{X}} (D_{X} ⊗ Ω_{X}^{−1}) = O. So (18) is canonically isomorphic to N as O-module. This isomorphism commutes with the action of vector fields. So (19) is canonically isomorphic to Hom_{D_{X}} (M, N). Theorem 3.3 is proved.

Placing to (16) N = C^{∞}(X) and using ⋆ ⋆ M = M we get

Corollary 3.4

DR(M ⊗_{O} C^{∞}(X))[−dimX] = RHom_{D_{X}} (⋆M, C^{∞}(X))  (20)

In particular

Hom_{D_{X}} (⋆D_{X}, D′(X)) ⟶ DR(D_{X}^{Ω} ⊗_{O} D′(X)) = D^{n}(X)

Corollary 3.5 For any A, B ∈ D_{coh}^{b}(X), C ∈ D^{b}(X) one has a canonical functorial isomorphism in D^{b}(Sh_{X}):

RHom_{D_{X}} (A, B ⊗_{O_{X}} C) = RHom_{D_{X}} (A ⊗_{O_{X}} ⋆B, C)

Proof. By the theorem above both parts are isomorphic to

RHom_{D_{X}} (O_{X}, ⋆A ⊗_{O_{X}} B ⊗_{O_{X}} C)

In particular for M ∈ D_{coh}^{b}(X) there is a canonical morphism of D_{X}-modules

i_{M} : O_{X} ⟶ ⋆M ⊗_{O_{X}} M  (21)

More precisely, there exists a canonical section over X of the sheaf H^{0}(RHom_{D_{X}} (O_{X}, ⋆M ⊗_{O_{X}} M), or, what is the same, a canonical morphism ℂ_{X} → RHom_{D_{X}} (O_{X}, ⋆M ⊗_{O_{X}} M) in D^{b}(Sh_{X}).

3. Functors between the derived categories of D-modules . Let Y → X be a morphism of varieties and d_{Y,X} := dimY − dimX. Let p^{+} be the naive inverse image functor on D-modules. Then p^{!} := Lp^{+}[d_{Y,X}]. If p : Y → X is smooth then p_{∗} can be computed via relative DeRham complex p_{∗}M = Rp_{•}(DR_{Y |X}(M)), where DR_{Y |X}(M) := Ω_{Y |X}^{•} ⊗ M[d_{Y,X}].

Lemma 3.6 Suppose p is smooth. Then there is a canonical isomorphism of functors on D_{coh}^{b}(D)

p^{∗} := ⋆p^{!}⋆ = p^{!}[−2d_{Y,X}]

Proof. See proof of proposition 9.13 in [Bo].

Let p_{!} := ⋆p_{∗}⋆.

Theorem 3.7 Suppose p is proper. Then p_{!} = p_{∗} on D_{coh}^{b}(D) and the functor p_{!} (resp. p^{∗}) is left adjoint to p^{!} ( resp p_{∗}), i.e.

RHom_{D}(p^{∗}M, N) = RHom_{D}(M, p_{∗}N)
RHom_{D}(p_{!}M, N) = RHom_{D}(M, p^{!}N)

Proof. See proof of theorem 9.12 in [Bo].

Lemma 3.8 Let p : X ⟶ ∗ be projection to the point and Δ : X ↪ X × X be the diagonal. Then Rp_{•}RHom_{D}(⋆M, N) = p_{∗}Δ^{!}(⋆M ⊠ N)

4 The Green class of M

“... we can say that there is only one formula (which we shall call “fundamental formula”) in the whole theory of partial differential equations,

no matter to which type they belong.”

J.Hadamard, Lectures on the Cauchy problem.

0. The definition. For any M ∈ D_{coh}^{b}(D_{X}) the identity map in Hom_{D_{X}} (M, M) provides a canonical element

G_{M} ∈ H^{dimX}(DR(⋆M ⊗_{O_{X}} M))  (22)

I will call it the Green class of M.

The right hand side of (22) is a sheaf on X, and G_{M} is a canonical section of this sheaf. A more concrete way to think about it is this. Choose a locally projective resolution M^{•} for M. Take a Cech covering {U_{i}} of X (in the classical or Zariski topology). Then there exists a section in the Cech complex C(U_{•}, DR(⋆M^{•} ⊗_{O_{X}} M^{•})) with coefficients in the complex of sheaves DR(⋆M^{•} ⊗_{O_{X}} M^{•}) which represents the Green class.

1. The Green class and the classical Green formula. Let P be a differential operator. Set

P = \frac{D_{X}}{D_{X} · P} and ˜⋆P = \frac{D_{X}^{Ω}}{P · D_{X}^{Ω}}

Here D_{X}^{Ω} is considered as a left D-module with respect to the second structure. Notice that Hom_{D}(D_{X}^{Ω}, C^{∞}(X)) = A^{n}(X). Let v ∈ A^{n}(X). According to the Green formula there exists an (n − 1)-form ω_{n−1}(φ; P; v) on X such that

P φ · v − φ · P^{∗}v = dω_{n−1}(φ; P; v)  (23)

Of cource neither (n−1)-form ω_{n−1}(φ; P; v) nor its cohomology class [ω_{n−1}(φ; P; v)] are defined canonically by (23). However there is a way to define a cohomology class in H^{n−1}(X, ℝ) starting form the Green formula. Namely, locally there exists an algebraic bidifferential operator

ω_{P} : C^{∞}(X) ⊗_{O} A^{n}(X) → A^{n−1}(X) such that dω_{P} = P ⊗ 1 − 1 ⊗ P^{∗}

so ω_{n−1}(φ; P; v) := ω_{P} (φ⊗v). For two different algebraic bidifferential operators ω_{P} and ω′_{P} there exists an algebraic bidifferential operator

ω′′_{P} : C^{∞}(X) ⊗_{O} A^{n}(X) → A^{n−2}(X) such that dω′′_{P} = ω_{P} − ω′_{P}

and so on. So choosing a covering and taking a partition of unity corresponding to it we get a well defined cohomology class [ω_{n−1}(φ; P; v)]. Below we explane how to get it without computaions in local coordinates, using the D-modules instead. (On the other hand the approach we scetched leads to an equivariant cohomology class of the group of diffeomorphisms of X).

Lemma 4.1 ⋆P[−1] is isomorphic to ˜⋆P, so the Green class G_{⋆P} is an element of H^{n−1}DR(˜⋆P ⊗_{O} P).

Proof. Let D_{X} ⟶^{P} D_{X} be the obvious free resolution for P. It is concentrated in degrees [-1,0]. So

RHom_{D_{X}} (P, D_{X}^{Ω}) = (D_{X}^{Ω} ⟶^{P^{∗}} D_{X}^{Ω})[n]

(the complex is concentrated in degrees [-n,-(n-1)]), where P^{∗} : Q → P ∘ Q. Recall that there is canonical involution on D_{X}^{Ω} interchanging two left D_{X}- structures. If we choose a volume form ω this involution sends P ⊗ ω^{−1} just to P^{t} ⊗ ω^{−1} where P^{t} is the transposed to P defined using ω. The lemma follows immediately from these remarks.

Let

R_{P} := D_{X} ⟶^{P} D_{X}, R_{P^{∗}} := D_{X}^{Ω} ⟶^{P^{∗}} D_{X}^{Ω}

be the resolutions for P and ˜⋆P. Their tensor product over O

D ⊗_{O} D^{Ω} ⟶^{P ⊗1} D ⊗_{O} D^{Ω}
R_{P} ⊗_{O} R_{P^{∗}} = 1 ⊗ P^{∗} ↑  ↑ −P^{∗} ⊗ 1
D ⊗_{O} D^{Ω} ⟶^{P ⊗1} D ⊗_{O} D^{Ω}

sits in degrees [−(n − 2), −n]. Let

... ⟶^{d} R_{P} ⊗_{O} R_{P^{∗}} ⊗_{O} Ω^{n−1} ⟶^{d} R_{P} ⊗_{O} R_{P^{∗}} ⊗_{O} Ω^{n}

be the de Rham complex DR(R_{P} ⊗_{O} R_{P^{∗}}). Its degree −(n − 1) part is

(D ⊗_{O} D^{Ω} ⊗_{O} Ω^{n} ⊕ D ⊗_{O} D^{Ω} ⊗_{O} Ω^{n}) ⊕ D ⊗_{O} D^{Ω} ⊗_{O} Ω^{n−1} (24)

Since D ⊗_{O} D^{Ω} ⊗_{O} Ω^{n} = D ⊗_{O} D, there is a canonical element (1 ⊗ 1, 1 ⊗ 1) in the left summand of (24).

Choose a covering {U_{i}} of X. Consider the Cech complex

C({U_{•}}, DR(R_{P} ⊗_{O} R_{P^{∗}}))

An −(n − 1)-cocycle G̃_{M} in this complex such that

the component in C(U_{i}, (D⊗_{O}D^{Ω}⊗_{O}Ω^{n} ⊕ D⊗_{O}D^{Ω}⊗_{O}Ω^{n})) is (1⊗1, 1⊗1)

represents the Green class. Its existence follows from general theory.

Let ω_{n−1}^{(i)} be the componenet of G̃_{M} in C(U_{i}, D ⊗_{O} D^{Ω} ⊗_{O} Ω^{n−1}), and ω_{n−2}^{(i,j)} the component in C(U_{i,j}, D ⊗_{O} D^{Ω} ⊗_{O} Ω^{n−2}). Then dω_{n−2}^{(i,j)} = ω_{n−1}^{(i)} − ω_{n−1}^{(j)}. To relate this cocycle with the discussion in s. 4.1 notice that ω_{n−1}^{(i)} can be vied as an ”algebraic bidifferential operator”.

2. The Green formula and the Bar construction. Let E^{1} and E^{2} be vector bundles over an n-dimensional manifold X and E^{1} ⟶^{P} E^{2} be a differential operator. Set V_{i} := E^{i}^{∗} ⊗ A^{n}. There are canonical pairings

Γ_{0}(X, E^{i}) ⊗ Γ(X, V_{i}) ⟶ ℝ (φ, g ⊗ ω) ⟶ ∫_{X}(φ, g)ω

So one has the adjoint operator V_{1} ⟵^{P^{∗}} V_{2}. It is a differential operator of the same order as P uniquely defined by the property (φ_{1}, P^{∗}v_{2}) = (P φ_{1}, v_{2}).

Now suppose we have a sequence (not necessarily a complex) of differential operators

E^{0} ⟶^{P_{1}} E^{1} ⟶^{P_{2}} ... ⟶^{P_{k}} E^{k}

Consider the sequence of adjoint differential operators

V_{0} ⟵^{P_{1}^{∗}} V_{1} ⟵^{P_{2}^{∗}} ... ⟵^{P_{k}^{∗}} V_{k}

Theorem 4.2 For any k there exists forms ω_{n−k}(φ_{0}; P_{1}, ..., P_{k}; v_{k}) such that ω_{n}(φ; 1; v) := φ · v and

dω_{n−k}(φ_{0}; P_{1}, ..., P_{k}; v_{k}) = ω_{n−k+1}(P_{1}φ_{0}; P_{2}, ..., P_{k}; v_{k})+

∑_{i=1}^{k−1}(−1)^{i}ω_{n−k+1}(φ_{0}; P_{1}, ..., P_{i}∘P_{i+1}, ..., P_{k}; v_{k})+(−1)^{k}ω_{n−k+1}(φ_{0}; P_{1}, ..., P_{k−1}; P_{k}^{∗}v_{k})

3. How to compute the Green class. Let us call a D-module M excellent if the object ⋆M is concentrated in just one degree, i.e. H^{i}(⋆M) = 0 for all i but one. By the Roose theorem this degree is −d_{M}. In this case set ˜⋆M := H^{−d_{M}}(⋆M). Consider a locally free resolution of a D-module M:

P^{•} = {P^{−k} ⟶ ... ⟶ P^{−2} ⟶ P^{−1} ⟶ P^{0}}

Let

⋆P^{•} = {∗(P^{0}) ⟶ ∗(P^{1}) ⟶ ∗(P^{2}) ⟶ ... ⟶ ∗(P^{k})}[d_{X}]

be the dual complex. Then E^{•} := Hom_{D}(P^{•}, C^{∞}(X)) is a complex of differential operators between vector bundles:

E^{0} ⟶^{P_{1}} E^{1} ⟶^{P_{2}} ... ⟶^{P_{k}} E^{k}

The adjoint complex

V_{•} := {V_{k} ⟶^{P_{k}^{∗}} V_{k−1} ⟶^{P_{k−1}^{∗}} ... ⟶^{P_{1}^{∗}} V_{0}}

is canonically isomorphic to Hom_{D}(⋆P^{•}, C^{∞}(X))[−d_{X} − k].

Suppose that a D-module M is excellent and admits a locally free resolution of the minimal possible length k = d_{M}. (This usually happen in integral geometry). Then

Sol(M, C^{∞}(X)) = KerP_{1} and Sol(˜∗M, D′(X)) = KerP_{k}^{∗}

Therefore for any φ_{0} ∈ KerP_{1} and v_{k} ∈ KerP_{k}^{∗} one has dω_{n−d_{M}}(φ_{0}; P^{•}; v_{k}) = 0

Theorem 4.3 The cohomology class of the form ω_{n−d_{M}}(φ_{0}; P^{•}; v_{k}) coincides with the Green class G_{M}(φ_{0}; v_{k}).

Proof. It is similar to the proof of lemma (4.1). Since P^{−i} is a locally free D-module, there is a canonical element 1_{i} in

P^{−i} ⊗_{O} Hom_{D}(P^{−i}, D^{Ω}) ⊗ Ω  (25)

Namely, locally P^{−i} = D ⊗_{ℂ} V, so (25) is

V ⊗_{ℂ} V^{∗} ⊗_{ℂ} D ⊗_{O} Hom_{D}(D, D^{Ω}) ⊗ Ω = End(V) ⊗_{ℂ} D ⊗_{O} D

and we take Id_{V} ⊗ 1 ⊗ 1. A 0-cycle in DR(P ⊗_{O} ⋆P) whose component in P ⊗_{O} ⋆P ⊗_{O} Ω is ∑ 1_{i} represents the Green class.

