Sheaf quantization in Weinstein symplectic manifolds
aa r X i v : . [ m a t h . S G ] J u l SHEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS
DAVID NADLER AND VIVEK SHENDE
Abstract.
Using the microlocal theory of sheaves, we associate a category to each We-instein manifold. By constructing a microlocal specialization functor, we show that exactLagrangians give objects in our category, and that the category is invariant under Weinsteinhomotopy.
Contents1. Introduction 2
2. Propagation and displacement 133. Relative microsupport 18
4. On the full faithfulness of nearby cycles 22 ∗ -pullback of external hom . . . . . . . . . . . . . . . . . . 254.4. The lower square of (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5. The upper square of (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5. Microlocal sheaves 296. Antimicrolocalization 32
7. Gapped specialization of microlocal sheaves 418. Microlocal sheaves on polarizable contact manifolds 43
9. The cotangent bundle of the Lagrangian Grassmannian 50
10. Maslov data and descent 54
References 57 Introduction
Principles.
For a particle traveling on a manifold M , geometric quantization relatessemiclassical states — half-densities on Lagrangian submanifolds L ⊂ T ∗ M — to quantumstates — half-densities on the base M . Sheaf quantization is a categorical analogue: afunctor from the category of locally constant sheaves on L to the category of sheaves on M .In both cases, singularities of the object on M are controlled by the asymptotics of L .The physical setting appropriate to sheaf quantization is the topological A-twist of the 2dsupersymmetric sigma model from [55]. In mathematical terms [15], one studies Lagrangiansubmanifolds L , L in a symplectic manifold ( X, ω ), and considers their transverse inter-sections L ∩ L . Instantons, i.e. pseudo-holomorphic strips running along L , L , provide adifferential on the space of functions on L ∩ L ; the resulting Floer cohomology HF ∗ ( L , L )is invariant under Hamiltonian isotopy [15]. Polygons running along multiple Lagrangiansprovide the structure constants of Fukaya categories [16, 43].Sheaf quantization arises from Fukaya categorical considerations as follows. Given anobject L of the Fukaya category of T ∗ M , for instance given by an exact Lagrangian car-rying a local system, one obtains a sheaf F L on M roughly by the prescription ( F L ) p =Hom F uk ( T ∗ M ) ( T ∗ p M, L ). Variants on this theme appear in [39, 34, 1, 51, 52, 21].More remarkably, it is possible to construct sheaf quantization in cotangent bundles en-tirely in sheaf theoretic terms, in particular with no reference to instantons [22]. Our pur-pose here is to do the same in a general Weinstein (in particular exact; ω = dλ ) symplecticmanifold. This will provide, for such manifolds, an alternative categorical framework forLagrangian intersection theory, constructed via topology rather than analysis.Returning to the cotangent bundle X = T ∗ M , consider a graphical Lagrangian Γ df ⊂ T ∗ M with primitive a Morse function f : M → R . Form the family L t = ϕ t ( L ) = Γ tdf ⊂ T ∗ M ,for t ∈ R > , and note the limit L = lim t → L t ⊂ T ∗ M is simply the zero-section. Theintersection L t ∩ M , for t = 0, is a finite set, while L ∩ M is the entire zero-section. Theseare not as different as they at first appear: for t = 0, functions on L t ∩ M underlie theMorse complex of f , while functions on the derived intersection L ∩ M are differential forms HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 3 on M . Imposing the Morse differential on the former and the de Rham differential on thelatter renders them quasi-isomorphic; in fact, the family provides an interpolation betweenthe two [54]. Nevertheless, while the differential for L ∩ L t involves global “instantons”, thede Rham differential does not.More generally, by Stokes’ theorem, pseudo-holomorphic strips along exact Lagrangianswhere λ | L = df , λ | L = df have area given by the change in the “action” f − f atintersection points. In particular, if all intersection points have the same action, there canbe no instanton corrections. To arrive at such a situation, one can attempt to move exactLagrangians λ | L = df so they are conic for the Liouville flow v λ = ( dλ ) − ( λ ) and hence haveconstant primitive functions ( df = 0). In particular, writing ϕ t = e log( t ) v λ : X → X , for t ∈ R > , for the Liouville flow obtained by integrating v λ , one can attempt to take limits L = lim t → ϕ t ( L ) ⊂ X Making sense of the categorical intersection theory in this limit would come with a substantialpayoff: the absence of instantons implies the theory should be local in nature.In general, and in contrast to the case of graphical Lagrangians in cotangent bundles, L will typically be very singular. No existing account of Lagrangian Floer theory is compatiblewith the singularities which arise. Instead, we construct by different means a category for L , and show it enjoys the familiar properties of Fukaya categories. Our construction willbe local in nature as suggested by the absence of instantons highlighted above.Our main tool will be the microlocal theory of sheaves, as pioneered by Kashiwara-Schapira[29]. This is originally formulated in the context of cotangent bundles; in order to adapt itto a general exact symplectic manifold X we embed the latter as a symplectic submanifoldin some contact cosphere bundle S ∗ N , and consider microlocal sheaves along its core c ( X ).While we will not use any Floer theory in our constructions, let us give a heuristic Floertheoretic discussion here to motivate certain expectations which we will later establish rigor-ously with sheaf theoretical methods. We restrict to the case when X ⊂ S ∗ N is a hypersur-face and c ( X ) ⊂ S ∗ N is a singular Legendrian. Were c ( X ) smooth, it would be natural toconsider the Legendrian contact homology of c ( X ) in S ∗ N [12]; most relevent for us wouldbe the positive augmentation category [40]. Homs in this category are generated by Reebchords from c ( X ) to itself, some of finite length and some infinitesimal. As this is a purelyheuristic discussion, let us imagine ourselves in possession of a (currently nonexistent) versionof Legendrian contact homology for singular Legendrians. We are interested only in the phenomena intrinsic to X , and wish to decouple these fromanything related to the embedding. In our hypothetical setup, it would be natural to proceedby considering the chord-length filtration on Hom spaces and restricting attention to the zerolength part. By analogy with typical behavior of quantum systems, it is plausible that this In fact the works [19, 20, 21], relying on the “antimicrolocalization” result of the present work, show thatthe category constructed here is equivalent to the Fukaya category. As presently written in those works thisrequires a stable normal polarization, but it may be expected that techniques similar to those appearing inSec. 10 of the present article, combined with the local-to-global principle of [20], will allow the removal ofthe polarizability assumption. For 1d Legendrians with tripod singularities, a combinatorial LCH has been constructed [3] and satisfiesthe expected comparison with sheaves [4]. In general one can replace the Legendrian contact homology withthe endomorphism algebra of the linking disks to c ( X ) as in [21, Sec. 6.4]; this is always sensible but it isno longer evident how to give a filtration whose associated graded is generated by self chords. DAVID NADLER AND VIVEK SHENDE “ground state” category remains constant under deformations of the inclusion c ( X ) ⊂ S ∗ N that are gapped in the sense that the length of the shortest chord is bounded below.The workhorse technical result of this article, proven entirely by sheaf theoretical methods,is that gapped specialization is indeed well behaved on categories of microlocal sheaves. This is applied to the problem of sheaf quantization as follows. Consider a compact exactLagrangian L ⊂ X . Once flowed a large but finite distance under the Liouville flow, it liftsto a Legendrian e L in the contactization X × ( − ǫ, ǫ ), which we may regard as embedded in S ∗ N . A contact lift of the Liouville flow of ˜ L limits to a subset of c ( X ); this flow is gappedbecause a short self-chord of e L would be a self-intersection of L . Our theorem on gappedspecialization implies the existence of a fully faithful functor from microsheaves on L tomicrosheaves on c ( X ). A similar argument establishes invariance of the category associatedto X under Weinstein homotopy. This is a highly nontrivial result: such homotopies willgenerally be accompanied by significant variation of the geometry of c ( X ).Sect. 1.2 immediately following contains a rough summary of the techniques we will usein our constructions. Afterwards, in Sect. 1.3, we state theorems in a manner intended tobe intelligible to a non-specialist in microlocal sheaf theory and calibrated for comparisonsto the Fukaya category.1.2. Methods.
After reviewing here some basic notions of the microlocal theory of sheaves,we will sketch in this section how we adapt and extend this theory to study general exactsymplectic manifolds.Let M be a smooth manifold. Denote by sh ( M ) sheaves on M valued in some symmetricmonoidal presentable stable ∞ -category, for instance the dg derived category of (unbounded)complexes of sheaves of Z -modules on M . In particular, an object F ∈ sh ( M ) assigns toany open submanifold U ⊂ M with smooth boundary ∂U ⊂ M an object (e.g. a complex of k -modules) F ( U ). Microsheaves on cotangent and cosphere bundles.
The microlocal study of sheavesmight begin with the question: for what isotopies of an open U t ⊂ M does the complex F ( U t ) remain homologically unchanged? Nonobviously, it is profitable to formulate theanswer in terms of conic subsets of the cotangent bundle T ∗ M → M .Indeed, to each sheaf F ∈ sh ( M ), there is a closed conic coisotropic subset ss ( F ) ⊂ T ∗ X called its microsupport. As long as the intersection ss ( F ) ∩ T + ∂U t M ⊂ T ∗ M of the singularsupport and outward conormal of the boundary lies in the zero-section M ⊂ T ∗ M , thesections F ( U t ) remain homologically unchanged (see e.g. [29, Proposition 2.7.2, Cor. 5.4.19]).In particular one finds that F is locally constant if and only if ss ( F ) lies in the zero-section.It is then possible to study sheaves “along their microsupports”. More precisely, given anopen subset Ω ⊂ T ∗ M , consider the full subcategory Null (Ω) ⊂ sh ( M ) of objects F withdisjoint singular support ss ( F ) ∩ Ω = ∅ . If we measure objects of sh ( M ) by the change intheir sections along the codirections lying in Ω, then Null (Ω) consists of objects for whichsuch measurements vanish. Thus from the viewpoint of Ω such objects are trivial, and we We remark that [20, Thm. 1.4] translated through [21] gives a different criterion (constancy of the contactcomplement); we do not know a comparison. In this article, all functors are derived, in particular F ( U ) means what is elsewhere written R Γ( U, F ).Our sheaf conventions are detailed in Sect. 1.5, with also some notes on the use of unbounded sheaves. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 5 may as well pass to the quotient category µ sh pre (Ω) = sh ( M ) / Null (Ω)This assignment provides a presheaf µ sh pre of dg categories on open subsets Ω ⊂ T ∗ M , butnot in general a sheaf: the hom spaces do not form a sheaf [29, Exercise VI.6], and moreoverit may happen that for some cover Ω = S i Ω i , there are objects F i ∈ µ sh pre (Ω i ) whoserestrictions to intersections are coherently isomorphic, but which nevertheless do not glue toany object F ∈ µ sh pre (Ω). In general, the results on µ sh pre in [29] are formulated in termsof its stalks. We consider its sheafification µ sh , and use that its properties can often bechecked stalkwise and hence reduce to results of [29].Since singular support is always conic, the sheaf µ sh only depends on the saturation R > · Ω ⊂ T ∗ M of open subsets Ω ⊂ T ∗ M . Thus we lose nothing by viewing µ sh as a sheafon the conic topology where opens Ω ⊂ T ∗ M are assumed to be saturated Ω = R > · Ω. Saidanother way, µ sh descends along the quotient map T ∗ M → T ∗ M/ R > , i.e., is the pullback ofa sheaf from the quotient. Away from the zero-section, the latter is just the cosphere bundle S ∗ M = ( T ∗ M \ M ) / R > . Thus the restriction of µ sh away from M ⊂ T ∗ M provides a sheaf(in the usual topology) on S ∗ M which we also denote by µ sh .Finally, for any subset X ⊂ T ∗ X (resp. X ⊂ S ∗ M ), there is a subsheaf µ sh X ⊂ µ sh offull dg subcategories of objects whose singular support (resp. projectivized singular support)lies in X . When X is closed, note that µ sh X is the pushforward of a sheaf on X which wewill also denote by µ sh X .1.2.2. Microsheaves on Liouville manifolds.
We define microlocal sheaves on more generalspaces by fixing an embedding into a cotangent bundle. Of course, we will need to understandthe dependence of our constructions on the choice of embedding.As a first example, suppose given a smooth manifold N embedded as a Legendrian in S ∗ M . A local calculation shows that µ sh N is locally isomorphic to the sheaf of (derived) localsystems on N . Globally, µ sh N is twisted by the Maslov class of N and related topologicaldata (for a precise formulation see Rem. 10.8). In particular, µ sh N is largely ignorant of thesubtle contact topology of the embedding; for example, it cannot tell whether N is loose.By contrast, the quotient category sh N ( M ) / Loc( M ) is sensitive to the contact topology;for instance, it vanishes for N loose. The sheafification of µ sh pre into µ sh has destroyedour access to such subtle global information. While this would be undesirable for defininginvariants of the embedding, it is well suited for defining invariants of N itself.Now consider a Liouville manifiold ( X, λ ) with Liouville vector field v λ = ( dλ ) − ( λ ) andcompact core c ( X ) ⊂ X . By an exact embedding of X into a cosphere bundle S ∗ M , wewill mean the germ along c ( X ) of a smooth embedding, such that there is a contact formon S ∗ M restricting to the Liouville form λ . At present, we can only prove good invarianceresults when c ( X ) ⊂ X is (possibly singular) Lagrangian, which we assume henceforth.Suppose we have an exact embedding of X as a (codimension one) hypersurface in M .Then λ | c ( X ) = 0, hence c ( X ) ⊂ S ∗ M is Legendrian. Let us consider the sheaf µ sh ofmicrolocal sheaves on S ∗ M , and its subsheaf µ sh c ( X ) ⊂ µ sh of microlocal sheaves supportedalong c ( X ) ⊂ S ∗ M . Ultimately, we are interested in its global sections Sh ( X ) := µ sh c ( X ) ( c ( X ))For example, when X = T ∗ N so that c ( X ) = N , an exact embedding provides a Leg-endrian embedding N ⊂ S ∗ M , and conversely, by a standard neighborhood theorem, a DAVID NADLER AND VIVEK SHENDE
Legendrian embedding N ⊂ S ∗ M extends to an exact embedding. In this case, as discussedabove, µ sh N is locally isomorphic to the sheaf of derived local systems on N .In complete generality, we do not know whether every Liouville manifold ( X, λ ) admits acodimension one exact embedding into a cosphere bundle S ∗ M . To get around this, one canimagine repeating the above constructions locally on X and then gluing them together. Wewill instead follow [45] in using higher codimensional exact embeddings, whose existence anduniqueness are completely controlled by Gromov’s h-principle for contact embeddings. Thenwe will choose a normally polarized thickening of X to a codimension one exact embeddingand proceed with the above constructions. This is explained in Sec. 8.1.Finally, we study the extent to which the resulting µ sh c ( X ) depends on such an embeddingand thickening. We will check the dependence factors through a choice of topological “Maslovdata” on X , which is in fact the same data required to define the Fukaya category on X .As a result we can relax the requirement of the existence of a normal polarization. This isexplained in Section 10.1.2.3. Lagrangian objects via nearby cycles.
Suppose given a Liouville manifold (
X, λ ) withLiouville vector field v λ = ( dλ ) − ( λ ), compact core c ( X ) ⊂ X , and equipped with Maslovdata. Then we have assigned the sheaf µ sh c ( X ) of microlocal sheaves along c ( X ), and thushave its global sections Sh ( X ) = µ sh c ( X ) ( c ( X )). We outline here how compact exact La-grangian submanifolds L ⊂ X give rise to objects F L ∈ Sh ( X ).Ultimately, this will depend on certain “secondary Maslov data”. To avoid this complica-tion in the introduction, suppose we have construced µ sh c ( X ) using a codimension one exactembedding of X into a cosphere bundle S ∗ M . By a standard neighborhood theorem, suchan exact embedding extends to the germ along c ( X ) × { } of a contact embedding of thecontactization X × R .Fix a compact exact Lagrangian L ⊂ X with primitive f : L → R with λ | L = df , and itsLegendrian lift Λ = Graph( f ) ⊂ L × R ⊂ X × R . For t ∈ R , apply the Liouville flow toobtain the family L t = e tv λ · L f t = e t f Λ t = Graph( f t )Note for any open neighborhood U ⊂ X of c ( X ), and ǫ >
0, there is t ∈ R such thatfor t < t , we have Λ t ⊂ U × ( − ǫ, ǫ ). Thus for t < t , we have a family of Legendrianembeddings Λ t ⊂ S ∗ M . As discussed above, since Λ t ⊂ S ∗ M is a smooth Legendrian,the subsheaf µ sh Λ t ⊂ µ sh of microlocal sheaves supported along Λ t is a twisted form ofthe sheaf of (derived) local systems on Λ t . Thus objects of the global section category µ sh Λ t (Λ t ) ⊂ µ sh ( S ∗ M ) can be identified with certain topological “decorations” on Λ t ≃ L .Moreover this category itself constant for 0 < t < t .Fix some object F t ∈ µ sh Λ t (Λ t ). We would like to take its limit as t → −∞ to obtain aconic global object “ lim t →−∞ F t ∈ µ sh c ( X ) ( c ( X )) = Sh ( X )”Such limits can be made sense of in sheaf theory using the formalism of nearby cycles. To ap-ply this notion in our microlocal setting, we construct an “antimicrolocalization” embeddingof µ sh Λ t (Λ t ) ֒ → Sh ( M ), whose image is characterized as sheaves microsupported on a cer-tain “double” of Λ t (Theorems 6.17 and 6.28). This antimicrolocalization is of independentinterest, being in particular a key step of the [19, 20, 21] programme to prove Kontsevich’slocalization conjecture [30] (see particularly [21, Sec. 6.6]). HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 7
In order that the antimicrolocalization can be performed uniformly in the family (hencecompatibly with the limit), it is necessary to impose that the Λ t are uniformly displaceablefrom themselves by a positive flow. That is, the length of chords for this flow must bebounded below as t → −∞ . We term such a family gapped .In fact, the notion of gappedness also appears naturally in the study of a more fundamentalquestion: when is the nearby cycles functor fully faithful? We give a criterion in Theorem4.2, in which gappedness is an essential ingredient.The antimicrolocalization construction and the criterion for full faithfullness of nearbycycles are major new technical results of this article. Combining them, we obtain in Theorem7.3 a fully faithful functor (“gapped specialization”) µ sh Λ (Λ) → µ sh c ( X ) ( c ( X )) = Sh ( X ) Remark 1.1.
A key consequence of full faithfulness is the following. Recall that an object
L ∈ µ sh Λ (Λ) can be locally identified with a local system; suppose it is rank 1. ThenEnd( L ) ∼ = H ∗ ( L ). By full faithfulness, the same holds for the image of L in Sh ( X ). Thus,exact Lagrangians (with trivial Maslov class) define objects whose endomorphisms are thecohomology of the Lagrangian, just as in the Fukaya category. Example 1.2.
Suppose we consider a smooth Lagrangian immersion L → X lifting to aLegendrian embedding Λ ⊂ X × R . Then the double points in the image correspond toself Reeb chords of Λ t ⊂ S ∗ M whose length shrinks to zero as t → −∞ . This violates thegapped condition. Example 1.3.
Consider a pair of smooth Lagrangians
L, L ′ ⊂ X intersecting in a singlepoint. The Legendrian lifts Λ , Λ ′ ⊂ X × R can be chosen either to intersect at a single point(in which case the gapped specialization theorem applies) or disjoint. In the latter case,note that objects of µ sh Λ ∪ Λ ′ (Λ ∪ Λ ′ ) supported on different components will be orthogonal.But given that L, L ′ intersect in a single point, we would certainly expect such objects tohave non-trivial homs between them! Again, we have violated the gapped condition: theintersection point L ∩ L ′ corresponds to self-Reeb chords of Λ t ∪ Λ ′ t whose length shrinks tozero as t → −∞ .We have been considering smooth L (and hence Λ), but in fact the gapped specializationtheorem µ sh Λ (Λ) → µ sh c ( X ) ( c ( X )) does not require smoothness (though the identificationof objects of µ sh Λ (Λ) in simple terms as topological brane structures does).In particular, we can apply gapped specialization to show that Sh ( X ) is in fact invari-ant under varying the exact symplectic primitive λ , at least as long as the variation isthrough primitives with isotropic core. Note that c ( X ) may undergo drastic changes duringthis process. The basic idea is the following. Denote the core for some variant primitive λ + df by c ′ ( X ), Then we may apply gapped specialization to get a fully faithful functors µ sh c ′ ( X ) ( c ′ ( X )) → µ sh c ( X ) ( c ( X )) and µ sh c ( X ) ( c ( X )) → µ sh c ′ ( X ) ( c ′ ( X )). Showing these func-tors compose to the identity can be done by studying an interpolation between the cores,and is done in Theorem 8.15.The gapped specialization can be applied more generally to noncompact but eventuallyconical Lagrangians, though in this case we must increase the core. That is, there is a fullyfaithful functor µ sh Λ (Λ) → µ sh c ( X ) ∪ R L ∞ ( c ( X ) ∪ R L ∞ ) =: Sh ( X ; L ∞ ) DAVID NADLER AND VIVEK SHENDE
We can compose this functor with the (not fully faithful) left adjoint to the inclusion Sh ( X ) → Sh ( X ; L ∞ ) to get a functor µ sh Λ (Λ) → Sh ( X ). This is not fully faithful, asshould be expected from the counterpart of using noncompact Lagrangians to define objectsof the wrapped Fukaya category. Remark 1.4.
The idea of using the gapped condition in microlocal sheaf theory was firstproposed in [37, 35], where a certain kind of deformation of Legendrians (“arborealization”)was constructed and shown to leave invariant the sheaf categories, and it was suggested thatthis could be demonstrated more abstractly under a gappedness hypothesis plus some sortof topological constancy. In [56] it was shown that Legendrian ribbotopies (deformations ofskeleta induced by isotopies of a Weinstein ribbon) induce equivalences on sheaf categories;note that a ribbotopy necessarily leaves invariant the homotopy type of the skeleton. (Strictlyspeaking, [56] does not in fact imply [35] as it is unknown if or when the arborealizationprocedure used in that paper is in fact a ribbotopy.) Here we show that absent any hypothesesof topological constancy or any hypotheses on the existence on Weinstein thickenings , thegapped condition implies full faithfulness. For ribbotopies this automatically upgrades toan equivalence since one can construct full faithful maps in both directions and interpolatebetween their composition and the identity. The current version is essential to show thatLagrangians give objects.1.3.
Results.
