Arborealization I: Stability of arboreal models
AARBOREALIZATION I: STABILITY OF ARBOREAL MODELS
DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER
Abstract.
We establish a stability result for the normal forms of arboreal singularities. Asour main application of the stability result we give a geometric characterization of arborealsingularities as the closure of the class of smooth germs of Lagrangian submanifolds under theoperation of taking iterated transverse Liouville cones. The parametric version of the stabilityresult implies that the space of germs of symplectomorphisms which preserve an arborealsingularity is weakly homotopy equivalent to the space of automorphisms of the correspondingsigned rooted tree. Hence the local symplectic topology of an arboreal singularity reduces tocombinatorics, even parametrically.
Contents
1. Introduction 12. Arboreal models 43. The stability theorem 17References 361.
Introduction
Main results.
This is the first in a series of papers by the authors on the arboreal-ization program. The goal of this program is to determine the extent to which a Weinsteinmanifold can be deformed so that its skeleton becomes a stratified Lagrangian with arborealsingularities, and to establish the uniqueness properties of such an arboreal skeleton. Beyondthis, the implications of the existence and uniqueness of an arboreal skeleton for the study ofWeinstein manifolds will also be pursued in future work of the authors.Arboreal singularities were first defined by the third author in the paper [N13], by meansof explicit local models. For the purposes of the arborealization program, it is advantageousto have a more geometric characterization of arboreal singularities. This characterizationis tailored to the strategy of proof employed by the authors to arborealize the skeleton ofa Weinstein manifold in [AGEN20b]. The main result of this paper is to establish such acharacterization, which we formulate as follows.
Date : January 26, 2021.DA was partially supported by NSF grant DMS-1638352 and the Simons Foundation.YE was partially supported by NSF grant DMS-1807270.DN was partially supported by NSF grant DMS-1802373. a r X i v : . [ m a t h . S G ] J a n DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER
First, we introduce some auxilliary notions. A closed subset of a symplectic or contactmanifold is called isotropic if it is stratified by isotropic submanifolds. It is called Lagrangianor Legendrian if it is isotropic and purely of the maximal possible dimension. The germ atthe origin of a locally simply-connected isotropic subset L ⊂ T ∗ R n of the cotangent bundlewith its standard Liouville structure λ = pdq admits a unique lift to an isotropic germ atthe origin (cid:98) L ⊂ J R n = T ∗ R n × R of the 1-jet bundle. Given an isotropic subset Λ ⊂ S ∗ R n of the cosphere bundle, its Liouville cone C (Λ) ⊂ T ∗ R , i.e. the closure of its saturation bytrajectories of the Liouville vector field Z = p ∂∂p , is an isotropic subset. Definition 1.1.
Arboreal Lagrangian (resp. Legendrian) singularities form the smallest classArb symp n (resp. Arb cont n ) of germs of closed isotropic subsets in 2 n -dimensional symplectic(resp. (2 n + 1)-dimensional contact) manifolds such that the following properties are satisfied:(i) (Invariance) Arb symp n is invariant with respect to symplectomorphisms and Arb cont n isinvariant with respect to contactomorphisms.(ii) (Base case) Arb symp contains pt = R ⊂ T ∗ R = pt .(iii) (Stabilizations) If L ⊂ ( X, ω ) is in Arb symp n , then the product L × R ⊂ ( X × T ∗ R , ω + dp ∧ dq ) is in Arb symp n +1 .(iv) (Legendrian lifts) If L ⊂ T ∗ R n is in Arb symp n , then its Legendrian lift (cid:98) L ⊂ J R n is inArb cont n .(v) (Liouville cones) Let Λ , . . . , Λ k ⊂ S ∗ R n be a finite disjoint union of arboreal Legen-drian germs from Arb cont n − centered at points z , . . . , z k ∈ S ∗ R n . Let π : S ∗ R n → R n be the front projection. Suppose- π ( z ) = · · · = π ( z k ).- For any i , and smooth submanifold Y ⊂ Λ i , the restriction π | Y : Y → R n is anembedding (or equivalently, an immersion, since we only consider germs).- For any distinct i , . . . , i (cid:96) , and any smooth submanifolds Y i ⊂ Λ i , . . . , Y i (cid:96) ⊂ Λ i (cid:96) ,the restriction π | Y i ∪···∪ Y i(cid:96) : Y i ∪ · · · ∪ Y i (cid:96) → R n is self-transverse.Then the union R n ∪ C (Λ ) ∪ · · · ∪ C (Λ k ) of the Liouville cones with the zero-sectionform an arboreal Lagrangian germ from Arb symp n .With the above classes defined, we can also allow boundary by additionally taking theproduct L × R ≥ ⊂ ( X × T ∗ R , ω + dp ∧ dq ) for any arboreal Lagrangian L ⊂ ( X, ω ), andsimilarly for arboreal Legendrians.In Section 3 we prove the Stability Theorem 3.5 for arboreal models. Rather than give theprecise formulation here, we state our main application: for fixed dimension n the class ofLagrangian (resp. Legendrian) arboreal singularities as defined above contains only finitelymany local models up to ambient symplectomorphism (resp. contactomorphism). More pre-cisely, to each member of the class Arb symp n , one can assign a signed rooted tree T = ( T, ρ, ε )with at most n + 1 vertices. Here T is a finite acyclic graph, ρ is a distinguished root vertex,and ε is a sign function on the edges of T not adjacent to ρ . This discrete data completelydetermines the germ: RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 3
Theorem 1.2.
