ON TWISTED MICRODIFFERENTIAL MODULES I. NON-EXISTENCE OF TWISTED WAVE EQUATIONS

NON-EXISTENCE OF TWISTED WAVE EQUATIONS

ANDREA D’AGNOLO AND PIERRE SCHAPIRA

Abstract. Using the notion of subprincipal symbol, we give a necessary con- dition for the existence of twisted D-modules simple along a smooth involutive submanifold of the cotangent bundle to a complex manifold. As an application, we prove that there are no generalized massless field equations with non-trivial twist on grassmannians, and in particular that the Penrose transform does not extend to the twisted case.

Introduction

Let T be a 4-dimensional complex vector space, P the 3-dimensional projective space of lines in T, and G the 4-dimensional grassmannian of 2-planes in T. Accord- ing to Penrose, G is a conformal compactification of the complexified Minkowski space. Denote by M _(h) the D _G -modules associated with the massless field equa- tions of helicity h ∈ ¹ ₂ Z. The Penrose correspondence realizes M (1+m/2) as the transform of the D _P -module associated with the line bundle O _P (m), for m ∈ Z. For λ ∈ C, O P (λ) makes sense in the theory of twisted sheaves. It is then a natural question to ask whether the Penrose correspondence extends to the twisted case.

In particular, are there “massless field equations” of complex helicity h / ∈ ¹ ₂ Z?

The D _G -modules M _(h) are simple along a smooth involutive submanifold V of the cotangent bundle to G which is given by the geometry of the integral transform.

In this paper we give a negative answer to the question raised above: for topological reasons, there are no simple D _G -modules along V with non-trivial twist. Indeed, this is a corollary of the following more general result.

Let X be a complex manifold, and V a conic involutive submanifold of its cotan- gent bundle. Denote by D _Ω

V /X

the ring of differential operators on V acting on relative half-forms and by D ^bic

Ω

(0) its subring of operators homogeneous of degree 0 and commuting with the functions which are locally constant on the bicharacter- istic leaves. The ring of microdifferential operators E X is endowed with the so-called V -filtration {F ^V _k E X } _k∈Z and by a result of Kashiwara-Oshima, there is a natural isomorphism of rings F ^V ₀ E _X /F ^V ₋₁ E _X − ^∼ → D ^bic

Ω

(0).

Let S be a stack of twisted sheaves on X, and consider the category of twisted microdifferential modules Mod(E X ; S). One says that a twisted microdifferential module is simple along V if it can be endowed with a good V -filtration whose associated graded module is locally isomorphic to O V (0).

Let Σ be a smooth bicharacteristic leaf of V . Recall that stacks of twisted sheaves on X are classified by H ² (X; C ^× X ), and denote by [S] the class of S. Our main result (see Theorem 7.1) consists in associating to [S] a class in H ² (Σ; C ^× Σ ) whose

1

triviality is a necessary condition for the existence of a globally simple module along V in Mod(E X ; S).

Let us briefly describe our construction. Let M be a globally simple module along V in Mod(E _X ; S). By definition, M has a good V -filtration, and we denote by M its associated graded module.

(i) By Kashiwara-Oshima’s result mentioned above, we consider M as an ob- ject of Mod(D ^bic _V (0); T). Here, T is a stack of twisted sheaves on V whose class [T] ∈ H ² (V ; C ^× V ) is the product of the pull back of [S] by the class of the stack containing the inverse relative half-forms Ω ^−1/2 _{V /X} .

(ii) The restriction M _Σ of M to Σ is a line bundle with flat connection in the category of twisted differential modules Mod(D _Σ ; U), where U is a stack of twisted sheaves on Σ whose class [U] ∈ H ² (Σ; C ^× Σ ) is the restriction of [T].

(iii) By the Riemann-Hilbert correspondence, M _Σ is associated with a local system of rank one in U(Σ). Since there are no local systems of rank one with non-trivial twist, the triviality of [U] is a necessary condition for the existence of a globally simple module along V in Mod(E _X ; S).

We would like to thank Masaki Kashiwara for extremely useful conversations and helpful insights.

1. Review of twisted sheaves

In this section we briefly review the notion of twisted sheaves. References are made to [7, 8], see also [2].

Let X be a complex analytic manifold, O X its structure sheaf, and denote by C X the constant sheaf with stalk C. If A is a sheaf of C-algebras on X, we de- note by Mod(A) the category of sheaves of A-modules on X and by Mod(A) the corresponding C-stack, U 7→ Mod(A| ^U ). We denote by A ^× the sheaf of invertible sections of A.

The short exact sequence of abelian groups

1 − → C ^× _X − → O ^× _X − → O _X ^× /C ^× X − → 1 induces the exact sequence

(1) H ¹ (X; C ^× X ) − →

α H ¹ (X; O _X ^× ) − →

β H ¹ (X; O ^× _X /C ^× X ) − →

δ H ² (X; C ^× X ).

Note that the isomorphism d log : O ^× _X /C ^× X

− ∼ → dO X induces an isomorphism (2) ι : H ¹ (X; O ^× _X /C ^× X ) − ^∼ → H ¹ (X; dO _X ).

The C-vector space structure of H ¹ (X; dO X ) thus gives a meaning to λ · c for c ∈ H ¹ (X; O ^× _X /C ^× X ) and λ ∈ C.

We will consider several characteristic classes with values in these cohomology groups.

• A local system is a C X -module locally free of finite rank. To a local system L of rank one corresponds a class [L] ∈ H ¹ (X; C ^× X ) which characterizes L up to isomorphisms of C ^X -modules.

