Using a result of J-M. Bony, we prove the weak involutivity of truncated
microsupports. More precisely, given a sheaf $F$ on a real manifold and an
integer $k$, if two functions vanish on the truncated microsupport $Ss_k(F)$,
then so does their Poisson bracket.Comment: 9 page

Fernandes, Teresa Monteiro

Kashiwara, Masaki

Schapira, Pierre

English

arXiv

Monteiro Fernandes, Térésa

Numérisation de Documents Anciens Mathématiques

Bull. Soc. math. France131 (2), 2003, p. 259–266INVOLUTIVITY OF TRUNCATED MICROSUPPORTSby Masaki Kashiwara, Te´re´sa Monteiro Fernandes& Pierre SchapiraAbstract. — Using a result of J.-M. Bony, we prove the weak involutivity of trun-cated microsupports. More precisely, given a sheaf F on a real manifold and k ∈ Z,if two functions vanish on SSk(F ), then so does their Poisson bracket.Re´sume´ (Involutivite´ des micro-supports tronque´s). — Utilisant un re´sultat deJ.-M. Bony, nous prouvons l’involutivite´ faible des micro-supports tronque´s. Pluspre´cise´ment, e´tant donne´s un faisceau F sur une varie´te´ re´elle et k ∈ Z, si deuxfonctions s’annulent sur SSk(F ), il en est de meˆme de leur crochet de Poisson.Texte rec¸u le 21 mai 2002, accepte´ le 24 septembre 2002Masaki Kashiwara, Research Institute for Mathematical Sciences, Kyoto University, Kyoto606-8502 (Japan) • E-mail : masaki@kurims.kyoto-u.ac.jpTe´re´sa Monteiro Fernandes, Centro de A´lgebra da Universidade de Lisboa, Complexo 2,2 Avenida Prof. Gama Pinto, 1699 Lisboa codex (Portugal)E-mail : tmf@ptmat.lmc.fc.ul.ptPierre Schapira, Universite´ Pierre et Marie Curie, case 82, Institut de Mathe´matiques,4 place Jussieu, 75252 Paris cedex 05 (France) • E-mail : schapira@math.jussieu.frUrl : http://www.institut.math.jussieu.fr/~schapira/2000 Mathematics Subject Classiﬁcation. — Primary: 35A27; Secondary: 32C38.Key words and phrases. — Sheaves, microsupport, involutivity.M.K.: Research partially supported by Grant-in-Aid for Scientiﬁc Research (B) 13440006,Japan Society for the Promotion of Science.T.M.F.: Research supported by FCT and Programa Cieˆncia, Tecnologia e Inovac¸a˜o doQuadro Comunita´rio de Apoio.BULLETIN DE LA SOCIE´TE´ MATHE´MATIQUE DE FRANCE 0037-9484/2003/259/$ 5.00c© Socie´te´ Mathe´matique de France260 KASHIWARA (M.), MONTEIRO FERNANDES (T.) & SCHAPIRA (P.)1. IntroductionThe notion of microsupport of sheaves was introduced by two of the presentauthors (M.K. and P.S.) in the course of the study of the theory of linear partialdiﬀerential equations. References are made to [3]. These authors also intro-duced a variant of this notion, that of “truncated microsupport” and developedits study with the third author in [2].A crucial result in the microlocal theory of sheaves is the involutivity of themicrosupport. This property does not hold true any more for the truncatedmicrosupport, but a weak form of it does. More precisely, we prove here thatthe truncated microsupport is stable by Poisson bracket, that is, if two functionsvanish on it, so does their Poisson bracket. The main technical tool is a resultof J.