If two arboreal Lagrangian singularities L ⊂ ( X, ω ) , L (cid:48) ⊂ ( X (cid:48) , ω (cid:48) ) have thesame dimension and signed rooted tree T , then there is (the germ of ) a symplectomorphism ( X, ω ) (cid:39) ( X (cid:48) , ω (cid:48) ) identifying L and L (cid:48) . Similarly, each member of the class Arb cont n is determined by an associated signed rootedtree T = ( T, ρ, ε ) with at most n + 1 vertices.As a representative for each signed rooted tree T one may take the local model L T ⊂ T ∗ R n detailed in Section 2, where n = | n ( T ) | is one less than the number of vertices in the tree. Themodel L T ⊂ T ∗ R n is given as the positive conormal to an explicit front H T ⊂ R n . In Section3 we also establish a parametric version of the Stability Theorem, whose main consequencecan be formulated as follows: Theorem 1.3.
Fix a signed rooted tree T = ( T, ρ, ε ) , set n = | n ( T ) | and consider thearboreal T -front H T ⊂ R n . Let D ( R n , H T ) be the group of germs at of diffeomorphisms of R n preserving H T as a front, i.e. as a subset along with its coorientation.Then the fibers of the natural map D ( R n , H T ) → Aut( T ) are weakly contractible. Hence the local symplectic topology of an arboreal singularity is completely characterizedby the combinatorics of the underlying signed rooted tree, even parametrically.1.2.
Arborealization program.
Arboreal singularities were introduced by the third authorin the paper [N13] as candidates for tractable singularities of skeleta of Weinstein manifolds.On the one hand, it is possible to calculate the symplectic invariants of arboreal singular-ities: this was shown in [N13] from the viewpoint of microlocal sheaves. Parallel results alsohold in Lagrangian intersection theory, for instance in the language of Fukaya categories onecan apply Lefschetz fibration calculations [Se08] to the plumbing characterization of [Sh18].Moreover, arboreal skeleta arise naturally in symplectic geometry and allow for efficientcalculations: for a basic example, the complex pair-of-pants (the complement of n + 2 generalhyperplanes in P n ) has a natural arboreal Lagrangian retract constructed in [FK14] andconfirmed to be a skeleton in [GN20]. As another concrete example, see [C20] for explicitarboreal skeleta (without boundary) associated to the links of plane curve singularities. Fora relationship between arboreal singularities and loose Legendrians see [M19].On the other hand, it was shown in [N15] that singularities of Whitney stratified La-grangians can always be locally deformed to arboreal Lagrangians in a non-characteristicfashion, i.e. without changing their microlocal invariants. But it remained an open ques-tion whether a global theory existed at the level of Weinstein structures. In two dimensionsthe story is classical: generic ribbon graphs provide arboreal skeleta. In four dimensions,Starkston proved arboreal skeleta always exist for the Weinstein homotopy class of any Wein-stein domain [St18]. In the paper [AGEN20b], the authors establish that, in any dimension, aWeinstein manifold X can be arborealized whenever it admits a polarization, which is a globalfield of Lagrangian planes in T X . Moreover, for a polarized Weinstein manifold it is shownin [AGEN20b] that the skeleton can be arranged to have only positive arboreal singularities,
DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER which is the distinguished subclass of arboreal singularities corresponding to signed rootedtrees with all signs equal to +1. Furthermore, the existence of a polarization is a necessaryand sufficient condition for existence of a positive arboreal skeleton. This result, as well asforthcoming work of the authors [AGEN21], relies crucially on the Ridgification Theoremproved by the authors in [AGEN19], as well as on the Stability Theorem 3.5 proved in thisthis paper.1.3.