• A line bundle is an O X -module locally free of rank one. To a line bundle

L on X corresponds a class [L] ∈ H ¹ (X; O _X ^× ) which characterizes L up to

isomorphisms of O _X -modules.

• A stack of twisted sheaves is a C-stack locally C-equivalent to Mod(C X ).

To a stack of twisted sheaves S corresponds a class [S] ∈ H ² (X; C ^× X ) which characterizes S up to C-equivalences. Objects of S(X) are called twisted sheaves.

Recall that [S] has the following description using Cech cohomology. Let X = S

i U _i be an open covering such that there are C-equivalences ϕ i : S| _U

− → Mod(C U

).

By Morita theory, the auto-equivalence ϕ _i ◦ ϕ ⁻¹ _j of Mod(C U

) are given by G 7→

G ⊗ L _ij for a local system L _ij of rank one. By refining the covering we may assume that L _ij ' C U

. The isomorphisms L _ij ⊗ L jk ' L ik on U _ijk are then multiplica- tion by locally constant functions c ijk ∈ Γ(U ijk ; C ^× X ). The class [S] is described by the Cech cocycle {c ijk }. A twisted sheaf F ∈ S(X) is then described by a family of sheaves F i ∈ Mod(C U

) and isomorphisms θ ij : F j | U

− → F i | U

satisfying θ ij ◦ θ jk = c ijk θ ik .

Let S be a stack of twisted sheaves on X and let A be a sheaf of C-algebras on X. We denote by Mod(A; S) the stack of left A-modules in S.

• A twisted line bundle is a pair (S, F ) of a stack of twisted sheaves S and an object F ∈ Mod(O _X ; S) locally free of rank one over O _X . To a twisted line bundle corresponds a class [S, F ] ∈ H ¹ (X; O ^× _X /C ^× X ) which characterizes it up to the following equivalence relation: two twisted line bundles (S, F ) and (T, G) are equivalent if there exist a C-equivalence ϕ : S − → T and an isomorphism ϕ(F ) ' G in T(X).

Let (S, F ) be a twisted line bundle and let X = S

i U i be an open covering such that there are C-equivalences ϕ ⁱ : S| U

− → Mod(C U

), and denote by {c ijk } the Cech cocycle of [S]. These induce equivalences ϕ i : Mod(O U

; S| U

) − → Mod(O U

) and F is described by a family of line bundles F i ∈ Mod(O U

) and isomorphisms θ ij : F j | U

− → F i | U

. By refining the covering, we may assume that there are nowhere vanishing sections s i ∈ Γ(U i ; F i ), so that F i ' O U

. Hence θ ij are mul- tiplications by the sections f ij = s i /θ ij (s j ) ∈ Γ(U ij ; O ^× _X ), so that f ij f jk = c ijk f ik . The class [S, F ] in H ¹ (X; C ^× X − → O ^× _X ) is thus described by the Cech hyper-cocycle {f ij , c ijk }.

The characteristic classes constructed above are related (up to sign) as follows, using the exact sequence (1):

(1) if L is a local system of rank one, then α([L]) = [L ⊗ O _X ], (2) if L is a line bundle, then β([L]) = [Mod(C X ), L],

(3) if (S, F ) is a twisted line bundle, then δ([S, F ]) = [S].

The next result will play an essential role in the proof of Theorem 7.1. It imme- diately follows from the Morita theory for stacks.

Proposition 1.1. A stack of twisted sheaves S is globally C-equivalent to Mod(C ^X ) if and only if there exists an object F ∈ S(X) locally free of rank one over C.

Example 1. For L an untwisted line bundle, and λ ∈ C, there is a twisted line bundle (S _L

, L ^λ ) whose class [S _L

, L ^λ ] is described as follows. Let X = S

i U i be

an open covering such that there are nowhere vanishing sections s i ∈ Γ(U i ; L), and

set g ij = s i /s j . Choose a determination f ij for the ramified function g ^λ _ij on U ij .

Then f ij f jk and f ik are different determinations of g ^λ _ik , so that f ij f jk = c ijk f ik for

some c _ijk ∈ Γ(U ijk ; C ^× X ). Then [S _L

, L ^λ ] is described by the Cech hyper-cocycle

{f ij , c ijk }. Since d log f ij = λd log g ij , we have

[S _L

, L ^λ ] = λ · β([L]) in H ¹ (X; O ^× _X /C ^× X ),

where the action of λ on β([L]) is induced by the isomorphism (2). Note that L ^λ is unique up to tensoring by a local system of rank one.

Consider two stacks S and S ⁰ of twisted sheaves on X (here, X is simply a topological space, or even a site). There are stacks of twisted sheaves S ~ S ⁰ and S ^~−1 on X such that if F ∈ S(X) and F ⁰ ∈ S ⁰ (X) are twisted sheaves, then F ⊗ F ⁰ ∈ (S ~ S ⁰ )(X) and if F is a local system of rank one, then F ⁻¹ ∈ S ^~−1 . Moreover,

[S ~ S ⁰ ] = [S] · [S ⁰ ] [S ^~−1 ] = ([S]) ⁻¹ .

If f : Y − → X is a morphism of topological spaces (or of sites), there exists a stack of twisted sheaves f ^~ S on Y such that if F ∈ S(X), then f ⁻¹ F ∈ (f ^~ S)(Y ).

Moreover,

[f ^~ S] = f ^] ([S]).

Here, for t, t ⁰ ∈ H ² (X; C ^× X ), we denote by t · t ⁰ and t ⁻¹ the product and the inverse in H ² (X; C ^× X ), respectively, and by f ^] t ∈ H ² (Y ; C ^× Y ) the pull-back.