-M. Bony which asserts the same property for the normal cone to a closedsubset.2. Notations and reviewWe will follow the notations of [3] and [2]. For the reader convenience, werecall some of them as well as the deﬁnition of the truncated microsupport.Let X be a real analytic manifold. We denote by τ : TX −→ X the tangentbundle to X and by π : T ∗X −→ X the cotangent bundle. For a smoothsubmanifold Y of X , TYX denotes the normal bundle to Y and T ∗YX theconormal bundle. In particular, T ∗XX is the zero section of T∗X . We setT˙ ∗X = T ∗X \X , and denote by π˙ : T˙ ∗X −→ X the restriction of π to T˙ ∗X .For a morphism f : X −→ Y of real manifolds, we denote the associatedmorphisms byfπ : X ×Y T ∗Y −→ T ∗Y and fd : X ×Y T ∗Y −→ T ∗X.For a subset A of T ∗X , we denote by Aa the image of A by the antipodalmap a : (x; ξ) → (x;−ξ). The closure of A is denoted by A.For a cone γ ⊂ TX , the polar cone γ◦ to γ is the convex cone in T ∗X deﬁnedbyγ◦ ={(x; ξ) ∈ T ∗X ;x ∈ π(γ) and 〈v, ξ〉 ≥ 0 for any (x; v) ∈ γ}.A closed convex cone γ is called proper if 0 ∈ γ and Int(γ◦) = ∅.Let k be a ﬁeld. One denotes by Mod(kX) the abelian category of sheavesof k-vector spaces and by Db(kX) its bounded derived category. One denotesby DbR-c(kX) the full triangulated subcategory of Db(kX) consisting of objectswith R-constructible cohomology. If X is a complex manifold, one denotes byDbC-c(kX) the full triangulated subcategory of Db(kX) consisting of objectswith C-constructible cohomology.If S is a locally closed subset of X , one denotes by kXS the sheaf on Xwhich is the constant sheaf with stalk k on S and 0 on X \S. If there is no riskof confusion, we may write kS instead of kXS .tome 131 – 2003 – no 2INVOLUTIVITY OF TRUNCATED MICROSUPPORTS 261For k ∈ Z, we denote as usual by D≥k(kX) (resp. D≤k(kX)) the full ad-ditive subcategory of Db(kX) consisting of objects F satisfying Hj(F ) = 0for any j < k (resp. Hj(F ) = 0 for any j > k).We denote by τ≤k : D(kX) −→ D≤k(kX) the truncation functor.Recall that for F ∈ D(kX) the morphism τ≤kF −→ F induces isomorphismsHj(τ≤kF ) ∼−→ Hj(F ) for j ≤ k and Hj(τ≤kF ) = 0 for j > k.If F is an object of Db(kX), SS(F ) denotes its microsupport, a closedR+-conic subset of T ∗X . For p ∈ T ∗X , Db(kX ; p) denotes the localizationof Db(kX) by the full triangulated subcategory consisting of objects F suchthat p /∈ SS(F ). A property holds “microlocally on a subset S of T ∗X” ifit holds in the category Db(kX ; p) for any p ∈ S.We recall the deﬁnition of involutivity of [3]. This notion makes use of thatof “normal cone” (see [3, Def. 4.1.1]). For a pair of subsets S1 and S2 of amanifold X , the normal cone Cp(S1, S2) at p ∈ X is deﬁned as follows: it isa closed cone in the tangent space TpX consisting of points v such that, for alocal coordinate