5 General linear maps and natural linear maps between solution spaces

Denote by RHom^{c}(·, ·) the RHom with compact supports in the category of sheaves. For M, N ∈ D^{b}(D_{X}) set

RHom_{D_{X}}^{c} (M, N) := RΓ_{c}(X; RHom_{D_{X}} (M, N))

We will define a canonical morphism

RHom_{D}^{c}(⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2})) ⊗ RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ (26)
RHom_{D}(M_{2}, D′(X_{2}))[−n]

Any linear map

RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ RHom_{D}(M_{2}, D′(X_{2}))[−n]  (27)

continuous in an appropriate topology is given by a unique element in

RHom_{D}^{c}(⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2}))  (28)

( this follows from proposition (5.1)), so the space (28) gives us the general linear maps (27). Our goal in this paper is to construct and study an interesting subspace in (28), the subspace of natural linear maps.

To make more clear the relation with natural functionals we will spell the construction of the map (26) using the Green class

G_{M} ∈ H^{dimX}(DR(⋆M ⊗_{O} M))  (29)

and using the canonical morphism in D^{b}(D_{X})

i_{M} : O_{X} ⟶ ⋆M ⊗_{O_{X}} M  (30)

They are, of cource, equivalent. To clarify the main point we will start from the case when X_{2} is a point.

1. The canonical pairing via the Green class. Let Δ_{X} be the orientation sheaf of X. Set D̃′(X) := D′(X) ⊗_{ℤ} Δ_{X}. We will define the canonical pairing

R^{i}Hom_{D_{X}}^{c} (⋆M, D̃′(X)) ⊗ R^{n−i}Hom_{D_{X}} (M, C^{∞}(X)) ⟶ ℂ

If A_{i}, B_{j} are sheaves on X, a ∈ R^{i}Hom(A_{1}, A_{2}) and b ∈ R^{j}Hom(B_{1}, B_{2}) then the tensor product over ℂ provides an element

a ⊗_{ℂ} b ∈ R^{i+j}Hom(A_{1} ⊗_{ℂ} B_{1}, A_{2} ⊗_{ℂ} B_{2})

If A_{i}, B_{j} are sheaves of O-modules on X we can make a tensor product over O:

a ⊗_{O} b ∈ R^{i+j}Hom(A_{1} ⊗_{O} B_{1}, A_{2} ⊗_{O} B_{2})

Suppose X is a smooth variety over ℝ of dimension n and

v ∈ R^{i}Hom_{D_{X}}^{c} (⋆M, D′(X)), f ∈ R^{n−i}Hom_{D_{X}} (M, C^{∞}(X))

Their tensor product over O_{X} is an element

v ⊗_{O} f ∈ R^{n}Hom_{D_{X}} (⋆M ⊗_{O} M, D̃′(X) ⊗_{O} C^{∞}(X))  (31)

So the multiplication m : D̃′(X) ⊗_{O_{X}} C^{∞}(X) ⟶ D̃′(X) leads to an element

m(v ⊗_{O} f) ∈ R^{i+j}Hom_{D_{X}} (⋆M ⊗_{O} M, D̃′(X))

Applying this element to the Green class (29) we get a cohomology class

m(v ⊗_{O} f)(G_{M}) ∈ H_{c}^{n}(DR(D̃′(X)[n]) = H_{c}^{n}(X̃, ℂ) = ℂ

where X̃ = X if X is orientable and it is a two fold covering given by the orientation class if it is not.

2. The canonical pairing via the morphism i_{M}.. Taking the Koszule resolution of the D-module O_{X} we see that the complex of sheaves RHom_{D_{X}} (O_{X}, D′(X)) is canonically quasiisomorphic to the De Rham complex of currents on X:

D′(X) ⟶^{d} D′(X) ⊗_{O_{X}} Ω_{X}^{1} ⟶^{d} ... ⟶^{d} D′(X) ⊗_{O_{X}} Ω_{X}^{n−1} ⟶^{d} D′(X) ⊗_{O_{X}} Ω_{X}^{n}

(the last group sitting in degree n). If we take the RHom’s with compact support we get the De Rham complex of currents with compact support.

There is the trace map given by integration over the fundamental class of X:

∫_{X} : R^{n}Hom_{D_{X}}^{c} (O_{X}, D̃′(X)) ⟶ ℂ

The composition of the morphism i_{M} (30) with the element m(v ⊗_{O} f) (31) gives

m ∘ i_{M}(v ⊗_{O} f) ∈ R^{n}Hom_{D_{X}}^{c} (O_{X}, D̃′(X))

Applying ∫_{X} we get a pairing

v ⊗ f ⟼ ∫_{X} m ∘ i_{M}(v ⊗_{O} f) ∈ ℂ

Recall that there is the Grothendieck duality theory for topological vector spaces [Gr]. In particular C^{∞}(X) has a natural topology of a Fréchet nuclear space, and D′_{0}(X) has a natural topology of a dual to a Fréchet nuclear space, so they are topologically dual. An immediate consequence of this is the following simple duality result.

Proposition 5.1 Suppose X is a smooth variety over ℝ of dimension n. Then

R^{i}Hom_{D_{X}} (M, C^{∞}(X))  (32)

has a topology of a Fréchet nuclear space and

R^{n−i}Hom_{D_{X}}^{c} (⋆M, D′(X))  (33)

has a topology of a dual to a Fréchet nuclear space. The spaces (32) and (33) are dual to each other.

Proof. Consider first the classical case of a single differential operator. Let P be a differential operator acting on smooth functions and P^{∗} the adjoint acting on the distributions with compact supports:

\begin{matrix}C^{∞}(X) & ⟶^{P} & C^{∞}(X) \\ D_{0}^{n}(X) & ⟵^{P^{∗}} & D_{0}^{n}(X) \end{matrix}

The canonical pairing boils down to the obvious duality between KerP and the closure of CokerP^{∗}, and the closure of CokerP and KerP^{∗}. The general statement for any M ∈ D^{b}(D_{X}) we get similarly taking a locally projective resolution.

3. A construction of the map (26). Let RHom^{c_{1}} be the RHom with compact supports along the factor X_{1}. Choose

K ∈ RHom_{D}^{c_{1}}(⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2})) f ∈ RHom_{D}(M_{1}, C^{∞}(X_{1}))

Their product K ⊗_{O_{X_{1}}} f over X_{1} belongs to

RHom_{D_{X_{1}×X_{2}}}^{c_{1}} (⋆M_{1} ⊗_{O_{X_{1}}} M_{1} ⊠ M_{2}, D̃′(X_{1}) ⊗_{O_{X_{1}}} C^{∞}(X_{1}) ⊠ D′(X_{2}))

Using the multiplication map

m_{X_{1}} : D̃′(X_{1}) ⊗_{O_{X_{1}}} C^{∞}(X_{1}) ⊠ D′(X_{2}) ⟶ D̃′(X_{1}) ⊠ D′(X_{2})

we get a class

m_{X_{1}}(K ⊗_{O_{X_{1}}} f) ∈ RHom_{D_{X_{1}×X_{2}}}^{c_{1}} (⋆M_{1} ⊗_{O_{X_{1}}} M_{1} ⊠ M_{2}, D̃′(X_{1}) ⊠ D′(X_{2}))

The canonical morphism i_{M_{1}} : O_{X_{1}} ⟶ ⋆M_{1} ⊗_{O_{X_{1}}} M_{1} provides an element

(m_{X_{1}} ∘ i_{M_{1}})(K ⊗_{O_{X_{1}}} f) ∈ RHom_{D_{X_{1}×X_{2}}}^{c_{1}} (O_{X_{1}} ⊠ M_{2}, D̃′(X_{1}) ⊠ D′(X_{2}))

Applying ∫_{X_{1}} : R^{n}Hom_{D_{X_{1}}}^{c} (O_{X_{1}}, D̃′(X_{1})) ⟶ ℂ we get

K̄(f) ∈ RHom_{D_{X_{2}}} (M_{2}, D′(X_{2}))

4. Natural functionals. Recall that R^{n−m}Hom^{c}(ℂ, Δ_{X}) = H_{m}(X, ℂ) and the Poincare duality is given by

R^{i}Hom(ℂ, ℂ) ⊗ R^{n−i}Hom^{c}(ℂ, Δ_{X}) ⟶ R^{n}Hom^{c}(ℂ, Δ_{X}) \underrightarrow{_{∫_{X}}} ℂ

The first arrow is the composition of Hom′s. Tensor product over ℂ provides a canonical map

R^{j}Hom^{c}(ℂ, Δ_{X}) ⊗ R^{i}Hom_{D_{X}} (M, D′(X)) ⟶ R^{i+j}Hom^{c}(M, D̃′(X))

Combining it with the pairing (??) we get a map

< ·, ·, · >_{M}:
H_{i+j}(X, ℂ) ⊗ R^{i}Hom_{D_{X}} (⋆M, D′(X)) ⊗ R^{j}Hom_{D_{X}} (M, C^{∞}(X)) ⟶ ℂ (34)

By definition the natural functionals on the space R^{j}Hom_{D_{X}} (M, C^{∞}(X)) are the functionals < γ, v, · >_{M} provided by a homology class γ ∈ H_{i+j}(X, ℂ) and v ∈ R^{i}Hom_{D_{X}} (⋆M, D̃′(X)).

5. Natural linear maps. There is a map

RΓ_{c}(X_{1}, Δ_{X_{1}}) ⊗ Hom_{D}(⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2})) ⟶

RHom_{D}^{c}(⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2}))

So we get a canonical morphism

RΓ_{c}(X_{1}, Δ_{X_{1}})⊗RHom_{D}(⋆M_{1}⊠M_{2}, D′(X_{1}×X_{2}))⊗RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶

RHom_{D}(M_{2}, D′(X_{2}))[−n]  (35)

In particular it induces a map

H_{i+j}(X_{1}, ℤ)⊗R^{i+k}Hom_{D_{X_{1}×X_{2}}} (⋆M_{1}⊠M_{2}, D′(X_{1}×X_{2}))⊗R^{j}Hom_{D}(M_{1}, C^{∞}(X_{1})) ⟶

R^{k}Hom_{D_{X_{2}}} (M_{2}, D′(X_{2}))  (36)

By definition a natural linear map

K̄_{γ} : R^{j}Hom_{D_{X_{1}}} (M_{1}, C^{∞}(X_{1})) ⟶ R^{k}Hom_{D_{X_{2}}} (M_{2}, D′(X_{2}))

is given by a ”kernel”

K = K(x_{1}, x_{2}) ∈ R^{i+k}Hom_{D_{X_{1}×X_{2}}} (⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2}))

and γ ∈ H_{i+j}(X_{1}, ℤ).

6. Examples. Suppose M is an excellent D_{X}-module, m := d_{M}. Recall that ˜⋆M := H^{−m}(⋆M) is the dual system to M. Taking k = m, i = 0, l = m we get

Hom_{D}(M, C^{∞}(X)) ⊗ Hom_{D}(˜⋆M, D′(X)) ⊗ H_{m}(X, ℝ) ⟶ ℝ

Let G̃_{M}(·, ·) be a cocycle in DR(˜⋆M ⊗_{O} M) representing the Green class. Let f(x) be a smooth solution of the system M. Choose a distributional solution v(x) of ˜⋆M. Then we get a closed differential form G̃_{M}(v(x), f(x)) of degree d_{M} on X. Choose a cycle γ of dimension d_{M} in X. Then

< γ, v, f >= ∫_{γ} G̃_{M}(v(x), f(x))

is a functional on smooth solutions of M.

Example 0. Suppose M = D_{X}. Then ⋆M = D_{X}^{Ω}[n] and ˜⋆M = D_{X}^{Ω}. Recall that Hom_{D}^{c}(D_{X}^{Ω}, D′(X)) = D_{0}^{n}(X). We get the usual paring C^{∞}(X) ⊗ D_{0}^{n}(X) ⟶ ℂ.

The following examples show a wider class of functionals on solution spaces then the natural functionals we just defined above. The point is that sometimes we can integrate the differential form G̃_{M}(v(x), f(x)) not only over cycles, but also over some chains (which do not represent a homology class in general sense) still getting a functional on smooth solutions of a system M.

Example 1. Let M be a D-module on ℝ^{n} corresponding to the system x_{1} · f = x_{2} · f = ... = x_{m} · f = 0, 0 < m < n. Then

RHom_{D}^{i}(M, C^{∞}(X)) = 0 for i ≠ m

RHom_{D}^{m}(M, C^{∞}(X)) = \frac{C^{∞}(X)}{(C^{∞}(X) · x_{i})}

and

RHom_{D}^{i}(M, D′(X)) = 0 for i ≠ 0
Hom_{D}(M, D′(X)) = δ(x_{1})δ(x_{2})...δ(x_{m}) · D′(X)

So there is a natural pairing

RHom_{D}^{m}(M, C^{∞}(X)) ⊗ Hom_{D}(M_{1}, D′(X)) ⟶ ℝ

It should correspond to the case i = 0, j = m, k = m. However H_{m}(ℝ^{n}, ℝ) = 0 in any topological sense.

Comparing the general and natural functionals. Let P be a differential operator on X. Recall that the general functionals on KerP we get from the closure of CokerP^{∗}, see the proof of proposition (5.1). The natural functionals we get in a different way. Take f ∈ KerP and v ∈ D′(X), v ∈ KerP^{∗}. Notice that if, for example, P is an operator with constant coefficients, then the restriction of KerP ∩ D′_{0}(X) = 0, so it is essential that v is not necessarily compactly supported. Then make the Green class [ω_{n−1}(v; P; f)] and integrate it over a homology class [γ]. A simplest example is given in the example 2 below.

An advantage of natural functionals on KerP is that they correspond to ”functions”, i.e. elements of the subspace KerP^{∗}, rather then to elements of the quotient CokerP^{∗}.

Example 2. Let M_{a} be the system (∑_{i=1}^{n} x_{i}∂_{x_{i}} − a)f(x) = 0 on ℝ^{n}\0. Then ˜⋆M_{a} = M_{−a−n}. Consider the following differential (n − 1)-form

σ_{n}(x, dx) := ∑_{i=1}^{n}(−1)^{i−1}x_{i}dx_{1} ∧ ... ∧ d̂x_{i} ∧ ... ∧ dx_{n}  (37)

Then

G̃_{M}(v(x), f(x)) = v(x)f(x)σ_{n}(x, dx)

Let γ be an (n − 1)-cycle generating H_{n−1}(ℝ^{n}\0) . Then

∫_{γ} v(x) · f(x)σ_{m}(x, dx)  (38)

provides a nondegenerate pairing between the smooth solutions of M_{a} and M_{−a−n}.