In this section, we give a sample formulation of our results intelligible withoutany familiarity with microlocal sheaf theory. We work over some fixed symmetric monoidalpresentable stable ∞ -category C . For technical foundations we use [32, 33]. The readerunfamiliar with these works may take C to be the dg derived category of Z -modules and willlose none of the new ideas.We say a map of sheaves of categories has some property (e.g. is fully faithful) if thisholds on all sections. If Y ⊂ X is a closed inclusion, we often omit the notation for thepushforward functor identifying sheaves on Y and sheaves on X supported in Y .We put off defining precisely the notion of “Maslov data” (and later “secondary Maslovdata”) until Sect. 10; suffice it to say here that it is certain topological data related totrivializing natural maps related to Lagrangian Grassmannians; the data required is thesame as is usually used in setting up the Fukaya category. Theorem 1.5.
Let ( X, λ ) be a Liouville manifold, equipped with Maslov data. (1) Locality.
For conic Λ ⊂ X , there is a sheaf of presentable stable ∞ -categories µ sh Λ on the conic topology on Λ . The restriction maps are continuous and co-continuous.In addition, for Λ ⊂ Λ ′ there is a fully faithful, continuous and cocontinuous em-bedding µ sh Λ → µ sh Λ ′ , functorial in composition of inclusions. In particular, thereis such a functor on global sections µ sh Λ (Λ) → µ sh Λ ′ (Λ ′ ) . (2) Removal.
Given an inclusion of closed conic Λ ⊂ Λ ′ , the adjoints to the maps abovefit into a localization sequence: µ sh Λ ′ (Λ ′ \ Λ) → µ sh Λ ′ (Λ ′ ) → µ sh Λ (Λ) → (The first morphism is not in general fully faithful.) (3) Involutivity.
The sheaf µ sh vanishes wherever Λ is not co-isotropic. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 9 (4)
Microstalks.
Given a smooth contractible Lagrangian open U ⊂ Λ and p ∈ U , thenatural restriction is an equivalence µ sh ( U ) ∼ −→ µ sh p ∼ = C where the isomorphism to the coefficient category is non-canonical. (5) K¨unneth.
For conic Λ ⊂ X and Ξ ⊂ Y , there is a fully faithful functor µ sh Λ ⊠ µ sh Ξ → µ sh Λ × Ξ which is an isomorphism at least whenever either Λ or Ξ is smooth Lagrangian. When X is a cotangent bundle, the construction of µ sh is essentially due to Kashiwaraand Schapira [29], though working technically with sheaves of homotopical categories requiressome foundations such as [32, 33]. We review the construction in Sect. 5. For cotangentbundles, (1) and (2) follow formally from the definition of µ sh and elementary properties ofmicrosuports, as does the existence and full faithfulness of the K¨unneth map. (3) follows fromthe deep theorem [29, 6.5.4] that microsupports are coisotropic. The facts about Lagrangiansin (4) and (5) are well known (see e.g. [22, 36]) and follow from a calculation which is easyafter contact transformation, as we recall in Lemmas 5.2 and 5.3 and Cor. 5.4.The new content of the theorem is the generalization to other Liouville X . We do this byembedding X in a cotangent bundle; once this is done the importation of facts about µ sh from the cotangent setting is immediate. This was described in [45]; we give more detailshere. The construction is in two stages; for X equipped with a stable normal polarization,the construction is in Sect. 8.1; in fact this already suffices for typical applications. In fullgenerality, we explain in Sect. 10 how to weaken the requirement of a normal polarizationto that of Maslov data. Remark 1.6.
For comparisons to the Fukaya category and applications in mirror symmetry,it is preferable to regard µ sh as a cosheaf by passing to the left adjoints of the restrictionmaps. The corestriction maps in the cosheaf are then left adjoints of left adjoints, and hencepreserve compact objects. The first morphism of (2) above is such a corestriction, and thesequence should be compared to the stop removal of [20]. Remark 1.7.
When Λ is “Lagrangian enough”, we may give compact generators for µ sh Λ .Indeed, by (4) we obtain from each Lagrangian point p a (noncanonical) map µ p : C → µ sh Λ (Λ). Let Λ ◦ ⊂ Λ be the Lagrangian locus, and assume that Λ \ Λ ◦ contains no co-isotropic subset (e.g. say Λ admits an isotropic Whitney stratification). Then by (2) wehave µ sh Λ (Λ ◦ ) → µ sh Λ (Λ) → µ sh Λ \ Λ ◦ (Λ) → µ sh Λ \ Λ ◦ (Λ) = 0 and hence µ sh Λ (Λ ◦ ) → µ sh Λ (Λ) is surjective.Because µ sh is a cosheaf, the former category is generated by the images of the µ p , hence sois µ sh Λ ( X ). A similar argument shows that if both Λ \ Λ ◦ and Ξ \ Ξ ◦ have no co-isotropicsubsets, then the K¨unneth map is an isomorphism.The cospecialization maps, hence also µ p , carry compact objects of C to compact objectsof µ sh Λ ( X ). Thus if Λ \ Λ ◦ is not co-isotropic and C is compactly generated, then µ sh Λ ( X )is compactly generated by the images of the µ p . More generally, in this situation all sections of the sheaf µ sh Λ are compactly generated. We may take adjoints to view µ sh as a cosheaf,and then pass to compact objects and obtain a cosheaf µ sh c Λ valued in small categories.These compact objects were called “wrapped microlocal sheaves” in [36]. There is nowrapping in their definition: the name indicated an expected comparison to the wrappedFukaya category, since established in [21]. It is also natural to study the dual sheaf ofpseudoperfect modules, µ sh pp Λ : U F un ex ( µ sh c Λ ( U ) , C c ). Remark 1.8.
Under appropriate tameness hypotheses on Λ, e.g. Λ subanalytic Lagrangian, µ sh pp Λ ( U ) is obtained by microlocalizing the bounded derived category of constructible sheaves(see [36] for a proof by “arborealization” or [21] for a more elementary argument when U is the cotangent bundle). Under similar hypotheses, it is possible to show that µ sh c (Λ) iscategorically smooth, in particular giving a natural inclusion µ sh pp (Λ) ⊂ µ sh c (Λ) (e.g. byarborealization, the sections are a finite colimit of representation categories of tree quivers;these being smooth and proper the colimit is smooth).Microlocal sheaves in the sense of Theorem 1.5 respect the conic topology on X hencedepend on the symplectic primitive λ . In particular, there is no a priori role for non-conicLagrangians. On the other hand, the Fukaya category does not depend on λ , and certainlycontains non-conic Lagrangians. In order to produce something like the Fukaya categoryfrom µ sh , we must resolve this tension; this is the main purpose of the present article.Recall that the core c ( X ) of a Liouville manifold ( X, λ ) is the locus which does not escapeunder the Liouville flow. Given Λ ⊂ ∂X a closed subset of the contact boundary ∂ ∞ X , letCone(Λ) ⊂ X denote its closed Liouville cone, and form the union c ( X, Λ) = c ( X ) ∪ Cone(Λ) ⊂ X In particular, we have c ( X, ∅ ) = c ( X ).Given L ⊂ X with conic end, we define L as the limit of L under the Liouville flow. Moreprecisely, we consider L t ⊂ X × (0 , ∞ ) ⊂ X × [0 , ∞ ) the images of L under the Liouvilleflow, and take L = S L t ∩ ( X × L ⊂ c ( X, L ∞ ).We will use the phrase “sufficiently isotropic” below to mean a certain collection of techni-cal properties which appear in the body of the text, all of which follow from for example theexistence of an isotropic Whitney stratification. We say a symplectic primitive is sufficientlyWeinstein if it gives rise to a sufficiently isotropic core. (See Def. 8.11.) Theorem 1.9.
For ( X, λ ) Liouville, equipped with Maslov data, and Λ ⊂ X ∞ we write Sh λ ( X ; Λ) := µ sh c ( X, Λ) ( X ) This category has the following properties: (1)
Lagrangian objects.
Let L ⊂ X be a smooth closed (not necessarily compact)exact Lagrangian L ⊂ X with conic end L ∞ . Fix secondary Maslov data on L , andlet Loc ( L ) be the (derived) category of local systems on L .If L is sufficiently isotropic (e.g. if c ( X, L ∞ ) is sufficiently isotropic), then thereis a fully faithful functor ψ : Loc ( L ) ֒ → Sh λ ( X, Λ ∪ L ∞ ) It is a cosheaf because colimits commute with cocompletion. We could not analogously get a sheaf ofcompact objects, since limits do not.
HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 11 (2)
Invariance. If λ, λ ′ are sufficiently Weinstein for ( X, Λ) , then there is a fully faithfulmorphism Sh λ ( X, Λ) ֒ → Sh λ ′ ( X, Λ) . If λ and λ ′ admit a sufficiently Weinsteinhomotopy, this morphism is an isomorphism. The major new technical ingredients in the proof of Theorem 1.9 are the criterion for fullfaithfulness of nearby cycles (Theorem 4.2) and the antimicrolocalization (Theorem 6.28).Indeed, the former is the ultimate source of the asserted full faithfulness, and the latteris how a Lagrangian (a fundamentally microlocal object) is turned into any sort of sheaf.These are combined into a construction of microlocal specialization (Theorem 7.3), which isthen applied in Corollary 8.9 to prove (1) and in Theorem 8.15 to prove (2). Of course weexpect the morphism of (2) to be an isomorphism in general, but at present our techniquesrequire the existence of a homotopy. (Strictly speaking, the above discussion is in the stablypolarizable case, but the required ideas to work with only Maslov data instead are orthogonal,and the constructions immediately generalize to that setting once it is set up in Sect. 10.)
Remark 1.10.
Note that full faithfulness means in particular for
E ∈
Loc ( L ), we haveEnd Sh λ ( X, Λ ∪ L ∞ ) ( ψ ( E )) ≃ C ∗ ( L ; End( E ))In other words, endomorphisms of an object on a given Lagrangian are given by the co-homology of that Lagrangian, as one expects for e.g. compact exact Lagrangians, or moregenerally noncompact Lagrangians in an infinitesimally wrapped Fukaya category. Remark 1.11. (Wrapped Lagrangian objects.) Given L as in (1) above, we may compose ψ with the (not fully faithful) “removal” functor ρ : Sh λ ( X, Λ ∪ L ∞ ) → Sh λ ( X, Λ) to obtaina (not fully faithful) functor ρφ : Loc ( L ) → Sh λ ( X, Λ)Let us consider this map in the case where L is conic (so ψ does nothing) and L is a disktransverse to c ( X ) at a Lagrangian point p , e.g. a critical co-core for a Weinstein Morsefunction. As L is a disk, Loc ( L ) ∼ = C (noncanonically), and a local calculation shows that ρψ : C → Sh ( X ; ∅ ) = µ sh c ( X ) ( c ( X )) is isomorphic to the map µ p : C → µ sh c ( X ) ( c ( X ))described in Rem. 1.7. (After contact transformation, the local calculation is just the factthat the stalk of a local system is given by Hom from the skyscraper sheaf.)When X = T ∗ M and L = T ∗ p M , one can check (e.g. because the cosheaf of local systemsis the constant cosheaf of stable categories, which, by an ∞ -version of the van Kampentheorem, is modules over chains on the based loop space) thatEnd Sh ( T ∗ M, ∅ ) ( ρψ ( E )) ≃ C −∗ (Ω M ; End( E ))Thus ρψ is very far from being fully faithful, but it does behave as one might expect forLagrangians in the wrapped Fukaya category. Remark 1.12.
In fact for any exact and sufficiently isotropic (but possibly singular) L withconic end, we construct a fully faithful functor from a category of microsheaves on L to Sh λ ( X, Λ ∪ L ∞ ). This construction is independent of the choice of secondary Maslov data;the role of which when L is smooth is only to identify the microsheaf category on L withlocal systems.Taking adjoints and passing to compact objects provides the analogue of the “Viterborestriction” functor of [2]. Remark 1.13.
Under suitable tameness hypotheses on the Liouville flow of L (e.g. if X is subanalytic Weinstein in the sense of [21, Sect. 6.8] and L is subanalytic), our construc-tions preserve boundedness and constructibility thus ψ : Loc ( L ) ֒ → Sh λ ( X, Λ ∪ L ∞ ) willcarry finite rank local systems to microlocalizations of constructible sheaves. As noted inRem. 1.7, under subanalyticity assumptions these are pseudoperfect modules and moreoverpseudoperfect modules are compact. It follows that ρψ carries pseudoperfect modules tocompact objects.1.4. Motivations.
While this article is wholly foundational, we mention here some motiva-tions for the present work.1.4.1.
Homological mirror symmetry.
The ideas of [39, 34] indicated that the ‘A’ side ofmirror symmetry, while usually defined using Fukaya categories, may sometimes be calculatedusing sheaf theory. The calculational strength of this approach was first illustrated by thedetermination of the category associated to the mirrors of a toric variety [14, 31]. Anotherinteresting calculation concerns cluster algebras associated to surfaces [47].More recent calculations have run somewhat ahead of the foundational work, insofar asthey calculate some well defined category of microlocal sheaves, sometimes with a somewhatad-hoc definition, which was nevertheless neither known to match the corresponding Fukayacategory nor even known to be a priori a symplectic invariant. Examples include calculationsof microlocal categories associated to very affine hypersurfaces, [36, 18], to Landau-Ginzburgmodels [38], Lagrangian mutations [46], and multiplicative quiver varieties [7] and multiplica-tive hypertoric varieties [17].One purpose of the present article is to remedy this situation. On the one hand, our mainresults construct an invariant category of microlocal sheaves which the prior calculationsdetermine in interesting situations. On the other, the antimicrolocalization lemma provenhere is a key ingredient in the comparison theorem to Fukaya categories anticipated by[30, 36] and proven in [21].1.4.2.
Geometric representation theory.
The use of stratifications to define categories ofsheaves has a long history in geometric representation theory. A paradigmatic exampleis the BGG category O (a certain category of modules of a semisimple complex Lie al-gebra g ) is identified (by Beilinson-Bernstein localization followed by the Riemann-Hilbertcorrespondence) with perverse sheaves on the g flag variety, constructible with respect tothe stratification by Schubert cells. As always, one can impose this constructibility via amicrosupport condition, namely the conormals to the Schubert cells. In the present contextthere is a good reason to do this. There is a braid group action on the derived categoryof O , given explicitly by sheaf kernels [42, Sect. 5]. However, the fact that this is a braidgroup action is not apparent a priori and must be calculated.It would be more satisfying to have on abstract grounds the action of some geometricallydefined group, which a posteriori is calculated to be the braid group. (See [25] for work inthis direction.) Using [23], one can see that loops of contactomorphisms of S ∗ Flag( g ) willact, which in fact is enough to construct the braid group action for sl . From our presentwork, one sees more generally that loops of gapped deformations will act; this may providea geometric construction in general. We pause to remark that we do not know whether theconormals to the Schubert cells meet S ∗ Flag( g ) in the skeleton of some Weinstein manifold;it would be of some interest to demonstrate this. Our methods can also be applied beyondcotangent bundles, e.g. to the analogous study of category O for symplectic resolutions as in HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 13 [28, 8]. Relatedly, it may be of interest to produce a sheaf-theoretic account of the symplecticKhovanov homology of [44].The next step beyond symplectic resolutions is the Hitchin moduli space associated to analgebraic curve C . Here the relevant category appears on one side of the “Betti geometricLanglands” of [6], and asserts an equivalence between sheaves on Bun G ( C ) with prescribedsingular support, and a category of coherent sheaves on the character variety of C . Inparticular, the latter being independent of the complex structure on C , the same is predictedfor the former. This is nonobvious: while the topology of Bun G does not vary with C , theprescribed singular support does. One may hope to approach this question as well using ourtheorem on gapped specialization.1.5. Sheaf conventions.
For a manifold M , we write sh ( M ) for the category of sheavesvalued in the symmetric monoidal stable ∞ -category C of the reader’s choice. For instance,the dg category of complexes of sheaves of abelian groups on M , localized along the acyclicsheaves. Another possibility for C is the stable ∞ -category of spectra.We appeal often to results from the foundational reference on microlocal sheaf theory [29].This work was written in the then-current terminology of bounded derived categories. Thehypothesis of boundedness can typically be removed, though one needs in some cases differentarguments, such as for proper base change, where to work in the unbounded setting one needsto use [49], and in one lemma pertaining to the noncharacteristic deformation lemma, whereto work in the unbounded setting one needs to use [41]. There are some instances whenboundedness is fundamental, e.g. whenever one wants to apply Verdier duality twice, butwe will never do this. Passing to the dg or stable ∞ setting is solely a matter of havingadequate foundations; in the dg setting one can use [11, 50] and in general [32, 33]. Thelatter also provide foundations for working with sheaves of categories more generally, whichwe will require as well.For a sheaf F ∈ sh ( M ), we write ss ( F ) ⊂ T ∗ M for its microsupport, i.e. the locus ofcodirections along which the space of local sections is nonconstant. For X ⊂ T ∗ M , we write sh X ( M ) for sheaves microsupported in X . Acknowledgements.
D. N. was supported by the grants NSF DMS-1802373, and in addi-tion NSF DMS-1440140 while in residence at MSRI during the Spring 2020 semester. V. S.was supported by the grant NSF CAREER DMS-1654545, and by the Simons Foundationand the Centre de Recherches Math´ematiques, through the Simons-CRM scholar-in-residenceprogram. 2.
Propagation and displacement
A fundamental tool in the arsenal of the microlocal sheaf theorist is the noncharacteristicpropagation lemma. Essentially it says that given a sheaf F and a family of open sets U t , if F ( U t ) changes as t varies, then it must change locally at some point at the “boundary” ofsome particular U t ; a precise statement can be found in [29, Lemma 2.7.2]. It is particularlyuseful in its microlocal formulation: Lemma 2.1. (Noncharacteristic propagation [29, Cor. 5.4.19] ) For a sheaf F on M and a C function φ : M → R proper on the support of F , if dφ / ∈ ss ( F ) over φ − [ a, b ) thenthe natural maps F ( φ − ( −∞ , b )) → F ( φ − ( −∞ , a )) → F ( φ − ( −∞ , a ]) are isomorphisms. Note that dφ over the boundary of a regular level set of of φ is just the outward conormalto the boundary. To apply noncharacteristic propagation, we will want to know when suchconormals can be displaced from ss ( F ). We introduce the following: Definition 2.2.
For a contact manifold V , a closed subset Λ ⊂ V is positively displaceablefrom legendrians (pdfl) if given any Legendrian submanifold L (compact in a neigborhood of V ), there is a 1-parameter positive family of Legendrians L t for t ∈ ( − ǫ, ǫ ) (constant outsidea compact set), such that L t is disjoint from Λ except at t = 0.To explain the term “positive” in the above definition, we briefly recall the notion ofpositive contact vector fields. On a contact manifold ( V, ξ ), a contact vector field is bydefinition one whose flow preserves the distribution ξ ; they are in natural natural bijectionwith sections of T V /ξ . When ξ is co-oriented, it makes sense to discuss positive vector fields;a contact form α identifies sections of T V /ξ with functions and positive vector fields withpositive functions. The Reeb vector field is the contact vector field associated to the constantfunction 1; in other words it is characterized by α ( v Reeb ) = 1 and dα ( v Reeb , · ) = 0. In factany positive vector field is a Reeb vector field, namely the vector field corresponding to f isthe Reeb field for the contact form f − α .Another way of expressing the above is in terms of the symplectization ˆ V = T + V ⊥ ξ , thespace of covectors ( n, α ) ∈ T ∗ V with α | ξ = 0 and α = cλ , with c >
0, for any coorientedlocal contact form α . For example, if V = S ∗ X , then ˆ V ≃ T ◦ X .We view ˆ V → V as an R > -bundle with R > -action given by scaling α . An R > -equivarianttrivialization ˆ N ≃ N × R > over N is the same data as an R > -equivariant function f : ˆ V → R > . Its Hamiltonian vector field v f is R > -invariant, preserves the level-sets f − ( c ) ≃ N ,for all c ∈ R > , and the corresponding Hamiltonian flow is a contact isotopy on each. Forexample, if N = S ∗ X , so that ˆ N = T ◦ X , a metric on X provides such a trivialization T ◦ X ≃ S ∗ X × R > with the function | ξ | : T ◦ X → R > given by the length of covectors andpositive contact isotopy given by normalized geodesic flow.By a positive contact isotopy, we mean the flow of a (possibly time dependent) positivecontact vector field. The typical example is the geodesic flow, which corresponds to the Reebvector field for the form pdq restricted to the cosphere bundle in a given metric. The imageof a given cosphere fiber under this flow is the boundary of some round ball. We recall thatany positive contact vector field also locally determines balls.A family of Legendrians is positive if it is the flow of a positive Legendrian under a positivecontact isotopy. (By Gray’s theorem, this can be checked just on the Legendrians withoutneed to explicitly construct an ambient positive isotopy.) An important example is thegeodesic flow of a cosphere. For time positive and smaller than the injectivity radius, theimage of the cosphere is the outward conormal to a ball; for time negative it will be theinward conormal. Generalizing this we have: Lemma 2.3.
Let η t : S ∗ M → S ∗ M be the time t contactomorphism of some positive contactflow. For small time t < T ( x ) , the cosphere η t ( S ∗ x M ) is the outward conormal to the boundaryof some open topological ball B t ( x ) , with closure B t ( x ) . For t < t ′ one has B t ( x ) ⊂ B t ′ ( x ) .For negative time, η t ( S ∗ x M ) will be similarly an inward conormal. A stickler might reserve the term ‘positively displaceable’ for a variant of the above definition when weconsider only t ∈ [0 , ǫ ). HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 15
Proof.
Because the projection of the cosphere to M is degenerate, it is convenient to changecoordinates before making a transversality argument. We may as well assume M = R n .There is a contactomorphism J S n − → T ∗ R n sending the zero section of J S n − to thecosphere over 0, which induces on front projections the “polar coordinates” map R × S n − → R n . Embeddedness of the front is a generic condition, so any sufficiently small isotopy of thezero section in J S n − will leave the front in R × S n − embedded. A positive isotopy will carrythe front to a subset of R > × S n − , hence its image under the embedding R > × S n − ֒ → R n will remain embedded. (cid:3) Definition 2.4.
In the situation of Lemma 2.3, it is always possible to choose a function f x : M → R ≥ with f − x ( t ) the front projection of η t ( S ∗ x M ) for small t . We term such afunction a neighborhood defining function at x for the flow η t .Note that a similar argument to Lemma 2.3 shows that the positive perturbation of theconormal to a submanifold will be the outward conormal to its boundary. Correspondingly,we may also speak of the neighborhood defining functions for a submanifold.By the injectivity radius at a point (or submanifold) we mean the first time T > η T applied to the conormal fails to have an embedded projection. The neighborhooddefining function is well defined for 0 < t < T , and we use it to give sense to the notion ofdistance to the point or submanifold. (For the Reeb flow for a metric, these are of coursethe usual notions of injectivity radius and distance.)The basic purpose of the pdfl condition is to ensure the following: Lemma 2.5.