Acknowledgements.
We are very grateful to Laura Starkston who collaborated withus on the initial stages of this project. The first author is grateful for the great workingenvironment he enjoyed at Princeton University and the Institute for Advanced Study, aswell as for the hospitality of CRM Montreal. The second author thanks RIMS Kyoto andITS ETH Zurich for their hospitality. The third author thanks MSRI for its hospitality.Finally, we are very grateful for the support of the American Institute of Mathematics, whichhosted a workshop on the arborealization program in 2018 from which this project has greatlybenefited. 2.
Arboreal models
Quadratic fronts.
Before we present the local models for arboreal singularities, weintroduce the quadratic fronts out of which the models will be built and discuss some of theirbasic properties.2.1.1.
Basic constructions.
For i ≥
0, define functions h i : R i → R by the inductive formula h = 0 h i = h i ( x , . . . , x i ) = x − h i − ( x , . . . , x i ) For example, for small i , we have h ( x ) = x h ( x , x ) = x − x h ( x , x , x ) = x − ( x − x ) Fix n ≥
0. For i = 0 , . . . , n , define smooth graphical hypersurfaces n Γ i = { x = h i } ⊂ R n +1 equipped with the graphical coorientation, and consider their union n Γ = (cid:83) ni =0 n Γ i Note the elementary identities n Γ i = i Γ i × R n − i i = 0 , . . . , n n Γ i ∩ n Γ = n − Γ i − i = 1 , . . . , n Let T ∗ R n denote the cotangent bundle with canonical 1-form pdx = (cid:80) ni =1 p i dx i where p = ( p , . . . , p n ) are dual coordinates to x = ( x , . . . , x n ). Let J R n = R × T ∗ R n denote the1-jet bundle with contact form dx + pdx = dx + (cid:80) ni =1 p i dx i . RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 5
Figure 2.1.
The hypersurfaces Γ i Figure 2.2.
The hypersurfaces Γ i .Given a function f : R n → R with graph Γ f = { x = f ( x ) } ⊂ R × R n , we have the conormalLagrangian of the graph L Γ f = { x = f ( x ) , p i = − p ∂f /∂x i } ⊂ T ∗ R n +1 , and the conormalLegendrian of the graph Λ Γ f = { x = f ( x ) , p = 1 , p i = − ∂f /∂x i } ⊂ J R n .For i = 0, let n L = R n ⊂ T ∗ R n denote the zero-section. For i = 1 , . . . , n , introduce theconormal Lagrangian n L i = L n − Γ i − ⊂ T ∗ R n of the graph n − Γ i − ⊂ R n , and consider their union n L = (cid:83) ni =0 n L i Similarly, for i = 0 , . . . , n , introduce the conormal Legendrian n Λ i = Λ n Γ i ⊂ J R n of the graph n Γ i ⊂ R n +1 , and consider their union n Λ = (cid:83) ni =0 n Λ i DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER
Note that the Liouville form vanishes on the conical Lagrangian n L i ⊂ T ∗ R n , hence its liftto J R n = R × T ∗ R n with zero primitive is a Legendrian. We have the following compatibility: Lemma 2.1.
The contactomorphism S : J R n (cid:47) (cid:47) J R n S ( x , x, p ) = ( x − p / , x + p / , x , . . . , x n , p , . . . , p n ) takes the Legendrian n Λ i isomorphically to the Legendrian { } × n L i , and thus the union n Λ isomorphically to the union { } × n L .Proof. Set h i, = h i − ( x , . . . , x i ) so that h i = x − h i, . Observe n Λ i ⊂ J R n is given by theequations x = h i pdx = − dh i = − h i dh i = − h i ( dx − h i, dh i, )so in particular p = − h i and (cid:80) ni =2 p i dx i = 4 h i h i, dh i, .If we write (ˆ x , ˆ x, p ) = S ( x , x, p ), for ( x , x, p ) ∈ n Λ i , then we haveˆ x = x − p / ± ( x − h i ) = 0 ˆ x = x + p / x − h i = x − ( x − h i, ) = h i, Now it remains to observe n L i ⊂ T ∗ R n is given by the equations x = h i, (cid:80) ni =2 p i dx i = − p dh i, = − p h i, dh i, This completes the proof. (cid:3)
Distinguished quadrants.