Let (S _F , F ) and (S _G , G) be twisted line bundles on X, and consider the associ- ated twisted line bundles (S _F

, F ⁻¹ ) and (S _{F ⊗}

O

G , F ⊗ _O G) on X, and (S f

F , f ^∗ F ) on Y . Then there are C-equivalences

S _F

' S ^~−1 _F , S _{F ⊗}

O

G ' S _F ~ S G , S f

F ' f ^~ S F .

2. Review of twisted differential operators

In this section we briefly review the notions of twisted differential operators.

References are made to [7, 1] (see also [2] for an exposition).

Let X be a complex analytic manifold, and D _X the sheaf of finite order differen- tial operators on X. Recall that automorphisms of D _X as an O _X -ring are described by closed one-forms.

• A ring of twisted differential operators (a t.d.o. ring for short) is a sheaf of O X -rings locally isomorphic to D _X . To a t.d.o. ring A corresponds a class [A] ∈ H ¹ (X; dO _X ) which characterizes A up to isomorphism of O _X -rings.

Let (S, F ) be a twisted line bundle. An example of t.d.o. ring is given by D _F = F ⊗ _O D X ⊗ _O F ⁻¹ ,

where F ⁻¹ = Hom _O (F , O X ). Notice that F ⁻¹ ∈ Mod(O X ; S ^~−1 ), so that D F is untwisted as a sheaf.

As we recalled, a twisted line bundle (S, F ) can be described by an open covering X = S

i U i , C-equivalences ϕ ⁱ : S| U

− → Mod(C U

), line bundles F i ∈ Mod(O U

), and isomorphisms θ _ij : F _j | U

− → F i | U

. For nowhere vanishing sections s _i ∈ Γ(U i ; F i ), and f ij = s i /θ ij (s j ) ∈ Γ(U ij ; O _X ^× ), sections of D _F are described by families s i ⊗ P i ⊗ s ⁻¹ _i , where P i ∈ Γ(U i ; D X ) and

(3) P _i = f _ji · P j · f ij in Γ(U _ij ; D _X ).

The isomorphism ι in (2) is then described by ι([S, F ]) = [D F ]. In particular, to any t.d.o. ring A is associated a twisted line bundle F , unique up to tensoring by a local system of rank one, such that A ' D F as an O X -ring.

Let (S, F ) be a twisted line bundle and T a stack of twisted sheaves on X. There is a C-equivalence

Mod(D _F ; T) − → Mod(D X ; S ^~−1 ~ T) (4)

M 7→ F ⁻¹ ⊗ _O M.

Denote by Θ X the sheaf of vector fields and by Ω X the sheaf of forms of maximal degree. We end this section by giving an explicit description, which will be of use later on, of the t.d.o. ring D _Ω

X

for λ ∈ C. Let v ∈ Θ X . Recall that the Lie derivative L(v) acts on differential forms of any degree, in particular on O _X , where L(v)(a) = v(a), and on Ω _X . Let ω be a nowhere vanishing local section of Ω _X . One checks that the morphism

L ^(λ) : Θ _X − → D _Ω

X

= Ω ^λ _X ⊗ _O D _X ⊗ _O Ω ^−λ _X (5)

v 7→ ω ^λ ⊗ (v + λ L(v)(ω)

ω ) ⊗ ω ^−λ

is well defined and independent from the choice of ω. (Here L(v)(ω)/ω = a, where a ∈ O _X is such that L(v)(ω) = aω.) Then D _Ω

X

is generated by O _X and L ^(λ) (Θ _X ) with the relations

L ^(λ) (av) = a · L ^(λ) (v) + λv(a), (6)

[ L ^(λ) (v), a] = v(a), (7)

[ L ^(λ) (v), L ^(λ) (w)] = L ^(λ) ([v, w]), (8)

for a ∈ O X , and v, w ∈ Θ X . Of course, L ⁽⁰⁾ (v) = v and L ⁽¹⁾ (v) = L(v).

3. Microdifferential operators on involutive submanifolds In this section we recall the notion of V -filtration on microdifferential operators.

References are made to [12, 11] (see also [6, 9, 13] for expositions).

Let W be a complex manifold. In this paper, by a submanifold of W , we mean a smooth locally closed complex submanifold.

Let X be a complex manifold, and denote by π : T ^∗ X − → X its cotangent bundle.

Identifying X with the zero-section of T ^∗ X, one sets ˙ T ^∗ X = T ^∗ X \ X.

The canonical 1-form α _X induces a homogeneous symplectic structure on T ^∗ X.

Denote by {f, g} ∈ O _T

_X the Poisson bracket of two functions f, g ∈ O _T

_X and by H : T ^∗ T ^∗ X − ^∼ → T T ^∗ X

the Hamiltonian isomorphism. For k ∈ Z, denote by O ^T

^X (k) ⊂ O T

X the subsheaf of functions ϕ homogeneous of order k, that is, satisfying eu(ϕ) = k · ϕ. Here, eu = −H(α X ) denotes the Euler vector field on T ^∗ X, the infinitesimal generator of the action of C ^× .

Denote by E X the ring of microdifferential operators on T ^∗ X. It is endowed with the order filtration {F _m E X } _m∈Z , where F _m E X is the subsheaf of microdifferential operators of order at most m. There is a canonical morphism

σ _m : F _m E X − → O T

X (m)

called the principal symbol of order m. This morphism induces an isomorphism of graded rings Gr E X ' L

k O T

X (k). If P ∈ F m E X , Q ∈ F l E X , one has σ _m+l (P Q) = σ _m (P )σ _l (Q),

(9)

σ m+l−1 ([P, Q]) = {σ m (P ), σ l (Q)}.