system, there exist a sequence {xn}n in S1, {yn}n in S2 anda sequence {an}n in R≥0 such that xn and yn converge to p and an(xn − yn)converges to v. For a subset S, Cp(S, {p}) is denoted by Cp(S).Definition 2.1. — (see [3, Deﬁnition 6.5.1]) Let S be a locally closed sub-set of T ∗X and let p ∈ S. One says that S is involutive at p if for anyθ ∈ T ∗p (T ∗X) such that the normal cone Cp(S, S) is contained in the hyper-plane {v ∈ Tp(T ∗X) ; 〈v, θ〉 = 0} one has H(θ) ∈ Cp(S). Here H : T ∗p (T ∗X) ∼−→Tp(T ∗X) is the Hamiltonian isomorphism.If S is involutive at each p ∈ S, one says that S is involutive.The involutivity theorem of [3, Theorem 6.5.4] asserts that the microsupportof sheaves is involutive.The following deﬁnition was introduced by the authors of [3] and developedin [2].Definition 2.2. — Let X be a real analytic manifold and let p ∈ T ∗X . LetF ∈ Db(kX) and k ∈ Z. The closed conic subset SSk(F ) of T ∗X is deﬁnedby p /∈ SSk(F ) if and only if the following condition is satisﬁed.(i) There exists an open conic neighborhood U of p such that for anyx ∈ π(U) and for any R-valued C1-function ϕ deﬁned on a neighborhoodof x such that ϕ(x) = 0, dϕ(x) ∈ U , one has(2.1) Hj{ϕ≥0}(F )x = 0 for any j ≤ k.We refer to [2] for equivalent deﬁnitions. In particular, it is proved in loc. cit.that one can replace the condition that ϕ is of class C1 by ϕ is of class Cα,where α ∈ Z≥1 ∪ {∞, ω}.This condition is also equivalent to:BULLETIN DE LA SOCIE´TE´ MATHE´MATIQUE DE FRANCE262 KASHIWARA (M.), MONTEIRO FERNANDES (T.) & SCHAPIRA (P.)(ii) there exists F ′ ∈ D≥k+1(kX) and an isomorphism F  F ′ in Db(kX ; p).3. Normal coneLet us recall the notion of the normal vector due to J.-M. Bony. Here we setfor r ≥ 0Br(x) ={y ∈ Rn ; ‖y − x‖ < r},the open ball with center x and radius r.Proposition 3.1. — Let X be a real manifold, S a closed subset of X, andα ∈ Z≥1 ∪ {∞, ω}. Then the following subsets of T ∗X are equal:(i) SS0(kS);(ii)α the closure of the set of points p = (x; ξ) ∈ T ∗X such that x ∈ S and thereis a Cα function ϕ deﬁned on a neighborhood U of x such that ϕ(x) = 0,dϕ(x) = ξ and S ∩ U ⊂ ϕ−1(R≥0);(iii) the closure of the set of points p = (x; ξ) ∈ T ∗X such that x ∈ S andCx(S) ⊂ {v ∈ TxX ; 〈v, ξ〉 ≥ 0}.If X is an open subset of Rn, then the above sets are equal to the followingset:(iv) the closure of the set of points p = (x; ξ) ∈ T ∗X such that x ∈ S and theopen ball Bt‖ξ‖(x− tξ) does not intersect S for some t > 0.Proof. — We may assume that X = Rn.(i) = (ii)α is an immediate consequence of Deﬁnition 2.2. In particular, theset (ii)α does not depend on α.(ii)α = (iv) for α ≥ 2 is obvious.(ii)α ⊂ (iii) is clear.(iii) ⊂ (iv). Let p = (x0; ξ0) ∈ T ∗X with x0 ∈ S andCx0(S) ⊂ {v ∈ TxX ; 〈v, ξ0〉 ≥ 0}.We have to prove that p belongs to the set (iv). The proof is very similar tothat of Lemma 3.3 of [2], but for the sake of completeness, we shall repeat it.If ξ0 = 0, then it is trivially true. Hence we may assume that ξ0 = 0.We shall show that for an arbitrary open conic neighborhood U of p, thereexists a point (x1; ξ1) ∈ U such that the open ball B‖ξ1‖(x1 − ξ1) satisﬁesx1 ∈ S, x1 /∈ B‖ξ1‖(x1 − ξ1) ∩ S.