We now specify some distinguished quadrants of the n Γ whichwe will use to define our arboreal models. Which of these quadrants are cut out by our signconventions will become clearer when the arboreal models are introduced.For 0 ≤ j < i ≤ n , set h i,j := h i − j ( x j +1 , . . . , x i )so in particular h i, = h i ( x , . . . , x i ) and h i,i − = h ( x i ) = x i .For fixed 0 ≤ i ≤ n , consider the collection of functions h i, , . . . , h i,i − Note the triangular nature of the linear terms of the collection: for all 0 ≤ j ≤ i −
1, thesubcollection h i,j − x j +1 , h i,j +1 , . . . , h i,i − is independent of x j +1 . Thus the level sets of the collection are mutually transverse.Fix once and for all a list of signs δ = ( δ , δ , . . . , δ n ), δ i ∈ {± } . Define the domainquadrant n Q δi ⊂ R n to be cut out by the inequalities δ h i, ≤ , . . . , δ i h i,i − ≤ RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 7
By the transversality noted above, n Q δi is a submanifold with corners diffeomorphic to R i ≥ × R n − i . Its codimension one boundary faces are given by the vanishing of one of thefunctions h i,j .Note n Q δi only depends on the truncated list δ , . . . , δ i . In particular, it is independent of δ which will enter the constructions next.Define the cooriented hypersurface n Γ i | δ ⊂ R n +1 to be the restricted signed graph n Γ i | δ = { x = δ h i }| n Q δi with the graphical coorientation.Thus n Γ i | δ is cut out by the equations x = δ h i , δ h i, ≤ , . . . , δ i h i,i − ≤ n Γ i | δ is graphical over n Q δi , it is also a submanifold with corners diffeomorphic to R i ≥ × R n − i . Likewise, its codimension one boundary faces are given by the vanishing of oneof the functions h i,j .Consider as well the union n Γ | δ = (cid:83) ni =0 n Γ i | δ Remark . Note that n Γ i = (cid:83) δ,δ =1 n Γ i | δ n Γ = (cid:83) δ,δ =1 n Γ | δ since x ∈ n Γ i implies x ∈ n Γ i | δ where for 1 ≤ j ≤ i , we set δ j = − sgn ( h i,j ( x )), when h i,j ( x ) (cid:54) = 0, and choose it arbitrarily otherwise. Remark . Note if we set δ (cid:48) = ( δ , . . . , δ n − , − δ n ), then the map R n +1 → R n +1 ,( x , . . . , x n − , x n ) (cid:55)→ ( x , . . . , x n − , − x n ), takes n Γ | δ isomorphically to n Γ | δ (cid:48) as a coorientedhypersurface. Thus we could always set δ n = 1 and not miss any new geometry.Note n Γ i ∩ { x < } , hence also n Γ i | δ ∩ { δ x < } , is empty since n Γ i is the graph of h i ≥ Lemma 2.4.
Fix δ = ( δ , . . . , δ n ) , and set δ (cid:48) = ( δ δ , δ , . . . , δ n ) . The homeomorphism s : δ R ≥ × R n (cid:47) (cid:47) δ R ≥ × R n s ( x , x , x , . . . , x n ) = ( x , δ δ ( x + δ √ δ x ) , x , . . . , x n ) gives a cooriented identification s ( n Γ i | δ ∩ { δ x ≥ } ) = δ R ≥ × n − Γ i − | δ (cid:48) < i ≤ n Proof.