(10)

Let V ⊂ T ^∗ X be a submanifold and denote by J V ⊂ O T

X its annihilating ideal. Recall that V is called homogeneous, or conic, if eu J V ⊂ J V . In this case, eu _V := eu | _V is tangent to V , and one defines O _V (k) ⊂ O _V similarly to O _T

X (k) ⊂ O T

X . A conic submanifold V ⊂ T ^∗ X is called involutive if for any pair f, g ∈ J _V of holomorphic functions vanishing on V , the Poisson bracket {f, g} vanishes on V . A conic involutive submanifold V is called regular if α _X | _V never vanishes.

Let V ⊂ T ^∗ X be a conic involutive submanifold, and set I V = {P ∈ F 1 E X | V ; σ 1 (P )| V = 0} ⊂ E X | V . Note that [I V , I V ] ⊂ I V .

Definition 3.1. Let V ⊂ ˙ T ^∗ X be a conic involutive submanifold. One denotes by E _V the subring of E _X | _V generated by I _V , and one sets F ^V _m E _X := F _m E _X | _V · E _V .

One easily checks that F ^V _m E X = E V · F m E X | V , and F ^V _m E X · F ^V _l E X ⊂ F ^V _m+l E X . In particular, {F ^V _k E X } _k∈Z is an exhaustive filtration of E X | V , called the V -filtration, and F ^V ₋₁ E X is a two-sided ideal of E _V = F ^V ₀ E X .

Example 2. Let (x) = (x 1 , . . . , x n ) be a local coordinate system on X and de- note by (x; ξ) = (x 1 , . . . , x n ; ξ 1 , . . . , ξ n ) the associated homogeneous symplectic lo- cal coordinate system on T ^∗ X. Recall that locally, any conic regular involutive submanifold V of codimension d may be written after a homogeneous symplectic transformation as:

V = {(x; ξ); ξ 1 = · · · = ξ d = 0}.

In such a case,

F ^V _m E _X ' (F _m E _X | _V )[∂ _x

, . . . , ∂ _x

].

4. Systems with simple characteristics

In this section we recall the notion of systems with simple characteristics. Ref- erences are made to [12, 11]. See also [13, 9] for an exposition.

Definition 4.1. Let M be a coherent E _X -module. A lattice in M is a coherent F ₀ E X -submodule M ₀ which generates M over E _X .

Recall that if an F 0 E X -submodule M 0 of M defined on an open subset of ˙ T ^∗ X is locally of finite type, then it is coherent. A lattice M 0 endows M with the filtration

F k M = F k E X · M 0 .

If M is endowed with a filtration {F k M} k , its associated symbol module is given by

Gr(M) := O f T

X ⊗ _Gr(E

X

) Gr(M), where Gr(M) = ⊕ _k∈Z (F k M/F k−1 M).

Definition 4.2. Let V ⊂ ˙ T ^∗ X be a conic involutive submanifold.

(a) A coherent E X -module M is simple along V if it is locally generated by a section u ∈ M, called a simple generator, such that denoting by I u

the annihilator ideal of u in E X , the symbol ideal f Gr(I u ) is reduced and coincides with the annihilator ideal J V of V in O T

X .

(b) A coherent E X -module M is globally simple along V if it admits a lattice M 0 such that E V M 0 ⊂ M 0 and M 0 /F ₋₁ M is locally isomorphic to O V (0).

Such an M 0 is called a V -lattice in M.

Lemma 4.1. If M is globally simple, then it is simple.

Proof. Let M 0 be a V -lattice. Choose a local section u ∈ M 0 whose image in M 0 /F ₋₁ M is a generator of O V (0). Then M 0 = F 0 E X u + F ₋₁ M and it follows that for all k ≤ 0

M 0 = F 0 E X u + F k M

Since the filtration on M is separated (see [12]), u generates M 0 over F 0 E X .  Let (t) ∈ C be a coordinate, and denote by (t; τ ) ∈ T ^∗ C the associated ho- mogeneous symplectic coordinate system. Let V ⊂ T ^∗ X be a conic involutive submanifold, non necessarily regular. The trick of the dummy variable consists in replacing V with the conic involutive submanifold e V = V × ˙ T ^∗ C, which is regular.

Let p ∈ V and q ∈ ˙ T ^∗ C. If Σ is the bicharacteristic leaf of V through p, then Σ × {q} is the bicharacteristic leaf of e V through (p, q).

Proposition 4.1. If M is a globally simple E X -module along V , then f M = E _X×C ⊗ _E

X

E

(M  E C ) is globally simple along e V . Proof. Let M 0 be a V -lattice in M, and set

M f ₀ = F ₀ E _X×C ⊗ _F

0

E

F

E

(M ₀  F 0 E _C ).

Clearly, f M 0 is a lattice in f M and moreover, E

V e M f 0 ⊂ f M 0 . Note that F ₋₁ M = F f 0 E _X×C ⊗ _F

0

E

F

E

(F ₋₁ M  F 0 E _C + M 0  F −1 E _C ).