For
F ∈ sh ( M ) , if ss ( F ) ⊂ T ∗ M is pdfl, then for any x ∈ M , there exists r ( x ) > so that for any < r < r ( x ) , the natural restrictions are isomorphisms Γ( B r ( x ) , F ) ∼ / / Γ( B r ( x ) , F ) ∼ / / F x Proof.
For φ the distance squared, φ − [0 , ǫ ) and φ − [0 , ǫ ] are the open and closed balls ofradius ǫ ; dφ over the boundary of these is the outward conormal to the boundary. Nowapply the noncharateristic propagation lemma. (cid:3) Remark 2.6.
Since we formulated the pdfl hypothesis including negative times, a similarformula holds for costalks. It follows from Lemma 2.5 (and its costalk version) that if ss ( F )is pdfl then F satisfies the first two conditions of “cohomological constructibility” of [29,Def. 3.4.1].We will want to formulate displaceability conditions in terms of chord lengths. Definition 2.7.
Fix a co-oriented contact manifold ( V , ξ ) and positive contact isotopy η t .For any subset Y ⊂ V we write Y [ s ] := η s ( Y ).Given Y, Y ′ ⊂ V we define the chord length spectrum of the pair to be the set of lengthsof Reeb trajectories from Y to Y ′ : cls ( Y → Y ′ ) = { s ∈ R | Y [ s ] ∩ Y ′ = ∅} We term cls ( Y ) := cls ( Y → Y ) the chord length spectrum of Y . Remark 2.8.
An entirely similar argument shows that for any (closed) submanifold N ⊂ M ,any small positive contact perturbation of S ∗ N M gives the outward conormal to a neighbor-hood of N . We also use the notion of neighborhood defining function in this context. Borrowing terminology from quantum mechanics or functional analysis, we say that thechord length spectrum is gapped if its intersection with some interval ( − ǫ, ǫ ) contains only 0. Definition 2.9.
Given a parameterized family of pairs ( Y b , Y ′ b ) in some V over b ∈ B , wesay it is gapped if there is some interval (0 , ǫ ) uniformly avoided by all cls ( Y b → Y ′ b ). In case Y = Y ′ , we simply say Y is gapped. Example 2.10.
One can use this notion meaningfully even if B is a point: Def. 2.2 can bereformulated as: Λ is pdfl if, for every smooth Legendrian L , there is some positive contactisotopy for which ( L, Λ) is gapped.Note that in the notion of gapped, some positive contact isotopy is assumed given andfixed. We will often specify this explicitly and say that the family is “gapped with respectto η t ”. In the typical case where η t is generated by a time independent positive Hamiltonianand hence is Reeb flow for some fixed contact form α , we may say that the family is “gappedwith respect to α ”.We also say:(1) Y is ǫ -chordless if (0 , ǫ ] ∩ cls ( Y ) = ∅ .(2) Y is chordless if cls ( Y ) = { } .(3) Y is locally chordless if for any point y ∈ Y , there exists ǫ > U of y such that Y ∩ U is ǫ -chordless.Note that a compact, locally chordless subset is ǫ -chordless for some ǫ . Example 2.11.
Smooth Legendrians are locally chordless for any positive isotopy. Thismay be seen by an argument similar to the proof of Lemma 2.3.
Example 2.12.
The curve selection lemma can be used to show that a subanalytic subsetwhich is Legendrian at all smooth points is locally chordless for the Reeb flow for an analyticmetric. (Corollary: such a subanalytic subset is pdfl. Proof: its union with any cosphereremains subanalytic, hence locally chordless; now apply Ex. 2.10.)
Example 2.13.
An example of a singular Legendrian which is not locally chordless for thegeodesic flow: consider the Legendrian in S ∗ R whose front projection is the union of the x -axis and the graph of e − /x sin( x ). Example 2.14.
A submanifold W ⊂ V is said to be a (exact) symplectic hypersurface if,for some choice of contact form λ , the restriction dλ | W provides a symplectic form. Such ahypersurface is locally chordless, since the Reeb flow is along the kernel of dλ .Such a hypersurface is said to be Liouville if λ | W gives W the structure of a Liouvilledomain or manifold. Liouville domains themselves are the subject of much inquiry; inparticular the subclass of Weinstein domains, see [9]. The elementary contact geometry ofsuch Liouville hypersurfaces is studied in e.g. [5, 13]. Example 2.15.
In particular, if Λ ⊂ V is a compact subset for which there exists a Liouvillehypersurface W ⊂ V with Λ = Core ( W ) (“Λ admits a ribbon”), then Λ is locally chordless.We note some properties and examples of positive flows. Generalizing the relationshipbetween jet and cotangent bundles, we have: Lemma 2.16.
Let
M, N be manifolds. Then T ∗ M × S ∗ N ⊂ T ∗ ( M × N ) is a contact level,which is identified (by real projectivization) with S ∗ ( M × N ) \ T ∗ N . HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 17
Given any positive contact isotopy φ t on S ∗ N , the product φ t × on T ∗ M × S ∗ N remainspositive. In particular, the Reeb vector field on T ∗ M × S ∗ N is just the pullback of the Reebvector field on S ∗ N .Proof. Pull back the contact hamiltonian by the projection map. (cid:3)
Remark 2.17.
The most typical appearance of this fact is when N = R , in which caseit provides the contactomorphism embedding J M ֒ → S ∗ ( M × R ). Note that the Reebdirection in J M happens only along the R factor; from this embedding we can see thatalong a certain subset of S ∗ ( M × R ), just moving in the R direction is a positive contactisotopy.We will use the lemma later in the opposite case M = R , for the purpose of learningsimilarly that using the “Reeb flow in the M direction” is positive along a certain subset of S ∗ ( M × R ).Consider now manifolds X, Y and positive contact isotopies φ t on S ∗ X , η t on S ∗ Y corre-sponding in the symplectization to R > -equivariant Hamilonians x : T ◦ X → R > , y : T ◦ Y → R . Observe we may define an R > -equivariant Hamiltonian by the formula x y := ( x + y ) / : T ◦ ( X × Y ) → R > Although x, y are only defined on T ◦ X, T ◦ Y , since they are R > -equivariant, their squares x , y smoothly extend to T ∗ X, T ∗ Y . Let us write ( φ η ) t for the flow on S ∗ ( X × Y ) generatedby x y , and note it is again a positive contact isotopy. Example 2.18. If x , y are the length of covectors for some metrics on X, Y , so that φ t , η t are normalized geodesic flow, then x y is the length of covectors for the product metric on X × Y , and ( φ η ) t is again normalized geodesic flow. Lemma 2.19.
Let
V, W ⊂ T ◦ X be conic subsets, and φ t the positive contact isotopy on S ∗ X generated by a time-independent R > -equivariant Hamiltonian f : T ◦ X → R > .Then the following are in length-preserving bijection: • Chords for φ t from V to W • Chords for ( φ φ ) t from the conormal of the diagonal to − V × W .Let us write ρ for the φ t distance, and ρ for the φ t φ t distance, and ∆ for the diagonal.When both are defined, ρ ( x , x ) = √ ρ (( x , x ) , ∆) . If X is compact (or the contacthamiltonian is bounded below), there is some ǫ ∈ R such that both distances are alwaysdefined in the ǫ neighborhood of the diagonal, hence B ǫ (∆) .Chords of length smaller than ǫ are then additionally in bijection with intersections in T ∗ ( B ǫ (∆)) between the graph dρ ( x , x ) and − V × W , with the chord length matching thevalue of ρ at the intersection.Proof. Let v f = ω − ( df ) denote the vector field generating φ t . Let t γ ( x, ξ ) t denote theintegral curve of v f through ( x, ξ ) ∈ T ◦ X .Write ( x , ξ , x , ξ ) for a point of T ◦ ( X × X ), and set h := f f = ( f ( x , ξ )+ f ( x , ξ )) / .By direct calculation, ( φ φ ) t is generated by the vector field v f f = h − ( f ( x , ξ ) v f ⊕ f ( x , ξ ) v f ). Although v f is only defined on T ◦ X , the scaling f v f smoothly extends to T ∗ X ,vanishing on the 0-section. Indeed, it is the Hamiltonian vector field for f and its integralcurve through ( x, ξ ) ∈ T ∗ X is given by t γ ( x, ξ ) t ′ where t ′ = f ( x, ξ ) t .Thus the integral curve of v f f through ( x , ξ , x , ξ ) , ∈ T ◦ ( X × X ) is given by t γ ( x , ξ ) t × γ ( x , ξ ) t where t = h − f ( x , ξ ) t , t = h − f ( x , ξ ) t . Note that h − f ( x , ξ ), h − f ( x , ξ ) are constant along the integral curve, so give a linear reparametrization of time.In particular, if we have f ( x , ξ ) = f ( x , ξ ) = 1, then the reparametrization constant is h − = √ / γ ( x , ξ ) t = ( x , ξ ) at t = ℓ , i.e. provides a chord of ρ -length ℓ from( x , ξ ) to ( x , ξ ). Suppose without loss of generality that f ( x , ξ ) = f ( x , ξ ) = 1 so that h = √
2. Set ( x ′ , ξ ′ ) = γ ( x , ξ ) ℓ/ . Then the integral curve t γ ( x ′ , − ξ ′ ) t × γ ( x ′ , ξ ′ ) t liesin the conormal to the diagonal at t = 0, and equals ( x , − ξ , x , ξ ) at t = ℓ √ , i.e. providesa chord of ρ -length ℓ √ from the conormal to the diagonal to ( x , − ξ , x , ξ ).Conversely, given an integral curve t γ ( x ′ , − ξ ′ ) t × γ ( x ′ , ξ ′ ) t , the integral curve t γ ( x ′ , ξ ′ ) t − ℓ/ provides the inverse construction.Finally, note that the level-sets of ρ are the fronts of the flow of the conormal to thediagonal. Thus dρ lies in a subset precisely when the flow of the conormal intersects thesubset. (cid:3) Remark 2.20.
In particular, Lemma 2.19 allows the chord length spectrum and the gappedcondition to be reformulated in terms of the graph of the derivative of the distance from thediagonal.
Remark 2.21.
We will want to use the above lemma (and remark) in connection with the“gapped” condition of Def. 2.9, and so henceforth we will only consider gappedness withrespect to a fixed contact form. This is solely due to the above lemma; if a similar resultcould be shown for time dependent flows then we could use them throughout.3.
Relative microsupport
Let π : E → B be a smooth fiber bundle. We write T ∗ π for the relative cotangent bundle,i.e. the bundle over E defined by the short exact sequence0 → π ∗ T ∗ B → T ∗ E Π −→ T ∗ π → T ∗ π are the relative cotangent spaces: for e ∈ E ,( T ∗ π ) e = T ∗ e ( E π ( e ) )We may also view T ∗ π as a fiber bundle over B , with fibers the cotangent bundles to thefibers of π : for b ∈ B , ( T ∗ π ) b = T ∗ ( E b )In this section, we present some lemmas concerning the microlocal theory of sheaves on E , on the fibers E b , and on the base B .Regarding the comparison with the base, we have the following consequence of the standardestimate on microsupport of a pushforward ([29, Proposition 5.4.4]): Definition 3.1.
We say Λ ⊂ T ∗ E is π -noncharacteristic, or synonymously B -noncharacteristic,if Λ ∩ π ∗ T ∗ B ⊂ T ∗ X . We say a sheaf F on X is π -noncharacteristic if its microsupport ss ( F )is π -noncharacteristic, and π is proper on the support of F . Lemma 3.2. If F is π -noncharacteristic, then π ∗ F has microsupport contained in the zerosection of B and hence is locally constant. The interaction with the fiber will require more subtle microsupport estimates, recalled inthe next subsection.
HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 19
Microsupport estimates.
We recall from [29, Chap. 6] the standard estimates onhow microsupport interacts with various functors.These are expressed in terms a certain operations on conical subsets. A special case is theˆ+ construction. It is an operation on conical subsets of T ∗ X , the result of which is largerthan the naive sum. It is defined in [29, Chap. 6.2] in terms of a normal cone construction.In practice, one uses the equivalent pointwise characterization [29, 6.2.8.ii], which we willjust take as the definition. Definition 3.3.
Given conical
A, B ⊂ T ∗ X , a point ( x, ξ ) ∈ A ˆ+ B if, in local coordinates,there are sequences ( a n , ζ n ) ∈ A and ( b n , η n ) ∈ B with a n , b n → x and η n + ζ n → ξ , satisfyingthe estimate | a n − b n || η n | → | a n − b n || ζ n | → a n , b n → x implies that | a n − b n | →
0. So certainly sequences with bounded ζ n satisfy the estimate. Of course, in this case passing to a subsequence gives ( a n , ζ n ) convergentin A , hence the element ( x, ξ ) would be already in A + B . The additional points of A ˆ+ B arise when | ζ n | → ∞ , in a manner controlled by | a n − b n || ζ n | → Remark 3.4.
Omitting the condition | a n − b n || ζ n | → | a n − b n || ζ n | → Example 3.5.
Consider the loci A and B given by the conormals to the x and y axes in R x,y . Then A ˆ+ B = A + B consists of the union of these conormals and the conormal to theorigin. Example 3.6.
Consider the loci A = { y = 0 } and B = { y = x } in R x,y . Then A + B isjust the union of these conormals, whereas A ˆ+ B is the union of these with the conormal tothe origin.A related operation arises in the context of a map f : Y → X . Denote the natural maps T ∗ Y df ←− f ∗ T ∗ X f π −→ T ∗ X . Definition 3.7.
Given conic A ⊂ T ∗ X and B ⊂ T ∗ Y , then ( y, η ) ∈ f ( A, B ) if (in co-ordinates) there is a sequence ( x n , ξ n ) × ( y n , η n ) ∈ A × B with y n → y , x n → f ( y ), and df y n ( ξ n ) − η n → η , while respecting the estimate | x n − f ( y n ) || ξ n | → f ( A ) := f ( A, T ∗ Y Y ). The ˆ+ construction is the special case A ˆ+ B = id ( A, − B ).Just as A + B ⊂ A ˆ+ B , we have − B + df ( f − π ( A )) ⊂ f ( A, B ) as the locus where the ξ n remain bounded. Example 3.8.
Of particularly frequent use is the case when f is (locally) a closed embed-ding. We take coordinates ( z, y, ζ , η ) on T ∗ X , with y the coordinates on Y , ζ the directionsconormal to Y , and η the directions cotangent to Y . Then a point ( y, η ) is in f ( A ) ⊂ T ∗ Y if there is a sequence ( z n , y n , ζ n , η n ) ∈ A ⊂ T ∗ X with ( y n , η n ) → ( y, η ) while z n → | z n || ζ n | → Lemma 3.9.
Some microsupport estimates: • [29, Theorem 6.3.1] For j : U → X an open inclusion, and F ∈ sh ( U ) ss ( j ∗ F ) ⊂ ss ( F ) ˆ+ N + U ss ( j ! F ) ⊂ ss ( F ) ˆ+ N − U In particular, if U is the complement of a closed submanifold Y ⊂ X , then [29,Proposition 6.3.2] ss ( j ∗ F ) | Y ⊂ ss ( F ) ˆ+ T ∗ Y X ss ( j ! F ) | Y ⊂ ss ( F ) ˆ+ T ∗ Y X • [29, Cor. 6.4.4] For f : Y → X , ss ( f ∗ F ) ⊂ f ( ss ( F )) ss ( f ! F ) ⊂ f ( ss ( F )) • [29, Cor 6.4.5] For sheaves F , G ss ( F ⊗ G ) ⊂ ss ( F ) ˆ+ ss ( G ) ss ( H om ( F , G )) ⊂ − ss ( F ) ˆ+ ss ( G )3.2. Relative microsupport.
We return to our smooth fiber bundle π : E → X . Recallwe write Π : T ∗ E → T ∗ π for the projection to the relative cotangent bundle. Definition 3.10.
For
F ∈ sh ( E ), we define the relative microsupport to be the conical locusin T ∗ π given by ss π ( F ) := Π( ss ( F ))The definition is motivated by the following connection with the ˆ+ construction. Lemma 3.11.
Let π : E → B be a fiber bundle, and let Λ ⊂ T ∗ E be conical. Let Π : T ∗ E → T ∗ π be the natural projection to the relative cotangent bundle. Then Λ ˆ+ π ∗ T ∗ B = Π − (Π(Λ)) Proof.
The assertion is local on E ; choose local coordinates ( x, e ) with x the coordinatesalong B and e the bundle coordinates. Let ( x, e, ξ, η ) be corresponding coordinates on T ∗ E .Then ( x, e, ξ, η ) ∈ Λ ˆ+ f ∗ T ∗ B iff there is a sequence ( x n , e n , ξ n , η n ) ∈ Λ and some ( x ′ n , e ′ n , ξ ′ n , x n , x ′ n → x and e n , e ′ n → e and ξ n + ξ ′ n → ξ and η n → η , and some estimate holds.But we may as well take x n = x ′ n and e n = e ′ n , so the estimate is vacuous. The condition on ξ n is vacuous as well, since ξ ′ n can be chosen arbitrarily. Thus the condition with content is η n → η . I.e., ( x, e, η ) is a limit point of Π(Λ). (cid:3) Suppose given in addition some submanifold α : A ⊂ B . We write π A : E A → A forthe restricted bundle, and Π A : T ∗ E A → T ∗ π A for the relative cotangent bundle. Note thenatural identification T ∗ π A = T ∗ π | E A . In this setting, we may write π B := π and Π B := Πfor clarity.We have the following estimate on images in the relative cotangent: Lemma 3.12.
For Λ ⊂ T ∗ E conical, we have Π A ( α Λ) ⊂ Π B (Λ) | E A .Proof. We take local coordinates ( y, z, e, γ, ζ , η ) on T ∗ E , where y are coordinates on A , the z are coordinates on B in the normal directions to A , the e are bundle coordinates, andthe γ, ζ , η are corresponding cotangent coordinates. We use corresponding notations forcoordinates on related spaces.Consider a point ( y, e, γ, η ) ∈ α Λ ⊂ T ∗ E A . By definition, there must be a sequence( y n , z n , e n , γ n , ζ n , η n ) ∈ Λ with ( y n , e n , γ n , η n ) → ( y, e, γ, η ) and z n → | z n || ζ n | → HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 21
The image of ( y, e, γ, η ) in T ∗ π A is ( y, e, η ). We must show this point is already inΠ B ( ss ( F )). Thus consider the image of the above sequence in T ∗ π B , i.e. ( y n , z n , e n , η n ).By the above hypotheses, this converges to ( y, , e, η ). (cid:3) This lemma implies that the relative microsupport interacts well with restriction to sub-manifolds of the base.
Lemma 3.13.
For
F ∈ sh ( E ) , we have ss π ( α ∗ F ) ⊂ ss π ( F ) | E A , and ss π ( α ! F ) ⊂ ss π ( F ) | E A .Proof. What is being asserted is that Π A ( ss ( α ∗ F )) ⊂ Π B ( ss ( F )) | E A and Π A ( ss ( α ! F )) ⊂ Π B ( ss ( F )) | E A . This follows from the previous lemma and the estimate ss ( α ∗ F ) ⊂ α ss ( F ). (cid:3) Corollary 3.14.
In particular, if P ∈ B is a point, ss ( F | E P ) ⊂ ss π ( F ) | E P . Corollary 3.15.
Let π , π : E × B E → B denote the projections to the factors. For sheaves F , F on E , consider the relative external hom H om E × B E ( π ∗ F , π !2 F ) ∈ sh ( E × B E ) Let ˜ π : E × B E → B be the structure map, and e Π : T ∗ ( E × B E ) → T ∗ ˜ π the projection to therelative cotangent. Then ss ˜ π ( H om E × B E ( π ∗ F , π !2 F )) ⊂ − ss π ( F ) ⊞ ss π ( F ) Proof.
This follows by applying Lemma 3.13 to δ : B → B × B . (Here we use the generalidentity f ! H om ( F, G ) = H om ( f ∗ F, f ! G ). This is Prop [29, 3.1.13], which is there statedunder some boundedness hypotheses on F, G , but the proof is a series of adjunctions whichhold in general.) (cid:3)
Microsupport of nearby cycles.
The nearby cycles functor in sheaf theory is auseful notion for taking limits of families of sheaves. We recall its construction. Let X be atopological space and π : X → R ≥ a continuous map. We consider the diagram X > π (cid:15) (cid:15) (cid:31) (cid:127) j / / X π (cid:15) (cid:15) X ? _ i o o π (cid:15) (cid:15) R > (cid:31) (cid:127) j / / R ≥ { } ? _ i o o where X > := π − ( R > ) and X = π − (0). The nearby cycles functor is by definition ψ = i ∗ j ∗ : sh ( X > ) / / sh ( X )By precomposing with j ∗ , we may also consider ψ = i ∗ j ∗ j ∗ : sh ( X ) → sh ( X ). In this settingwe also have the vanishing cycles functor, φ := Cone ( i ∗ → i ∗ j ∗ j ∗ ).Consider the case when X is a manifold, and X = X × R ≥ and π is the projection tothe second factor. X × R > π (cid:15) (cid:15) (cid:31) (cid:127) j / / X × R ≥ π (cid:15) (cid:15) X ? _ i o o π (cid:15) (cid:15) R > (cid:31) (cid:127) j / / R ≥ { } ? _ i o o In this case it makes sense to ask about the microsupport of the nearby cycles.
Recall our notation for the cotangent sequence0 / / π ∗ T ∗ R ≥ / / T ∗ ( X × R ≥ ) Π / / T ∗ π / / T ∗ π | ≃ T ∗ X . Lemma 3.16.
For
F ∈ sh ( X × R > ) , we havess ( ψ ( F )) ⊂ Π( ss ( F )) ∩ T ∗ X Proof.
Follows from Lemma 3.9 and Lemma 3.11. (cid:3)
Definition 3.17.
Given a subset Λ ⊂ T ∗ ( X × R > ), we define its nearby subset as ψ (Λ) := Π(Λ) ∩ T ∗ X so as to write the statement of the previous lemma as ss ( ψ ( F )) ⊂ ψ ( ss ( F ))4. On the full faithfulness of nearby cycles
Main statement.