Recall n Γ i | δ is defined by x = δ h i δ h i, ≤ , . . . , δ i h i,i − ≤ DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER in particular x = δ h i δ h i, = δ h i ≤ h i, , . . . , h i,i − are independent of the coordinates x , x .When δ x ≥ δ h i ≤
0, the equation x = δ h i is equivalent to √ δ x = − δ h i .Expanding this in terms of the definitions, we can rewrite this in the form x + δ √ δ x = h i − ( x , . . . , x i ) Thus since δ (cid:48) = δ δ , we see s takes n Γ i | δ ∩ { δ x ≥ } into δ R ≥ × { x = δ (cid:48) h i − } .Moreover, the additional functions h i, , . . . , h i,i − cutting out n − Γ i − | δ (cid:48) ⊂ { x = δ (cid:48) h i − } pull back to the same functions h i, , . . . , h i,i − cutting out n Γ i | δ .Finally, the coorientations of n Γ i | δ , n − Γ i − | δ (cid:48) are positive on respectively ∂ x , ∂ x . Ob-serve the ∂ x -component of s ∗ ∂ x is in the direction of ∂ x , and hence s gives a coorientedidentification. (cid:3) Alternative presentation.
For compatibility with inductive arguments, it is useful tointroduce an alternative sign convention and alternative presentation of the local models.Fix signs ε = ( ε , . . . , ε n ). Consider the involution σ ε : R n → R n defined by σ ε ( x , . . . , x n ) = ( ε x , . . . , ε n x n ).Define the domain quadrant n R εi ⊂ R n cut out by the inequalities ε ε h i, ◦ σ ε ≤ , . . . , ε i − ε i h i,i − ◦ σ ε ≤ n Γ εi ⊂ R n +1 to be the restricted signed graph n Γ εi = { x = ε h i ◦ σ ε }| n R εi with the graphical coorientation. Thus n Γ εi is cut out by the equations x = ε h i ◦ σ ε ε ε h i, ◦ σ ε ≤ , . . . , ε i − ε i h i,i − ◦ σ ε ≤ n Γ ε = (cid:83) ni =0 n Γ εi Remark . A simple but important observation: n Γ εi in fact only depends on ε , . . . , ε i − and not ε i . This is because h i,i − = x i and so ε i − ε i h i,i − ◦ σ ε = ε i − x i . In particular, theunion n Γ ε is independent of ε n .We have the following adaption of Lemma 2.4. Lemma 2.6.
Fix ε = ( ε , . . . , ε n ) , and set ε (cid:48) = ( ε , . . . , ε n ) . The homeomorphism s : ε R ≥ × R n (cid:47) (cid:47) ε R ≥ × R n s ( x , x , x , . . . , x n ) = ( x , x + ε √ ε x ) , x , . . . , x n ) RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 9 gives a cooriented identification s ( n Γ εi ∩ { ε x ≥ } ) = ε R ≥ × n − Γ ε (cid:48) i − < i ≤ n Proof.
Recall n Γ εi is defined by x = ε h i ◦ σ ε ε ε h i, ◦ σ ε ≤ , . . . , ε i − ε i h i,i − ◦ σ ε ≤ x = ε h i ◦ σ ε ε ε h i, ◦ σ ε = ε ε h i ◦ σ ε ≤ h i, , . . . , h i,i − are independent of the coordinates x , x .When ε x ≥ ε ε h i ◦ σ ε ≤
0, the equation x = ε h i ◦ σ ε is equivalent to √ ε x = − ε ε h i ◦ σ ε . Expanding this in terms of the definitions, we can rewrite this in the form x + ε √ ε x = ε h i − , ◦ σ ε (cid:48) Thus we see s takes n Γ εi ∩ { ε x ≥ } into ε R ≥ × { x = ε h i − , ◦ σ ε (cid:48) } .Moreover, the additional functions h i, , . . . , h i,i − cutting out n − Γ ε (cid:48) i − ⊂ { x = ε h i − , ◦ σ ε (cid:48) } pull back to the same functions h i, , . . . , h i,i − cutting out n Γ εi .Finally, the coorientations of n Γ εi , n − Γ ε (cid:48) i − are positive on respectively ∂ x , ∂ x . Observe the ∂ x -component of s ∗ ∂ x is in the direction of ∂ x , and hence s gives a cooriented identification. (cid:3) Here is a useful corollary that “explains” the geometric meaning of the signs ε . Corollary 2.7.