(3.1)This implies Bt‖ξ1‖(x1 − tξ1) ∩ S = ∅ for 0 < t 1.Let us take an open neighborhood V and a proper closed convex cone γ suchthat ξ0 ∈ Int(γ◦), V × (γ◦ \{0}) ⊂ U and Int(γ) = ∅. Since Cx0(S)∩γa ⊂ {0},there is ρ > 0 such thatH− :={x ∈ Rn ; 〈x− x0, ξ0〉 > −ρ}tome 131 – 2003 – no 2INVOLUTIVITY OF TRUNCATED MICROSUPPORTS 263satisﬁes H−∩(x0+γa) ⊂ V and S∩H−∩(x0+γa) = {x0}. Set S0 := S∩H−.Let us deﬁne the function ψ(x) on Rn byψ(x) = dist(x, γa) := inf{‖y − x‖ ; y ∈ γa}.It is well known that ψ is a continuous function on Rn, and C1 on Rn \ γa.More precisely for any x ∈ Rn \ γa, there exists a unique y ∈ γa such thatψ(x) = ‖x− y‖. Moreover dψ(x) = ‖x− y‖−1(x − y) ∈ γ◦ \ {0}. FurthermoreBψ(x)(y) is contained in {z ∈ Rn;ψ(z) < ψ(x)}.For ε > 0, we setγaε ={x ∈ Rn ; ψ(x) < ε}.Then γaε is an open convex set. Moreover γaε + γa = γaε .Let us take v ∈ Int(γ), and δ > 0 such that x0 − δv ∈ H−. Then take ε > 0such that γaε − δv ⊂ Int(γa), (x0 + γaε ) ∩H− ⊂ V and (x0 + γaε ) ∩ S0 ⊂ H−.Set Wt = x0 + γaε + tv for t ∈ R. Then one hasWt =⋃t′<tWt′ and Wt =⋂t′>tWt′ =⋂t′>tWt′ ,(3.2)x0 ∈ Wt ∩ S for t ≥ 0, and Wt ∩H− = ∅ for t 0,(3.3)Wt ∩ S0 ⊂ H− ∩ V for t ≤ 0.(3.4)Hence, for any closed subset K of H− and t ∈ R such that K ∩Wt = ∅, thereexists t′ > t such that K ∩Wt′ = ∅.Let us set c = sup{t ; Wt ∩ S0 = ∅}. Then c ≤ 0 and Wc ∩ S0 = ∅. By theabove remark, one hasWc∩S0 = ∅. Take x1 ∈ S0∩∂Wc. Here ∂Wc :=Wc \Wcis the boundary of Wc. The point x1 belongs to H− ∩ V .As seen before, there exists an open ball Bε(y) such that Bε(y) ⊂ Wc,‖x1 − y‖ = ε and x1 − y ∈ γ◦. Hence Bε(y) ∩ S ∩ H− = ∅, while H− is aneighborhood of x1. Hence (x1;x1−y) belongs to U and satisﬁes condition (3.1).Definition 3.2. — For a closed subset S of X , the closed subset givenin Proposition 3.1 is called the 0-conormal cone of S and denoted by N∗0 (S).Note that N∗0 (S) is a closed cone and it satisﬁesπX(N∗0 (S))= πX(N∗0 (S) ∩ T ∗XX)= S.Note also that one has N∗0 (S) = SS0(kS) ⊂ SS(kS).Example 3.3. — (i) If S = X , then N∗0 (S) = T∗XX .(ii) If S is a closed submanifold, then N∗0 (S) = T ∗SX .(iii) If X = R2 and S = {(x, y) ∈ X ; x ≥ 0 or y ≥ 0}. ThenN∗0 (S) ={(x, y; 0, 0) ∈ T ∗XX ; (x, y) ∈ S}∪{(x, y; ξ, η) ∈ T ∗X ; y = 0, x ≤ 0, η ≥ 0 and ξ = 0}∪{(x, y; ξ, η) ∈ T ∗X ; x = 0, y ≤ 0, ξ ≥ 0 and η = 0}.BULLETIN DE LA SOCIE´TE´ MATHE´MATIQUE DE FRANCE264 KASHIWARA (M.), MONTEIRO FERNANDES (T.) & SCHAPIRA (P.)One has the microlocal isomorphismskXS kX on {(x, y; 0, 0) ; x > 0 or y > 0},k{y=0} on {(x, 0; 0, η) ; x < 0 and η > 0},k{x=0} on {(0, y; ξ, 0) ; y < 0 and ξ > 0},k{(0,0)}[−1] on {(0, 0; ξ, η) ; ξ > 0 and η > 0}.Note that (see Theorem 4.5 below)SS(kXS) = SS1(kXS) = N∗0 (S) ∪{(0, 0; ξ, η) ∈ T ∗X ; ξ, η ≥ 0}and this set is diﬀerent from N∗0 (S).Proposition 3.4. — (i) Let f : X −→ Y be a morphism of manifolds and let Sbe a closed subset of X. Assume that f(S) is a closed subset of Y . ThenN∗0(f(S)) ⊂ fπf−1d(N∗0 (S)).If f is moreover a closed embedding, then one has N∗0 (f(S)) = fπf−1d (N∗0 (S)).