Let M be a manifold, and consider the diagram: U = M × R =0 π (cid:15) (cid:15) (cid:31) (cid:127) j / / N = M × R π (cid:15) (cid:15) M ? _ i o o π (cid:15) (cid:15) R =0 (cid:31) (cid:127) j / / R { } ? _ i o o Consider the functor ψ = i ∗ j ∗ : sh ( U ) / / sh ( M )(This is nearby cycles if we restrict attention to sheaves supported always over R ≥ , whichwe will do in all applications. It is evident that all results of Sect. 3.3 hold in the presentsituation.)In particular, given F , G ∈ sh ( U ), there is an induced map between the following sheaveson M :(1) ψ : ψ H om ( F , G ) / / H om ( ψ F , ψ G )We are interested to know when (1) induces an isomorphism on global sections. (Note wemay replace R with any neighborhood of 0.)We factor (1) as:(2) i ∗ j ∗ H om ( F , G ) j ∗ / / i ∗ H om ( j ∗ F , j ∗ G ) i ∗ / / H om ( i ∗ j ∗ F , i ∗ j ∗ G )Since j is an open embedding, the counit of adjunction is an equivalence j ∗ j ∗ ∼ → id, andhence the arrow above labelled j ∗ is always an isomorphism.Thus we may as well begin with F , G ∈ sh ( N ), with the properties F = j ∗ F | U and G = j ∗ G| U , and inquire when the functorial map(3) i ∗ : i ∗ H om ( F , G ) / / H om ( i ∗ F , i ∗ G )induces an isomorphism on global sections. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 23
Example 4.1.
The following example shows the global sections of (3) can easily fail to bean isomorphism for sheaves not of the form F = j ∗ F U , G = j ∗ G U .Set M = pt , so N = R . Take F = k , G = k R . Then we have i ∗ H om N ( F , G ) ≃ k [ − H om M ( i ∗ F , i ∗ G ) ≃ k so (3) does not induce an isomorphism on global sections. Theorem 4.2.
Let F U , G U be sheaves on U = M × R =0 . Assume • ss ( F U ) and ss ( G U ) are R =0 -noncharacteristic; • ψ ( ss ( F U )) and ψ ( ss ( G U )) are pdfl; • The family of pairs in S ∗ M determined by ( ss π ( F U ) , ss π ( G U )) is gapped for somefixed contact form on S ∗ M .Set F = j ∗ F U , G = j ∗ G U . Then the natural map i ∗ : Γ( M, i ∗ H om N ( F , G )) → Hom M ( i ∗ F , i ∗ G ) is an isomorphism.Proof. Here we will reduce the proof to three technical results whose proofs occupy theremainder of the section.Let δ M : M → M × M be the diagonal embedding. We write i for a base-change of i : { } → R . Consider the diagram:(4) M δ M (cid:15) (cid:15) i / / M × R δ M (cid:15) (cid:15) M × M i × id (cid:15) (cid:15) i / / ( M × M ) × R δ R (cid:15) (cid:15) ( M × R ) × M id × i / / ( M × R ) × ( M × R )We write H := H om ( M × R ) × ( M × R ) ( p ∗ F , p !2 G ) ∈ sh ( M × R × M × R )Note H om M × R ( F , G ) = δ ! M δ ! R H . A diagram chase shows we may factor (3) as:Γ( M, i ∗ H om N ( F , G )) = Γ( M, i ∗ δ ! M δ ! R H ) → Γ( M, δ ! M i ∗ δ ! R H ) → Γ( M, δ ! M ( i × id ) ! ( id × i ) ∗ H ) → Hom M ( i ∗ F , i ∗ G )We must check that the arrows are in fact isomorphisms. Each is an example of a base changeformula which does not hold in general. They will however be valid under the microsupporthypotheses we have imposed. The first arrow, we check in Lemma 4.14 below. For thesecond, we will check below that the map i ∗ δ ! R H → ( i × id ) ! ( id × i ) ∗ H is an isomorphism, inLemma 4.13. The third arrow is an isomorphism by Cor. 4.11. (cid:3) Microlocal criterion for comparing hyperbolic restrictions.
Let X be a man-ifold, and Y , Y ⊂ X transverse submanifolds. Let Y = Y ∩ Y so we have a Cartesiandiagram(5) Y i (cid:15) (cid:15) i / / Y y (cid:15) (cid:15) Y y / / X We will write p : Y → pt for the map to a point.We will be interested in “hyperbolic restriction” from sheaves on X to sheaves on Y . Thereis a natural map of functors(6) i ∗ y !1 / / i !2 y ∗ More specifically, we will be interested in (6) after taking global sections(7) p ∗ i ∗ y !1 / / p ∗ i !2 y ∗ Example 4.3.
In general, (6) and (7) are not equivalences. Take X = R ≥ , Y = R ≥ × { } , Y = { } × R ≥ , Y = { (0 , } . Take F = k ∆ > ∗ to be the standard extension of the constantsheaf on the diagonal ∆ > = { ( t, t ) | t ∈ R > } . Then i ∗ y !1 F ≃
0, while i !2 y ∗ F ≃ k Y .The following is a natural “correction” of the issue arising in the prior example. Lemma 4.4.
Take X = R ≥ , Y = R ≥ × { } , Y = { } × R ≥ , Y = { (0 , } .Suppose H ∈ sh ( X ) is locally constant on the open submanifold ˚ X = R > near to Y . Thenthe base-change map is an equivalence (8) i ∗ y !1 H ∼ / / i !2 y ∗ H Proof.
It is elementary to check the base-change map is an equivalence for H supported on Y or Y . Thus we may assume H is the standard extension of a sheaf on ˚ X = R > . Withthe given assumption, it is elementary to verify this case as well. (cid:3) Let us reduce the general situation of (7) to the model situation of the lemma.
Proposition 4.5.
Let X be a manifold, and Y , Y ⊂ X transverse submanifolds. Let Y = Y ∩ Y , and denote maps: (9) Y i (cid:15) (cid:15) i / / Y y (cid:15) (cid:15) Y y / / X Assume that Y is compact, or more generally the support of a sheaf H ∈ sh ( X ) is compact.Fix a positive contact flow on S ∗ X . For i = 1 , , let f i : X → R ≥ be associated neigbor-hood defining functions (Def. 2.4) for Y i . Consider the map f = f × f : X → R ≥ .Suppose for some ǫ , over < f , f < ǫ , the intersection span( df , df ) ∩ ss ( H ) ⊂ T ∗ X lies in the zero-section. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 25
Then the natural map on global sections is an equivalence (10) Γ(
Y, i ∗ y !1 H ) ∼ / / Γ( Y, i !2 y ∗ H ) Proof.
We compute the map on global sections after first pushing forward along f . Weassumed supp ( H ) compact; in particular f is proper on it, hence we may invoke properbase-change to express (7) in the form(11) i ∗ y !1 f ∗ / / i !1 y ∗ f ∗ The above hypothesis on span( df , df ) ensures that singular support ss ( f ∗ H ) ∈ T ∗ R ≥ liesin the zero-section over R > . This implies local constancy of this pushforward in this region.We now apply the prior lemma to f ∗ H and conclude that (11), and hence (7), evaluated on H is an equivalence. (cid:3) Note that as we have only been interested in the geometry of X near Y , one is freeanywhere to replace X by a small neighborhood of Y .4.3. Second factor ∗ -pullback of external hom. Here we will consider a very specialcase of our general nearby cycle setup. Consider manifolds
V, W . We write π V and π W forthe operations of projecting out the V or W factor in various products. We write i : 0 → R for the inclusion, or anything base changed from it. Note by smooth basechange we mayfreely commute most operations past any pushforward along π V , π W . Lemma 4.6.
For
F ∈ sh ( V ) and G ∈ sh ( W × R ) , suppose that G ( W × ( − ǫ, ǫ )) is essentiallyconstant. Then the natural map i ∗ : i ∗ H om V × W × R ( π ∗ W × R F , π ! V G ) → H om V × W ( π ∗ W F , π ! V i ∗ G ) induces an isomorphism on global sections.Proof. Taking global sections means applying π V ∗ and π W ∗ . After applying the naturaladjunctions f ∗ H om ( f ∗ · , · ) = H om ( · , f ∗ · ) f ∗ H om ( · , f ! · ) = H om ( f ! · , · )and various base changes, we are reduced to studying i ∗ H om R ( π ∗ R π V ! F, π W ∗ G ) → Hom ( π V ! ( F ) , i ∗ π W ∗ G )To clarify what is at stake, let us write F = π V ! F for the relevant module and F for theconstant sheaf on R with this stalk. Then we are asking when the following map is anisomorphism H om R ( F , π W ∗ G ) → Hom ( F , ( π W ∗ G ) )By hypothesis, the projective system whose limit is ( π W ∗ G ) is in fact essentially constant,so the above identity would hold for any module A . (cid:3) Corollary 4.7.
For
F ∈ sh ( V ) and G ∈ sh ( W × R ) , suppose that for any w ∈ W , there isa basis of opens w ∈ W w,α such that for fixed α , the projective system G ( W w,α × ( − ǫ, ǫ )) isessentially constant.Then the natural map i ∗ : i ∗ H om V × W × R ( π ∗ W × R F , π ! V G ) → H om V × W ( π ∗ W F , π ! V i ∗ G ) is an isomorphism. Proof.
It suffices to check that this is an isomorphism on stalks, hence on arbitrary U × W w,α for U ⊂ V open. This is the same as checking that the original map is an isomorphism onglobal sections after replacing V → U and W → W w,α . We conclude by Lemma 4.6. (cid:3) We now develop a criterion for verifying the hypothesis of the corollary. For a givenpositive flow on S ∗ W , for x ∈ W and r >
0, let B r ( x ) ⊂ W , B r ( x ) ⊂ W , S r ( x ) ⊂ W denotethe respective open ball, closed ball and sphere around x of size r in the sense of 2.3.Consider the inclusions of cylinders c : C r,δ ( x ) = B r ( x ) × ( − δ, δ ) / / M × R c : C r,δ ( x ) = B r ( x ) × ( − δ, δ ) / / M × R and their projections p : C r,δ ( x ) = B r ( x ) × ( − δ, δ ) / / ( − δ, δ ) p : C r,δ ( x ) = B r ( x ) × ( − δ, δ ) / / ( − δ, δ )Note is c is an open embedding and p is proper. Lemma 4.8.
Suppose Λ ⊂ T ∗ ( W × R =0 ) is R =0 -noncharacteristic, and φ (Λ) ∞ ⊂ T ∞ ( W × is pdfl.Fix x ∈ W , and a positive flow displacing S ∗ x from φ (Λ) . Then there exists r ( x ) > and δ ( x, r ( x )) > so that for all < r < r ( x ) , < δ < δ ( x, r ( x ))) , the Legendrian at infinity Λ ∞ ⊂ T ∞ ( W × R ) is disjoint from the outward conormal Legendrian of the cylinder C r,δ ( x ) .Proof. Note the closure C r,δ ( x ) is a manifold with corners and its outward conormal Legen-drian is a union of several pieces: there are the two codimension one faces ∂ r = S r ( x ) × ( − δ, δ ) ∂ δ = B r ( x ) × {± δ } and the corner ∂ r,δ = S r ( x ) × {± δ } The assertion for ∂ δ is immediate from the R =0 -noncharacteristic hypothesis.We write Π : T ∗ ( W × R ) → ( T ∗ W ) × R for the projection to the relative cotangent. Forfixed r >
0, if Λ ∞ intersects the outward conormal along ∂ r , any point in the intersectiondefines a point in Π(Λ) ∞ . Similarly, by the R =0 -noncharacteristic hypothesis, if Λ ∞ intersectsthe outward conormal along ∂ r,δ , any point in the intersection also defines a point in Π(Λ) ∞ .Taking the limit of such points as δ → φ (Λ) ∞ with theoutward conormal of B r ( x ).But these conormals were chosen disjoint from φ (Λ) ∞ , this having been possible becausethis locus was assumed pdfl. (cid:3) Corollary 4.9.
For
G ∈ sh ( W × R =0 ) , suppose ss ( G ) ⊂ T ∗ ( W × R ) is R =0 -noncharacteristicand ψ ( ss ( G )) is pdfl.Then for any x ∈ W , there exists r ( x ) > and δ ( x, r ( x )) > so that for all < r < r ( x ) , < δ < δ ( x, r ( x ))) , the following restriction maps are isomorphisms. p ∗ c ∗ j ∗ G ∼ −→ p ∗ c ∗ j ∗ G Γ( C r,δ ( x ) , j ∗ G ) ∼ / / Γ( C r,δ ( x ) , j ∗ G ) ∼ / / j ∗ G x Proof.
Noncharacteristic propagation. (cid:3)
HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 27
Remark 4.10.
Note that Corollary 4.9 implies the functor p ∗ c ∗ commutes with standardoperations: it commutes with !-pullbacks and ∗ -pushforwards (since c is smooth) and p ∗ c ∗ commutes with !-pushforwards and ∗ -pullbacks (since p is proper). Corollary 4.11.
For
G ∈ sh ( W × R =0 ) such that ss ( G ) ⊂ T ∗ ( W × R ) is R =0 -noncharacteristicand ψ ( ss ( G )) is pdfl, and for any F ∈ sh ( V ) , the natural map i ∗ : i ∗ H om V × W × R ( π ∗ W × R F , π ! V G ) → H om V × W ( π ∗ W F , π ! V i ∗ G ) is an isomorphism.Proof. We use Cor. 4.9 to verify the hypothesis of Cor. 4.7. (cid:3)
Remark 4.12.
The conclusion of Cor. 4.11 also holds if F is cohomologically constructible.Indeed, in this case we may use Verdier duality:(1 V ⊠ i ∗ ) H om V × W × R ( π ∗ W × R F , π ! V G ) = (1 V ⊠ i ∗ )( D F ⊠ G )= D F ⊠ i ∗ G = H om V × W ( π ∗ W F , π ! V i ∗ G )Looking back through the logic of this section, recall at some moment we were to con-sider H om R ( F , π W ∗ G ) → Hom ( F , ( π W ∗ G ) ). Previously we argued it was an isomorphismbecause of some essential constancy of π W ∗ G near zero. However, we could have also con-cluded that it was an isomorphism if F were a compact object. But we do not want toimpose any finiteness conditions on stalks, in particular because it would prevent us fromlater considering the category of ‘wrapped microlocal sheaves’.4.4. The lower square of (4) . For the lower square of (4), we do not need the gappedhypothesis, but do need displaceability from legendrians.
Lemma 4.13.
Let F U , G U be sheaves on U = M × R =0 . Assume ss ( F U ) , ss ( G U ) are R =0 -noncharacteristic, and ψ ( ss ( F U )) , ψ ( ss ( F U )) are pdfl.Set F = j ∗ F U , G = j ∗ G U . Then the natural map (12) i ∗ δ ! R H / / ( i × id ) ! ( id × i ) ∗ H arising from the lower square of (4) is an isomorphism.Proof. We will check (12) is an isomorphism on the stalk at a point ( x , x ) ∈ M × M .Let Λ ⊂ T ∗ ( M × M ) denote the singular support of either side of (12). By standardestimates it is contained in − ψ ( ss ( F U )) ⊠ ψ ( ss ( G U )). Because ψ ( ss ( F U )) and ψ ( ss ( G U )) arepdfl, the Legendrian at infinity Λ ∞ ⊂ T ∞ ( M × M ) is disjoint from the outward conormalLegendrian of the polyball B r ( x ) × B r ( x ), for all small r > x , x ) by taking sections over the polyball B r ( x ) × B r ( x ) which is the special fiber of the polycylinder C r,δ ( x ) × C r,δ ( x ). We preservefrom the previous subsection (see above Lemma 4.8) the notation for inclusion and projectionof cylinders.Writing this in terms of standard operations, we seek to show the induced map( p × p ) ∗ ( c × c ) ∗ i ∗ δ ! R H / / ( p × p ) ∗ ( c × c ) ∗ ( i × id ) ! ( id × i ) ∗ H is an isomorphism.For the right hand side, applying Lemma 4.8 and Corollary 4.8, in view of the identitiesof Remark 4.10, we find ( p × p ) ∗ ( c × c ) ∗ ( i × id ) ! ( id × i ) ∗ H = ( i × id ) ! ( id × i ) ∗ ( p × p ) ∗ ( c × c ) ∗ H = ( i × id ) ! ( id × i ) ∗ H om R × R ( p ! c ! j ∗ F U , p ∗ c ∗ j ∗ G U )= ( i × id ) ! ( id × i ) ∗ H om R × R ( j ∗ p ! c ! F U , j ∗ p ∗ c ∗ G U )Similarly, for the left hand side, repeating the arguments of Lemma 4.8 and Corollary 4.8,and thus deducing analogous identities to Remark 4.10, we find( p × p ) ∗ ( c × c ) ∗ i ∗ δ ! R H = i ∗ δ ! R H om R × R ( j ∗ p ! c ! F U , j ∗ p ∗ c ∗ G U )Thus we seek to show the natural map i ∗ δ ! R H om R × R ( j ∗ p ! c ! F U , j ∗ p ∗ c ∗ G U ) / / ( i × id ) ! ( id × i ) ∗ H om R × R ( j ∗ p ! c ! F U , j ∗ p ∗ c ∗ G U )is an isomorphism.A final application of Corollary 4.8, to replace p with the proper map p , shows by non-characteristic propagation that p ! c ! F U , p ∗ c ∗ G U are locally constant on (( − δ, δ ) \ { } ) . Thusit remains to verify the assertion in the case when M is a point. This is an elementaryexercise. (cid:3) The upper square of (4) . For the upper square of (4), we do not require the (pdfl)condition on microsupports or its “constructibility” consequences. However, we do need thegappedness of the pair ( F , G ). Lemma 4.14.
Let F , G ∈ sh ( M × B ) be B -noncharacteristic, and assume the pair ( ss π ( F ) , ss π ( G )) is gapped (Def. 2.9) over B \ b . Then the natural map Γ( M, i ∗ δ ! M ( δ ! B H )) → Γ( M, δ ! M i ∗ ( δ ! B H )) associated to (13) M δ M (cid:15) (cid:15) i / / M × B δ M (cid:15) (cid:15) M × M i / / ( M × M ) × B is an isomorphism.Proof. Let m : T ◦ M → R be the linear conic Hamiltonian for the positive flow exhibit-ing gappedness (recall this Hamiltonian is assumed time independent). Let b be the conichamiltonian for Reeb flow on T ∗ B \ B (i.e. the norm of the cotangent coordinate). On T ◦ ( M × M × B ) we consider the hamiltonian h := ( m ⊕ m ⊕ b ) / ; being linear and conicit determines a contact flow on S ∗ ( M × M × B ).We will apply Proposition 4.5 to the sheaf δ ! B H . In the notation there we should consider X = ( M × M ) × B Y = M × B Y = M × M Y = M Let f and f be neighborhood defining functions for M × B and M × M respectively.To verify the hypotheses of Proposition 4.5 we must show that for some ǫ , above { f , f ∈ (0 , ǫ ) } , the locus ss ( δ ! B H ) is disjoint from the span of df , df .Evidently df is pulled back from the cotangent to B , so the disjointness from df followsbecause F , G are B -noncharacteristic. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 29
We now consider df . Because df is contained in T ∗ B , it suffices to check that ss ( δ ! B H )is disjoint from df in some neighborhood f − (0 , ǫ ) after projecting to T ∗ X/T ∗ B . That is,we should study the relative microsupport ss ˜ π ( δ ! B H ).The relevant estimate is Cor. 3.15, which tells us that in the fiber over some particularpoint b ∈ B , i.e. inside ( T ∗ X/T ∗ B ) b = T ∗ ( M × M ), we have ss ˜ π ( δ ! B H ) b ⊂ − ss π ( F ) b ⊞ ss π ( G ) b We therefore study intersections of df with the RHS above. Per Lemma 2.19, thesecorrespond to chords whose length is the value of f at the intersection point. From thegapped hypothesis, we may once and for all choose ǫ small enough that there are no such in f − (0 , ǫ ). (cid:3) Microlocal sheaves
Let M be a manifold, and C a symmetric monoidal presentable stable ∞ -category C . ∞ -category. (The reader will not learn less from this article by taking C to be the dg derivedcategory of modules over a commutative ring, say Z .)The category of sheaves on M valued in C microlocalizes over the cotangent bundle T ∗ M in the sense that it is the global sections of a sheaf of categories on T ∗ M . This sheaf ofcategories is defined as follows. Recall that for V ⊂ T ∗ M , we write sh V ( M ) for the categoryof sheaves on M microsupported in V . For an open subset U ⊂ T ∗ M , we set µ sh pre ( U ) := sh ( M ) / sh M \U ( M )For U ⊂ V , we evidently have sh M \U ( M ) ⊃ sh M \V ( M ), and thus there are restriction maps µ sh pre ( U ) → µ sh pre ( V ); it is easy to see that these make µ sh pre into a presheaf of symmetricmonoidal presentable stable ∞ -categories. We write µ sh for its sheafification (see Rem. 5.1for a discussion of this construction in the present context).While µ sh is sensible in the usual topology on T ∗ M , it is in fact pulled back from theconic topology, in which the open sets are all invariant under the R > -action. We will oftenbe interested in its restriction to the the complement of the zero section T ∗ M \ M , whereit is pulled back from a sheaf on the cosphere bundle ( T ∗ M \ M ) / R = S ∗ M , which we alsodenote by µ sh .For F ∈ µ sh ( U ), there is a well defined microsupport, ss ( F ) ⊂ U . For Λ ⊂ M we write µ sh Λ ( U ) for the full subcategory of objects microsupported in Λ ∩ U . Note that µ sh Λ is asubsheaf of µ sh , and the pushforward of a sheaf on Λ. Remark 5.1.