Set M ₋₁ = F ₋₁ M, M 0 = M 0 /M ₋₁ , and consider the commutative exact diagram of F ₀ E _X  F 0 E _C -modules:

0



0



0



0 // _M

 F

E

//



M

 F

E

//



M

 F

E

//



0 0 // M

 F

E

//



M

 F

E

//



M

 F

E

//



0 0 // M

 (F

E

/F

E

) //



M

 (F

E

/F

E

) //



M

 (F

E

/F

E

) //



0

0 0 0

It follows that the sequence

0 − → M ₋₁  F 0 E _C + M ₀  F −1 E _C − → M ₀  F 0 E _C − → M ₀  F 0 E _C /F ₋₁ E _C − → 0 is exact. Since F ₀ E _X×C is flat over F ₀ E X  F 0 E _C , we locally have

F 0 M/F f −1 M ' F f 0 E _X×C ⊗ _F

0

E

F

E

(M 0  F ⁰ E _C /F −1 E _C ) ' F 0 E _X×C ⊗ _F

0

E

F

E

(O V (0)  O T ^˙

C (0)) ' O

V e (0).

 Remark. Let S be a C-stack of twisted sheaves on X. Then Definition 4.2, Lemma 4.1 and Proposition 4.1 extend to objects of Mod(E X ; π ^~ S).

5. Differential operators on involutive submanifolds

We recall here the construction of the ring of homogeneous twisted differential operators invariant by the bicharacteristic flow.

Let V ⊂ T ^∗ X be a conic regular involutive submanifold and denote by T V ^⊥ ⊂ T V the symplectic orthogonal to T V . Denote by Θ ^⊥ _V ⊂ Θ V the sheaf of sections of the bundle T V ^⊥ − → V , and let

O _V ^bic := {a ∈ O V ; v(a) = 0 for any v ∈ Θ ^⊥ _V }, O _V ^bic (k) := O _V ^bic ∩ O _V (k).

Then O _V ^bic is the sheaf of holomorphic functions locally constant along the bichar- acteristic leaves of V . Consider the ring

D ^bic _V := {P ∈ D _V ; [a, P ] = 0 for any a ∈ O _V ^bic }, and the subring of operators homogeneous of degree zero

D _V ^bic (0) := {P ∈ D ^bic _V ; [eu _V , P ] = 0}.

Example 3. Let (x; ξ) = (x ₁ , . . . , x _n ; ξ ₁ , . . . , ξ _n ) be a local homogeneous symplectic coordinate system on T ^∗ X and assume that

V = {(x; ξ); ξ ₁ = · · · = ξ _d = 0}.

Set x ⁰ = (x ₁ , . . . , x _d ), x ⁰⁰ = (x _d+1 , . . . x _n ), and similarly set ξ = (ξ ⁰ , ξ ⁰⁰ ). One has (x ⁰ , x ⁰⁰ , ξ ⁰⁰ ) ∈ V , and the bicharacteristic leaves of V are the submanifolds defined by

Σ = {(x ⁰ , x ⁰⁰ ; ξ ⁰⁰ ); (x ⁰⁰ ; ξ ⁰⁰ ) = (x ⁰⁰ ₀ ; ξ ₀ ⁰⁰ )}.

The Euler field eu V is given by eu V =

n

X

d+1

ξ i ∂ ξ

= ξ ⁰⁰ ∂ ξ

.

Hence a function locally constant along the bicharacteristic leaves depends only on (x ⁰⁰ , ξ ⁰⁰ ). A section of O _V (0) is a holomorphic functions in the variable (x ⁰ , x ⁰⁰ , ξ ⁰⁰ ), homogeneous of degree 0 with respect to ξ ⁰⁰ . Moreover a section of D ^bic _V (0) is uniquely written as a finite sum

(11) X

α∈N

a _α ∂ ^α _x

, with a _α ∈ O V (0).

Let j _Σ : Σ − → V be the embedding of a bicharacteristic leaf. Assume that V is regular, and that Σ is a locally closed submanifold of V . Denote by J _Σ ^bic (0) the annihilator ideal of Σ in O ^bic _V (0), and note that O Σ ' O _V ^bic (0)/J _Σ ^bic (0)| Σ . Since O ^bic _V (0) is in the center of D _V ^bic (0), there is a restriction map

j _Σ ^∗ : Mod(D ^bic _V (0)) − → Mod(D Σ ) M 7→ C Σ ⊗ _O

V

(0)|

M| Σ .

We will be interested in the twisted analogue of the above construction. Namely, set

D ^bic

Ω

:= {P ∈ D _Ω

; [a, P ] = 0 for any a ∈ O ^bic _V }, D ^bic

Ω

(0) := {P ∈ D ^bic

Ω

; [L ^(1/2) (eu V ), P ] = 0}.

For p ∈ Σ, the quotient T _p V /T _p Σ ' T _p V /T _p V ^⊥ is a symplectic space. Hence j _Σ ^∗ Ω _V ' Ω _Σ . Thus, there is a restriction morphism

(12) j _Σ ^∗ : Mod(D ^bic

Ω

(0)) − → Mod(D _Ω

).

6. Subprincipal symbol

In this section we recall the notion of subprincipal symbol, and prove the regular involutive analogue of an isomorphism obtained in [10, Lemma 1.5.1] for the La- grangian case. References are made to [11, 10, 6, 9] (see [4] for the corresponding constructions in the C ^∞ case).

As we will recall, the subprincipal symbol is intrinsically defined for microdiffer- ential operators twisted by half-forms. We will thus consider here the ring

E _Ω

= π ⁻¹ Ω ^1/2 _X ⊗ _π

O E X ⊗ _π

O π ⁻¹ Ω ^−1/2 _X ,

instead of E X . All the notions recalled in Section 3 extend to this ring. In particular, its V -filtration is defined by

I ^Ω

1/2 X

V = {P ∈ F 1 E _Ω

| V ; σ 1 (P )| V = 0}

' π ⁻¹ _V Ω ^1/2 _X ⊗ _π

V

O I V ⊗ _π

V

O π _V ⁻¹ Ω ^−1/2 _X , F ^V _m E _Ω

X

= π ⁻¹ _V Ω ^1/2 _X ⊗ _π

V

O F ^V _m E X ⊗ _π

V

O π ⁻¹ _V Ω ^−1/2 _X , E _V,Ω

X

= F ^V ₀ E _Ω

, where π V = π| V .