Example 3. Consider ℂ^{n} as a real manifold. Let M_{a,b} be the following system in ℂ^{n}\0:

(∑_{i=1}^{n} z_{i}∂_{z_{i}} − a)f(z, z̄) = 0, (∑_{i=1}^{n} z̄_{i}∂z̄_{i} − b)f(z, z̄) = 0

Then ˜⋆M_{a,b} = M_{−a−n,−b−n}. Then

G̃_{M}(v(z, z̄), f(z, z̄)) = v(z, z̄) · f(z, z̄) · σ_{n}(z, dz) ∧ σ_{n}(z̄, dz̄)

Let Γ be a chain intersecting any one dimensional subspace in ℂ^{n} with multiplicity one. Then

∫_{Γ} v(z, z̄)f(z, z̄)σ_{m}(z, dz) ∧ σ_{m}(z̄, dz̄)  (39)

provides a pairing between the solutions of M_{a,b} and M_{−a−n,−b−n}. However H_{2n−2}(ℂ^{n}\0) = H_{2n−2}(S^{2n−1}) = 0!

A chain Γ can be considered as a discontinuous ”section ” of the Hopf bundle ℂ^{n}\0 ⟶ ℂP^{n−1}. A better way to think about this integral is the following. The form we integrate can be pushed down to ℂP^{n−1}, so we integrate over the fundamental cycle.

7. Composition of natural maps between smooth solution spaces. We can not define in general a morphism

RHom_{D}(M_{1}, D′(X_{1})) ⟶ RHom_{D}(M_{2}, D′(X_{2}))

using distributional kernels because of the lack of multiplication of distributions, and a priory there is no way to compose operators

RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ RHom_{D}(M_{2}, D′(X_{2}))

and

RHom_{D}(M_{2}, C^{∞}(X_{2})) ⟶ RHom_{D}(M_{3}, D′(X_{3}))

However the natural linear maps constructed using smooth kernels can be composed. Namely, suppose

K_{12} ∈ RHom_{D}(⋆M_{1} ⊠ M_{2}, C^{∞}(X_{1} × X_{2})), γ_{1} ∈ H_{l_{1}}(X_{1}, ℝ)
K_{23} ∈ RHom_{D}(⋆M_{2} ⊠ M_{3}, C^{∞}(X_{2} × X_{3})), γ_{2} ∈ H_{l_{2}}(X_{2}, ℝ)

They define the corresponding natural maps

K̄_{12}^{γ_{1}} : RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ RHom_{D}(M_{2}, C^{∞}(X_{2}))
K̄_{23}^{γ_{2}} : RHom_{D}(M_{2}, C^{∞}(X_{2})) ⟶ RHom_{D}(M_{3}, C^{∞}(X_{3}))

Their composition is given by the data

K_{23} ∘ K_{12} ∈ RHom_{D}(⋆M_{1} ⊠ M_{3}, C^{∞}(X_{1} × X_{3})), γ_{1} ∈ H_{l_{1}}(X_{1}, ℝ)

where the kernel K_{23} ∘ K_{12} is constructed as follows. Let

Δ_{2} : X_{1} × X_{2} × X_{3} ↪ X_{1} × X_{2} × X_{2} × X_{3}

be the diagonal imbedding and π_{2} : X_{1}×X_{2}×X_{3} ⟶ X_{1}×X_{3} be the projection. Then

Δ_{2}^{!}C^{∞}(X_{1} × X_{2} × X_{2} × X_{3}) = C^{∞}(X_{1} × X_{2} × X_{3})[−d_{X_{2}}]

and

H_{l}(X_{2}, ℤ) ⟶ R^{d_{X_{2}} −l}Hom_{D}(π_{2}_{∗}C^{∞}(X_{1} × X_{2} × X_{3}), C^{∞}(X_{1} × X_{3}))

Therefore one has canonical morphism H_{l}(X_{2}, ℤ) ⟶ R^{−l}Hom_{D}(π_{2}_{∗}Δ_{2}^{!}C^{∞}(X_{1} × X_{2} × X_{2} × X_{3}), C^{∞}(X_{1} × X_{3})) According to lemma 3.8 one has G_{M} ∈ p_{∗}Δ^{!}(M ⊠ ⋆M). Therefore

⋆M_{1} ⊠ M_{3} ⟶^{id⊠G_{M}⊠id} π_{2}_{∗}Δ_{2}^{!}(⋆M_{1} ⊠ M_{2} ⊠ ⋆M_{2} ⊠ M_{3})  (40)

There is a canonical map RHom_{D}(⋆M_{1} ⊠ M_{2}, C^{∞}(X_{1} × X_{2})) ⊗ RHom_{D}(⋆M_{2} ⊠ M_{3}, C^{∞}(X_{2} × X_{3})) =

RHom_{D}(⋆M_{1} ⊠ M_{2} ⊠ ⋆M_{2} ⊠ M_{3}, C^{∞}(X_{1} × X_{2} × X_{2} × X_{3})) ⟶^{G_{M}×γ_{2}}

RHom_{D}(⋆M_{1} ⊠ M_{3}, C^{∞}(X_{1} × X_{3}))

provided by the morphism (40) and morphism

γ_{2} ∈ R^{−l_{2}}Hom_{D}(π_{2}_{∗}Δ_{2}^{!}C^{∞}(X_{1} × X_{2} × X_{2} × X_{3}), C^{∞}(X_{1} × X_{3})) Theorem 5.2 The kernel K_{23} ∘ K_{12} coincides with (G_{M} × γ_{2})(K_{23} ⊗ K_{12}).

Proof. Follows immediately from the definitions.

Example. Suppose M_{i} are excellent D_{X_{i}}-modules. In this case usually the natural smooth kernels are just functions

K_{12}(x_{1}, x_{2}) ∈ Hom_{D}(˜⋆M_{1} ⊠ M_{2}, C^{∞}(X_{1} × X_{2}))

and

K_{23}(x_{2}, x_{3}) ∈ Hom_{D}(˜⋆M_{2} ⊠ M_{3}, C^{∞}(X_{1} × X_{2}))

and the composition is defined by the natural kernel K_{13}(x_{1}, x_{3}) = ∫_{γ_{2}} G_{M_{2}}(K_{12}(x_{1}, x_{2}), K_{23}(x_{2}, x_{3}))

6 Natural linear maps for excellent D-modules.

1. The general scheme. Let M and N be excellent D-modules on manifolds X and Y, i.e. ˜⋆M := (⋆M)[−d_{M}] is a D-module. Let c_{M} := codimSSM = dimX − d_{M}. Then solutions

f ∈ Hom_{D}(M, C^{∞}(X)) and g ∈ Hom_{D}(˜⋆M, D′(X))

provide a homomorphism

H^{c_{M}}DR(˜⋆M ⊗_{O} M) ⟶^{f̄⊗ḡ} H^{c_{M}}DR(C^{∞}(X) ⊗_{O} D′(X)) ⟶^{m} H^{d_{M}}D^{′•}(X)

The Green class of M goes under this map to a cohomology class of degree d_{M} on X. Recall that we put DR(M) in degrees [−dimX, 0], while the smooth de Rham complex A^{•}(X) is sitting in degrees [0, dimX].

Let us define a natural linear map

I : Sol(M, C^{∞}(X)) ⟶ Sol(N , D′(Y))

by a kernel

K_{I}(x, y) ∈ Sol(˜⋆M ⊠ N , D′(X × Y))  (41)

and a cycle γ_{X} of dimension d_{M} in X as follows. Let G̃_{M}(·, ·) be a cocycle in the Cech complex of a covering of of X with coefficients in DR(˜⋆M ⊗_{O} M) representing the Green class. (In integral geometry one may usually take a cocycle in the complex DR(˜⋆M⊗_{O}M)). Using solutions K_{I}(x, y) of ˜⋆M (where y is considered as a parameter) and f(x) of M we get a closed differential form G̃_{M}(K_{I}(x, y), f(x)) of degree d_{M} on X. Set

f(x) ⟼ ∫_{γ} G̃_{M}(K_{I}(x, y), f(x)) ∈ Sol(N , D′(Y))  (42)

Under certain assumption on the wave front of the kernel K_{I}(x, y), which we will assume below, the integral over cycle γ makes sense and the image of (41) lies in C^{∞}(Y). Then a (natural) inverse for I is an integral transformation

J : Sol(N , C^{∞}(Y)) ⟶ Sol(M, C^{∞}(X))
J : φ(x) ⟼ ∫_{γ_{Y}} G̃_{N} (K_{J}(x, y), φ(y))  (43)

defined via a certain d_{N} -cycle γ_{Y} in Y and a kernel

K_{J}(x, y) ∈ Sol(M ⊠ ˜⋆N , D′(X × Y))  (44)

This data defines also a transformation

J^{t} : Sol(˜⋆M, C^{∞}(X)) ⟶ Sol(˜⋆N , C^{∞}(Y))

g(x) ⟼ ∫_{γ_{X}} G̃_{M}(g(x), K_{J}(x, y))

There is a canonical map

< ·, ·, · >_{M} : Sol(˜⋆M, C^{∞}(X))⊗Sol(M, C^{∞}(X))⊗H_{d_{M}}(X, ℝ) ⟶ ℝ (45)
< g, f, γ_{X} >_{M} := ∫_{γ_{X}} G̃_{M}(g(x), f(x))

So if we choose a homology class γ_{X} we get a pairing

< g, f >_{M} :=< g, f, γ_{X} >_{M}  (46)

and a similar one for N.

Theorem 6.1 (the Plancherel formula) . Let J be a natural inverse for I: J ∘ I = id_{X}. Then for f ∈ Sol(M, C^{∞}(X)), g ∈ Sol(˜⋆M, C^{∞}(X)) one has

< g, f, γ_{X} >_{M} = < J^{t}g, If, γ_{Y} >_{N}

Proof. < g, f, γ_{X} >_{M} = < g, J ∘ If, γ_{Y} >_{N}. So the theorem follows from

Lemma 6.2 Let φ ∈ Sol(N , C^{∞}(Y)) and g ∈ Sol(˜⋆M, C^{∞}(X)). Then

< g, Jφ, γ_{X} >_{M} = < J^{t}g, φ, γ_{Y} >_{N}  (47)

Proof. The Green class is multiplicative with respect to the ⊠ - product. So we can set G̃_{M⊠N} := G̃_{M} ⊠ G̃_{N}. Consider the following solutions

g(x) ⊠ φ(y) ∈ Sol(˜⋆M ⊠ ˜⋆N , C^{∞}(X × Y))
K_{J}(x, y) ∈ Sol(M ⊠ N , D′(X × Y))

They are solutions to the dual systems. So there is a pairing

< g(x) ⊠ φ(y), K_{J}(x, y), γ_{X} × γ_{Y} >_{M⊠N}

We can evaluate it computing first the pairing along X and then along Y. In this case we get the right-hand side of (47). Computing first pairing along Y and then along X we get the left-hand side of (47).

The kernel K_{J} is a much more simple (and fundamental) object then the actual integral transformation J. The reasons are the following:

1) The kernel K_{J} is a canonically defined distribution, while the formula for Jφ(x) depends on a cocycle G̃_{N} representing the Green class.

2) Explicit calculation of cocycle G̃_{N} can be a nontrivial problem and so the final formula for the right-hand side of (43) could be quite complicated even for a very simple kernel K_{J}.

So the problem of inversion of the transformation I splits on 3 steps:

Step 1. Find a distribution (44).

Step 2. Compute a cocycle G̃_{N} for the Green class.

Step 3. Find a cycle γ_{Y}.

The distribution (44) should be uniquely defined if exist. However it may not exist. The Green class always exist. Different cocycles representing it together with different choices of cycles γ_{Y} provides the diversity of concrete inversion formulas. I will demonstrate below how this general scheme works in the simplest concrete problems.

2. The Fourier transform of homogeneous functions and the Radon transform. As everybody knows the Fourier transform in an n-dimensional real vector space V_{n} is defined by the formula

S(V_{n}) ⟶ S(V_{n}^{∗}); f(x) ⟶ f̃(ξ) := ∫ f(x)e^{2πi<x,ξ>}d^{n}x

The inverse operator is f(ξ) ⟶ ∫ f(ξ)e^{−2πi<x,ξ>}d^{n}ξ. Using the Plancherel formula one can define the Fourier transform of generalized functions.

Let Φ_{λ}^{+}(ℝP^{n−1}) (resp. Φ_{λ}^{−}(ℝP^{n−1})) be space of even (resp. odd) smooth homogeneous functions f(x) on ℝ^{n}\0 of degree λ: f(ax) = |a|^{λ}f(x), a > 0. Also let Ψ_{λ}(V_{n}) be the space of homogeneous distribution of degree λ in V_{n}, and Ψ_{λ}(V_{n}) = Ψ_{λ}^{+}(V_{n}) ⊕ Ψ_{λ}^{−}(V_{n}) is the decomposition on even and odd parts.