Let us give some technical remarks regarding sheaves of ∞ -categories, inparticular sheafification; throughout, we rely upon the foundations provided by [32, 33]. Toease the exposition, when possible, we will say category in place of ∞ -category.In general, when discussing presheaves of categories, it would be necessary to specifyin which category of categories we are working; specific natural choices would include thecategory Cat of all categories, and also the categories
P r L and P r R of presentable categories See [32, Ch. 5.5] for a foundational treatment of presentable ∞ -categories. In particular: the adjointfunctor theorem for presentable ∞ -categories implies that being a left adjoint is equivalent to preservingcolimits, and being a right adjoint is equivalent to being accessible and preserving limits. Note that acolimit preserving functor is certainly accessible, hence a functor which preserves colimits and limits hasboth adjoints. See [33, Ch. 1] for a foundational treatment of stable ∞ -categories. with continuous or cocontinuous morphisms. On the other hand, to determine whethera presheaf is a sheaf, this is immaterial: limits in P r L or P r R exist and are computedby the corresponding limits in the category of categories [32, Chap. 5.5]. In particular,given a presheaf of presentable categories for which all restriction maps are continuous andcocontinuous, its sheafifications in P r L and P r R and Cat all agree. Indeed, the universalproperty of sheafification guarantees natural maps between these. As sheafification preservesstalks and isomorphisms of sheaves is determined stalkwise, the sheafifications agree, and inparticular this common sheafification has continuous and cocontinuous restriction maps.Now let us return specifically to µ sh . As the microsupport of a limit or colimit is containedin the union of the microsupports of the terms, the full subcategory sh M \U ( M ) ⊂ sh ( M ) isclosed under limits and colimits. It follows (see e.g. [33, Prop A.8.20, Rem A.8.19]) thatthe quotient map sh ( M ) → sh ( M ) / sh M \U ( M ) is continuous and cocontinuous. Similarly,all restriction maps in µ sh pre are continuous and cocontinuous. Thus we need not concernourself with choosing between P r L , P r R , Cat in defining µ sh , and the restriction maps forthis sheaf of categories are continuous and cocontinous. Similarly, since all categories insight are stable and all functors exact, the sheafification is automatically a sheaf of stablecategories.Next, let us recall some properties of µ sh available in the literature. While µ sh is notexplicitly considered in [29], µ sh pre appears in [29, Sec. 6.1], where µ sh pre ( U ) (with bound-edness assumptions) is called D b ( M ; U ). The stalks of µ sh pre , which are also the stalks of µ sh , also appear (again with boundedness assumptions) in [29] under the name D b ( M, p ). Inconstructing a morphism of sheaves, it is enough to do so for their corresponding presheaves;to check properties of a morphism (in particular, when it is fully faithful or an isomorphism)it is enough to check on stalks. Thus the results of [29] serve well for these purposes. Mostfundamentally, the µhom functor of [29] gives the sheaf of Homs of objects in µ sh . (Indeed,in [29, Theorem 6.1.2], it is shown that there is an isomorphism on stalks, and the prooffirst constructs a natural morphism on presheaves.) The map µ sh pre ( U ) → µ sh ( U ) is notgenerally an isomorphism, but in case U = T ∗ U one has Sh ( U ) = µ sh pre ( T ∗ U ) = µ sh ( T ∗ U ).A basic tool to study µ sh is the microlocal theory of quantized contact transformationsdeveloped in [29, Chap. 7]. A key result is that a contactomorphism induces local isomor-phisms on µ sh . More precisely, a contactomorphism φ between a germ of x ∈ S ∗ M and of y ∈ S ∗ N induces an isomorphism , respecting microsupports, of µ sh x and φ ∗ µ sh y [29, Cor7.2.2].In order to globalize this, we note the evident: Lemma 5.2.
Suppose e Λ ⊂ T ∗ ( M × R ) is the product of Λ ⊂ T ∗ M and T ∗ R n . Then pullbackalong i : M × n → M × N induces an isomorphism i ∗ : i ∗ µ sh e Λ → µ sh Λ , and pullback along π : M × N → M induces an isomorphism π ∗ : µ sh e Λ → π ∗ µ sh Λ . (cid:3) By contact transformation we have:
Lemma 5.3.
Suppose the germ of e Λ ⊂ S ∗ M is contactomorphic to the germ of Λ × N ⊂U × T ∗ N for some contact U , by a map restricting to f : e Λ ∼ = Λ × N .Let π : Λ × N → Λ be the projection. Then for λ ∈ Λ , there is an isomorphism µ sh e Λ | λ ∼ = µ sh Λ | π ◦ f ( λ )9 The isomorphism is unique up to a choice of invertible object in the coefficient category.
HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 31
Proof.
The statement is local on e Λ, so by contact transformation we are reduced to Lemma 5.2. (cid:3)
That is, µ sh e Λ is locally constant in the N direction. In particular, by taking Λ a point,one has: Corollary 5.4.
Let X ⊂ S ∗ M be a smooth Legendrian. Then µ sh X is locally isomorphic tothe category of local systems on X . That is, µ sh X is a sheaf of categories of twisted local systems. The twistings are relatedto the Maslov obstruction and similar homotopical considerations, and are studied in [22,27, 24, 26]. We will also consider them below.We now consider contact isotopies. Recall that a hamilonian isotopy φ t on S ∗ M determinesa Lagrangian Φ ⊂ T ∗ M × T ∗ M × T ∗ R . Using Φ as a correspondence gives a map (we alsodenote it Φ) from subset of T ∗ M to subsets of T ∗ M × T ∗ R , and Φ is characterized by theproperty φ t ( X ) is the symplectic reduction of Φ( X ) over t ∈ R . Definition 5.5.
For a contact isotopy on S ∗ M , we obtain similarly a map Φ from subsetsof S ∗ M to subsets of S ∗ M × T ∗ R (e.g. by viewing it as a conic Hamiltonian isotopy). For X ⊂ S ∗ M we term Φ( X ) ⊂ S ∗ M × T ∗ R ⊂ S ∗ ( M × R ) as the contact movie of X . Lemma 5.6.
Let φ t be a contact isotopy on S ∗ M , and Λ ⊂ S ∗ M any subset. Then there isan isomorphism φ t ∗ µ sh Λ ∼ = µ sh φ t (Λ) .Proof. Note there is a contactomorphism (Φ( X ) , N bd (Φ( X ))) ∼ = ( X × R , N bd ( X ) × T ∗ R ), sowe may apply Lemma 5.3 to conclude that µ sh Φ(Λ) is constant in the R direction. As Φ(Λ)is noncharacteristic for the inclusion of any M × t , pullback along such an inclusion inducesan isomorphism µ sh Φ(Λ) | M × t → µ sh φ t (Λ) . (cid:3) We recall the stronger result:
Theorem 5.7. [23]
For any contact isotopy φ t of S ∗ M there is a unique sheaf K Φ ∈ sh Φ ( M × M × R ) such that K Φ | M × M × is the constant sheaf on the diagonal. The relation of this theorem to the above discussion is that K Φ | M × M × t gives an integralkernel which on microsupports away from the zero section applies the contact transformation φ t . Its real strength has to do with the fact that one obtains equivalences of categories ofsheaves, not just (as in Lemma 5.6) microsheaves away from the zero section.As this result will be important to us, let us sketch the proof. Uniqueness can be seen asfollows: consider the functor sh Φ ( M × M × R ) → µ sh Φ (Φ). Note the latter is a categoryof local systems. Thus if two candidate K Φ are isomorphic at M ×
0, hence microlocallyisomorphic along Φ | , they must be microlocally isomorphic everywhere away from the zerosection. Thus the cone between them is a local system; as it vanishes at M × t (for t small) isa conormal to the boundary of a neighborhood of the diagonal. The corresponding K Φ canbe taken as the constant sheaf on this (open or closed according as the isotopy is positive ornegative) neighborhood.We will mainly use this result through its following consequence: Corollary 5.8.
Let η : M × R → R be the projection. Fix a contact isotopy φ t : S ∗ M → S ∗ M and any X ⊂ S ∗ M . Assume M is compact. Then the sheaf of categories (on R ) givenby η ∗ sh Φ( X ) is locally constant. Pullback to t ∈ R induces an equivalence ( η ∗ sh Φ( X ) ) t ∼ = sh φ t ( X ) ( M ) . The hypothesis of compactness may as usual be replaced by an assumption that in thecomplement of a compact set, (
M, X, φ t ) are the product of some structures on a compactmanifold with constant structures on a noncompact one.6. Antimicrolocalization
Because µ sh is a quotient of sheaves of categories, hence suffers in its definition a sheafi-fication , it is nontrivial to compute in µ sh directly. In particular, for X ⊂ S ∗ M , it is notgenerally true that the map sh X ( M ) → µ sh X ( X ) is a quotient. One can often neverthelessreduce problems of microsheaf theory to problems of sheaf theory by finding some larger X ′ ⊂ S ∗ M for which the natural map sh X ′ ( M ) → µ sh X ( X ) has a right inverse. We termsuch an inverse an antimicrolocalization .When X projects finitely to M , a local version of this problem can be solved directly usingthe “refined microlocal cutoff” of [29]; we give an account in Section 6.1; similar results canbe found in [53, 22].When Λ is a smooth Legendrian in a jet bundle, Λ ⊂ J M ⊂ T ∗ ( M × R ), Viterbo observedthat Λ ∪ Λ ǫ (the latter being a small Reeb pushoff) should provide an antimicrolocalization[51, 52]. His argument was Floer-theoretic: there is an (exact) Lagrangian L ∼ = Λ × R with ∂L = Λ ∪ Λ ǫ ; now µ sh L ( L ) is local systems on L , and the map µ sh L → sh Λ ∪ Λ ǫ ( M × R ) isobtained by sending a given local system L to the sheaf organizing the Floer theory of ( L, L )with cotangent fibers.A direct sheaf-theoretical construction was later given by Guillermou [22]. Here we willprove an analogous result for a singular Legendrian Λ in an arbitrary cosphere bundle dis-placed by an arbitrary positive flow, and a generalization to the case when Λ is only locallyclosed. Even in the case of a jet bundle and smooth Λ, the proof is new. Remark 6.1.
The fact that antimicrolocalization should exist in this generality is partiallymotivated by the results on “stop doubling” of [20, Ex. 8.6]. Indeed, these results show thatgiven a Weinstein manifold W with skeleton Λ, and an embedding as an exact hypersurface W → S ∗ M , then there is a fully faithful functor F uk ( W ) ֒ → F uk ( T ∗ M ; Λ ∪ Λ ǫ ). In [21] itis shown that F uk ( T ∗ M ; Λ ∪ Λ ǫ ) ∼ = sh Λ ∪ Λ ǫ ( M ), and one could imagine running an analogueof Viterbo’s construction above – if one knew that µ sh (Λ) ∼ = F uk ( W ). In fact [21] usesthis idea in the reverse direction, applying the antimicrolocalization in order to reduce thegeneral problem of showing µ sh (Λ) ∼ = F uk ( W ) to the case of cotangent bundles. This resultwas originally conjectured in [36]. Remark 6.2.
Note that antimicrolocalization is the special case of gapped specializationcorresponding to taking the natural (Λ × R ) ⊂ T ∗ M whose boundary is Λ ∪ Λ ǫ , and flowingdown by the Liouville flow. Guillermou makes a local construction which he then proves glues. A similar construction will work forsingular Λ in a jet bundle, but a difficulty of adapting this to the case of an arbitrary positive flow is thataside from the Reeb flow in the jet bundle, it is not clear how to construct a cover on the base M compatiblewith the effect of the contact flow on S ∗ M . We instead provide a global construction. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 33
Local antimicrolocalization.
Let us recall from [29] the refined microlocal cutoff.The assertions are local so from the start we will work with X = R n a vector space andfocus on the origin x = 0. Let X ∗ ≃ R n denote the dual vector space so that T ∗ X ≃ X × X ∗ . Proposition 6.3. ( [29, Proposition 6.1.4] ) Let K ⊂ X ∗ be a proper closed convex cone, and U ⊂ K an open cone. Fix F ∈ sh ( X ) and W ⊂ X ∗ a conic neighborhood of K ∩ ss ( F ) | \ .Then there exists F ′ ∈ sh ( X ) and a map u : F ′ → F along with a neighborhood B ⊂ X of ∈ X such that (1) ss ( F ′ ) | B ⊂ B × U and ss ( F ′ ) | ⊂ W ∪ . (2) u induces an isomorphism in µ sh pre ( B × U ) . For further developments of the microlocal cutoff, in particular to treat the complex set-ting, see also [10, 53].
Corollary 6.4.
In the situation above, after possibly shrinking B , we have (1 ′ ) ss ( F ′ ) | B ⊂ B × ( W ∪ .Proof. If not, there exists a sequence ( x i , ξ i ) ∈ ss ( F ′ ), with x i →
0, but ξ i W ∪
0. Since W and ss ( F ′ ) are conic, we may assume | ξ i | = 1 and the sequence converges ξ i → ξ ∞ . Since ss ( F ′ ) is closed, we have (0 , ξ ∞ ) ∈ ss ( F ′ ) | ⊂ W ∪ | ξ ∞ | = 1, hence ξ ∞ ∈ W . But W is open so ultimately ξ i ∈ W . (cid:3) The proof given in [29, Proposition 6.1.4] involves
F ∈ sh ( X ) only through its singularsupport ss ( F ) ⊂ T ∗ X , and the sheaf F ′ is constructed functorially. In addition the cutofffunctor makes sense for any F ; only its properties depend on the asserted condition on themicrosupport of F . In other words what their argument actually shows is: Proposition 6.5.
Suppose given • K ⊂ X ∗ a proper closed convex cone • U ⊂ K an open cone • Λ ⊂ T ∗ X a closed conic subset • W ⊂ X ∗ a conic neighborhood of K ∩ Λ | \ .Then there is a map φ : sh ( X ) → sh ( X ) with a natural transformation u : φ → id , such thatfor a small enough neighborhoods B of ∈ X (1) For any F , ss ( φ ( F )) | B ⊂ B × ( W ∪ . (2) If ss ( F ) ∩ ( B × U ) = Λ ∩ ( B × U ) , then u : φ ( F ) → F induces an isomorphism in µ sh pre ( B × U ) . Corollary 6.6.
In the situation of Proposition 6.5, suppose in addition W ⊂ U . Then forany ∈ A ⊂ X , and for any small enough B ∋ , the microlocal cutoff induces a functor φ : µ sh pre Λ ( A × U ) / / sh Λ ( B ) /sh T ∗ B B ( B ) commuting with the natural projections on both sides to µ sh pre Λ ( B × U ) .Proof. We apply Proposition 6.5 with X replaced by its subset A . We use the superscript ∞ to denote the real projectivization of the complement of the zero section of a subset of T ∗ X .Now for any F , we have ss ∞ ( φ ( F )) | B ⊂ B × W ∞ ⊂ B × U ∞ by property (1) In [29] it is stated “induces an isomorphism on U ”. However from the proof, it is clear that the presentassertion is what is meant. If ss ∞ ( F ) ∩ ( B × U ∞ ) = Λ ∩ ( B × U ∞ ) then also ss ∞ ( φ ( F )) ∩ ( B × U ∞ ) = ss ∞ ( F ) ∩ ( B × U ∞ ) by property (2)Thus under this microsupport condition, ss ∞ ( φ ( F )) ∩ ( B × U ∞ ) = ss ∞ ( F ) ∩ ( B × U ∞ ).In particular if ss ∞ ( F ) ∩ ( A × U ∞ ) = ∅ , then ss ∞ ( φ ( F )) | B = ∅ , so φ factors through µ sh pre ( A × U ) as stated. Additionally if ss ∞ ( F ) ∩ ( A × W ) ⊂ Λ then ss ∞ ( φ ( F )) | B ⊂ B × Λ,giving the microsupport condition on the image.Finally, the assertion that φ commutes with the projections follows from the fact that u : φ → id induces an equivalence on B × U . (cid:3) Now let us use this to prove the following. (See [53, Sect. 5.1] for similar arguments in thecomplex setting.)
Lemma 6.7.
Fix a closed subset Λ ⊂ S ∗ X . Suppose for a point x ∈ X , the fiber Λ | x =Λ ∩ S ∗ x X is a single point λ . Then the natural map of sheaves of categories q : sh Λ /sh T ∗ X X / / π ∗ µ sh Λ is an isomorphism at x .Proof. The assertion is local so we may assume X = R n , x = 0. Thus T ∗ X ≃ X × X ∗ and S ∗ X = X × X ∞ where X ∗ is the dual of X , and X ∞ = ( X ∗ \ / R > is its projectivization.We will Proposition 6.5 with U = W ⊂ X ∗ an open convex conic neighborhood of thecoray R > · λ ⊂ X ∗ , and K = U = W ⊂ X ∗ the closure. Note we are able to make thesechoices precisely due to the assumption that Λ | x = λ .For any open neighborhood A ⊂ X of 0 ∈ X , and small enough open neighborhood B ⊂ X of 0 ∈ X , we have from Cor. 6.6 a functor φ : µ sh pre Λ ( A × U ) / / sh Λ ( B ) /sh T ∗ X X ( B )The rest of the argument follows the complex setting detailed in [53, Sect. 5.1]. Next, asin [53, Lemma 5.1.4], one observes the composed functor µ sh pre Λ ( A × U ) φ / / sh Λ ( B ) /sh T ∗ X X ( B ) / / ( sh Λ /sh T ∗ X X ) | x factors through the natural map µ sh pre Λ ( A × U ) / / π ∗ µ sh pre Λ | x For this, let us write φ U,A to convey its dependence on the choices. Given smaller U ′ ⊂ U , A ′ ⊂ A , by construction there is a natural map φ U ′ ,A ′ → φ U,A which induces an isomorphismin ( sh Λ /sh T ∗ X X ) | x since its cone must have singular support in the zero-section.Finally, one checks the resulting map φ x : π ∗ µ sh pre Λ | x / / ( sh Λ /sh T ∗ X X ) | x is an inverse to q x . First, we have q x ◦ φ x ≃ id thanks to property (2). Second, we have φ x ◦ q x ≃ id since the cone of the natural morphism relating them must have singular supportin the zero-section. (cid:3) Remark 6.8.
The Λ to which this is applied in practice will typically have enough tamenessthat µ sh pre Λ | x is computed by some particular µ sh pre Λ ( B × U ), i.e. one need not pass to thelimit. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 35
Now let us use the lemma to prove the following generalization.
Proposition 6.9.
Fix a closed singular Legendrian Λ ⊂ S ∗ X . Suppose for a point x ∈ X ,the fiber Λ | x = Λ ∩ S ∗ x X is a finite set of points λ , . . . , λ k . Then the natural map of sheavesof categories q : sh Λ /sh T ∗ X X / / π ∗ µ sh Λ is an isomorphism at x .Proof. Let Λ , . . . , Λ k ⊂ Λ be the connected components through the respective points λ , . . . , λ k ∈ Λ | x . Note the evident direct sum decomposition π ∗ µ sh Λ | x ≃ ⊕ ki =1 π ∗ µ sh Λ i | x Thus by the prior lemma, it suffices to show the natural map ⊕ ki =1 sh Λ i /sh T ∗ X X | x / / sh Λ /sh T ∗ X X | x is an equivalence.Set Λ ′ = ∪ k − i =1 Λ i , Λ ′′ = Λ k . By induction, it suffices to show the natural map i ′ ⊕ i ′′ : sh Λ ′ /sh T ∗ X X | x ⊕ sh Λ ′′ /sh T ∗ X X | x / / sh Λ /sh T ∗ X X | x is an equivalence. Let us first show the images of i ′ , i ′′ are orthogonal to each other. For thenatural map q ′ ⊕ q ′′ : sh Λ /sh T ∗ X X | x / / π ∗ µ sh Λ ′ | x ⊕ π ∗ µ sh Λ ′′ | x we have q ′ ◦ i ′ , q ′′ ◦ i ′′ are equivalences, and im( i ′ ) = ker( q ′′ ) , im( i ′′ ) = ker( q ′ ). This impliesthe desired orthogonality since for exampleHom( i ′ F ′ , i ′′ F ′′ ) ≃ Hom( i ′ F ′ , ( q ′′ ) r q ′′ i ′′ F ′′ ) ≃ Hom( q ′′ i ′ F ′ , q ′′ i ′′ F ′′ ) ≃ q ′′ ) r denotes the right adjoint to q ′′ .Finally, it remains to show i ′ ⊕ i ” is essentially surjective. For F ∈ sh Λ /sh T ∗ X X | x , theabove identities imply we have a triangle i ′ ( i ′ ) r F / / F / / ( q ′′ ) r q ′′ F where ( i ′ ) r denotes the right adjoint to i ′ . Since im(( q ′′ ) r ) = im( i ′′ ), this implies i ′ ⊕ i ” isessentially surjective. (cid:3) Definition 6.10.
We say Λ ⊂ S ∗ X is in finite position over x ∈ X if it satisfies the hypothesisof Proposition 6.9 over x . When this is true at all x , we say Λ is in finite position . If Λ maybe perturbed to have finite projection by a contact isotopy, we say Λ is perturbable to finiteposition. Finally, for Λ a closed subset equipped with the germ of an embedding into a contact man-ifold, we say Λ is perturbable to finite position if every codimension zero contact embeddingof Λ into a cosphere bundle is perturbable to finite position.
Remark 6.11.
It is easy to see that a subset with an isotropic Whitney stratification isperturbable to finite position.
Cusps and doubling.
Let M be a manifold, and consider Λ ⊂ S ∗ M . Fix a positivecontact isotopy φ t on S ∗ M . For s ≥
0, let e φ ± s = φ ± s / , andΛ ± s := ˜ φ ± s (Λ) ⊂ S ∗ M Λ ≺ = e Φ + (Λ) ∪ e Φ − (Λ) ⊂ S ∗ ( M × R )Note the contact reduction of Λ ≺ over s ≥ + s ∪ Λ − s , and is empty for s <
0. Also observethe germ of S ∗ ( M × R ) around Λ ≺ is locally contactomorphic to Λ × R ⊂ N bd (Λ) × T ∗ R .In particular, µ sh Λ ≺ is the pullback of µ sh Λ .If Λ was a point in S ∗ R and we take the Reeb flow, then the front projection of Λ ≺ is thestandard cusp y = x . This example is studied in some detail by elementary arguments in[48, Sec. 3.3] and [40, Sec. 7.3.3].We note the following key fact: Lemma 6.12.
Objects in sh Λ ≺ ( M × R ) are locally constant over M × R ≤ .Proof. This is obvious over M × R < . Over M ×
0, note that Λ ≺ has no conormals inthe ds directions, so noncharacteristic propagation shows that stalks agree with those over M × R < . (cid:3) We write sh Λ ≺ ;0 ⊂ sh Λ ≺ for the sheaf of full subcategories of sheaves (on M × R , micro-supported at infinity in Λ ≺ ) with vanishing stalks near M × −∞ (and hence over M × R ≤ ). Lemma 6.13.
Let j : M × R > → M × R be the inclusion. Then the natural map of sheavesof categories j ! j ! : sh Λ ≺ ;0 → j ∗ j ∗ sh Λ ≺ ;0 (induced on objects from the usual sheaf functor j ! j ! ) is an equivalence.Proof. There is nothing to prove except along M ×
0. Along this locus the result followsfrom the fact that the natural morphism on sheaves j ! j ! F → F is zero for any sheaf whichvanishes over M × R ≤ . This vanishing holds for the sheaves in question by Lemma 6.12. (cid:3) We now consider the map η : M × R → M and the sheaf of categories on R given by η ∗ sh Λ ≺ ;0 . Proposition 6.14.