Fix ε = ( ε , . . . , ε n ) .For i = 0 , . . . , n − , we have ε i = ± if and only if n Γ i +1 is on the ± -side of n Γ i withrespect to the graphical dx -coorientation.Moreover, for i = 1 , . . . , n − , we have ε i = ± if and only if n Γ i +1 ∩ n Γ is on the ± -sideof n Γ i ∩ n Γ with respect to the graphical dx -coorientation.Proof. For i = 0, the first assertion is immediate from the definitions n Γ = { x = 0 } and n Γ = { x = ε ( ε x ) = ε x , ε ε ( ε x ) = ε x ≤ } .For i >
0, both assertions follow by induction from Lemma 2.6. (cid:3)
Fix signs ε = ( ε , . . . , ε n − ). For i = 0, let n L ε = R n ⊂ T ∗ R n denote the zero-section. For i = 1 , . . . , n , introduce the positive conormal bundles n L εi = T + n − Γ εi − R n ⊂ T ∗ R n determined by the graphical coorientation, and consider their union n L ε = (cid:83) ni =0 n L εi Fix signs ε = ( ε , . . . , ε n ). For i = 0 , . . . , n , introduce the Legendrian n Λ εi ⊂ J R n projecting diffeomorphically to the front n Γ εi ⊂ R n +1 , and consider their union n Λ ε = (cid:83) ni =0 n Λ εi We have the following compatibility of the above Lagrangians and Legendrians analogousto Lemma 2.1.
Lemma 2.8.
Fix signs ε = ( ε , . . . , ε n ) , and set ε (cid:48) = ( ε , . . . , ε n ) . The contactomorphism S ε : J R n (cid:47) (cid:47) J R n S ε ( x , x, p ) = ( x − ε p / , x + ε p / , x , . . . , x n , p , . . . , p n ) takes the Legendrian n Λ εi isomorphically to the Legendrian { } × n L ε (cid:48) i , and thus the union n Λ ε isomorphically to the union { } × n L ε (cid:48) .Proof. The proof is the same as that of Lemma 2.1 with the following observations. Considerthe additional equations ε ε δ h i, ◦ σ δ ≤ , . . . , ε i − ε i h i,i − ◦ σ ε ≤ ε ε h i, ◦ σ ε ≤
0, when p = − ε ε h i, ◦ σ ε , we then have p = − ε ε h i, ◦ σ ε ≥ h i, , . . . , h i,i − are independent of x , x . Thus S ε indeed takes n Λ εi to { } × n L εi . (cid:3) Remark . By the lemma, we see the Legendrian n Λ εi ⊂ J R n is independent of the initialsign ε so only depends on ε (cid:48) = ( ε , . . . , ε n ).It is also useful to record the following relationship of n Γ ε with the extended model n Γ. Lemma 2.10.
Fix signs ε = ( ε , . . . , ε n ) .Given a contactomorphism J R n → J R n restricting to a closed embedding n Λ ε ⊂ ε · n Λ with n Λ εi ⊂ ε · n Λ i , for all i , consider the front Υ = π ( n Λ ε ) ⊂ ε · n Γ .Then either the involution σ ε or its composition with x n (cid:55)→ ± x n takes Υ to n Γ ε .Proof. Note we have n Λ ε = ε · n Λ = n Λ . Consider the intersection Υ (cid:48) = π (( n Λ ε \ n Λ ) ∩ n Λ )as a front inside of π ( n Λ ) = n Γ = { x = 0 } . By induction, either the involution σ ε or itscomposition with x n (cid:55)→ ± x n takes Υ (cid:48) to n − Γ ε (cid:48) where ε (cid:48) = ( ε , . . . , ε n ). So we may assumeΥ (cid:48) = n − Γ ε (cid:48) . Now observe n Γ ε is the unique way to extend n − Γ ε (cid:48) within σ ε ( ε · n Γ) compatiblewith coorientations. (cid:3)
We also have the following observation about signs.
Lemma 2.11.
Let ν be the vertical polarization of T ∗ R n → R n .Then we have ε ( n L ε , ν , n L ε ) = ε .Proof. Recall n L ε is the positive conormal to the graph n − Γ ε = { x = 0 } , and n L ε is thepositive conormal to the graph n − Γ ε = { x = (cid:15) x } . Since ε x is an ε -definite quadraticform in x , the assertion follows. (cid:3) RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 11
Arboreal models.
We now present the local models for arboreal singularities.2.2.1.
Signed rooted trees.
Definition 2.12.