Then Φ_{λ}(ℝP^{n−1}) ⊂ Ψ_{λ}(V_{n}). This inclusion is not an isomorphism for integral λ = −k, k ≥ n. One has

Ψ_{−k}(V_{n})/Φ_{−k}(ℝP^{n−1}) = S^{k−n}(V_{n}) = {δ − functions of degree −k at 0}

The Fourier transform of generalized functions provides an isomorphism

F̃_{λ} : Ψ_{−λ−n}^{±}(V_{n}) ⟶ Ψ_{λ}^{±}(V′_{n})

Restricting to Φ_{−λ−n}^{±}(ℝP^{n−1}) we get a map

F_{λ} : Φ_{−λ−n}^{±}(ℝP^{n−1}) ⟶ Φ_{λ}^{±}((ℝP^{n−1})′)

It is remarkable that there is another way to define the operator F_{λ}. Let me recall that the space of homogeneous degree λ generalized function on ℝ is 2-dimensional and splits on the even and odd components (with respect to the involution x → −x) generated by

\frac{|x|^{λ}}{Γ(\frac{λ+1}{2})} and \frac{|x|^{λ}sgnx}{Γ(\frac{λ+2}{2})}

They are both analytic on λ on the whole complex plane. One has

\frac{|x|^{λ}}{Γ(\frac{λ+1}{2})}|_{λ=−2k−1} = \frac{(−1)^{k}k!}{(2k)!} · δ^{(2k)}(x);  (48)

\frac{|x|^{λ}sgnx}{Γ(\frac{λ+2}{2})}|_{λ=−2k} = \frac{(−1)^{k}(k − 1)!}{(2k − 1)!} · δ^{(2k−1)}(x)  (49)

Let γ_{n−1} be a cycle generating H_{n−1}(ℝ^{n}\0; ℤ). The kernel

K_{λ}^{+}(ξ, x) := \frac{| < ξ, x > |^{λ}}{Γ(\frac{λ+1}{2})}

and the cycle γ_{n−1} defines the operator

I_{λ}^{+} : Φ_{−λ−n}(ℝP^{n−1}) ⟶ Φ_{λ}((ℝP^{n−1})′)

I_{λ}^{+} : f(x) ⟶ \frac{1}{2} ∫_{γ_{n−1}} f(x)\frac{| < ξ, x > |^{λ}}{Γ(\frac{λ+1}{2})} σ_{n}(x, dx)

The odd kernel

K_{λ}^{−}(ξ, x) := \frac{| < ξ, x > |^{λ} · sgn(< ξ, x >)}{Γ(\frac{λ+2}{2})}

defines an integral transformation

(I_{λ}^{−}f)(ξ) = ∫_{γ_{n−1}} f(x)K_{λ}^{−}(ξ, x)σ_{n}(x, dx)

Proposition 6.3 F_{λ}^{+} = π^{1/2+λ}Γ(\frac{−λ}{2}) · I_{λ}^{+}, F_{λ}^{−} = i · π^{1/2+λ}Γ(\frac{−λ}{2}) · I_{λ}^{−}.

Set

f_{λ}(x) := π^{λ/2} \frac{|x|^{λ}}{Γ(\frac{λ+1}{2})}, g_{λ}(x) := π^{λ/2} \frac{|x|^{λ}sgn(x)}{Γ(\frac{λ+2}{2})}

Lemma 6.4

F(f_{λ}(x)) = f_{−1−λ}(ξ), F(g_{λ}(x)) = i · g_{−1−λ}(ξ)  (50)

Proof. See p. 173 in [GS] for an equivalent formula.

Proof of the proposition. Using the polar coordinates x = r · s where s ∈ S^{n−1}, |s| = 1, we have d^{n}x = r^{n−1}drds_{n−1} where ds_{n−1} is the standard volume form on the unit sphere in ℝ^{n}. Then for f(x) ∈ Φ_{−λ−n}(ℝP^{n−1}) we have

∫_{ℝ^{n}} f(x)e^{2πi<ξ,x>}d^{n}x = \frac{1}{2} ∫_{−∞}^{∞} |r|^{−λ−n}e^{2πir·(ξ,s)}r^{n−1}dr ∫_{S^{n−1}} f(e)ds_{n−1} =

\frac{1}{2}π^{λ+1/2}Γ(\frac{−λ}{2}) ∫_{S^{n−1}} f(e)\frac{| < ξ, s > |^{λ}}{Γ(\frac{λ+1}{2})} ds_{n−1} = π^{−λ−1/2}Γ(\frac{−λ}{2})I_{λ}^{+}f

The proof in the case of odd functions is completely similar.

Corollary 6.5

I_{−λ−n}^{+} ∘ I_{λ}^{+} = \frac{π^{n}}{Γ(\frac{−λ}{2})Γ(\frac{λ+n}{2})} · Id

I_{−λ−n}^{−} ∘ I_{λ}^{−} = \frac{π^{n}}{Γ(\frac{−λ}{2})Γ(\frac{λ+n}{2})} · Id

In particularly using (48) we see that I_{−1} is just a projectively invariant version of the Radon transform:

(I_{−1}f)(ξ) = ∫_{γ_{m}} f(x)δ(< ξ, x >)σ_{m+1}(x, dx)

and the inversion formula looks as follows. When n is even

f(y) = \frac{(−1)^{\frac{n−2}{2}}}{2(2π)^{n−2}} ∫_{γ_{n−1}} f̂(ξ)δ^{(n−2)}(< ξ, y >)σ_{n}(ξ, dξ)

When n is odd

f(y) = \frac{(−1)^{m/2−1}(n − 2)!}{2(2π)^{n−1}} ∫_{γ_{n−1}} f̂(ξ)(< ξ, y >)^{−n+1}σ_{n}(ξ, dξ)

The operator I_{−λ−n}^{+} is defined on the space of all homogeneous degree λ functions. However it is zero on the subspace of odd functions. The reason is this. A sphere in ℝ^{n}\0 representing the generator in H_{n−1}(ℝ^{n}\0) has canonical coorientation ”out of the origin”. The involution x → −x preserves it. So it acts on the class γ_{n−1} in the same way as it acts on the orientation class of ℝ^{n} and hence on the form σ_{n−1}(x, dx): by multiplication by (−1)^{n}. So if f(x) is an odd function the integral ∫_{γ_{n−1}} f(x)σ_{n}(ξ, dξ) vanish because the contributions of the opposite parts of the sphere cancel each other.

From our point of view these results looks as follows. Let

L_{λ} := ∑_{i=1}^{n} x_{i}∂_{x_{i}} − λ

be the Euler operator. Denote the corresponding D-module by L_{λ}. Then Φ_{λ}(ℝP^{n−1}) is the space of smooth even solutions of L_{λ}.

It follows from lemma (4.1) that ⋆L_{λ} = L_{−λ−n}[1] and the Green class of L_{λ} is

G_{L_{λ}}(φ; v) = φ · v · σ_{n}(x, dx)

So pairing (46) looks in this case as follows:

Φ_{λ}(ℝP^{n−1}) ⊗ Φ_{n−λ}(ℝP^{n−1}) ⟶ ℝ

f(x) ⊗ g(x) ⟼ ∫_{γ_{n−1}} f(x)g(x)σ_{n}(x, dx)

Notice that

K_{λ}^{±}(x, ξ) ∈ Sol(L_{λ} ⊠ L_{λ}, D′(ℝ^{n}\0 × ℝ^{n}\0))^{±}  (51)

One has L_{λ} = ˜⋆L_{−λ−n}, so the integral transformation I_{λ}^{±} is just the natural linear map provided by the kernel (51).

3. The complex space. Let λ and μ be complex numbers such that n := λ − μ is an integer. Let

Φ_{λ,μ}(ℂP^{m}) := {f | f(az, āz̄) = a^{λ}ā^{μ}f(z, z̄)}

be the space of smooth homogeneous function in ℂ^{m+1}\0 of the bidegree λ, μ. Consider the kernel

K_{λ}^{ℂ}(x, ξ) = \frac{< ξ, x >^{λ} · < ξ̄, x̄ >^{μ}}{Γ(\frac{s+|n|+2}{2})}

where s = λ+μ. It is a homogeneous generalized function. It defines an integral transformation

I_{λ,μ}^{ℂ} : Φ_{−λ−m−1,−μ−m−1}(ℂP^{m}) ⟶ Φ_{λ,μ}((ℂP^{m})′)
f(z, z̄) ⟶ ∫_{ℂP^{m}} f(z, z̄)K_{λ}^{ℂ}(z, ξ)σ_{m}(z, dz) ∧ σ_{m}(z̄, dz̄)

Here the integral has the following meaning. The form we integrate can be pushed down to ℂP^{m}, so we integrate over the fundamental cycle. One has (see [GGV]).

K_{λ}^{ℂ}(z, ξ)|_{λ=−k−1,μ=−l−1} = \frac{π(−1)^{k+l+1}j!}{k!l!}δ^{k,l}(z, z̄)

where j = min(k, l). In particularly applying the above results to the case k = 0, l = 0 we come to the Radon transform of smooth homogeneous functions of degree (−m, −m) in ℂP^{m}:

I_{m}^{ℂ} : Φ_{−m,−m}(ℂP^{m}) ⟶ Φ_{−1,−1}((ℂP^{m})′)
(I_{m}^{ℂ}f)(ξ) = f̂(ξ) = (i/2)^{m} ∫_{ℂP^{m}} f(x)δ(< ξ, x >)σ_{m}(x, dx) ∧ σ_{m}(x̄, dx̄)

The projectively invariant inversion formula is

(J_{m}^{ℂ}f)(y) = c_{m}^{ℂ} ∫_{ℂP^{m}} f̂(ξ)δ^{(m−1,m−1})(< ξ, y >)σ_{m}(ξ, dξ) ∧ σ_{m}(ξ̄, dξ̄)

where c_{m}^{ℂ} = (−1)^{m−1}(m − 1)!(π)^{−2m+2}(i/2)^{m}.

7 Integral geometry on the family of spheres

1. The integral transformation. Let

S^{m} = {x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2} = 0}/ℝ^{∗}

be a sphere in ℝP^{m+1}. The stereographic projection identifies the family of its hyperplane sections with the family of all spheres in ℝ^{m}.

Let Q_{m+1} := {x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2} = 0} be a cone in ℝ^{m+2}\0. It has two connected components: Q_{m+1}^{+} in the half space x_{m+2} > 0 and its opposite Q_{m+1}^{−}.

Denote by Φ_{λ}^{+}(S^{m}) (resp. Φ_{λ}^{−}(S^{m})) the space of all even (resp. odd) smooth homogeneous functions of degree λ on the cone Q_{m+1}.

Let β_{m} be a hyperplane section of Q_{m+1}^{+} which is isomorphic to a sphere. The orientation of ℝ^{m+2} provides an orientation of β_{m}: the cycle β_{m} is cooriented out of the origin in the cone, and the cone itself coorientated outside of the convex component in ℝ^{m+2}. Let β_{m}^{+} be an oriented this way cycle. Its homology class is a generator of H_{m}(Q_{m+1}^{+}, ℤ).

The group O(m + 1, 1) has 2 connected component when m is odd and 4 when m is even. Namely in the latter case the involution −Id has positive determinant and interchanges the cones Q_{m+1}^{+} and Q_{m+1}^{−}.

Let ε(1) be the 1-dimensional O(m+1, 1)-module which is trivial on the connected component of the unity such that the elements −Id and diag(−1, 1, ..., 1) acts by multiplication by −1.

Lemma 7.1 There is a nondegenerate O(m + 1, 1) - invariant pairing

< ·, · >_{S^{m}}: Φ_{−λ−m}^{+}(S^{m}) ⊗ Φ_{λ}^{−}(S^{m}) ⟶ ε(1)

defined by the formula

< f, g >_{S^{m}}:= ∫_{β_{m}^{+}} δ(x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2})f(x)g(x)σ_{m+2}(x, dx)

Here we integrate the closed m-form on Q_{m+1}^{+}. By definition it is the restriction to Q_{m+1}^{+} of any form α_{m} satisfying the condition

d(x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2}) ∧ α_{m} = f(x)g(x)σ_{m+2}(x, dx)

The restriction is well defined on Q_{m+1}^{+}.

Proof. The SO(m+1, 1)_{0} - invariance is obvious. The involution diag(−1, 1, ..., 1) multiplies the class [β_{m}^{+}] as well as the form σ_{m+2}(x, dx) by (−1) and changes orientation of the space. The involution −Id multiplies the orientation of the space and the form σ_{m+2}(x, dx) by the same factor (−1)^{m+2} and multiplies f ·g by (−1). In particularly this integral is zero if f(x)g(x) is an even function.

Let ξ_{1}, ..., ξ_{m+2} be coordinates in (ℝ^{m+2})′ dual to x_{i} and < ξ, x >= ∑ ξ_{i}x_{i}. Consider the kernel

K_{λ}^{+}(ξ, x) := \frac{| < ξ, x > |^{λ}}{Γ(\frac{λ+1}{2})}  (52)

Set

Δ := ∂_{ξ_{1}}^{2} + ... + ∂_{ξ_{m+1}}^{2} − ∂_{ξ_{m+2}}^{2}; L_{λ} := ∑_{i=1}^{m+2} ξ_{i}∂_{ξ_{i}} − λ

Let us denote by M_{λ} the D-module on ℝ^{m+2} corresponding to the system

M_{λ} : L_{λ}f = 0, (x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2})f = 0

and by N_{λ} the D-module on (ℝ^{m+2})′corresponding to the system of differential equations

N_{λ} : L_{λ}φ = 0, Δφ = 0

Then

K_{λ}^{+}(ξ, x) ∈ Sol(M_{λ} ⊠ N_{λ}, D′(ℝ^{m+2} × (ℝ^{m+2})′))^{+}

is an even solution of this system. Notice that M_{λ} = ˜⋆M_{−λ−m}. So the kernel K_{λ}^{+}(ξ, x) defines an operator

I_{λ}^{+} : Φ_{−λ−m}^{+}(S^{m}) ⟶ Sol(N_{λ})^{+};  (53)

(I_{λ}^{+}f)(ξ) = ∫_{β_{m}^{+}} δ(x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2})f(x)K_{λ}^{+}(ξ, x)σ_{m+2}(x, dx)

Consider the following domain:

Γ̃_{0} := {ξ|ξ_{1}^{2} + ... + ξ_{m+1}^{2} − ξ_{m+2}^{2} = 0}  Γ̃_{1} := {ξ|ξ_{1}^{2} + ... + ξ_{m+1}^{2} > ξ_{m+2}^{2}}

Remark. The functions I_{λ}^{±}f(ξ) are a priory smooth only in the complement to the cone Γ̃_{0}. Indeed, the integral transform I_{λ}^{+}, for instance, is written in affine coordinates as

I_{λ}^{+}(f)(ξ) = ∫ f(x_{1}, ..., x_{m+1})δ(x_{1}^{2} + ... + x_{m+1}^{2} − 1)\frac{| < ξ′, x > +s|^{λ}}{Γ(\frac{λ+1}{2})} d^{m+1}x

where ξ = (ξ′, s) and < ξ′, x >= ∑ ξ_{i}x_{i}. Set ξ_{1} = 1, ξ_{i} = 0 for i > 1. Then

I_{λ}^{+}(f)(1, 0, ..., 0; s) = ∫ f̃(x_{1})\frac{|x_{1} + s|^{λ}}{Γ(\frac{λ+1}{2})} dx_{1}

where f̃(x_{1}) := ∫ f(x)δ(x_{1}^{2} + ... + x_{m+1}^{2} − 1)dx_{2}...dx_{m+1}. The function f̃(x_{1}) vanishes outside of the segment [−1, 1], smooth inside of it but not smooth near x_{1} = ±1. The integral ∫ |x|^{λ}f(x)dx is regularized near x = 0 in assumption that the function f(x) is smooth near zero.