(ii) For a closed subset S1 of X1 and a closed subset S2 of X2, one hasN∗0 (S1 × S2) = N∗0 (S1)×N∗0 (S2).(iii) If a closed subset S of X satisﬁes N∗0 (S) ⊂ T ∗XX, then S is an opensubset.(iv) Let f be a C 1-function on X, let S a closed subset, and let c ∈ R. Assumethat f S : S −→ R is proper, f(S) ⊂ [a,∞) for a  0. If df(x) /∈ N∗0 (S)for x such that f(x) < c, then f(S) ⊂ [c,∞).(v) Let γ be a proper closed convex cone of Rn, Ω an open subset of Rn suchthat Ω + γa = Ω, and S a closed subset of Ω such that S is relativelycompact in Rn. If N∗0 (S) ∩ (S × Int(γ◦)) = ∅, then S is an empty set.Proof. — (i), (ii), (iii) are easy exercises.(iv) If f S takes its minimal value at x ∈ S, then df(x) belongs to N∗0 (S).(v) is nothing but Lemma 3.3 in [2].The following property of involutivity is due to J.-M. Bony [1].Theorem 3.5. — Let S be a closed subset of X and f , g two C 1-functionson T ∗X such that N∗0 (S) is contained in the zeroes’s set of f and that of g.Then N∗0 (S) is contained in the zeroes’s set of the Poisson bracket {f, g}.4. Involutivity of truncated microsupportsTheorem 4.1 (Weak involutivity of truncated microsupports)Let F ∈ Db(kX), k ∈ Z and let f , g be two C 1-functions on T ∗X such thatSSk(F ) is contained in the zeroes’s set of f and that of g. Then SSk(F ) iscontained in the zeroes’s set of the Poisson bracket {f, g}.tome 131 – 2003 – no 2INVOLUTIVITY OF TRUNCATED MICROSUPPORTS 265Proof. — Assume that SSk(F ) ⊂ {f = 0} ∩ {g = 0} and let p = (x0; ξ0) suchthat {f, g}(p) = 0. We have to show that for any function ϕ with ϕ(x0) = 0and dϕ(x0) = ξ0, the local cohomology Hk{ϕ≥0}(F )x0 vanishes.By induction on k we may assume that p /∈ SSk−1(F ). Hence we may assumethat F ∈ D≥k(kX). Then Hk{ϕ≥0}(F )x0  Γ{ϕ≥0}(Hk(F ))x0 . Assume thats ∈ Γ{ϕ≥0}(Hk(F ))x0 does not vanish. There exists an open neighborhoodV of x0 such that s extends to s˜ ∈ Γ{ϕ≥0}(V ;Hk(F )). Then S := supp(s˜)satisﬁes N∗0 (S) ⊂ SSk(F ). Hence p /∈ N∗0 (S) by Theorem 3.5, which is acontradiction.Remark 4.2. — The truncated microsupport is not involutive in the senseof Deﬁnition 2.1. Indeed, for X = R and Z = {x ∈ X ;x > 0}, one hasSS0(kZ) ={(x; ξ) ; ξ = 0, x ≥ 0}.Hence one has Cp(SS0(F ), SS0(F )) ⊂ {−dξ = 0} with p = (0; 0), but −∂/∂x =H(−dξ) /∈ Cp(SS0(F )).Corollary 4.3. — Let S be a locally closed submanifold of T ∗X such thatTpS is not involutive for any p ∈ S. Then S ∩ SSk(F ) ⊂ SSk(F ) \ S.Theorem 4.4. — Let F ∈ Db(kX), k ∈ Z and let S be a subanalytic subset ofT ∗X of dimension smaller than dimX. Then SSk(F ) = SSk(F ) \ S.Proof. — We shall argue by the induction on dimS. There is a closed subana-lytic subset S1 of S such