The sheaf η ∗ sh Λ ≺ ;0 is zero on ( −∞ , and constant on [0 , c ) , where c is a (universal positive function of the) length of the shortest chord for the flow of Λ under φ t .Proof. Vanishing over ( −∞ ,
0) is obvious. Over (0 , c ) we may apply a contact isotopy tomove Λ ± s in order to trace out Λ; note this becomes impossible exactly at c . Thus localconstancy follows from Cor 5.8. Finally, as we saw in Lemma 6.13, objects in sh Λ ≺ ;0 are all!-extensions of from objects in sh Λ ≺ ;0 ( M × (0 , ∞ )). So constancy at zero follows from fullfaithfullness of the !-extension. (cid:3) Corollary 6.15. If Λ is compact, then for any < s ′ < s < c , the following naturalrestriction maps are all isomorphisms. sh Λ ≺ ;0 ( M × R ≤ s ) ∼ −→ sh Λ ≺ ;0 ( M × R
Proper base change (of sheaves of categories). (cid:3)
HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 37
Let π : S ∗ M → M be the projection, and likewise ˜ π : S ∗ ( M × R ) → M × R . Lemma 6.16.
Assume Λ is in finite position. Then the natural map of sheaves of categorieson M × R sh Λ ≺ ;0 → ˜ π ∗ µ sh Λ ≺ is an equivalence along M × .Proof. It is enough to check at stalks. Evidently if Λ is in finite position, then Λ ≺ is in finiteposition over 0. By Prop 6.9, the stalk of µ sh is the orthogonal complement to constantsheaves in a neighborhood. Along M ×
0, this orthogonal complement is sh Λ ≺ ;0 by Lemma6.13. (cid:3) Theorem 6.17.
Assume that Λ is perturbable to finite position. For s < c , the composition sh Λ ≺ ;0 ( M × R ≤ s ) → sh Λ s ∪ Λ − s ( M ) → µ sh Λ − s (Λ − s ) is an equivalence. In particular, the second map has a right inverse.Proof. By [23] the statement is invariant under contact perturbation (of Λ, before formingΛ ≺ ), so we may assume Λ is in finite position.We have the commutative diagram(14) sh Λ ≺ ;0 ( M × R ≤ s ) sh Λ s ∪ Λ − s ( M ) µ sh Λ − s (Λ − s ) sh Λ ≺ ;0 | M × ( M ) ˜ π ∗ µ sh Λ ≺ | M × ( M ) µ sh Λ ≺ (Λ ≺ ) Cor. 6.15 ∼ Cor. 6.15 ∼ Lemma 6.16 ∼ ∼
The labelled arrows are fully faithful or equivalences by the noted previous results, or inthe case of comparisons between µ sh categories, by Lemma 5.3. The result follows. (cid:3) It follows from Thm 6.17 that the image of the map sh Λ ≺ ;0 ( M × R ≤ s ) → sh Λ s ∪ Λ − s ( M ) isthe subcategory generated by microstalks on Λ − s . We now discuss how to give a more explicitcharacterization (which however we will not require in the sequel). By inspection of the [23]construction, the image is always contained in the locus of sheaves whose microsupport iscontained in the locus U s := R > Λ s ∪ R > Λ − s ∪ [ − s ≤ r ≤ s π (Λ r ) ⊂ T ∗ M Example 6.18.
It is not always the case that the map to sh U s is surjective. Indeed, consider M = R and Λ both conormals to 0. Now µ sh Λ is two copies of the coefficient category, while sh U s is representations of the A quiver. An object not in the image is the skyscraper at 0.The above is essentially the only difficulty, and in its absence we can characterize theimage. Proposition 6.19.
Assume that Λ is in finite position, and c > . Assume in additionthat Λ is the closure of the locus of points where the front projection is an injection, or that M = N × R and Λ ⊂ J N ⊂ S ∗ ( N × R ) . Then the map sh Λ ≺ ;0 ( M × R ≤ s ) → sh U s ( M ) is an isomorphism for small s . Proof.
By Cor. 5.8, there is an equivalence sh Λ s ∪ Λ − s ( M ) ∼ −→ sh Λ ≺ ( M × (0 , s ]) we composewith !-pushforward to get a map to sh Λ ≺ ∪ T ∗ M × ( M × R ≤ s ) ( M × R ≤ s ).In the jet bundle case, the support of the extension to sh Λ ≺ ( M × (0 , s ]) of an objectin sh U s ( M ) will be ‘between’ the front projections of sh Λ t ∪ Λ − t ( M ) for every < t < s .(Indeed, applying the kernel for the Reeb flow will just shrink the front projection in the R direction.) Thus the !-extension to sh Λ ≺ ( M × R ≤ s ) will have no support in M × M ×
0) as desired.In the case when Λ is in general position, note that for s small, the complement of S − s ≤ r ≤ s π (Λ r ) will have the same connected components as the complement of π (Λ). Asthis moreover evidently contains the corresponding locus for all smaller s , we can cutoff thecontact isotopy so it is constant above some locus in the complement of each component of π (Λ). It follows that any object in the image of sh U s ( M ) → sh Λ ≺ ∪ T ∗ M × ( M × R ≤ s ) ( M × R ≤ s ) isalso zero in the complement of π (Λ). Since Λ is in generic position, then on a dense locusits image is a (smooth) hypersurface with ‘only one conormal’ direction. It follows that anyobjects also must vanish along said image, hence be microsupported in the complement of adense set in the Legendrian subvariety Λ, hence be zero. (cid:3) Relative doubling.
Previously we have considered closed Λ ⊂ S ∗ M . We now gener-alize this to the case of locally closed Λ. We write ∂ Λ := Λ \ Λ (as opposed to Λ minus itsrelative interior). The need for a generalization is illustrated by the following example:
Example 6.20.
Recall that the microsupport of a sheaf can never be a manifold withnontrivial boundary.Consider M = R , and Λ ⊂ J R ⊂ S ∗ ( R ) with front projection the open interval(0 , × { } ⊂ R . Then while µ sh Λ (Λ) is the coefficient category, we have sh U s ( R ) = 0 bythe above recollection. Thus the assertion of Theorem 6.17 is false for this Λ (the argumentfails because Lemma 6.16 required Λ to be closed).While we could apply Theorem 6.17 to the closure of Λ, this would not be useful as µ sh Λ (Λ) = 0 (again by the recollection above).The solution is to cap off the boundaries of the doubled object. Definition 6.21.
A contact collar of a subset Λ ⊂ S ∗ M is a contact manifold U with a subset L , and an embedding ( L × (0 , ǫ ); U × T ∗ ( − ǫ, ǫ )) → (Λ , S ∗ M ) carrying L × ∂ Λ.We say a contact flow on S ∗ M is compatible with the collar if its restriction to U × T ∗ ( − ǫ, ǫ )is the product of a flow on U and the constant flow on T ∗ ( − ǫ, ǫ ).For collared Λ, we define (Λ , ∂ Λ) ≺ ⊂ S ∗ ( M × R ) as the union of Λ ≺ with a ‘boundarycapping off’ component contained entirely inside U × T ∗ ( − ǫ, ǫ )) × T ∗ R . This component isconstructed as follows. Note that Λ ≺ in this region is L × [0 , ǫ ) times a parabola P ⊂ T ∗ R .We fix some standard capping of the parabola in T ∗ R to a Legendrian half-paraboloid ˜ P ⊂ T ∗ ( − ǫ, × T ∗ R , so that ˜ P ∪ P ([0 , ǫ ) × P ) ⊂ T ∗ ( − ǫ, ǫ ) × T ∗ R is a manifold. We define(15) (Λ , ∂ Λ) ≺ := Λ ≺ ∪ ( L × ˜ P )Note that by construction Λ ≺ ⊂ (Λ , ∂ Λ) ≺ , and in the fiber over M × It is enough for our purposes that this be a C manifold; thus the construction can be made in thesubanalytic category if desired. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 39
Example 6.22. If M = R and Λ is a Legendrian closed interval whose front projectionis an embedding, then (Λ , ∂ Λ) ≺ is a disk whose front projection is (half of) a somewhatsquashed flying saucer. Remark 6.23.
The reason for the delicate construction of the capping is that we want toverify local constancy of µ sh along the boundary by simply invoking Lemma 5.3. The localconstancy would still be true with a less delicate capping, but one would have to argue forit directly.Let us write (Λ , ∂ Λ) s ⊂ S ∗ M for the contact reduction of (Λ , ∂ Λ) ≺ ⊂ S ∗ ( M × R ) over s ∈ R . Lemma 6.24.
The restriction of (Λ , ∂ Λ) ≺ to S ∗ ( M × R > ) is the legendrian movie of asymplectic isotopy acting on (Λ , ∂ Λ) s . Analogues of the results of the previous section hold for (Λ , ∂ Λ) ≺ . Again writing η : M × R → M , we have Proposition 6.25.
The sheaf η ∗ sh (Λ ,∂ Λ) ≺ ;0 is zero on ( −∞ , and constant on [0 , c ) , where c is the length of the shortest nonzero chord for the flow of Λ under φ t .Proof. Same as Proposition 6.14, with the additional observation that the capping off at theboundary adds no chords. (cid:3)
Corollary 6.26. If Λ is relatively compact, then for any < s ′ < s < c , the followingnatural restriction maps are all isomorphisms. sh (Λ ,∂ Λ) ≺ ;0 ( M × R ≤ s ) ∼ −→ sh (Λ ,∂ Λ) ≺ ;0 ( M × R
Proper base change (of sheaves of categories). (cid:3)
Lemma 6.27.
Assume Λ is collared and in finite position. Then the natural maps of sheavesof categories on M × R sh (Λ ,∂ Λ) ≺ ;0 → ˜ π ∗ µ sh (Λ ,∂ Λ) ≺ → ˜ π ∗ µ sh Λ ≺ are both equivalences along M × .Proof. Note that since Λ is in finite position, so is (Λ , ∂ Λ) ≺ over 0. Now the first map isan isomorphism by the same proof as Lemma 6.16. The second map is an isomorphismbecause (1) over M × ≺ ⊂ (Λ , ∂ Λ) ≺ agree except along the boundary where oneis topologically L × (0 , ǫ ) and the other L × [0 , ǫ ). and (2) µ sh is locally constant along the[0 , ǫ ) direction by Lemma 5.3. (cid:3) Theorem 6.28.
Assume Λ is collared, relatively compact, and perturbable to finite position.For s < c , the composition sh (Λ ,∂ Λ) ≺ ;0 ( M × R ≤ s ) → sh (Λ ,∂ Λ) s ( M ) → µ sh Λ − s (Λ − s ) is an equivalence. In particular, the second map has a right inverse.Proof. Similar to the proof of Theorem 6.17. (cid:3)
Proposition 6.29.
Proposition 6.19 holds for any locally closed, relatively compact, andcollared Λ , with U s := R > (Λ , ∂ Λ) s ∪ R > (Λ , ∂ Λ) − s ∪ [ − s ≤ r ≤ s π ((Λ , ∂ Λ) r ) ⊂ T ∗ M Proof.
Same as Proposition 6.19. (cid:3)
Two sided double.
Our previous constructions involved M × R or M × [0 , ǫ ], andmicrosupport going to infinity or the boundary. While these suffice for our purposes in thisarticle, we give here a variant of the doubling construction which avoids this defect. This canbe technically convenient when invoking theorems stated for compact manifolds; and willbe the version of antimicrolocalization invoked in [21]. This subsection owes its existence todiscussions with the authors of that article.The prototype of the construction is to begin with a Legendrian point in the contactmanifold R , and produce the standard Legendrian unknot, rather than simply half of it asbefore. We just use two copies of our construction above: first begin forming Λ ≺ or (Λ , ∂ Λ) ≺ as before, but near some time s < c , smoothly cutoff the Reeb pushoff so that in the region t ∈ ( s − ǫ, s ] one has (Λ , ∂ Λ) ± t independent of t . Finally, reverse the process so that theLegendrian for t > s is just the reflection of the Legendrian when t < s .We denote the resulting Legendrian as (Λ , ∂ Λ) ≺≻ ⊂ S ∗ ( M × R ). (Despite the notation,the corresponding front will be connected and rounded at the top and bottom.) Note this issupported over a compact subset of R ; fixing an inclusion R ⊂ S we may view (Λ , ∂ Λ) ≺≻ ⊂ S ∗ ( M × S ). As before we write Sh (Λ ,∂ Λ) ≺≻ ( M × S ) for the full subcategory of sheaves withvanishing stalks over S \ R . Theorem 6.30.
The category Sh (Λ ,∂ Λ) ≺≻ ( M × S ) is the (left and/or right) orthogonalcomplement to the category of local systems on M × S . Moreover, the following naturalmorphisms are equivalences: Sh (Λ ,∂ Λ) ≺≻ ( M × S ) ∼ −→ µ sh pre (Λ ,∂ Λ) ≺≻ ((Λ , ∂ Λ) ≺≻ ) ∼ −→ µ sh Λ (Λ) Proof.
Let us first see that any
F ∈ Sh (Λ ,∂ Λ) ≺≻ ( M × S ) is orthogonal to all local systems.By construction F is supported over some interval in S ; thus morphisms between F anda local system will factor through some G which is a local system on M times the constantsheaf on an open or closed interval I . Away from the zero section, the microsupport of G isentirely in the dt direction ( t the coordinate on S ), above ∂I . Shrinking I to an intervalentirely disjoint from the support of F and correspondingly propagating G is a noncharac-teristic homotopy since (Λ , ∂ Λ) ≺≻ is by construction S -noncharacteristic, i.e. disjoint from dt covectors. Thus we see that Sh (Λ ,∂ Λ) ≺≻ ( M × S ) is contained in both the left and rightorthogonal complements to the local systems. For the converse inclusion, consider now any F ∈ Sh (Λ ,∂ Λ) ≺≻ ( M × S ) . Let L F be the local system on M given by restricting F to M × ( S \ R ). Then the same noncharacteristic propagation argument shows that the iden-tity induces morphisms in both directions between F and L F , which are zero only if L F vanishes.Regarding the morphisms in the second assertion of the proposition, the above orthogonal-ity implies that the first is an isomorphism. Meanwhile the composite Sh (Λ ,∂ Λ) ≺≻ ( M × S ) → µ sh Λ (Λ) factors through Sh (Λ ,∂ Λ) ≺≻ ( M × S ) → Sh (Λ ,∂ Λ) ≺ ( M × R ≤ ǫ ) ∼ −→ µ sh Λ (Λ) HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 41 where we learned that the second morphism is an isomorphism in Theorem 6.28. Thusit suffices to show the first morphism is an isomorphism. If (Λ , ∂ Λ) ≺≻ is supported above[0 , T ] ⊂ R ⊂ S , then the same argument as in Lemma 6.13 shows that Sh (Λ ,∂ Λ) ≺≻ ( M × S ) is restricts by an equivalence to [0 , T ), and the same argument as in Corollary 6.15 showsthat this is in turn restricts by an equivalence to [0 , ǫ ). (cid:3) Gapped specialization of microlocal sheaves
We are interested here in limits of contact isotopies.
Definition 7.1.
Let Y be a manifold and Z t a family of subsets of Y defined for t ∈ (0 , Z := lim t → Z t := [ t × Z t ∩ (0 × Y )where the closure and intersection are taken in [0 , × Y . Remark 7.2.
We will be interested in the case when φ t is a contact isotopy, and the Z t are φ t ( Z ). In this case we could also form a similar construction on the contact movie of φ t .The symplectic reduction of the zero fiber of the closure of the movie is contained in, butnot in general equal to, the limit above.Using the results on antimicrolocalization, we now give a microlocal version of the theoremon gapped specialization (Theorem 4.2). Theorem 7.3.
Consider Λ ⊂ S ∗ M , which is either compact or locally closed, relativelycompact, and collared. Let φ t : S ∗ M → S ∗ M be a contact isotopy for t ∈ (0 , , compatiblewith the collar. Let e Λ ⊂ S ∗ M × T ∗ (0 , ⊂ S ∗ ( M × (0 , be the contact movie, and let Λ = lim t → φ t (Λ ) ⊂ S ∗ M . Assume: (1) For some contact form on S ∗ M , the family Λ t (including t = 0 ) is gapped. (2) Both Λ and Λ may be perturbed to project finitely to M . (3) Λ is positively displaceable from legendrians.Then antimicrolocalization and nearby cycles induce a fully faithful functor µsh Λ (Λ ) → µsh Λ (Λ ) Proof.
We will use the gapped hypothesis both in constructing the functor, and in provingfull faithfulness. Let η s be the Reeb flow for the given contact form. Let e η s be the lift of η s to S ∗ M × T ∗ (0 , φ compatible with the collar, Λ is also collared (in a compatible way).We form ( e Λ , ∂ e Λ) ≺ ⊂ S ∗ ( M × (0 , × R ) and (Λ , ∂ Λ ) ≺ ⊂ S ∗ ( M × R ). Note that (Λ , ∂ Λ ) ≺ is the closure at zero of the projection of ( e Λ , ∂ e Λ) ≺ to S ∗ (( M × (0 , × R . We have thediagram:(16) µ sh Λ (Λ ) sh (Λ ,∂ Λ ) ≺ ;0 ( M × R ≤ ǫ ) sh (Λ ,∂ Λ ) ǫ ( M ) µ sh e Λ ( e Λ) sh ( e Λ ,∂ e Λ) ≺ ;0 ( M × (0 , × R ≤ ǫ ) sh ( e Λ ,∂ e Λ) ǫ ( M × (0 , µ sh Λ (Λ ) sh (Λ ,∂ Λ ) ≺ ;0 ( M × R ≤ ǫ ) sh (Λ ,∂ Λ ) ǫ ( M ) ∼∼ ψ ∼ ψ ∼∼ The upward arrows are isomorphism by [23] (the leftmost one alternatively by Lemma 5.3).The top and bottom right horizontal arrows are the restriction at ǫ , and are fully faithfulby Cor. 6.26. The top and bottom left horizontal arrows are equivalences by Theorem 6.28.(If Λ is closed we may use the easier Cor. 6.15 and Theorem 6.17.) Note we are usinggappedness including at Λ in order to apply antimicrolocalization to Λ using the same ǫ as for the family.The downward arrows are induced by nearby cycles, and the image sheaves have the statedmicrosupports by the standard estimate (Lemma 3.16). By Theorem 4.2, the right downwardarrow is fully faithful; it follows formally that the middle is as well. Following around thediagram we find the desired fully faithful functor µ sh e Λ ( e Λ) ֒ → µ sh Λ (Λ ). (cid:3) Remark 7.4.
In the above diagram, we use the top row to avoid quoting theorems for themiddle row due to its noncompactness and also because we do not want to check hypothesesfor ˜Λ. We use the middle column to avoid explicitly characterizing the images in the rightcolumn; alternatively we could use Proposition 6.29. We use the right column because thegappedness is more evident there than in the middle column.
Remark 7.5.
Note that gappedness of Λ t including at t = 0 is equivalent to gappedness ofthe family Λ t for t = 0 plus ǫ -chordlessness of Λ by itself, these all being considered for thesame isotopy. Remark 7.6. (On the isotropicity hypotheses.) The hypotheses (2) and (3) on Λ roughlymean it is isotropic. We will see later that this comes at considerable cost: as a consequencewe will be forced to consider only those Liouville manifolds with isotropic skeleton, andalso only be able to prove invariance under homotopies of such manifolds, as opposed toarbitrary exact symplectomorphism. Since we know on other grounds (e.g. [21]) that theselatter constraints are in fact unnecessary, it is natural to expect that these hypotheses maybe relaxed. Let us at least recall here precisely how the hypotheses arise, and speculate onhow they may be relaxed. This discussion will occupy the remainder of this section, andnothing in the present article depends upon it .Before we begin let us note that in contrast to hypotheses (2) and (3), the gappednesshypothesis (1) does not imply isotropicity (the core of any Liouville manifold would do)and additionally seems to be fundamental on philosophical grounds as discussed in theintroduction.The hypothesis (3) is inherited ultimately from Lemma 4.8 and Lemma 4.13, where it isused to control the limiting behavior of a family using only facts about the limit geometry.However, in our setting we are not fully ignorant of the nature of the family; plausibly thiscould be used in place of the hypothesis on the limit.We will discuss hypothesis (2) in more detail. It comes from the fact that we use an-timicrolocalization in order to define the specialization functor on microlocal sheaves, andantimicrolocalization ultimately relies on Lemma 6.7 which requires a finite front projec-tion. On the one hand, perhaps this lemma may be improved – it derives from the refinedmicrolocal cutoff [29, Proposition 6.1.4], which has no explicit isotropicity hypotheses.On the other hand, it may seem odd that we use antimicrolocalization to define thespecialization functor at all: why not simply argue that nearby cycles respects microsupportshence factors through microlocalization? Let us develop this idea somewhat in order toexpose a subtle difficulty. Let π : E → B be a smooth fiber bundle. The results of Section3.2 make it natural to define a sheaf of categories of relative microsheaves on T ∗ π as follows. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 43
For an open U ⊂ T ∗ π , take N ull ( U ) = {F ∈ sh ( E ) | ss π ( F ) ∩ U = ∅} ⊂ sh ( E )and define µ sh π as the sheafification of the presheaf given by µ sh preπ ( U ) = sh ( E ) /N ull ( U )One virtue of this construction is that for a submanifold A ⊂ B , it follows from the estimatesLemma 3.13 and Remark 3.14. that there is a restriction map µ sh T ∗ π | T ∗ π A → µ sh T ∗ π A . Thesame would not be true if we simply took the usual microsheaves µ sh T ∗ E on T ∗ E andpushed this sheaf of categories to Π ∗ µ sh T ∗ E on T ∗ π . Indeed, while one has the equality ofpresheaves µ sh preπ = Π ∗ µ sh preT ∗ E , the sheafification does not commute with the pushforward:i.e. the natural map µ sh T ∗ π = (Π ∗ µ sh preT ∗ E ) shf → Π ∗ (( µ sh preT ∗ E ) shf ) = Π ∗ µ sh T ∗ E is is not usually an isomorphism, and there is no obvious map between the left and rightterms in Π ∗ µ sh | T ∗ π A ← µ sh T ∗ π | T ∗ π A → µ sh T ∗ π A To understand the interaction of relative microsheaves with nearby cycles, it thereforeremains to consider an open j : B ′ ⊂ B and contemplate inducing a functor on relativemicrosheaves from j ∗ .Let π ′ : X ′ → B ′ be the restriction, and similarly denote the various analogous mapswith ′ . We want to consider the relationship between µ sh T ∗ π ′ and µ sh T ∗ π . We write J ∗ forpushing forward sheaves or presheaves of categories by j . We claim there always exists thefollowing diagram:(17) µ sh preT ∗ π (cid:15) (cid:15) J ∗ µ sh preT ∗ π ′ (cid:15) (cid:15) j ∗ o o µ sh T ∗ π ( J ∗ µ sh preT ∗ π ′ ) shj ∗ o o / / J ∗ µ sh T ∗ π ′ Above, j ∗ on sheaves induces the top line by microsupport estimates. The lower line isinduced because sheafification is a functor, and a map from a presheaf to a sheaf factorsthrough the sheafification. However note that we have not succeeded in constructing afunctor “ j ∗ : J ∗ µ sh T ∗ π ′ → µ sh T ∗ π ”, since the natural map ( J ∗ µ sh preT ∗ π ′ ) sh → J ∗ µ sh T ∗ π ′ neednot be an isomorphism. One difficulty arises by considering the possibility of sheaves withcomponents of the microsupport accumulating near the boundary of B ′ .Plausibly by imposing appropriate hypotheses for the microsupport near ∂B ′ (e.g. fixingsome Λ ′ ⊂ T ∗ π ′ ), one can ensure that ( J ∗ µ sh pre Λ ′ ) sh → J ∗ µ sh Λ ′ is an isomorphism, and thenuse this construction to define the microlocal specialization functor. It would be of interestto determine for what Λ ′ this holds, and more generally to develop the relative microlocalsheaf theory.8. Microlocal sheaves on polarizable contact manifolds
Following [45], we explain how a category of microlocal sheaves can be associated topolarizable contact manifolds or their closed subsets, such as the skeleton of a Weinsteinmanifold. We then apply Theorem 7.3 to show that Lagrangians define objects, and that the category associated to (the skeleton of a) Weinstein manifold is invariant under deformationof the symplectic primitive. Note such deformations cause dramatic changes in the geometryof the skeleton.The polarizability hypothesis is a very strong version of asking for Maslov obstructions tovanish. We will later relax the former hypothesis to the latter.8.1.