Let (x) be a local coordinate system on X, and denote by (x; ξ) the associated homogeneous symplectic coordinate system on T ^∗ X. A microdifferential operator P ∈ F m E _Ω

X

is then described by its total symbol {p k (x; ξ)} k≤m , where p k ∈ O T

X (k). The functions p k depend on the local coordinate system (x) on X, except the top degree term p m = σ m (P ) which does not. Recall that the subprincipal symbol

σ ⁰ _m−1 : F _m E _Ω

−

→ O _T

_X (m − 1) given by

σ _m−1 ⁰ ((dx) ^1/2 ⊗ P ⊗ (dx) ^−1/2 ) = p m−1 (x, ξ) − 1 2

X

i

∂ x

∂ ξ

p m (x, ξ), does not depend on the local coordinate system (x) on X. For P ∈ F m E _Ω

X

, Q ∈ F l E _Ω

X

, one has

σ ⁰ _m+l−1 (P Q) = σ _m (P )σ ⁰ _l−1 (Q) + σ _m−1 ⁰ (P )σ _l (Q) + 1

2 {σ _m (P ), σ _l (Q)}, (13)

σ ⁰ _m+l−2 ([P, Q]) = {σ m (P ), σ _l−1 ⁰ (Q)} + {σ _m−1 ⁰ (P ), σ _l (Q)}.

(14)

Let V ⊂ T ^∗ X be a conic involutive submanifold. For f ∈ O T

X , denote by H f = H(df ) ∈ T T ^∗ X its Hamiltonian vector field. Recall that H induces an isomorphism

(15) H : T _V ^∗ T ^∗ X − ^∼ → T V ^⊥ .

In particular, H f | V is tangent to V for f ∈ J V . With notations (5), consider the transport operator

L ⁰ _V : I ^Ω

1/2 X

V − → F 1 D _Ω

, (16)

P 7→ L ^(1/2) (H _σ

_{(P )} | V ) + σ ₀ ⁰ (P )| _V .

Using the above relations, one checks that the morphism L ⁰ _V does not depend on the choice of coordinates, and satisfies the relation

L ⁰ _V (AP ) = σ ₀ (A)L ⁰ _V (P ), L ⁰ _V (P A) = L ⁰ _V (P )σ ₀ (A), L ⁰ _V ([P, Q]) = [L ⁰ _V (P ), L ⁰ _V (Q)], for P, Q ∈ I ^Ω

1/2 X

V and A ∈ F 0 E _Ω

(see [11, §2] or [9, §8.3]). It follows that L ⁰ _V extends as a ring morphism

(17) L V : E _V,Ω

X

−

→ D _Ω

by setting L V (P 1 · · · P r ) = L ⁰ _V (P 1 ) · · · L ⁰ _V (P r ), for P i ∈ I ^Ω

1/2 X

V .

Theorem 6.1. Let V ⊂ ˙ T ^∗ X be a conic regular involutive submanifold. The morphism (17) induces a ring isomorphism

(18) L V : E _V,Ω

X

/F ^V ₋₁ E _Ω

− ∼ → D ^bic

Ω

(0).

It is possible to show that the above statement holds even without the as- sumption of regularity for V (for example, the Lagrangian case is obtained in [10, Lemma 1.5.1]).

Proof. The statement is local. We may thus assume that Ω X ' O X and Ω V ' O V , so that we are reduced to prove the isomorphism

L V : E V /F ^V ₋₁ E X

− ∼ → D _V ^bic (0).

Moreover, since V is regular we may assume that we are in the situation of Exam- ple 3. By Example 2, sections of E _V are uniquely written as finite sums

(19) X

α∈N

A α ∂ _x ^α

, with A α ∈ F 0 E X | V . One concludes using (11) since, by definition of L _V ,

L V ( X

α

A α ∂ _x ^α

) = X

α

σ 0 (A α )∂ _x ^α

.

 Let Ω _{V /X} = Ω _X ⊗ _O π ^∗ _V Ω _X be the sheaf of relative forms. Recall from Example 1 that S _Ω

V /X

denotes a stack of twisted sheaves such that Ω ^−1/2 _{V /X} ∈ Mod(O V ; S _Ω

).

Corollary 6.1. Let V ⊂ ˙ T ^∗ X be a conic regular involutive submanifold, and T be a stack of twisted sheaves on V . Then there is an equivalence of categories

Mod(E V /F ^V ₋₁ E X ; T) ' Mod(D _V ^bic (0); T ~ S _Ω

V /X

).

7. Statement of the result

We can now state our main result. For a submanifold Σ ⊂ ˙ T ^∗ X, set π _Σ = π| _Σ , and denote by π _Σ ^] : H ² (X; C ^× X ) − → H ² (Σ; C ^× Σ ) the pull-back.

Theorem 7.1. Let V ⊂ ˙ T ^∗ X be a conic involutive submanifold, Σ ⊂ V a bichar- acteristic leaf, and T a stack of twisted sheaves on X. Assume that Σ is a locally closed submanifold of V , and that there exists a globally simple module along V in Mod(E X ; π ^~ T). Then

π ^] _Σ ([T]) = [S _Ω

] in H ² (Σ; C ^× Σ ).