that dimS1 < dimS and S0 := S \S1 is non singular.Since TpS0 is not involutive for any p ∈ S0, SSk(F ) ⊂ S1 ∪ SSk(F ) \ S. Henceone has SSk(F ) \ S = SSk(F ) \ S1, and the induction proceeds.Theorem 4.5. — Let F ∈ DbR-c(kX). Let {Yα}α∈A be a locally ﬁnite familyof real analytic submanifolds subanalytic in X, and let Λα be an open subsetof T ∗YαX subanalytic in T∗X, such that SS(F ) ⊂ ⋃α∈A Λα. Let Kα ∈ Db(k)and assume that F is microlocally isomorphic to kYα ⊗Kα at every point of Λα.SetAk :={α ∈ A;Kα /∈ D>k(k)}.Then SSk(F ) =⋃α∈Ak Λα for any k ∈ Z.Proof. — Set S =⋃α∈A(Λα \ Λα). Then dimS < dimX . If α ∈ Ak, thenΛα ⊂ SSk(F ) and if α ∈ Ak, then SSk(F ) ∩ Λα = ∅. Hence SSk(F ) \ S =(⋃α∈Ak Λα) \ S. Hence Theorem 4.4 implies the desired result.The following corollary is proved in [2] when k = C by a diﬀerent method.Let X be a complex manifold. Recall that F ∈ DbC-c(kX) is perverse ifcodimSupp(Hk(F )) ≥ k and codimSupp(Hk(RHom (F,kX))) ≥ kfor any k ∈ Z.BULLETIN DE LA SOCIE´TE´ MATHE´MATIQUE DE FRANCE266 KASHIWARA (M.), MONTEIRO FERNANDES (T.) & SCHAPIRA (P.)Corollary 4.6. — Let X be a complex manifold. Let F ∈ DbC-c(kX) and let{Xα}α∈A be a family of complex submanifolds such that Xα and Xα \Xα areclosed complex analytic subsets and SS(F ) =⋃α∈A T∗XαX.(i) If F is a perverse sheaf, then one hasSSk(F ) =⋃codimXα≤kT ∗XαX for any k.(4.1)(ii) Conversely if F ∈ Db(kX) satisﬁesSSk(F ) ∪ SSk(RHom (F,kX)) ⊂⋃codimXα≤kT ∗XαX for any k,(4.2)then F is a perverse sheaf.Proof. — Recall (see [3, Theorem 10.3.12]) that F is perverse if and only if Fis microlocally isomorphic to a ﬁnite direct sum of copies of kXα [− codimXα]at a generic point of T ∗XαX for any α.(i) Assume F is perverse. Using the notations of Theorem 4.5, we getAk ={α ; codimXα ≤ k}and the result follows from this theorem.(ii) The proof goes as [2, Corollary 6.10]. For the sake of completeness,we repeat it. Recall that µY (·) denotes the Sato’s microlocalization functor.By [3], F is isomorphic to kXα [− codimXα] ⊗K at a generic point of T ∗XαXfor some K ∈ Db(k). Since µXα(F ) must be in D≥codimXα(kT∗XαX) andµXα(F )  kT∗XαX [− codimXα]⊗K,one has K ∈ D≥0(k). Similarly,µXα(RHom (F,kX))  kT∗XαX [− codimXα]⊗RHom (K,k)implies K ∈ D≤0(k).BIBLIOGRAPHY[1] Bony (J-M.) – Quasi-analycite´ et unicite´ du proble`me de Cauchy pour lessolutions d’e´quations aux de´rive´es partielles, in E´quations aux de´rive´es par-tielles et analyse fonctionnelle, Se´minaire Goulaouic-Schwartz, E´cole Poly-technique Paris, 1971, Exp. 10.[2] Kashiwara (M.), Monteiro Fernandes (T.) & Schapira (P.) –Truncated microsupport and holomorphic solutions of D-modules, ArXivmath.AG/0203091, Annales Sci. E´cole Normale Sup., to appear.[3] Kashiwara (M.) & Schapira (P.) – Sheaves on manifolds, Grundlehrender Math. Wiss., vol. 292, Springer Verlag, 1990.tome 131 – 2003 – no 2

Involutivity of truncated microsupports

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