The embedding trick.
Following [45], we use high codimension embeddings to defineglobal categories of microlocal sheaves. The existence of the requisite embeddings followsfrom Gromov’s h-principle for contact embeddings, which implies in particular that for anycontact manifold U , there’s a nonempty space of embeddings U ⊂ R N +1 ≫ , which can bemade as connected as desired by increasing N .Such an embedding gives U its stable symplectic normal bundle ν U ; as a stable symplec-tic bundle, it is the negative of the tangent bundle. We (symplectomorphically) identifya a tubular neighborhood of U in its embedding with neighborhood of the zero section inthis normal bundle. By a thickening , we mean the preimage under this identification ofthe total space of a Lagrangian subbundle of ν U . Evidently this is determined by a sec-tion σ of LGr ( ν U ); we term such a section a “stable normal polarization”, and denote thecorresponding thickening by U σ .Note the following facts about stable normal polarizations: Lemma 8.1.
Let ( M, dλ ) be an exact symplectic manifold, and ( M × R , λ + dt ) its contac-tization. The following are equivalent: • A section of
LGr ( T M ⊕ R n ) → M for n ≫ . • A section of
LGr (ker( λ + dt ) ⊕ R n ) → M × R for n ≫ Let ( V, η ) be any co-oriented contact manifold. Then the following are equivalent: • A section of
LGr (ker( η ) ⊕ R n ) → V for n ≫ • A section of the Lagrangian Grassmannian bundle of the stable normal bundle of V Proof.
From M one forms M → BU classifying its stable symplectic tangent bundle, andcomposes with the projection BU → B ( U/O ) to get the map φ : M → B ( U/O ) classifyingthe Lagrangian Grassmannian. The Lagrangian Grassmanian of ker( λ + dt ) over M × R islikewise classified by a map e φ : M × R → B ( U/O ), which moreover satisfies e φ | M × = φ .Now if ( V, λ ) is any contact manifold, and e φ : V → B ( U/O ) classifies
LGr (ker( η ) ⊕ R n ),then the Lagrangian Grassmannian of the stable normal bundle to V is classified by − e φ .As U/O is an infinite loop space, hence a homotopy group, a section of the stable La-grangian Grassmannian is the same as a trivialization of it, hence given by a null-homotopyof, respectively, φ, e φ, − e φ in the above cases. Evidently these are equivalent. (cid:3) In any case, fixing a stable normal polarization, we may define microlocal sheaves on anycontact manifold.
Definition 8.2.
We write µ sh U ; σ := µ sh U σ | U . Remark 8.3.
Note that thickening by a lagrangian in the normal bundle ensures localconstancy along the normal directions (Lemma 5.3).Let us consider on what this invariant may depend. The h -principle implies that differentembeddings are themselves isotopic, so by another application of Lemma 5.3 we see that µ sh U ; σ does not depend on the embedding of U , save perhaps through the dimension 2 N + 1 HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 45 of the target. Similarly, homotopic choices of normal polarization give equivalent categories.(We will see in Section 10 below that in fact the category depends only on the image of thenormal polarization under a certain map.)To show independence of the embedding dimension, it suffices to observe that the categorywe defined is preserved under replacing U by U × T ∗ [0 , T ∗ [0 ,
1] direction, e.g. thezero section.The objects representing objects in µ sh U have a (micro)support in U σ , constant in thenormal bundle directions. We define their corresponding microsupport in U to be the re-striction to U . This newly defined microsupport is evidently co-isotropic (since the originalwas by [29]). Definition 8.4.
For Λ ⊂ U , we write µ sh Λ ,σ for the subsheaf of µ sh U ,σ consisting of fullsubcategories on the objects microsupported in Λ.Obviously µ sh Λ ,σ only depends on the ambient U through its germ along Λ. Remark 8.5.
When U = S ∗ M is a cosphere bundle, there are now two notions of microlocalsheaves on U . The first is the original µ sh S ∗ M . The second is obtained from what we havejust introduced, by observing that the cosphere fibers induce a polarization of the cospherebundle, so in particular a stable normal polarization. It follows from the above discussionof invariance that the original µ sh S ∗ M is canonically isomorphic to the new µ sh S ∗ M, fibers , assheaves of categories on S ∗ M .When, as for S ∗ M , the polarization is understood, or e.g. all polarizations in question areobtained by restricting some (possibly unspecified) fixed polarization on an ambient space,we may omit the polarization from the notation.8.2. Quantization of Legendrians and Lagrangians.
Given a set X equipped withthe germ of a contact embedding X → ˆ U X , by a stabilized embedding of X we mean acodimension zero embedding of ( X × (0 , n , ˆ U X × T ∗ (0 , n ) into some contact manifold. Definition 8.6.
We say X has a property ‘universally’ if every stabilized embedding of X has the property. Theorem 8.7.
Let U be a contact manifold equipped with a stable normal polarization τ .Let Λ ⊂ U a compact Legendrian, φ t : U × (0 , → U a compactly supported contact isotopy,and φ = id U . Let Λ t := φ t (Λ) and let Λ := lim t → Λ t ⊂ X . Assume: • The family Λ t (including at t = 0 ) is gapped (Def. 2.9) for some contact form on U . • Λ and Λ are universally perturbable to finite position (Def. 6.10). • Λ is universally positively displaceable from legendrians (Def: 2.2).Then there is a fully faithful functor ψ : µ sh Λ ,τ (Λ ) → µ sh Λ ,τ (Λ ) More generally, the same holds for Λ relatively compact and collared along its boundary,and φ t compatible with the collar.Proof. We will embed the problem and then appeal to Theorem 7.3. More precisely, fix an(high codimension) embedding
U ⊂ J ( R N ) ⊂ S ∗ ( R N +1 ), and thicken U , X, Λ , etc. in thedirection of the polarization. Extend isotopies from e U to its neighborhood in the embedding by pulling back the contact Hamiltonian by the projection of the tubular neigborhood =normal bundle to e U . Note the resulting flows are compatible with thickening and do notintroduce new chords in flows of thickenings. In particular Λ τt remains gapped and Λ τ is itslimit. We have assumed that Λ is universally pdfl, which by definition means that Λ τ ispdfl. Likewise, Λ τ , Λ τ are perturbable to finite position.Thus we may appeal to Theorem 7.3 to deduce the existence and full faithfulness ofsome map ψ τ : µ sh Λ τ (Λ τ ) → µ sh Λ τ (Λ τ ). As the natural restriction maps µ sh Λ τ (Λ τ ) → µ sh Λ ; τ (Λ ) and µ sh Λ τ (Λ τ ) → µ sh Λ ; τ (Λ ) are equivalences (essentially by definition, andindeed before passing to global sections), we obtain a fully faithful functor ψ : µ sh Λ ; τ (Λ ) → µ sh Λ ; τ (Λ ). (cid:3) Remark 8.8.
For X ⊃ L we may compose with the natural inclusion to obtain a fullyfaithful map ψ : µ sh Λ ,τ (Λ ) → µ sh X,τ ( X ).An exact symplectic manifold ( W, λ ) (or any subset thereof) is canonically embedded inits contactization W × R z , which we take with contact form dz − λ . In particular we maynaturally speak of microsheaves on subsets of exact symplectic manifolds (with stable normalpolarizations). Note also that if v λ is the Liouville flow, then v λ + z ddz is a contact vectorfield lifting v λ ; in case W is Liouville, this contact vector field retracts the contactization tothe core c ( W ). Corollary 8.9.
Let W be Liouville with stable normal polarization τ , and let Λ ⊂ W × R beunversally perturbable to finite position, and assume the projection Λ → W is an embeddingonto a subset L which is either compact or has conic end.Let L be the limit of L under the Liouville flow, and assume L is universally pdfl anduniversally perturbable to finite position. Then there is a fully faithful functor: µ sh Λ ; τ (Λ ) → µ sh L ; τ ( L ) ⊂ µ sh c ( W ) ∪ L ; τ ( c ( W ) ∪ L ) Proof.
Let Λ t be the flow of Λ under v λ + z ddz , parameterized to live over t in (0 , t → Λ t = L . We want to apply Theorem 8.7. Gappedness is automatic: the Λ t have noself-chords at all, as these would project to self-intersections of (the image under Liouvilleflow of) Λ , which we have assumed do not exist. The other hypotheses of the theorem holdby assumption. (cid:3) Remark 8.10.
We expect that similar techniques can be applied in the case when L hasconic end which, rather than being constant, is itself varying in a family which is gappedwith respect to some contact form on the contact end of W . Definition 8.11.
We say a subset of a contact manifold is sufficiently isotropic if it isuniversally pdfl and universally perturbable to finite position. We say a subset of an exactsymplectic manifold is sufficiently isotropic if it is sufficiently isotropic in the contactization.(See defs. 2.2, 6.10, 8.6 for the terminology.)
Remark 8.12.
A sufficient condition ensuring ‘sufficiently isotropic’ is the existence of aWhitney stratification by isotropics. Note also any subset of something sufficiently isotropicis sufficiently isotropic.
Remark 8.13. If W has sufficiently isotropic core and L is a smooth compact exact La-grangian, then the hypotheses of Corollary 8.9 are readily verified. Indeed, L is smoothLegendrian hence universally perturbable to finite position, and L ⊂ c ( W ), the latter beingby assumption sufficiently isotropic. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 47
In the corollary, we did not require that L was smooth. In particular, consider varyingthe symplectic primitive λ to some λ ′ = λ + df . Denote the respective cores by c and c ′ . Ifboth give sufficiently isotropic cores, then using Corollary 8.9 for ( W, λ ), we obtain a fullyfaithful functor µ sh c ′ ,τ ( c ′ ) ֒ → µ sh c ,τ ( c ); likewise using Corollary 8.9 for ( W, λ ′ ) we obtain µ sh c ,τ ( c ) ֒ → µ sh c ′ ,τ ( c ′ ). Definition 8.14.
We say a Liouville form λ on W is sufficiently Weinstein if the resultingcore is sufficiently isotropic. We say λ, λ ′ are sufficiently Weinstein cobordant if there existsa symplectic primitive η for the stabilization W × T ∗ R , such that η restricts to λ + pdq near −∞ and to λ ′ + pdq near + ∞ .More generally, for Λ ⊂ W ∞ , we say λ is sufficiently Weinstein for the pair ( W, Λ) ifthe relative core is sufficiently isotropic; correspondingly we speak of sufficiently Weinsteincobordisms.
Theorem 8.15.
Let λ, λ ′ be two sufficiently Weinstein forms for W which are sufficientlyWeinstein cobordant. Then the functor µ sh c λ ( W ) ( c λ ( W )) → µ sh c λ ′ ( W ) ( c λ ′ ( W )) is an equiva-lence.The same holds in the relative case, i.e. with W replaced by a pair ( W, Λ ⊂ W ∞ ) .Proof. The argument is identical in the absolute and relative cases, we content ourselves todiscuss the former.For notational clarity we write X = c λ ( W ) and X ′ = c λ ′ ( W ). We work inside W × T ∗ R .We use q for the coordinate on R , and p for the cotangent coordinate. On W × T ∗ R , weconsider the three Liouville forms e λ = λ + pdq e λ ′ = λ ′ + pdq η The cores for these Liouville forms are evidently X × T ∗ R R , X ′ × T ∗ R R , and some Y whichprojects to T ∗ R R under the projection W × T ∗ R → T ∗ R and interpolates between X for q ≪ X ′ for q ≫ µ sh X × T ∗ R R ( X × T ∗ R R ) ⇆ µ sh Y ( Y ) ⇆ µ sh X ′ × T ∗ R R ( X ′ × T ∗ R R )Restriction at q = −∞ gives canonical maps µ sh X × T ∗ R R ( X × T ∗ R R ) ∼ −→ µ sh X ( X ) ← µ sh Y ( Y )These maps evidently commute with the µ sh X × T ∗ R R ( X × T ∗ R R ) ⇆ µ sh Y ( Y ), since thosemaps are induced by flows constant near −∞ . It follows formally that the composition µ sh X × T ∗ R R ( X × T ∗ R R ) → µ sh Y ( Y ) → µ sh X × T ∗ R R ( X × T ∗ R R ) is the identity, hence that thesecond map µ sh Y ( Y ) → µ sh X × T ∗ R R ( X × T ∗ R R ) is essentially surjective. As we already knewit was fully faithful, it must be an equivalence; hence so must its left inverse. Likewise welearn that µ sh Y ( Y ) ⇆ µ sh X ′ × T ∗ R R ( X ′ × T ∗ R R ) are equivalences.Finally, consider the commutative diagram Note this is a not a Liouville manifold, but rather an “open Liouville sector” in the sense of [19]. µ sh X × T ∗ R R ( X × T ∗ R R ) / / | + ∞ (cid:15) (cid:15) µ sh Y ( Y ) | + ∞ (cid:15) (cid:15) µ sh X ( X ) / / µ sh X ′ ( X ′ )Here, the vertical maps are the restriction at q = + ∞ . The top horizontal map is thenearby cycles for flowing X × T ∗ R R to Y using the Liouville form η , and the lower horizontalmap is the restriction of this to q = + ∞ . Note this agrees with the nearby cycle for flowing X to X ′ using λ ′ , by construction. The diagram commutes because nearby cycles is a functorof sheaves of categories, hence commutes with restriction. Finally, we have seen that thetop, left, and right arrows are equivalences, hence so is the bottom one. (cid:3) Remark 8.16.
It is not completely obvious that the functor obtained by composing theflowdowns µ sh X × T ∗ R R ( X × T ∗ R R ) → µ sh Y ( Y ) → µ sh X ′ × T ∗ R R ( X ′ × T ∗ R R ) agrees with whatwould be obtained directly by flowing µ sh X × T ∗ R R ( X × T ∗ R R ) → µ sh X ′ × T ∗ R R ( X ′ × T ∗ R R ). Wedid not use this assertion in the above argument, though it does follow from what we haveshown. Remark 8.17.
Intuitively one would expect that if there is a one-parameter family ofsufficiently Weinstein forms λ t interpolating between λ and λ ′ , then one can find a sufficientlyWeinstein cobordism η . We do not know if this is true as stated (or for what appropriatestrengthening of “sufficiently isotropic” it is true). However if one assumes that the λ t areall in fact Weinstein (for possibly generalized Morse function), then it is known that one canform η Weinstein as well [9]. Note in [21, Sec. 6.8] it is explained how to make Weinsteinmanifolds subanalytic; doing this as well will ensure sufficiently isotropic cores.8.3.
Conicity on symplectic hypersurfaces.
As the construction of Cor. 8.9 is builtfrom the Liouville flow, the microsheaves it produces are automatically conic. Let us givesome evidence that in fact all microsheaves on exact symplectic manifolds are automaticallyconic.Recall the following stabilizations: • If (
W, λ ) is exact symplectic, then it canonically embeds as the zero section of itscontactization ( W × R z , dz − λ ). Conversely, any codimension one symplectic hyper-surface in a contact manifold has a local neighborhood with this local model. • A contact manifold with contact form (
V, α ) determines an exact symplectic manifold( V × R t , e t α ).Recall we say a subset of a symplectic manifold Z ⊂ ( W, ω ) is termed co-isotropic if T z Z contains its orthogonal under the symplectic form, i.e. T z Z ⊥ ω ⊂ T z Z , and we say a smoothsubset of a contact manifold Y ⊂ ( V, α ) is co-isotropic if T y Y ⊥ dα ∩ ker α ⊂ T y Y . (This notiondepends on the choice of the contact form α only through the contact distribution ker α .) Lemma 8.18.
Let ( W, λ ) be an exact symplectic manifold, and X ⊂ W a smooth submani-fold. • If τ := T X ⊥ ω ⊂ T W | X , then τ ⊂ T X and λ | τ = 0 . • X is conic coisotropic. • X × ⊂ ( W × R z , dz − λ ) is contact coisotropic. • X × × R t ⊂ ( W × R z × R t , e t ( dz − λ )) is coisotropic. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 49
Proof.
We will show the first property is equivalent to all others.The submanifold X ⊂ W is coisotropic iff τ ⊂ T X . Conicity of X ⊂ W is equivalentto asking that the Liouville vector field v = ( dλ ) − λ is contained in T X along X . Beingcontained in T X is the same as being orthogonal to τ , so a coisotropic is conic iff 0 = dλ ( v, τ ) = λ | τ .That X × W × R z , dz − λ ) is asking that ker( dz − λ ) ∩ ker( dλ | X ) ⊥ ,W × R dλ ⊂ T X . Evidently ker( dλ | X × ) ⊥ ,W × R dλ = τ ⊕ ∂∂z . We are asking when( τ ⊕ ∂∂z ) ∩ ker( dz − λ ) ⊂ T X
As already τ ⊂ T X and no vector with a nonzero component in ∂∂z will be contained in
T X ,this happens if and only if λ | τ = 0.Finally let us see what it means for X × × R t ⊂ ( W × R z × R t , e t ( dz − λ )) to becoisotropic. We must study the orthogonal complement of T X ⊕ R ∂∂t with respect to theform e t ( dtdz − dtλ − dλ ). The orthogonal complement of T X is τ ⊕ R ∂∂z ⊕ R ∂∂t . The orthogonalcomplement of R ∂∂t is the kernel of dz − λ . Thus we are asking when( τ ⊕ R ∂∂z ⊕ R ∂∂t ) ∩ ker( dz − λ ) ⊂ T X ⊕ R ∂∂t This happens iff ker( dz − λ ) ∩ ( τ ⊕ R ∂∂z ) ⊂ τ , which in turn happens iff λ | τ = 0. (cid:3) Remark 8.19.
In particular, note that for X a Lagrangian in W , we have that X is conicLagrangian iff dλ vanishes on X iff X × W × R z , dz − λ ). Remark 8.20.
Recall from [29] that for a closed subset X of a smooth manifold W , thereare two notions of tangent cone, C ( X ) ⊂ C ( X, X ) ⊂ T W | X . When X is smooth, C ( X ) = C ( X, X ) =
T X . If W is symplectic, with form ω , one says X coisotropic (involutive in [29])if C ( X, X ) ⊥ ω ⊂ C ( X )Meanwhile, conicity interacts well with C ( X ): according to [29, Lemma 6.5.3], a vectorfield preserves X if it is contained in C ( X ) along X .However, at least the proof of Lemma 8.18 does not appear to generalize to arbitrarycoisotropic subsets. More precisely, if we substitute τ → C ( X, X ) ⊥ dλ and elsewhere T X → C ( X ), then the first and fourth conditions remain equivalent. However, λ | τ = 0 no longerseems to imply that X is conic. Indeed, λ | τ = 0 ensures that the Liouville vector field isin τ ⊥ = ( C ( X, X ) ⊥ ) ⊥ . But in general ( C ( X, X ) ⊥ ) ⊥ ) C ( X ), and it is in the latter spacewhere we need the Liouville vector field to live. Corollary 8.21.
For a Liouville manifold W with normal polarization τ , and any object F ∈ µ sh W,τ ( W ) , the locus ss ( F ) is conic coisotropic whenever it is smooth.Proof. In an embedding W × R → S ∗ M , a local representative for F has by [29, Theorem6.5.4] conic co-isotropic microsupport in T ∗ M . By Lemma 8.18, since this microsupport iscontained in W × W . (cid:3) Remark 8.22.
One could ask for even a stronger version of conicity, as follows. On thecontactization ( W × R z , dz − λ ) of ( W, λ ), one has the contact vector field v + z ddz whichlifts the Liouville vector field v . By quantization of contact isotopy, one has an isomorphismΦ t : e tv ∗ µ sh ∼ = µ sh . One could ask for objects F which are conic in the sense that there exist isomorphisms F ∼ = Φ t ( F ). Note that the full subcategory spanned by these objects is thesame the Liouville-equivariant microsheaves, since R is contractible.In fact however any object with conic microsupport is also conic in this sense. Indeed, thequantization of the isotopy would produce an object in W × T ∗ R whose microsupport wascontained in (Φ t ( ss ( F )) , t, t direction. Remark 8.23.
When W = T ∗ M is a cotangent bundle, there are now two notions ofmicrolocal sheaves on W . The first is the original µ sh T ∗ M . The second is what we havejust defined, using the fiber polarization. In fact these are the same: µ sh T ∗ M = µ sh T ∗ M, fibers as sheaves of categories. One can see this by considering the image of the natural map sh ( M ) π ∗ −→ sh ( M × (0 , ∞ )) i ∗ −→ sh ( M × R ). In particular all objects of the “new” µ sh areindeed conic.9. The cotangent bundle of the Lagrangian Grassmannian
We will later want to universally trivialize Maslov obstructions by passing to the relativecontangent bundle of the Lagrangian Grassmannian bundle. In this section we discuss therelevant elementary symplectic geometry.9.1.
Symplectic structures on relative cotangent bundles.