Proof. The proof follows the same lines as in [10, §I.5.2]. Let us first reduce to the regular involutive case by the trick of the dummy variable. Let p : e X = X × C − → X be the projection. With the notations of Proposition 4.1, replace X with e X, T with T e = p ^~ T, V with e V = V × ˙ T ^∗ C, M with f M, and Σ with e Σ = Σ × {(0; 1)}. Under the isomorphism H ² (Σ; C ^× Σ ) ' H ² (e Σ; C ^×

Σ e ) one has π ^] _Σ ([T]) = π ^]

Σ e ([e T]). Hence we may assume that V is regular involutive.

Let M be a globally simple module along V in Mod(E _X ; π ^~ T), and let M ₀ be a V -lattice in M. Then M 0 = M 0 /F ₋₁ M ∈ Mod(E V /F ^V ₋₁ E X ; π _V ^~ T) is locally isomorphic to O _V (0). By Corollary 6.1, M ₀ ∈ Mod(D ^bic _V (0); T ~ S _Ω

V /X

).

By the restriction functor (12) and the equivalence (4), j _Σ ^∗ (M 0 ) is an object of Mod(D _Σ ; π ^~ _Σ T ~ S _Ω

Σ/X

) locally isomorphic to O _Σ . Hence its solution sheaf Hom _D

Σ

(j ^∗ _Σ (M 0 ), O Σ ) ∈ (π ^~ _Σ T ~ S _Ω

Σ/X

) ^~−1 (Σ) is a local system of rank 1. It follows by Proposition 1.1 that the class [(π ^~ _Σ T ~S _Ω

Σ/X

) ^~−1 ] = [S _Ω

]·π ^] _Σ ([T]) ⁻¹

is trivial in H ² (Σ; C ^× Σ ). 

Remark. Let us say that a coherent E X -module M is globally r-simple along V if it admits a lattice M 0 such that E V M 0 ⊂ M 0 and M 0 /F ₋₁ M is locally isomorphic to O V (0) ^r . Theorem 7.1 extends to globally r-simple modules as follows. If there exists a globally r-simple module along V in Mod(E X ; π ^~ T), then

π ^] _Σ ([T]) ^r = ([S _Ω

]) ^r in H ² (Σ; C ^× Σ ).

The proof goes along the same lines as the one above, recalling the following fact.

Let S be a stack of twisted sheaves on X, and let F ∈ S(X) be a local system of rank r. Then det F is a local system of rank 1 in S ^~r (X), so that S ^~r is globally C-equivalent to Mod(C X ).

Corollary 7.1. Let V ⊂ ˙ T ^∗ X be a conic involutive submanifold, Σ ⊂ V a bichar- acteristic leaf, and T a stack of twisted sheaves on X. Assume that Σ is a lo- cally closed submanifold of V , that π ^] _Σ : H ² (X; C ^× X ) − → H ² (Σ; C ^× Σ ) is injective, that [S _Ω

Σ/X

] = 1 in H ² (Σ; C ^× Σ ), and that there exists a globally simple module along V

in Mod(E X ; π ^~ T ). Then T is globally C-equivalent to Mod(C ^X ).

Proof. By Theorem 7.1, π ^] _Σ ([T]) = 1 in H ² (Σ; C ^× Σ ). Since π _Σ ^] is injective, [T] = 1 in H ² (X; C ^× X ), and this implies that the stack T is globally C-equivalent to Mod(C X ).



8. Application: non existence of twisted wave equations Let T be an (n + 1)-dimensional complex vector space, P the projective space of lines in T, and G the Grassmannian of (p + 1)-dimensional subspaces in T. Assume n ≥ 3 and 1 ≤ p ≤ n − 2. The Penrose correspondence (see [5]) is associated with the double fibration

(20) P ← −

f F − →

g G

where F = {(y, x) ∈ P × G; y ⊂ x} is the incidence relation, and f , g are the natural projections. The double fibration (20) induces the maps

T ˙ ^∗ P ← −

p

T ˙ _F ^∗ (P × G) − →

q

T ˙ ^∗ G, where T ^∗

F (P × G) ⊂ T ^∗ (P × G) denotes the conormal bundle to F, and p and q are the natural projections. Note that p is smooth surjective, and q is a closed embedding. Set

V = q( ˙ T _F ^∗ (P × G)).

Then V is a closed conic regular involutive submanifold of ˙ T ^∗ G, and q identifies the fibers of p with the bicharacteristic leaves of V .

For m ∈ Z, let O P (m) be the line bundle on P corresponding to the sheaf of homogeneous functions of degree m on T, and denote by N (m) := D _P ⊗ _O O _P (−m) the associated D _P -module. Denote by Dg ∗ and Df ^∗ the direct and inverse image in the derived categories of D-modules and consider the family of D _G -modules

M (1+m/2) := H ⁰ (Dg ∗ Df ^∗ N (m) ).

For n = 3 and p = 1, Penrose identifies G with a conformal compactification of the complexified Minkowski space, and the D _G -module M _(1+m/2) corresponds to the massless field equation of helicity 1 + m/2.

By [3], the microlocalization E _G ⊗ _π

D

π ⁻¹ M _(1+m/2) of M _(1+m/2) is globally simple along V .

Theorem 8.1. Let S be a stack of twisted sheaves on G and M an object of Mod(D _G ; S) whose microlocalization E _G ⊗ _π

D

π ⁻¹ M is globally simple along V . Then S is globally C-equivalent to Mod(C G ), so that Mod(D _G ; S) is C-equivalent to Mod(D _G ).

In other words, M is untwisted.