Similarly the kernel

K_{λ}^{−}(ξ, x) := \frac{| < ξ, x > |^{λ} · sgn(< ξ, x >)}{Γ(\frac{λ+2}{2})}

is an odd solution of the system M_{λ} ⊠ N_{λ}. It defines an intertwiner operator for the group O(m + 1, 1):

I_{λ}^{−} : Φ_{−λ−m}^{−}(S^{m}) ⟶ Sol(N_{λ})^{−};

Notice that

Φ_{λ}^{±}(S^{m}) = Sol(M_{λ}|_{ℝ^{m+2}\0}, C^{∞}(ℝ^{m+2}\0))^{±}

So the operators I_{λ}^{±} are natural linear operators between smooth solution spaces.

In this chapter we will work with the restriction of the functions I_{λ}^{±} to the domain Γ̃_{1}. Our first goal is to invert the operator

I_{λ}^{±} : Φ_{−λ−m}^{±}(S^{m}) ⟶ Sol(N_{λ}|Γ̃_{1})^{±};

2. The Green class. Now we make the crucial step. Consider the following m-form:

ω_{m}(φ; v) :=  (54)

∑_{1≤i<j≤m+2}(−1)^{i+j−1}(ξ_{i}·ε_{j}(v·φξ_{j}^{′}−vξ_{j}^{′}·φ)−ξ_{j}·ε_{i}(v·φξ_{i}^{′}−vξ_{i}^{′}·φ))dξ_{1}∧...d̂ξ_{i}...d̂ξ_{j}...∧dξ_{m+2}

Here ε_{m+2} = −1 and ε_{j} = 1 if j ≠ m + 2. Let ω_{m+1}(φ; Δ; v) be the Green form for the Laplacian Δ:

ω_{m+1}(φ; Δ; v) = ∑_{1≤j≤m+2}(−1)^{j−1}ε_{j}(φ_{ξ_{j}} · v − φ · v_{ξ_{j}})dξ_{1} ∧ ...d̂ξ_{j}... ∧ dξ_{m+2} (55)

Then (54) is the contraction of the Green form (55) with the Euler vector field L:

ω_{m}(φ; v) = −\frac{1}{2}i_{L}ω_{m+1}(φ; Δ; v)

Remark. More generally, for any homogeneous differential operator P with constant coefficients in ℝ^{n} the Green form for the system P f = 0, L_{a}f = 0 is equal to −\frac{1}{2}i_{L}ω_{n−1}(φ; Δ; v).

It can be also written as follows:

ω_{m}(φ; v) = [ξv, ε · \frac{∂}{∂ξ} φ, dξ, ..., dξ] − [ξφ, ε · \frac{∂}{∂ξ} v, dξ, ..., dξ]

Here [ξv, ε · \frac{∂}{∂ξ} φ, dξ, ..., dξ] means the determinant of the following matrix:

( \begin{matrix}ξ_{1}v & ε_{1} · φξ_{1}^{′} & dξ_{1} & ... & dξ_{1} \\ ξ_{2}v & ε_{2} · φξ_{2}^{′} & dξ_{2} & ... & dξ_{2} \\ ... & ... & ... & ... & ... \\ ξ_{m+2}v & ε_{m+2} · φξ_{m+2}^{′} & dξ_{m+2} & ... & dξ_{m+2} \end{matrix} )

Lemma 7.2 The form ω_{m}(φ; v) can be pushed down to Γ

Proof. An easy calculation.

Theorem 7.3 a) N_{a} = ˜⋆N_{b} where a + b + m = 0.

b) The form ω_{m}(φ; v) represents the Green class G_{N_{a}}(φ; v) of the system N_{a}.

Corollary 7.4 The form ω_{m}(φ; v) is closed if the functions φ and v satisfy the following systems of differential equations:

L_{a}φ = 0, Δφ = 0 and L_{b}v = 0, Δv = 0 where a + b + m = 0.

In the rest of this chapter we will use extensively this corollary (but not the fact that ω_{m}(φ; v) represents the Green class). So let me first give a straightforward proof independent of the proof of theorem (7.3).

Proof. One has

dω_{m}(v; φ) = ∑_{1≤j≤m+2}(−1)^{j−1}(ξ_{j}(v · Δφ − Δv · φ)−

ε_{j} · L_{−m−1}(v · φξ_{j}^{′} − vξ_{j}^{′} · φ))dξ_{1} ∧ ... ∧ d̂ξ_{j} ∧ ... ∧ dξ_{m+2}

Indeed, applying \frac{∂}{∂ξ_{i}} dξ_{j} to ω_{n−2}(v; φ) we get

∑_{1≤i<j≤m+2}(−1)^{j−1}(ξ_{j}(v · φξ_{i}ξ_{i}^{′′} − vξ_{i}ξ_{i}^{′′} · φ) − ξ_{i} \frac{∂}{∂ξ_{i}} ε_{j} · (v · φξ_{j}^{′} − φ · vξ_{j}^{′})−

−(j − 1)ε_{j} · (v · φξ_{j}^{′} − φ · vξ_{j}^{′}))dξ_{1} ∧ ... ∧ d̂ξ_{j} ∧ ... ∧ dξ_{m+2}

Similarly we compute the contribution of \frac{∂}{∂ξ_{j}} dξ_{j} and take the sum.

Proof of theorem (7.3). Consider a complex of D - modules D ⟶^{d} D^{2} ⟶^{d} D sitting in degrees [-2,0] (d has degree +1) which we visualize as:

D
Δ ↗  ↘^{L_{a}}
D  D
−L_{a−2} ↘  ↗_{Δ}
D

One has [L_{a}, Δ] = −2Δ, so ΔL_{−1} + (−L_{−3})Δ = 0, i.e. we get a complex. This is a resolution of the D- module N_{a}. Indeed, consider a filtration on D

such that the degree of x and \frac{∂}{∂x} is +1. Then both L_{a} and Δ have degree +2. Shifting the filtration in the second term of the resolution down by 2 and in the third down by 4 we get a filtered complex. The associated graded quotient complex is a Koszule resolution. So our complex is also a resolution. The part a) follows easyly from this..

To calculate the Green class we use theorem (??) for this resolution. The complex ∗P^{•} := Hom_{D}(P^{•}, D^{Ω})[m+2] is concentrated in degrees [−(m+2), −m] and looks as follows:

D^{Ω}
Δ ↙  ↖^{L_{a}^{∗}}
D^{Ω}  D^{Ω}
L_{a−2}^{∗} ↖  ↙_{Δ}
D^{Ω}

A homomorphism of D-modules D_{X} ⟶ C^{∞}(X) is determined by its value at 1 ∈ D_{X}. So one can represent the complex Hom_{D}(P^{•}, C^{∞}(ℝ^{m+2})) by the following picture where φ_{0}, φ_{1}, φ′_{1}, φ_{2} are the values of the corresponding homomorphisms at 1:

φ_{1}

Δ ↙  ↖^{L_{a}}
φ_{2}  φ_{0}
−L_{a−2}^{∗} ↖  ↙_{Δ}

φ1^{′}

Similary one can make a picture for Hom_{D}(⋆P^{•}, C^{∞}(ℝ^{m+2})):

v_{m+1}

Δ ↗  ↘^{L_{a}^{t}}
v_{m}  v_{m+2}
−L_{a−2}^{t} ↘  ↗_{Δ}

vm+1^{′}

Recall that Hom_{D}(D^{Ω}, C^{∞}(X)) = A^{m}(X). So v′s are forms of top degree. Then

(L_{a}φ_{0} · v_{m+1} − φ_{0} · L_{a}^{t}v_{m+1}) + (Δφ_{0} · vm+1^{′} − φ_{0} · Δvm+1^{′}) =

dω_{m+1}(φ_{0}; v_{m+1}, vm+1^{′})

and

(Δφ_{1} · v_{m} − φ_{1} · Δv_{m}) + ((−L_{a−2})φ1^{′} · v_{m} − φ1^{′} · (−L_{a−2})^{t}v_{m}) =

dω_{m+1}(φ_{1}, φ′_{1}; v_{m})

where

ω_{m+1}(φ_{0}; v_{m+1}, vm+1^{′}) := φ_{0} · v_{m+1}σ_{m+2}(ξ, dξ)+

∑_{i=1}^{n} ε_{i}((\frac{∂φ_{0}}{∂ξ_{i}} · vm+1^{′} − φ_{0} · \frac{∂vm+1^{′}}{∂ξ_{i}} )(−1)^{i−1}dξ_{1} ∧ ... ∧ d̂ξ_{i} ∧ ... ∧ dξ_{m+2}

and

ω_{m+1}(φ_{1}, φ1^{′}; v_{m}) := −φ1^{′} · v_{m}σ_{m+2}(ξ, dξ)+

∑_{i=1}^{m+2} ε_{i}(\frac{∂φ_{1}}{∂ξ_{i}} v_{m} − φ_{1} \frac{∂v_{m}}{∂ξ_{i}} )(−1)^{i−1}dξ_{1} ∧ ... ∧ d̂ξ_{i} ∧ ... ∧ dξ_{m+2}

Further,

ω_{m+1}(φ_{0}; Δv_{m}, −L_{a−2}^{t}v_{m}) + ω_{m+1}(L_{a}φ_{0}, Δφ_{0}; v_{m}) = dω_{m}(φ_{0}; v_{m})

where ω_{m}(φ; v) is the Green form (54).

3. Construction of the inverse operator. We have defined in s. 7.1 the domain Γ̃_{1} = {ξ_{1}^{2} + ... + ξ_{m+1}^{2} > ξ_{m+2}^{2}}. Let Γ_{1} = Γ̃_{1}/ℝ_{+}^{∗} be the manifold of all oriented rays inside Γ̃_{1}. Its closure Γ = Γ_{0} ∪ Γ_{1} parametrizes oriented hyperplane sections of the sphere S^{m} (here Γ_{0} = Γ̃_{0}/ℝ_{+}^{∗}).

Γ = S^{m+1}\D_{+} ∪ D_{−} where D_{+} is a ball {ξ_{1}^{2} + ... + ξ_{m+1}^{2} < ξ_{m+2}^{2}}/(ℝ^{∗})^{+} and D_{−} = −D_{+}. Therefore H_{m}(Γ, ℤ) = ℤ. Consider the cycle γ_{m} of rays in the hyperplane ξ_{m+2} = 0. It is cooriented by the function ξ_{m+2} (or, more invariantly, by the choice of one of the balls D_{+}). So orientation of ℝ^{m+2} provides an orientation of the cycle. Denote by γ_{m}^{+} the oriented this way cycle. Its homology class is a generator of H_{m}(Γ, ℤ).

There is a nondegenerate pairing

< ·, · >_{N_{λ}}: Sol(N_{λ})^{+} ⊗Sol(N_{−λ−m})^{−} ⟶ ε(1); < φ, v >_{N_{λ}}:= ∫_{γ_{m}^{+}} ω_{m}(φ, v)

Indeed, the involution ξ ⟼ −ξ multiplies the form ω_{m}(φ, v) by (−1)^{m+2} and the cycle γ_{m} by (−1)^{m+1}. The involution diag(−1, 1, ..., 1) multiplies by (−1) the orientation of the space, cycle γ_{m}^{+} and the form ω_{m}(φ, v). This provides the extra factor ε(1) in the above formulas.

Let K be a compact hypersurface in Γ. Its homology class [K] ∈ H_{n}(Γ) is equal to d(K) · [γ_{M}^{+}]. The integer d(K) is the intersection number of the class [K] with the cycle consisting of all oriented spheres passing through a given point x ∈ S^{m} and tangent to a given hyperplane in T_{x}S^{m}.

According to part a) of theorem (7.3) ˜⋆N_{λ} = N_{−m−λ}. Notice that Φ_{λ}^{−}(S^{m})⊗ ε(1) = Φ_{λ}^{+}(S^{m}). So by the general philosophy the kernel K_{λ}^{−}(ξ, x) define integral operators

J_{λ}^{+} : Sol_{C^{∞}}(N_{−λ−m})^{+} ⟶ Φ_{λ}^{−}(S^{m}) ⊗ ε(1) = Φ_{λ}^{+}(S^{m})  (56)

(J_{λ}^{+}φ)(ξ) = \frac{1}{2} ∫_{K} ω_{m}(φ; K_{λ}^{−}(ξ, x))

and similarly the even kernel K_{λ}^{+}(ξ, x) provides operators

J_{λ}^{−} : Sol_{C^{∞}}(N_{−λ−m})^{−} ⟶ Φ_{λ}^{+}(S^{m}) ⊗ ε(1) = Φ_{λ}^{−}(S^{m})

Notice that J_{λ}^{+} is defined by an odd kernel, and J_{λ}^{−} by an even kernel. Further, there are operators

(J_{−λ−m}^{±})^{t} : Φ_{λ}^{∓}(S^{m}) ⟶ Sol_{C^{∞}}(N_{−λ−m})^{∓}

(J_{−λ−m}^{±})^{t}(g) = ∫_{β_{m}^{+}} δ(x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2})g(x)K_{λ}^{∓}(x, ξ)σ_{m+2}(x, dx)

Theorem 7.5 a) These operators are intertwiners for the group O(m + 1, 1). b) For any m-cycle K ∈ Γ one has

d(K)· < f, g >_{S^{m}}= c(λ) · ∫_{K} ω_{m}(I_{λ}^{±}f; (J_{−λ−m}^{±})^{t}g)

where

c(λ) = \frac{π^{m+1}}{Γ(\frac{−λ}{2})Γ(\frac{λ+m+2}{2})}

c) In particular d(K) · J_{−λ−m}^{±} ∘ I_{λ}^{±} = c(λ) · Id.

The part b) can be viewed as the universal form of the Plancherel theorem for the integral transformation I_{λ}^{±}.