Let f : E → M be a fiberbundle of smooth manifolds. We will denote points of E by pairs ( m, q ) where m ∈ M , and q ∈ f − ( m ).Let π : T ∗ E → E denote the cotangent bundle. We will be interested in the relativetangent and cotangent bundles, which are bundles on E characterized by the short exactsequences 0 / / T f ι / / T E f ∗ / / f ∗ T M / / / / f ∗ T ∗ M f ∗ / / T ∗ E ι ∨ / / T ∗ f / / π f : T ∗ f → E for the natural projection, and introduce the composition F = f ◦ π f : T ∗ f / / M We denote points of T ∗ f by triples ( m, q, p ) where ( m, q ) ∈ E , and p ∈ π − f ( m, q ), and identify E with the zero-section { p = 0 } ⊂ T ∗ f .We are interested in having a “fiberwise canonical one-form” on the relative cotangentbundle F : T ∗ f → M . To construct this, we fix a connection on f , which we will formulate asa splitting of the first exact sequence0 / / T f ι / / T E s x x f ∗ / / f ∗ T M t v v / / / / f ∗ T ∗ M f ∗ / / T ∗ E t ∨ u u ι ∨ / / T ∗ f / / s ∨ v v ξ ∈ T ∗ E and v ∈ T E , we have h ξ, v i E = h ξ, ( ιs + t f ∗ )( v ) i E = h ι ∨ ( ξ ) , s ( v ) i f + h t ∨ ( ξ ) , f ∗ v i M HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 51 where brackets with subscripts indicate the canonical pairings between dual bundles.Let θ can ∈ Ω ( T ∗ E ) denote the canonical one-form on the cotangent bundle T ∗ E . Let ω can = dθ can ∈ Ω ( T ∗ E ) denote the canonical symplectic form, and v can ∈ Vect( T ∗ E ) theLiouville vector field characterized by i v can ( ω can ) = θ can . Recall that v can is the Euler vectorfield generating the dilation. Lemma 9.1.
For any m ∈ M , the restriction ( s ∨ ) ∗ θ can | T ∗ E m is the canonical one-form on T ∗ E m .Proof. Consider the natural commutative diagram with compatible sections T ∗ E ι ∨ / / T ∗ f s ∨ u u T ∗ E | E m q (cid:15) (cid:15) (cid:15) (cid:15) ?(cid:31) i O O ι ∨ / / T ∗ f | E m s ∨ t t ?(cid:31) i O O T ∗ E m ssssssssss ssssssssss The canonical one-form θ can,m ∈ Ω ( T ∗ E m ) is characterized by q ∗ θ can,m = i ∗ θ can . Usingthe section s ∨ , it can be calculated as θ can,m = ( s ∨ ) ∗ i ∗ θ can . By commutativity of the diagram,( s ∨ ) ∗ i ∗ θ can = i ∗ ( s ∨ ) ∗ θ can . (cid:3) We denote this “fiberwise canonical one-form” as θ f := ( s ∨ ) ∗ θ can Note that θ can vanishes on all vectors based along the zero-section E ⊂ T ∗ E , and so θ f vanishes on all vectors based along the zero-section E ⊂ T ∗ f . Example 9.2.
When M = pt , we have T ∗ f = T ∗ E is the absolute cotangent bundle, and θ f = θ can is the usual canonical one-form.Next, consider the short exact sequence(18) 0 / / T F I / / T ( T ∗ f ) F ∗ / / F ∗ T M / / T ∗ f , where by definition T F := ker( F ∗ ). Lemma 9.3.
The two-form ω f := dθ f ∈ Ω ( T ∗ f ) is non-degenerate on T F ⊂ T ( T ∗ f ) . Itskernel K := ker( ω f ) ⊂ T ( T ∗ f ) provides a splitting of (18) in the sense that F ∗ restricts to anisomorphism (19) F ∗ | K : K ∼ / / F ∗ T M
Proof.
We have ω f = d (( s ∨ ) ∗ θ can ) = ( s ∨ ) ∗ dθ can = ( s ∨ ) ∗ ω can . We have T F | E m = T ( T ∗ f | E m ) = T ( T ∗ E m ). From Lemma 9.1, the restriction of ( s ∨ ) ∗ ω can to each such fiber is the canonicalsymplectic form, hence non-degenerate. On the other hand, ω f clearly vanishes on f ∗ T M . (cid:3) Lemma 9.4.
Suppose f : E → M is a fiber bundle of smooth manifolds with splitting s ∨ : T ∗ f → T ∗ E and associated one-form θ f := ( s ∨ ) ∗ θ can ∈ Ω ( T ∗ f ) .If ω M ∈ Ω ( M ) is a symplectic form, then ω f := F ∗ ω M + dθ f ∈ Ω ( T ∗ f ) is a symplecticform. If L ⊂ M is Lagrangian, then so is E | L ⊂ T ∗ f . Proof.
We have the splitting T ( T ∗ f ) ≃ F ∗ T M ⊕ K where K = ker( dθ f ). By the non-degeneracy of ω M , we also have F ∗ T M = ker( F ∗ ω M ). Since ω M is non-degenerate on K =ker( dθ f ), and dθ f is non-degenerate on F ∗ T M = ker( F ∗ ω M ), we conclude that ω f is non-degenerate.Finally, if L ⊂ M is Lagrangian, then ω M | L = 0; on the other hand θ f | E = 0, hence ω f | E | L = 0. (cid:3) Lemma 9.5.
Suppose f : E → V is a fiber bundle of smooth manifolds with splitting s ∨ : T ∗ f → T ∗ E and associated one-form θ f := ( s ∨ ) ∗ θ can ∈ Ω ( T ∗ f ) .If λ V ∈ Ω ( V ) is a contact form, then λ f := F ∗ λ V + θ f ∈ Ω ( T ∗ f ) is a contact form. If Λ ⊂ V is Legendrian, then so is E | Λ ⊂ T ∗ f .Proof. We must show dλ f = F ∗ dλ V + dθ f ∈ Ω ( T ∗ f ) is non-degenerate on ξ = ker( λ f ) ⊂ T ( T ∗ f ), or other words, that ker( dλ f ) ∩ ξ = { } . Observe that ker( dλ f ) ⊂ K = ker( dθ f )since ω f = dθ f is non-degenerate on T F ⊂ T ( T ∗ f ). Recall also F ∗ | K : K ∼ → f ∗ T M , henceker( dλ f ) ⊂ K ∩ ker( f ∗ λ M ). But since dλ M is non-degenerate on ker( λ M ), we obtain theassertion. (cid:3) Next, suppose (
M, ω M = dλ M ) a Liouville manifold with v M ∈ Vect( M ) the Liouvillevector field characterized by i v M ( ω M ) = Λ M .On the one hand, let ˜ v M ∈ K = ker( dθ f ) ⊂ T ∗ f denote the lift of v M ∈ T M under theisomorphism(20) F ∗ | K : K ∼ / / F ∗ T M
On the other hand, note that dilations of T ∗ E preserve the image s ∨ ( T ∗ f ) ⊂ T ∗ E of thebundle splitting s ∨ : T ∗ f → T ∗ E . Denote by ˜ v can ∈ Vect( T ∗ f ) the vector field such that( s ∨ ) ∗ ˜ v can = v can | s ∨ ( T ∗ f ) Thus v can lies in the kernel of π f ∗ : T ( T ∗ f ) → T E , hence the kernel of F ∗ = f ∗ ◦ π f ∗ : T ( T ∗ f ) → T M .Finally, recall that T ∗ f equipped with the one-form λ f = F ∗ λ V + θ f ∈ Ω ( T ∗ f ) is an exactsymplectic manifold. Denote by v f ∈ Vect( T ∗ f ) the Liouville vector field characterized by i v f ( ω f ) = λ f . Lemma 9.6.
The Liouville vector field v f ∈ Vect( T ∗ f ) is given by the formula v f = ˜ v M + ˜ v can Proof.
Note the identities i ˜ v M ( F ∗ dλ M ) = F ∗ i F ∗ ˜ v M ( dλ M ) = F ∗ i v M ( dλ M ) = F ∗ λ M i ˜ v M ( dθ f ) = 0with the latter due to the fact that ˜ v M ∈ K = ker( dθ f ).Note the additional identity i ˜ v can ( F ∗ dλ M ) = 0due to the fact that F ∗ ˜ v can = 0.Thus we must show i ˜ v can ( dθ f ) = θ f HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 53
We calculate i ˜ v can ( dθ f ) = i ˜ v can (( s ∨ ) ∗ dθ can ) = ( s ∨ ) ∗ i ( s ∨ ) ∗ ˜ v can ( dθ can ) = ( s ∨ ) ∗ i v can ( dθ can ) = ( s ∨ ) ∗ θ can = θ f and we’re done. (cid:3) Finally, when (
M, ω M = dλ M ) is a Liouville (resp. Weinstein) manifold, we have likewisethat ( T ∗ f , ω f = dλ f ) is a Liouville (resp. Weinstein) manifold. Proposition 9.7.
Let ( M, ω M = dλ M ) be a Liouville (resp. Weinstein) manifold.Let f : E → M be a proper fiber bundle of smooth manifolds, with relative cotangent bundle π f : T ∗ f → E , and total projection F = f ◦ π f : T ∗ f → M .For any splitting s ∨ : T ∗ f → T ∗ E , with associated one-form θ f = ( s ∨ ) ∗ θ can ∈ Ω ( T ∗ f ) , wehave the following: (1) The relative cotangent bundle T ∗ f equipped with the one-form λ f = F ∗ λ M + θ f ∈ Ω ( T ∗ f ) is a Liouville (resp. Weinstein) manifold. (2) The Liouville vector field v f ∈ Vect( T ∗ f ) is given by v f = ˜ v M + ˜ v can , where ˜ v M ∈ ker( dθ f ) is the lift of the Liouville vector field v M ∈ Vect( M ) , and ˜ v M ∈ ker( π f ∗ ) generates dilations. (3) If L ⊂ M is the stable set of ( M, ω M = dλ M ) , then E | L ⊂ T ∗ f is the stable set of ( T ∗ f , ω f = dλ f ) . (4) If L ⊂ M is (exact) Lagrangian, then so is E | L ⊂ T ∗ f . (5) If v M ∈ Vect( M ) satisfies dϕ ( v M ) > over an open U ⊂ M (resp. is gradient-like)for a function ϕ : M → R , then v f ∈ Vect( T ∗ f ) satisfies d Φ( v f ) > over the open F − ( U ) ⊂ T ∗ f (resp. is gradient-like) for the function Φ = F ∗ ϕ + Q : T ∗ f → R , forany positive definite quadratic form Q on the bundle f : T ∗ f → E .Proof. In prior lemmas, we have seen ω f = dλ f is a symplectic form with Liouville vector field v f as asserted in (2), with (exact) Lagrangians satisfying (4). The description of v f in (2)immediately implies (3) since we assume f : E → M is proper, hence may lift any integralcurve of v M to an integral curve of ˜ v M . The description of v f in (2) also immediately implies(5) which in turn implies (1). (cid:3) Example 9.8.
Let (
M, ω M = dλ M ) be a Liouville (resp. Weinstein) manifold with a com-patible almost complex structure J M . Then we can take E to be the unitary frame bundleof M and choose any principal connection. Furthermore, for any compact manifold F witha unitary action, in particular any homogenous space, we can take E to be the associatedbundle. For example, we can take E to be the Lagrangian Grassmannian bundle of M .9.2. Polarizations.Lemma 9.9.
Let ( M, ω ) be a symplectic manifold, f : LGr ( T M ) → M the bundle ofLagrangian Grassmannians of its symplectic tangent bundle, and T ∗ f the total space ofthe relative cotangent bundle. A choice of almost complex structure on M determines asymplectic structure on T ∗ f along with a canonical Lagrangian distribution, i.e. section of LGr ( T T ∗ f ) → T ∗ f .Proof. We saw above that such a complex structure determines a connection on f and thence asymplectic structure on T ∗ f . We also a splitting of bundles on T ∗ f (induced by the connection) T ( T ∗ f ) = T F ⊕ F ∗ T M , where F : T ∗ f → M was the bundle of relative cotangent bundles.Note T F | E m = T T ∗ E m , and that T F contains as a sub-bundle π ∗ f T ∗ f . Recall we describe points of T ∗ f as ( m, q, p ) where m is a point in M , q is a point in theLagrangian Grassmannian of T m M , and p ∈ π − ( m, q ) is a relative cotangent vector. Nowover a point ( m, q, p ), we can take the direct sum of the subspace of F ∗ T M named by q withthe fibre of π ∗ f T ∗ f summand. Evidently this is isotropic for ω f , and half-dimensional, henceLagrangian. (cid:3) Lemma 9.10.
Let V be a contact manifold with contact form λ , let f : LGr ( ker λ ) → M the bundle of Lagrangian Grassmannians of the contact distribution, and T ∗ f the total spaceof the relative cotangent bundle. A choice of almost complex structure on ker( λ ) determinesa contact form λ f on T ∗ f along with a canonical Lagrangian distribution in ker( λ f ) .Proof. Similar to the previous lemma. (cid:3)
Maslov data and descent
A local system of categories over the Lagrangian Grassmannian. If M is anysmooth manifold and τ a stable polarization on T ∗ M , then µ sh M, − τ is a locally constantsheaf of categories, locally isomorphic to the (symmetric monoidal) coefficient category C .Note the restriction maps of µ sh , being built ultimately from the six operations, are C -linear. It follows that the monodromy is C -linear, hence given by tensor product with someinvertible element of C . That is, the sheaf of categories µ sh M, − τ is classified by some map − τ : M → BP ic ( C ). The minus sign is to remind that we define µ sh using a stable normalpolarization, which we can obtain by negating a tangent polarization. Remark 10.1.
Note that stable polarizations form a group, and that
M ap ( M, BP ic ( C ))has an evident group structure. It will follow from our later considerations that the map frompolarizations to this mapping space is a group homomorphism. Indeed, stable polarizationson T ∗ M are classified by M ap ( M, LGr ), and we will now construct a group homomorphism
LGr → BP ic ( C ). Here LGr is the stable Lagrangian Grassmannian.We write
LGr ( n ) for Lagrangian Grassmannian, parameterizing Lagrangian subspacesin a 2n dimensional symplectic vector space. The exact symplectic manifold T ∗ LGr ( n )carries its natural polarization as a cotangent bundle. We may however create another stablepolarization: inside T ∗ LGr ( n ) × R n , we may take the polarization given by the cotangentfiber in the first factor, and by the Lagrangian named by the given point in the second. Wewrite χ for this polarization; there is a corresponding map − χ : LGr ( n ) → BP ic ( C ). On theother hand we can also consider the tautological Lagrangian L n ⊂ T ∗ LGr ( n ) × R n . Usingnow the trivial (cotangent times constant) polarization on T ∗ LGr ( n ) × R n , we consider µ sh L n | LGr ( n ) . This defines another map χ n : LGr ( n ) → BP ic ( C ). (Once we substantiatethe above remark, it follows that these two maps are inverse as their notation indicates.)There is a natural inclusion LGr ( n ) ⊂ LGr ( n + 1), given by direct sum with some fixedLagrangian R ⊂ R . Observe that over a neighborhood of LGr ( n ), the total Lagrangian L n +1 is a trivial thickening (i.e. a product with some R k ⊂ T ∗ R k ) of L n . It follows that µ sh L n +1 | L n ∼ = µ sh L n and therefore that χ n +1 | LGr ( n ) ∼ = χ n . This compatible family determines a map χ : LGr → BP ic ( C )To show this map is a group homomorphism amounts to checking a series of compatibilities;we will describe the first but all the others are checked by the same principle. There is an HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 55 embedding
LGr ( n ) × LGr ( m ) → LGr ( n + m ) induced from direct sum of Lagrangians.We should show there is a natural isomorphism χ n + m | LGr ( n ) × LGr ( m ) ∼ = χ n ⊠ χ m . Again, L n + m | LGr ( n ) × LGr ( m ) is a trivial thickening of L n ×L m , so we have µ sh L n + m | L n ×L m ∼ = µ sh L n ×L m .To conclude we need to know that this latter is µ sh L n ⊠ µ sh L m . But there is always a (fullyfaithful) morphism µ sh X ⊠ µ sh Y → µ sh X × Y , which by Lemma 5.3 is an isomorphism if eitherfactor is a smooth Lagrangian.We summarize this discussion in the following: Theorem 10.2.
Microsheaves on the tautological Lagrangian form a locally constant sheafof categories classified by a group homomorphism χ : LGr → BP ic ( C ) . Forgetting the polarization.
We now relax the requirement of the existence of apolarization σ . Note that polarizations always exist locally, so for any contact U we shouldnaturally obtain a sheaf of categories over LGr ( ν U ), locally constant in the Grassmanniandirection. A clean way to see this is to form the Lagrangian Grassmannian of the contactdistribution, LGr U , and the total space of its relative cotangent bundle, e U . Then per Lemma9.10, e U has a canonical polarization, thus we may consider µ sh LGr U . Similarly for Λ ⊂ U ,we have the full subsheaf of subcategories µ sh LGr Λ Lemma 10.3.
Locally on Λ , fix a choice of isomorphism of the contact distribution witha trivial symplectic bundle, and correspondingly an isomorphism LGr Λ ∼ = Λ × LGr . Then µ sh LGr Λ ∼ = µ sh Λ ⊠ χ , where χ is the local system of categories constructed in the previoussection, and µ sh Λ is defined by a choice of local stable polarization (which locally exists andis unique up to contractible choice).Proof. As noted above, there is always a natural morphism µ sh X ⊠ µ sh Y → µ sh X × Y , whichis an isomorphism when one factor is a manifold inside its cotangent bundle. (cid:3) In order to organize these local isomorphisms, we consider the following composite map.
U → BU → BLGr Bχ −−→ B P ic ( C )Here the map U → BU classifies the symplectic tangent bundle, the map BU → BLGr isinduced from the map sending a symplectic vector bundle to the fiberwise Lagrangian Grass-mannian, and the final map is the delooping of the map χ : LGr → BP ic ( C ) constructedin the previous section (which exists and is unique because we showed that said map is agroup homomorphism). We will write BP ic ( C ) U for the BP ic ( C ) bundle classified by thiscomposite. Lemma 10.4.
The sheaf of categories µ sh LGr U descends to a sheaf of categories µ sh BP ic ( C ) U on BP ic ( C ) U , which is locally constant along the fiber.Proof. The homomorphism χ : LGr → BP ic ( C ) classifies a character P ic ( C )-bundle on LGr ; to avoid confusion, we will denote it here by F χ where we previously wrote χ . (Infor-mally speaking, the character structure means the pullback m ∗ F χ along the multiplication m : LGr × LGr → LGr is equipped with an isomorphism to F χ ⊠ F χ along with highercoherence.) We also have the canonical character P ic ( C )-bundle F can on BP ic ( C ) classifiedby the identity homomorphism BP ic ( C ) → BP ic ( C ), along with a canonical equivalence ofcharacter bundles F χ ≃ χ ∗ F can .We know by Theorem 10.2 and Lemma 10.3 that the sheaf µ sh LGr U is ( LGr, F χ )-equivariant.(Informally speaking, the equivariance means the pullback a ∗ µ sh LGr U along the action a : LGr × LGr U → LGr U is equipped with an isomorphism to F χ ⊠ µ sh LGr U along with highercoherence.) Such equivariance precisely means µ sh LGr U descends to a ( BP ic ( C ) , F can )-equivariant sheaf on the induced P ic ( C )-bundle BP ic ( C ) U . In particular, the descent isimplemented by taking LGr -invariants in the sheaf F can ⊠ µ sh LGr U on BP ic ( C ) × LGr U . (cid:3) Definition 10.5.
We call the composite map
U → B P ic ( C ) the ( C -valued) Maslov obstruc-tion . By
Maslov data we mean a choice of a null-homotopy of this map.A polarization on U is a null-homotopy of U → BU → BLGr , so determines by compo-sition a choice of Maslov data. Moreover, a choice of polarization τ gives a section τ of thebundle LGr U → U . By construction, τ ∗ µ sh LGr U ∼ = µ sh U ,τ .More generally, Definition 10.6.
Given a choice of Maslov data τ , we define µ sh U ; τ := τ ∗ µ sh BP ic ( C U ) , where τ : U →
BP ic ( C ) U is the section classified by the Maslov data.Evidently if τ came from a polarization, µ sh U ; τ defined this way agrees with the previousdefinition. Moreover if two (possibly different) polarizations determine the same Maslovdata, then they determine the same category. Proposition 10.7.
Theorem 8.7, Corollary 8.9 and Theorem 8.15 hold as stated with τ understood as Maslov data rather than as a polarization.Proof. In the proof of those theorems, replace all objects with the Lagrangian Grassmannianbundles over them. (cid:3)
Remark 10.8.
Consider in particular the case of a Weinstein manifold W and a smoothexact Lagrangian L ⊂ W . Fixing Maslov data τ on W , we have by Cor. 8.9 a fullyfaithful functor µ sh L,τ ( L ) → µ sh W,τ ( W ). Evidently this is similar to the statement thata Lagrangian (equipped with appropriate structures) determines an object in the Fukayacategory. To make a more direct comparison, note that in a neighborhood (of the contactlift of) L , there is a canonical polarization ℓ , coming from the structure as a jet bundle.We know that µ sh L,ℓ ( L ) is nothing other than the category of local systems on L . Thus anisomorphism ℓ ∼ = τ of Maslov data determines Loc ( L ) ∼ = µ sh L,ℓ ( L ) ∼ = µ sh L,τ ( L ) → µ sh W,τ ( W )We term such an isomorphism “secondary Maslov data”. (It has elsewhere been called‘brane data’ [27].) Note the obstruction to the existence of this data is the difference ℓ − τ ∈ [ L, BP ic ( C )]. In the case when the coefficient category C is the category of Z -modules, P ic ( C ) = Z ⊕ B ( Z / L, BP ic ( C )] = H ( L, Z ) ⊕ H ( L, Z / W = T ∗ M and τ is the fiber polarization, then the obstruction to the existence of secondaryMaslov data would be the Maslov class of L in the first factor, and w ( L, M ) in the second,though in fact these are known to always vanish. However, the same calculation applies forLegendrians in J ( M ), which can have nontrivial such classes. Remark 10.9.
Work of Jin [24, 26] shows that the map χ : LGr → BP ic ( C ) is in factthe topologists’ J-homomorphism in the universal case when C is the category of spectra.The specialization to the case of C the category of Z modules is essentially in [22]. We do not depend on these results for the abstract setup discussed above, but they are of courseinvaluable for actually constructing or characterizing Maslov obstructions and Maslov datain practice. HEAF QUANTIZATION IN WEINSTEIN SYMPLECTIC MANIFOLDS 57
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Department of Mathematics, UC Berkeley, Evans Hall, Berkeley, CA 94720
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