Proof. Let us start by recalling the microlocal geometry underlying the double fibration (20). There are identifications

T ^∗ P = {(y; η); y ⊂ T, η ∈ Hom (T/y, y)},

T ^∗ G = {(x; ξ); x ⊂ T, ξ ∈ Hom (T/x, x)},

T _F ^∗ (P × G) = {(y, x; τ ); y ⊂ x ⊂ T, τ ∈ Hom (T/x, y)}.

The maps p and q are described as follows:

T ˙ ^∗ P oo p T ˙ _F ^∗ (P × G)

q // ˙T ^∗ G (y; τ ◦ j) oo  (y, x; τ )  // (x; i ◦ τ),

where i : y  x and j : T/y  T/x are the natural maps. We thus get V = {(x; ξ); rk(ξ) = 1},

where rk(ξ) denotes the rank of the linear map ξ. In order to describe the bicharac- teristic leaves of V , denote by P ^∗ the dual projective space consisting of hyperplanes z ⊂ T, and consider the incidence relation

A = {(y, z) ∈ P × P ^∗ ; y ⊂ z ⊂ T}.

Then

T ˙ _A ^∗ (P × P ^∗ ) = {(y, z; θ); y ⊂ z ⊂ T, θ : T/z − ^∼ → y}.

There is an isomorphism

T ˙ _A ^∗ (P × P ^∗ ) − ^∼ → ˙ T ^∗ P (y, z; θ) 7→ (y; θ ◦ k).

where k : T/y  T/z is the natural map. Set y = im ξ, z = x + ker ξ, and consider the commutative diagram of linear maps

T/y

j //

k

""

T/x

 `

ξ // y ⁱ // x

T/z.

ξ e

>>

We thus get the following description of the composite map

p : e V ^∼ // ˙T _F ^∗ (P × G) ^p // ˙T ^∗ P ^∼ // ˙T _A ^∗ (P × P ^∗ ) (x; ξ)  // (im ξ, x; ξ)  // (im ξ; ξ ◦ j)  // (im ξ, x + ker ξ; ξ), e It follows that the bicharacteristic leaf Σ _(y,z,θ) := e p ⁻¹ (y, z, θ) of V is given by

Σ _(y,z,θ) = {(x; ξ); y = im ξ, z = x + ker ξ, θ ◦ ` = ξ}

(21)

= {(x; ξ); y ⊂ x ⊂ z, ξ = θ ◦ `},

where ` : T/x  T/z is the natural map. Thus, Σ (y,z,θ) is the Grassmannian of p-dimensional linear subspaces in the (n − 1)-dimensional vector space z/y.

Let us fix a point (y, z, θ) ∈ ˙ T _A ^∗ (P × P ^∗ ), and set Σ = Σ _(y,z,θ) . In order to apply Corollary 7.1, we need to compute the map π ^] _Σ and the class [S _Ω

Σ/G

].

The universal bundle U _G − → G is the sub-bundle of the trivial bundle G×T whose fiber at x ∈ G is the (p + 1)-dimensional linear subspace x ⊂ T itself. Consider the line bundle D _G = det U _G , and denote by O _G (−1) the sheaf of its sections. Recall the isomorphisms

H ¹ (G; C ^× _G ) ' H ² (G; O _G ^× ) ' 0,

H ¹ (G; O _G ^× ) ' Z with generator [O G (−1)],

H ¹ (G; O _G ^× /C ^× _G ) ' H ¹ (G; O ^× _G /C ^× _G ) ' C with generator [Mod(C G ), O _G (−1)],

so that the sequence of abelian groups H ¹ (G; C ^× _G ) − →

α H ¹ (G; O ^× _G ) − →

β H ¹ (G; O ^× _G /C ^× _G ) − →

δ H ² (G; C ^× _G ) − → H ² (G; O ^× _G ), is isomorphic to the sequence of additive abelian groups

0 − → Z − →

β C − →

δ C/Z − → 0.

Similar results hold for Σ, which is also a grassmannian.

By Lemma 8.1 below one has π _Σ ^∗ O _G (−1) ' O Σ (−1). Hence π _Σ ^] is the isomor- phism

π _Σ ^] : H ² (G; C ^× _G ) ' C/Z ' H ² (Σ; C ^× Σ ).

There are isomorphisms

Ω _G ' O _G (−n − 1), Ω Σ ' O Σ (−n + 1).

Again by Lemma 8.1, we thus have

π ^∗ _Σ Ω _G ' π _Σ ^∗ O _G (−n − 1) ' O _Σ (−n − 1).

It follows that Ω _Σ/G ' O Σ (2), and thus

[Ω _Σ/G ] = 2 in Z ' H ¹ (Σ; O ^× _Σ ).

Therefore

[S _Ω

, Ω ^1/2 _Σ/G ] = 1 in C ' H ¹ (Σ; O ^× _Σ /C ^× Σ ), so that

[S _Ω

] = δ([S _Ω

, Ω ^1/2 _Σ/G ]) = 0 in C/Z ' H ² (Σ; C ^× Σ ).

The statement follows by Corollary 7.1. 

Lemma 8.1. There is a natural isomorphism π _Σ ^∗ O _G (−1) ' O _Σ (−1).

Proof. Recall that D _G denotes the determinant of the universal bundle on G. Ge- ometrically, we have to prove that there is an isomorphism δ : D _Σ − ^∼ → D _G | _Σ .

Recall the description (21), and let (x; ξ) ∈ Σ for p = (y, z, θ) ∈ ˙ T ^∗ P. Then (D _Σ ) _(x;ξ) = det(x/y), (D _G ) _(x;ξ) = det x, and δ is obtained by a trivialization of

det y ' C. 

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