Proof. a) The operator J_{λ}^{±} is an intertwiner for the following three reasons.

1. A group element g ∈ SO(m + 1, 1)_{0} sends form ω_{m}(φ, v) to the form ω_{m}(g · φ, g · v). Indeed, the form ω_{m} is a cocycle representing the Green class for the system N_{λ}. This system as well as the volume form in ℝ^{m+2} is invariant under the action of the group SO(m + 1, 1)_{0}.

2. A connected Lie group acts trivially on the homology.

3. One can show that J_{λ}^{±} commutes with −Id and diag(−1, 1, ..., 1).

In the definition of the inverse operator J_{λ}^{−} we can integrate over an mcycle K̃ ⊂ (ℝ^{m+2})′ projecting to K. So J_{λ}^{±} apriory defined for any smooth function φ(ξ). However it commutes with the group action only on the subspace Sol(N_{λ}, C^{∞}(ℝ^{m+2})). Indeed, g moves the cycle K̃ to a different cycle gK̃ homologous to the initial one. To compare the integrals we use the Stokes formula for the form ω_{m}(φ; K_{λ}(ξ, x)). The integrals will be the same only if the form is closed. This happened only if φ(ξ) ∈ Sol_{C^{∞}}(N_{λ}).

b). Let n = (0 : .. : 0 : 1 : 1) be the “North pole”in S^{m}. The variety Γ_{n} parametrizing the hyperplane sections of the sphere S^{m} passing through the point n is a hyperplane given by equation ξ_{m+1} + ξ_{m+2} = 0.

It is sufficient to prove these formulas for one cycle K. The following lemma shows that for K = Γ_{n} they reduce to the Plancherel theorem and the inversion formula for the Radon transform in the projective space (see s. 6.2-6.3). Set ξ′ = (ξ_{1}, ..., ξ_{m+1}).

Lemma 7.6 . The restriction of the form ω_{m}(φ; v) to Γ_{n} is equal to

ω_{m}(φ; v)|_{Γ_{n}} = (−1)^{m+1}(v·(∂_{ξ_{m+1}}−∂_{ξ_{m+2}})φ−φ·(∂_{ξ_{m+1}}−∂_{ξ_{m+2}})v)σ_{m}(ξ′, dξ′)∧dξ_{m+1} Integrating by parts we get 2 · (−1)^{m}(φ · (∂_{ξ_{m+1}} − ∂_{ξ_{m+2}})v)σ_{m}(ξ′, dξ′)

4. An example: the Radon transform along the hyperplane sections of a sphere . The generalized functions (52) has no poles on λ. One has

K_{−(2k+1)}^{+}(ξ, x) = \frac{(−1)^{k}k!}{(2k)!} · δ^{(2k)}(< ξ, x >);

K_{−2k}^{−}(ξ, x) = \frac{(−1)^{k}(k − 1)!}{(2k − 1)!} · δ^{(2k−1)}(< ξ, x >)

K_{−2k}^{+}(ξ, x) = \frac{(−1)^{k}2^{k}\sqrt{π}}{(2k − 1)!!} < ξ, x >^{−2k}

K_{−(2k+1)}^{−}(ξ, x) = \frac{(−1)^{k}2^{k}\sqrt{π}}{(2k − 1)!!} < ξ, x >^{−2k−1}

So we get the following integral transformation. For f ∈ Φ_{1−m}^{+}(S^{m}) set

If(ξ) = ∫_{β_{m}^{+}} δ(x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2})f(x)δ(< ξ, x >)σ_{m+2}(x, dx)

The function If(ξ) is zero outside Γ. Consider the following kernel:

K_{−(m−1)}(ξ, x) := δ^{(m−2)}(< ξ, x >) for odd m and < ξ, x >^{−(m−1)} for even m

It defines an integral transformation acting on g ∈ Φ_{−1}^{−}(S^{m}):

(J^{t})g(ξ) = ∫_{β_{m}^{+}} δ(x_{1}^{2} + ... + x_{m+1}^{2} − x_{m+2}^{2})g(x)K_{−(m−1)}(< ξ, x >)σ_{m+2}(x, dx)

Theorem 7.7 a) For any m-cycle K ∈ Γ one has

d(K)· < f, g >_{S^{m}}= c_{m} · ∫_{K} ω_{m}(If; J^{t}g)

where −c_{m} = \frac{(−1)^{(m−1)/2}}{(2π)^{m−1}} for odd m and \frac{(−1)^{m/2}(m−1)!}{(2π)^{m}} for even m.

b) In particular

d(K) · f(x) = c_{m} · ∫_{K} ω_{m}(If; K_{−(m−1)}(ξ, x))  (57)

So the inversion formula is local for odd m and nonlocal for even m.

Theorem (7.7) is a special case of theorem (7.5).

The inverce operator J provided by the kernel K_{−(m−1)}(ξ, x) looks as follows:

(Jφ)(x) := ∫_{K} ∑_{1≤i<j≤n}(−1)^{i+j−1}(φ(ξ)(ξ_{i}x_{j} − ξ_{j}x_{i})δ^{(m−1)}(< ξ, x >)−

(ξ_{i}φξ_{j}^{′} − ξ_{j}φξ_{i}^{′})δ^{(m−2)}(< ξ, x >))dξ_{1} ∧ ... ∧ d̂ξ_{i} ∧ ... ∧ d̂ξ_{j} ∧ ... ∧ dξ_{n}

for odd m and

(Jφ)(x) := ∫_{K} ∑_{1≤i<j≤n}(−1)^{i+j−1}(φ(ξ)(ξ_{i}x_{j} − ξ_{j}x_{i}) < ξ, x >^{−m} −

(ξ_{i}φ_{ξ_{j}} − ξ_{j}φ_{ξ_{i}}) < ξ, x >^{−(m−1)})dξ_{1} ∧ ... ∧ d̂ξ_{i} ∧ ... ∧ d̂ξ_{j} ∧ ... ∧ dξ_{n}

for even m.

5. Admissible families of spheres. Restricting the integral operator I_{λ}^{±} to a family K of oriented spheres we get an integral transformation

I_{λ,K}^{±} : Φ_{λ}^{±}(S^{m}) ⟶ Ψ_{−λ−m}^{±}(K)

A priory the restriction of the form ω_{m}(φ; v) to a hypersurface K depends not only on the restriction of the functions φ and v on K, but on their first derivatives in the normal direction to K. Therefore for general K the right hand side of (57) can not be computed if know only I_{λ,K}^{±}(f). So it does not give an inversion formula for the integral transformation I_{λ,K}^{±}.

Definition 7.8 A hypersurfaces K ⊂ Γ is called admissible if the restriction of the form ω_{m}(φ; v) to K depends only on the restrictions of smooth solutions φ ∈ Sol_{C^{∞}}(N_{λ}), v ∈ Sol_{C^{∞}}(N_{λ}) to K.

This means that there exists a bidifferential operator ν : C^{∞}(K)^{⊗2} ⟶ A^{m}(K) such that for any φ, v as above ω_{m}(φ; v)|_{K} = ν(φ|_{K}, v|_{K}).

It is worth to compare this definition of admissibility with the one usually used in integral geometry, see [G3].

Let C be a submanifold in S^{m}. Consider the family Γ_{C} of oriented hyperplane sections of the sphere S^{m} tangent to C. For example if C is a point then d(Γ_{C}) = 1.

Lemma 7.9 For any C ⊂ S^{m} the hypersurface Γ_{C} is admissible.

Proof. For C = n this follows from the lemma (7.6). Indeed, the vector field (∂_{ξ_{m+1}} − ∂_{ξ_{m+2}}) is tangent to the hyperplane Γ_{n}.

In general we proceed as follows. The form ω_{m}(φ; v) is given by a bidifferential operator of first order (see (54)), so its restriction to K is determined by the restriction of the functions φ and v to the 1-st infinitesimal neighborhood of K. Let η ∈ Γ_{C} and t(η) be the tangency point of the hyperplane < η, x >= 0 with C. Then the tangent space to Γ_{K} at a point t(η) coincides with Γ_{t(η)}.

6. Inversion of the integral transform related to an admissible family. The restriction of the form ω_{m}(I_{λ}^{±}(f); K_{λ}^{∓}(ξ, x)) to Γ_{C} depends only I_{λ,Γ_{C}}^{±} f. So one can expect the inversion formula

d(Γ_{C})f(x) = c(λ) · ∫_{Γ_{C}} ω_{m}(I_{λ}^{±}(f); K_{λ}^{∓}(ξ, x)))  (58)

similar to (57). However the cycle Γ_{C} lies in the closure Γ of Γ_{0}, while the function I_{λ}^{±}(f) was well defined only inside of Γ_{0}. For the same reason the form ω_{m}(I_{λ}^{±}(f); K_{λ}^{∓}(ξ, x)) is closed only inside of Γ_{0} (and outside of Γ ). So it is a priory unclear whether the formula makes sense and is it possible to use the Stokes theorem.

To avoid this trouble we consider the integral transformation I_{λ,Γ_{C}}^{±} only on the subspace C^{∞}(S^{m}, C) of the functions vanishing in a very small neighborhood of the subvariety C in S^{m}.

Let Ĉ ∈ Γ be the subvariety of spheres of radius zero with center at points of C. Let Ψ_{λ}^{±}(Γ_{C}; Ĉ) be the subspace of Ψ_{λ}^{±}(Γ_{C}) consisting of functions smooth near Ĉ. Then I_{λ}^{±}f is smooth in a neighborhood of Ĉ. So we get an integral transformation

I_{λ,Γ_{C}}^{±} : Φ_{−λ−m}^{±}(S^{m}, C) ⟶ Ψ_{λ}^{±}(Γ_{C}; Ĉ)

Now we may apply the Stokes formula near Ĉ. Assuming this let us perturbate the cycle Γ_{C} near the boundary of Γ by moving it a little bit inside of Γ. Geometrically this means that we replace small spheres tangent to C by close to them small spheres which are not tangent to C.

Remarks. 1. The cycle K becomes homologous to 0 in the sphere S^{m+1} parametrizing all oriented hyperplanes.

2. One can deform smoothly the cycle K out of the domain Γ. However doing this we must cross all the points of the boundary Γ_{1} of Γ. Therefore we can not use the Stokes formula to compare

∫_{K} ω_{m}(I_{λ}^{±}(f); K_{λ}^{∓}(ξ, x))) and ∫_{K′} ω_{m}(I_{λ}^{±}(f); K_{λ}^{∓}(ξ, x)))

where K is inside Γ_{0} and K′ outside Γ. This is very natural: otherwise we would prove that they are equal, and so equal to zero since the cycle K′ is homologous to zero in the complement to Γ.

So we can reduce the investigation of the integral to the study of a similar integral over a cycle K inside Γ_{0}, which was done above. Therefore we come to the following conclusion:

Theorem 7.10 For an admissible family Γ_{C} the operator J_{λ}^{±} provides an operator

J_{λ,Γ_{C}}^{±} : Ψ_{λ}^{±}(Γ_{C}, Ĉ) ⟶ Φ_{−λ−m}^{±}(S^{m}, C)

such that

c(λ) · I_{λ,Γ_{C}}^{±} ∘ J_{λ,Γ_{C}}^{±} = d(K) · Id

7. Geometry of the family of spheres. The group SO(m + 1, 1) acts on the family of all spheres in S^{m}. A remarkable fact is that a bigger symmetry group, SO(m + 1, 2), acts as a group of contact transformations on the family of all spheres (including the points, which are spheres of zero radius!).

Namely, let

X_{m+1} := {η_{1}^{2} + ... + η_{m+1}^{2} − η_{m+2}^{2} − η_{m+3}^{2} = 0}/ℝ_{+}^{∗}

be the m+1-dimensional quadric of signature (m+1, 2). Its affine part η_{m+3} ≠ 0 is isomorphic to the hyperboloid Γ_{0} = {ξ_{1}^{2} + ... + ξ_{m+1}^{2} − ξ_{m+2}^{2} = 1}. The complement to the affine part is the projectivization of the cone {ξ_{1}^{2} + ... + ξ_{m+1}^{2} − ξ_{m+2}^{2} = 0}, i.e. it is a sphere {ξ_{1}^{2} + ... + ξ_{m+1}^{2} = 1}. The quadric X_{m+1} parametrizes all oriented hyperplane sections of the sphere S^{m}. The hyperboloid Γ_{0} parametrizes all oriented spheres of non zero radius.

Let A ⊂ S^{m}×X_{m+1} be the incidence subvariety. Consider the double bundle corresponding to this family and its symplectization:

A  N_{A}^{∗}(S^{m} × X_{m+1})
p_{1} ↙  ↘ p_{2}  π_{1} ↙  ↘ π_{2}
S^{m}  X_{m+1}  T^{∗}S^{m}  T^{∗}X_{m+1}

Let

Σ := π_{2}(N_{A}^{∗}(S^{m} × X_{m+1})) ⊂ T^{∗}X_{m+1}

Then Σ_{ξ} := T_{ξ}^{∗}X_{m+1} ∩ Σ is a nondegenerate quadratic cone in the cotangent bundle to ξ. This cone is dual to the cone in the tangent space to the quadric at the point ξ given by intersection of the quadric with the hyperplane in the projective space tangent to the quadric at ξ.

The hypersurface Σ is foliated on curves: bicharacteristics. This foliation is invariant under the action of the multiplicative group ℝ^{∗} on T^{∗}X_{m+1}.

Lemma 7.11 a) Projection along the bicharaceristics gives the ℝ^{∗} equivariant fibration

π_{Σ} : (Σ\ {zero section}) ⟶ (T^{∗}S^{m}\ {zero section})

b) The projection of a bicharacteristic to X_{m+1} consists of all spheres tangent to a given hyperplane at a given point.

So the manifold of all bicharacteristics is identified with the projectivization of the cotangent bunlde to S^{m}.

Geometrically P (Σ\ {zero section}) is the set of all pairs

{ a contact element h at a point x ∈ S^{m} , a sphere tangent to h at x }

The group SO(m+1, 2) acts on X_{m+1} and hence on Σ. Thanks to the lemma the group SO(m + 1, 2) acts as a group of homogeneous symplectomorphisms on T^{∗}S^{m}. It preserves the family of homogeneous Lagrangian subvarieties given by the conormal bundles to spheres (including the spheres of zero radius).

8. The Hamilton-Jacoby method for description of admissible families of spheres. A hypersurface K′ ⊂ X_{m+1} is characteristic if its conormal bundle in X_{m+1} is contained in Σ, i.e. for any nonsingular ξ ∈ K′ the tangent plane T_{ξ}K′ is tangent to the “light cone” Σ_{ξ}^{∗} ⊂ T_{ξ}X_{m+1}.

Proposition 7.12 An irreducible hypersurface K′ ⊂ X_{m+1} is admissible if and only if it is characteristic.

Proof. We already proved in lemma (7.9) that if K′ is characteristic then it is admissible. Let us prove the converse statement. Since ω_{m}(φ; v) is given by a bidifferential operator of order (1, 1) it is enough to check that the restriction of the differential form ω_{m}(φ; v) to any noncharacteristic hyperplane does depend on the derivatives of φ and v in the direction transversal to this hyperplane. The group SO(m + 1, 1) acts transitively on the variety of noncharacteristic hyperplanes in the tangent spaces T_{ξ}X_{m+1}. So it is sufficient to check the statement above for the hyperplane ξ_{m+2} = 0. One has

ω_{m}(φ; v)|_{ξ_{m+2}=0} = ∑_{1≤i≤m+1}(−1)^{i+m−1}ξ_{i}·(v·φξ_{m+2}^{′}−vξ_{m+2}^{′}·φ)dξ_{1}∧...d̂ξ_{i}...∧dξ_{m+1}

The proposition follows.

The following lemma is well known

Lemma 7.13 Any algebraic irreducible homogeneous Lagrangian subvariety in T^{∗}X is isomorphis to the conormal bundle to an algebraic irreducible subvariety Y ⊂ X

Theorem 7.14 Any admissible hypersurface in Γ is a piece of a hypersurface Γ_{C} for a certain C ⊂ S^{m}.

Proof. We may assume that K′ is irreducible. According to proposition (7.12) N_{K′}^{∗}X_{m+1} is a Lagrangian subvariety in Σ, so π_{Σ} it projects it down to a Lagrangian subvariety in T^{∗}S^{m}, which by the above lemma must have form N_{C}^{∗}S^{m}.

8 Holonomic kernels and their composition: the bicategory of D-modules

1. Motivations. As we emphasized before the composition of natural linear maps defined by distributional kernels not always exists. However when it is defined we come to the problem of computation of the composition. Many important problems of analysis can be considered as special cases of this one. For instance in integral geometry both the integral transformation and its inverse should be treated as natural linear maps between solution spaces of D-modules, so to invert an integral transformation we should be able to compute the composition of natural linear maps.

Let us assume for a moment that M_{i} are excellent D-modules. Then usually the natural kernels are distributions satisfying holonomic system of differential equations. This means that the image of homomorphism

˜⋆M_{1} ⊠ M_{2} ⟶ D′(X_{1} × X_{2})  (59)

provided by the kernel

K_{12}(x_{1}, x_{2}) ∈ Hom_{D}(˜⋆M_{1} ⊠ M_{2}, D′(X_{1} × X_{2}))

is a holonomic D-module. Let us denote it by K_{12} and by ˜⋆M_{1} ⊠ M_{2} ⟶^{α_{12}} K_{12} the corresponding morphism of D-modules. So (59) is a composition

˜⋆M_{1} ⊠ M_{2} ⟶^{α_{12}} K_{12} ↪ D′(X_{1} × X_{2})

The idea to keep only the first arrow suggests the following definition

Definition 8.1 A holonomic kernel on X_{1}×X_{2} is a collection (M_{1}, M_{2}, K_{12}; α) where

M_{1} ∈ D_{coh}^{b}(D_{X_{1}}), M_{2} ∈ D_{coh}^{b}(D_{X_{2}}), K_{12} ∈ D_{hol}^{b}(D_{X_{1}×X_{2}})

and

α ∈ RHom_{D_{X_{1}×X_{2}}} (⋆M_{1} ⊠ M_{2}, K_{12})

A holonomic kernel is a finer algebraic version of a holonomic distribution on X_{1} × X_{2} then the D-module which this distribution satisfies.

Example. Suppose that M_{i} = D_{X_{i}} for i = 1, 2. Then ˜⋆D_{X_{1}} = D_{X_{1}} and D_{X_{1}} × D_{X_{2}} = D_{X_{1}×X_{2}}. Morphisms of D-modules D_{X_{1}×X_{2}} ⟶ K are defined by their value on the generating section 1 and correspond just to the sections of K_{12}.

For instance, if X_{1} = X_{2} = 𝔸^{1} and K_{12} is the D-module of delta functions on the diagonal the morphisms above correspond to sections f(x)δ^{(k)}(x − y).

It seems that the notion of a bicategory is the appropriate language to discuss the holonomic kernels and their composition.

2. Bicategories. A complete definition of (lax) bicategory see in [Be] or p.200 [KV]. In particular a notion of bicategory C includes the following data:

a set ObC of objects;

for any 2 objects a set of 1-morphisms from A to B;

for any two 1-morphisms α_{1}, α_{2} between A and B a set of 2-morphisms between α_{1} and α_{2}.

For any 2 objects A_{1} and A_{2} of a bicategory there is a category Mor_{1}(A_{1}, A_{2}) of 1-morphisms from A_{1} to A_{2}. The objects in this category are 1-morphisms from A_{1} to A_{2}; the morphisms between given two 1-morphisms from A_{1} to A_{2} are given by the 2-morphisms between these 1-morphisms.

The composition of 1-morphisms provides a bifunctor

Mor_{1}(A_{1}, A_{2}) × Mor_{1}(A_{2}, A_{3}) ⟶ Mor_{1}(A_{1}, A_{3})

The archetypal example is the bicategory of all categories. Its objects are categories and for any two categories A and B the category Mor_{1}(A, B) is the category of functors from A to B.

3. A bicategory of D-modules. Below we work in the derived category. In particular all morphisms are morphisms in the derived category.

The objects of the bicategory are pairs (X, M) where X is an algebraic variety over a field k (char k = 0) and M ∈ D_{coh}^{b}(D_{X}).

By definition 1-morphisms between the 2 objects (X, M) and (Y, N) are holonomic kernels

⋆M ⊠ N ⟶^{α} K

It is the composition of 1-morphisms which makes the whole story relevant to integral geometry. Roughly speaking it answers to the question “what system of differential equations satisfies the kernel of the composition of 2 natural maps ?” and motivated by s. 5.7 above.

Let Δ_{2} : X_{1} × X_{2} × X_{3} ↪ X_{1} × X_{2} × X_{2} × X_{3} be the diagonal embedding of X_{2} and π_{2} : X_{1} × X_{2} × X_{3} → X_{1} × X_{3} be the projection. Consider the objects (X_{i}, M_{i}) where i = 1, 2, 3.

Definition 8.2 The composition of 1-morphisms

⋆M_{1} ⊠ M_{2} ⟶^{α_{12}} K_{12} and ⋆ M_{2} ⊠ M_{3} ⟶^{α_{23}} K_{23}

is the 1-morphism

⋆M_{1} ⊠ M_{3} ⟶^{α_{13}} K_{13}

where

K_{13} = K_{12} ∘ K_{23} := π_{2}_{∗}Δ_{2}^{!}(K_{12} ⊠ K_{23})

and the morphism α_{13} is the composition of the morphisms id ⊠ G ⊠ id and α_{12} ⊠ α_{23}:

⋆M_{1}⊠M_{3} ⟶^{id⊠G⊠id} π_{2}_{∗}Δ_{2}^{!}(⋆M_{1}⊠M_{2}⊠⋆M_{2}⊠M_{3}) ⟶^{α_{12}⊠α_{23}} π_{2}_{∗}Δ_{2}^{!}(K_{12}⊠K_{23})

A 2-morphism between 1-morphisms

⋆M ⊠ N ⟶^{α_{1}} K_{1} and ⋆ M ⊠ N ⟶^{α_{2}} K_{2}

is a morphism φ_{12} : K_{1} ⟶ K_{2} making the following diagram commutative:

⋆M_{1} ⊠ M_{3}
α_{1} ↙  ↘ α_{2}
K_{1}  ⟶^{φ_{12}}  K_{2}

A 2-morphism between a holonomic kernel α′_{13} : ⋆M_{1} ⊠ M_{3} → K′_{13} and the composition K_{12} ∘ K_{23} of holonomic kernels α_{12} : ⋆M_{1} ⊠ M_{3} → K_{12} and α_{23} : ⋆M_{2} ⊠ M_{3} → K_{23} is provided by the following commutative diagram

⋆M_{1} ⊠ M_{3} ⟶^{id⊠G⊠id} Rπ_{2}_{∗}Δ_{2}^{!}(⋆M_{1} ⊠ M_{2} ⊠ ⋆M_{2} ⊠ M_{3})
↓α′_{13}  ↓Rπ_{2}_{∗}Δ_{2}^{!}(α_{12} ⊠ α_{23})
K′_{13}  ⟶  Rπ_{2}_{∗}Δ_{2}^{!}(K_{12} ⊠ K_{23})

The composition of 2-morphisms is defined in an obvious way.

The identity 1-morphism Id_{M}. For any M ∈ D_{coh}^{b}(D_{X}) there is a canonical morphism

I_{M} : ⋆M ⊠ M ⟶ δ_{Δ}[d_{X}]

corresponding via (??) to the identity map Id ∈ Hom_{D_{X}} (⋆M, ⋆M).

We will say that the 1-morphism α_{23} is weakly inverse to the 1-morphism α_{12} (see (??)) if there is a 2-morphism from the identity 1-morphism Id_{M} to the composition of 1-morphisms α_{23} ∘ α_{12}. This means that the following diagram is commutative:

⋆M_{1} ⊠ M_{1} ⟶^{id⊠G⊠id} Rπ_{2}_{∗}Δ_{2}^{!}(⋆M_{1} ⊠ M_{2} ⊠ ⋆M_{2} ⊠ M_{1})
↓I_{M_{1}}  ↓Rπ_{2}_{∗}Δ_{2}^{!}(α_{12} ⊠ α_{23})
δ_{Δ}  ⟶^{φ}  Rπ_{2}_{∗}Δ_{2}^{!}(K_{12} ⊠ K_{23})

Remark. These definitions make sense for any (not necessarily holonomic) K_{ij} ∈ D_{coh}^{b}(D_{X}).

3. On composition of holonomic kernels. Recall that for M, N ∈ D_{coh}^{b}(D_{X}) one has

M ⊗^{!} N := Δ^{!}(M ⊠ N) = M ⊗_{O} N [−d_{X}]

where Δ : X ↪ X × X is the diagonal embedding.

Definition 8.3 Let K ∈ D_{coh}^{b}(D_{X_{1}×X_{2}}). Then it defines a functor

K̄ : D_{coh}^{b}(D_{X_{1}}) ⟶ D_{coh}^{b}(D_{X_{2}}) K̄(M) := p_{2}_{∗}(K ⊗^{!} p_{1}^{!}M)

This is motivated by the following proposition (compare with s. 8.2): Proposition 8.4

RHom_{D_{X_{1}×X_{2}}} (⋆M_{1} ⊠ M_{2}, K_{12}) = RHom_{D_{X_{2}}} (M_{2}, K_{12}(M_{1})) Proof. Let p_{i} : X_{1} × X_{2} ⟶ X_{i} be natural projections. We have

RHom_{D}(⋆M_{1}⊠M_{2}, K_{12}) = RHom_{D}(p_{1}^{!}(⋆M_{1})[−d_{X_{2}}]⊗_{O}p_{2}^{!}(M_{2})[−d_{X_{1}}], K_{12}) = RHom_{D}(p_{2}^{!}(M_{2})[−d_{X_{1}}], ⋆p_{1}^{!} ⋆ (M_{1})[d_{X_{2}}] ⊗_{O} K_{12}) =

RHom_{D}(p_{2}^{∗}(M_{2})[d_{X_{1}}], K_{12} ⊗_{O} p_{1}^{!}(M_{1})[−d_{X_{2}}]) =

RHom_{D}(M_{2}, p_{2_{∗}}(K_{12} ⊗_{O} p_{1}^{!}(M_{1})[−d_{X_{1}} − d_{X_{2}}]) = RHom_{D}(M_{2}, K_{12}(M_{1})) Proposition 8.5 There is natural isomorphism of functors:

K̄_{23} ∘ K_{12} = K̄_{23} ∘ K̄_{13}

Proof.

Consider the following diagrams:

X_{1} × X_{2} × X_{3}

π_{3} ↙  ↓ π_{2}  ↘ π_{1}
X_{1} × X_{2}  X_{1} × X_{3}  X_{2} × X_{3}
p_{1} ↙  ↘ p_{2}  q_{1} ↙  ↘ q_{2}
X_{1}  X_{2}  X_{3}

and

X_{1} × X_{3}
r_{1} ↙  ↘ r_{2}
X_{1}  X_{3}

Let Δ_{2} : X_{1} × X_{2} × X_{3} ↪ X_{1} × X_{2} × X_{2} × X_{3} is the diagonal imbedding of X_{2}.

Lemma 8.6 Let K_{12} ∈ D_{coh}^{b}(D_{X_{1}×X_{2}}) and K_{23} ∈ D_{coh}^{b}(D_{X_{2}×X_{3}}). Then

K_{12} ∘ K_{23} = π_{2}_{∗}(π_{3}^{!}K_{12} ⊗_{O} π_{1}^{!}K_{23})

Proof. Follows immediately from π_{3} = τ_{12} ∘ Δ_{2} and π_{1} = τ_{23} ∘ Δ_{2}. One has

K_{23}(K_{12}(M)) = q_{3}_{∗}(q_{2}^{!}p_{2}_{∗}(p_{1}^{!}M_{1} ⊗^{!} K_{12}) ⊗^{!} K_{23}) =

q_{3}_{∗}(π_{1}_{∗}π_{3}^{!}(p_{1}^{!}M_{1} ⊗^{!} K_{12}) ⊗^{!} K_{23}) = q_{3}_{∗}(π_{1}_{∗}(π_{1}^{!}M_{1} ⊗^{!} π_{12}^{!}K_{12}) ⊗^{!} K_{23}) =^{1}

q_{3}_{∗}π_{1}_{∗}(π_{1}^{!}M_{1} ⊗^{!} π_{3}^{!}K_{12} ⊗^{!} π_{1}^{!}K_{23}) = π_{3}_{∗}(π_{1}^{!}M_{1} ⊗^{!} π_{3}^{!}K_{12} ⊗^{!} π_{1}^{!}K_{23}) =

r_{3}_{∗}π_{2}_{∗}(π_{1}^{!}M_{1} ⊗^{!} π_{3}^{!}K_{12} ⊗^{!} π_{1}^{!}K_{23}) =

r_{3}_{∗}π_{13}_{∗}(π_{2}_{!}r_{1}^{!}M_{1} ⊗^{!} π_{3}^{!}K_{12} ⊗^{!} π_{1}^{!}K_{23}) =^{2} r_{3}_{∗}(r_{1}^{!}M_{1} ⊗^{!} K_{12} ∘ K_{23})

Here (1) is by the base change for the diagram

X_{1} × X_{2} × X_{3}
π_{3} ↙  ↘ π_{1}
X_{1} × X_{2}  X_{2} × X_{3}
p_{2} ↘  ↙ q_{2}
X_{2}

and (2) is by projection formula f_{∗}(A ⊗^{!} f^{!}B) = f_{∗}A ⊗^{!} B.

Lemma 8.7 The composition of 1-morphisms

(M_{1}, M_{2}, K_{12}; α_{12}) on X_{1}×X_{2}  and  (M_{2}, M_{3}, K_{23}; α_{23}) on X_{2}×X_{3}

is a 1-morphisms

(M_{1}, M_{3}, K_{13}; α_{13}) on X_{1} × X_{3}

where K_{13} := K_{23} ∘ K_{12} and α_{23} is the composition:

M_{3} ⟶ K̄_{23}(M_{2}) ⟶^{K̄_{23}(τ_{12})} K̄_{23}(K̄_{12}(M_{1})) = K̄_{13}(M_{1})

4. Natural linear maps provided by algebraic kernels. Let M_{i} ∈ D_{coh}^{b}(D_{X_{i}}), i = 1, 2, and K_{12} ∈ D_{coh}^{b}(D_{X_{1}×X_{2}}). Suppose we are given the following data:

1) an algebraic kernel

α_{12} ∈ RHom_{D_{X_{2}}} (M_{2}, K_{12}(M_{1})); K_{12} ∈ RHom_{D_{X_{1}×X_{2}}} (K_{12}, O_{X_{1}×X_{2}})

2) an element

γ ∈ RHom_{D}(p_{2}_{∗}O(X_{1} × X_{2}), O(X_{2}))

We will construct a linear map

RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ RHom_{D}(M_{2}, C^{∞}(X_{2}))

related to this data. Namely, by the functoriality

RHom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ RHom_{D}(p_{1}^{!}M_{1}, p_{1}^{!}C^{∞}(X_{1})) ⟶

RHom_{D}(p_{1}^{!}M_{1}, C^{∞}(X_{1} × X_{2})[d_{X_{2}}]) ⟶^{K_{12}⊗}

RHom_{D}(K_{12} ⊗_{O} p_{1}^{!}M_{1}, C^{∞}(X_{1} × X_{2})[d_{X_{2}}]) ⟶

RHom_{D}(p_{2}_{∗}(K_{12} ⊗_{O} p_{1}^{!}M_{1}), p_{2}_{∗}C^{∞}(X_{1} × X_{2})[d_{X_{1}}]) ⟶^{γ}

RHom_{D}(K_{12}(M_{1}), C^{∞}(X_{2})[−d_{X_{2}}])

The morphism α_{12} provides the last arrow

RHom_{D}(K_{12}(M_{1}), C^{∞}(X_{2})[−d_{X_{2}}]) ⟶ RHom_{D}(M_{2}, C^{∞}(X_{2})[−d_{X_{2}}])

If

α_{12} ∈ R^{f}Hom_{D}(M_{2}, K_{12}(M_{1})) , K_{12} ∈ R^{k}Hom_{D}(K_{12}, O)

and

γ ∈ R^{d_{X_{1}} −l}Hom_{D}(p_{2}_{∗}C^{∞}(X_{1} × X_{2}), C^{∞}(X_{2}))

then we get a linear map

R^{j}Hom_{D}(M_{1}, C^{∞}(X_{1})) ⟶ R^{j+k+l−d_{X_{1}}} Hom_{D}(M_{2}, C^{∞}(X_{2}))

5. Algebraic version of the Radon transform of (holonomic) functions. Any 1-morphism γ : A_{2} ⟶ A_{3} provides a functor

F_{β} : Mor_{1}(A_{1}, A_{2}) ⟶ Mor_{1}(A_{1}, A_{3}) α ⟼ β ∘ α

There is an object ⋆ corresponding to the one-dimensional vector space considered as a D-module over a point.

The category Mor_{1}(⋆, (X, M)) looks as follows. Its objects are pairs: a holonomic complex of D-modules L on X and a morphism α : M → L. The morphisms are provided by φ : L_{1} ⟶ L_{2} making the corresponding diagram commutative. We will call it the category of D-modules under M on X.

Therefore the 1-morphisms (α, K) : (X, M) → (X, N) provide functors from the category of D-modules under M on X to the category of D-modules under N on Y.

6. Examples. Let me first discuss the analytic properties of the Radon transform in ℝ^{2}.

φ(x, y) ⟼ φ̂(ξ_{1}, ξ_{2}, s) := ∫ φ(x, y)δ(ξ_{1}x + ξ_{2}y − s)dxdy

the 1-form

κφ̂(ξ_{1}, ξ_{2}) := φ̂′_{s}(ξ_{1}dξ_{2} − ξ_{2}dξ_{1})

is closed on the subvariey ξ_{1}x+ξ_{2}y −s = 0. Here (x, y) is a given point. Integral of this 1-form over any cycle in (ξ_{1}, ξ_{2}) plane is zero.

Consider the line through the point (x, y) corresponding to ξ = (ξ_{1}, ξ_{2}). On a line minus a point ((x, y) in our case),there is canonical multiplicatively invariant measure (\frac{dt}{t}). Let L(ξ) := ∫ φ(x − ξ_{2}t, y + ξ_{1}t)\frac{dt}{t} be the integral over this measure. Then

∫_{ξ}^{η}(κφ̂)(ξ_{1}, ξ_{2}) = L(η) − L(ξ)  (60)

where we integrate over any path connecting points ξ and η.

In particular in the affine picture

φ(x, y) ⟼ ∫_{∞}^{∞} φ(x, ax+b)dx, κφ̂ = φ̂′_{b}da ∫ φ̂′_{b}(a, y−ax)da = ∫_{−∞}^{∞} φ(x, y)\frac{dy}{y}

(61)

So The 1-form κφ̂ is exact on the image of functions vanishing at the point (x, y). For example

κ(I(xφ)) = (I(xφ))′_{b}da = (Iφ)′_{a}da = d(Iφ)

Formula (60) follows immediately from this.

Now let us turn to the D-module picture. Set

X_{1} = {(x, y)} = ℝ^{2}, X_{2} = {(a, b)} = ℝ^{2}, X_{3} = {(x′, y′)} = ℝ^{2}

and M_{X_{i}} = D_{X_{i}}. Notice that ⋆D_{X_{i}} ⊠ D_{X_{i+1}} = D_{X_{i}×X_{i+1}}[d_{X_{i}}]. Set

δ(A) = δ(y − ax − b) and δ(A′) = δ(y′ − ax′ − b)
K_{12} := D_{X_{1}×X_{2}} · δ(A), K_{23} := D_{X_{2}×X_{3}} · δ(A′)
α_{12}[−2] : 1_{X_{1}×X_{2}} ⟼ δ(A), α_{12}[−2] : 1_{X_{2}×X_{3}} ⟼ δ^{(1)}(A′)

The D-module K_{13} has a more complicated structure which can be described as follows.

0 ⟶ O_{X_{1}×X_{3}} ⊕ δ_{Δ_{13}} ⟶ K_{13} ⟶ δ_{V} ⟶ 0  (62)

Here Δ_{13} ⊂ X_{1} ×X_{2} is the diagonal and V is the divisor of pairs of points (p, p′) with x = x′ (I.e. the vertical line through p contain the point p′). Let

j : X_{1} × X_{3}\V ↪ X_{1} × X_{3} i : V ↪ X_{1} × X_{3} f : V \Δ_{13} ↪ V

Then (62) is the Bauer sum of the following two standard extensions:

0 ⟶ O_{X_{1}×X_{3}} ⟶ j_{∗}j^{∗}O_{X_{1}×X_{3}} ⟶ δ_{V} ⟶ 0

and

i_{∗}(0 ⟶ δ_{Δ_{13}} ⟶ f_{!}f^{∗}O_{V} ⟶ O_{V} ⟶ 0)

To see this consider the variety A := {p, l, p′} ⊂ X_{1} ×X_{2} ×X_{3} such that p, p′ ∈ l and its closure Ā in X_{1} × X̄_{2} × X_{3}. Notice that Ā is the blow up of the diagonal Δ_{13} in X_{1} × X_{3}.

Then K_{13} = π_{2}_{∗}O_{A}. One has π̄_{2}_{∗}OĀ = O_{X_{1}×X_{3}} ⊕δ_{Δ_{13}}. Further, notice that Ā\A projects isomorphically to V. So one has

0 ⟶ OĀ ⟶ g_{∗}O_{A} ⟶ δ_{V} ⟶ 0

Taking direct image of this extension to X_{1} × X_{3} we get (62)).

Proposition 8.8 The formula

δ(x − x′)δ(y − y′) ⟼ δ(y − ax − b) ⊗ δ^{(1)}(y′ − ax′ − b)dadb  (63)

defines a homomorphism of D-modules δ_{Δ} ⟶ K_{13} and hence a 2-morphism Id_{D_{X_{1}}} => (α_{13}, K_{13}).

Proof. We have to show that applying to the right hand side of (63) any differential equations which the left hand side satisfy, we will get exact 2-form in the de Rham complex with respect to (a, b) variables. This follows from the formulas

(x − x′) · δ(A) ⊗ δ^{(1)}(A′)dadb = d(δ(A) ⊗ δ(A′)(xda + db))
(y − y′) · δ(A) ⊗ δ^{(1)}(A′)dadb = d(δ(A) ⊗ δ(A′)a(xda + db))
(∂_{x} + ∂_{x′})δ(A) ⊗ δ^{(1)}(A′)dadb = d(δ(A) ⊗ δ^{(1)}(A′)ada)
(∂_{y} + ∂_{y′})δ(A) ⊗ δ^{(1)}(A′)dadb = d(δ(A) ⊗ δ^{(1)}(A)da)

It is amazing to see the structure of the extension (62) from this point of view. Namely, δ(A)⊗δ(A′)dadb is a generator of K_{13}, so (x−x′)δ(A)⊗δ(A′)dadb is the generator of the submodule O_{X_{1}×X_{3}} and δ(A) ⊗ δ^{(1)}(A′)dadb generates the submodule δ_{Δ_{13}}.

The Radon transform over the lines in the space.

X_{1} = {(x, y, z)} = 𝔸^{3}, X_{2} = {(a_{1}, a_{2}, b_{1}, b_{2})} = 𝔸^{4}, X_{3} = {(x′, y′, z′)} = 𝔸^{3}

and

M_{X_{i}} = D_{X_{i}} for i = 1, 3; M_{X_{2}} = D_{X_{2}} · (\frac{∂^{2}}{∂a_{1}∂b_{2}} − \frac{∂^{2}}{∂a_{2}∂b_{1}} )

Notice that ⋆M_{X_{2}} = M_{X_{2}}[3].

Let A ⊂ X_{1} × X_{2} be the correspondence {y − a_{1}x − b_{1} = 0, z − a_{2}x − b_{2} = 0} defining our family of lines. Set

δ(A) = δ(y−a_{1}x−b_{1})·δ(z−a_{2}x−b_{2}), δ(A′) = δ(y′−a_{1}x′−b_{1})·δ(z′−a_{2}x′−b_{2})

Then

K_{12} := D_{X_{1}×X_{2}} · δ(A) K_{23} := D_{X_{2}×X_{3}} · δ(A′)
α_{12} : 1_{X_{1}×X_{2}} ⟼ δ(A)[−3] α_{23} : 1_{X_{2}×X_{3}} ⟼ δ(A′)[−3]

Proposition 8.9 The formula

δ(x − x′)δ(y − y′)δ(z − z′) ⟼ G_{M_{2}}(δ(A′), δ(A))

defines a homomorphism of D-modules δ_{Δ} ⟶ K_{23}∘K_{12} providing a 2-morphism Id_{D_{X_{1}}} => (α_{23}, K_{23}) ∘ (α_{12}, K_{12})

Here

G_{M_{2}}(g, f) = \frac{1}{2}((g′_{a_{1}} · f − g · f′_{a_{1}})da_{1} ∧ da_{2} ∧ db_{1} − (g′_{a_{2}} · f − g · f′_{a_{2}})da_{1} ∧ da_{2} ∧ db_{2}

(g′_{b_{1}} · f − g · f′_{b_{1}})da_{1} ∧ db_{1} ∧ db_{2} − (g′_{b_{2}} · f − g · f′_{b_{2}})da_{2} ∧ db_{1} ∧ db_{2})

Relation with the ”form κ” of [GGS]. The Green class can be represented by another cocycle

κ_{M} := δ(A) ⊗ (\frac{∂}{∂b_{1}} δ(A′)da_{1} + \frac{∂}{∂b_{2}} δ(A′)da_{2}) ∧ (xda_{1} + db_{1}) ∧ (xda_{2} + db_{2})

Its great advantage is ”locality”: it is a cocycle in the de Rham complex with support in the incidence subvariety A.

The expression \frac{∂f}{∂b_{1}} da_{1} + \frac{∂f}{∂b_{2}} da_{2} is the famous ”1-form κ” ([GGS]). It is a 1-form on the incidence subvariety A.

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Max-Planck-Institute fur mathematik, Bonn, 53115, Germany; sasha@mpimbonn.mpg.de

Dept. of Mathematics, Brown Univercity, Providence, RI 02912 USA.