We will use the following terminology throughout:(i) A tree T is a nonempty, finite, connected acyclic graph.(ii) A rooted tree T = ( T, ρ ) is a pair of a tree T and a distinguished vertex ρ called the root .(iii) A signed rooted tree T = ( T, ρ, ε ) is a rooted tree (
T, ρ ) and a decoration ε of a sign ± T not adjacent to the root ρ . Figure 2.3.
A signed rooted tree.Given a signed rooted tree T = ( T, ρ, ε ), we write v ( T ) for the set of vertices, e ( T ) for theset of edges, and n ( T ) = v ( T ) \ ρ for the set of non-root vertices. We regard v ( T ) as a posetwith unique minimum ρ , and in general α ≤ β ∈ v ( T ) when the shortest path connecting β and ρ contains α . We call a non-root vertex β a leaf if exactly one edge of T is adjacent to β ,and write (cid:96) ( T ) ⊂ v ( T ) for the set of leaf vertices. Remark . Throughout what follows, for a finite set S , we write R S for the Euclideanspace of S -tuples of real numbers. One may always fix a bijection S (cid:39) { , , . . . , n } , for some n ≥
0, and hence an isomorphism R S (cid:39) R n , but it will be convenient to avoid choosing suchidentifications when awkward. We will most often consider S = n ( T ) the non-root vertices forsome rooted tree T = ( T, ρ ). Here if one prefers to fix a bijection b : n ( T ) ∼ → { , , . . . , | n ( T ) |} ,we recommend choosing b to be order-preserving: if α ≤ β , then one should ensure b ( α ) ≤ b ( β ).This will allow for a clear translation of our constructions. Definition 2.14.
A signed rooted tree T = ( T, ρ, ε ) is called positive if the decoration ε consists of signs +1.We will associate to any signed rooted tree T = ( T, ρ, ε ), a multi-cooriented hypersurface,conic Lagrangian, and Legendrian H T ⊂ R n ( T ) L T ⊂ T ∗ R n ( T ) Λ T ⊂ J R n ( T )2 DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER where as usual we write n ( T ) = v ( T ) \ ρ for the set of non-root vertices.By definition, the latter two will be determined by the first as follows:(i) L T is the union of the zero-section R n ( T ) and the positive conormal to H T .(ii) Λ T is the Legendrian lift of L T with zero primitive.2.2.2. Type A trees. Let us first consider the distinguished case of A n +1 -trees with extremalroot. Definition 2.15.
For n ≥
0, a linear signed A n +1 -rooted tree is a signed rooted tree A n +1 =( A n +1 , ρ, a ) with vertices v ( A n +1 ) = { , , . . . , n } , edges v ( A n +1 ) = { [ i, i +1] | i = 0 , . . . , n − } ,and root ρ = 0.By definition, the sign a is a length n − a [1 , , . . . , a [ n − ,n ] ). Let us set ε = ( ε , . . . , ε n − ) = ( a [1 , , . . . , a [ n − ,n ] ,
1) to be the length n list of signs where we pad a byadding a single 1 at the end. Definition 2.16.
The models for A n -type arboreal singularities are given as follows:(i) The arboreal A -front is the empty set H A = ∅ inside the point R .For n ≥
1, the arboreal A n +1 -front is the cooriented hypersurface H A n +1 = n − Γ ε ⊂ R n introduced in Section 2.1.3.(ii) For n ≥
0, the arboreal A n +1 -Lagrangian is the union of the zero-section and positiveconormal L A n +1 = R n ∪ T + R n H A n +1 ⊂ T ∗ R n (iii) For n ≥
0, the arboreal A n +1 -Legendrian is the liftΛ A n +1 = { } × L A n +1 ⊂ J R n Figure 2.4.
The two A fronts with positive and negative sign. RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 13
Figure 2.5.
Two A fronts with different choices of signs. The other twofronts can be obtained from these two by reflections. Remark . Following Remark 2.5, the arbitrary choice of the last sign ε n − = 1 does notaffect the arboreal A n +1 -models.Recall the linear signed A n +1 -rooted tree A n +1 = ( A n +1 , ρ, a ) has vertices v ( A n +1 ) = { , , . . . , n } with root ρ = 0, and so the non-root vertices form the set n ( A n +1 ) = { , . . . , n } .In the above definition, we should more invariantly view the ambient Euclidean space R n inthe form R n ( A n +1 ) where the ordering of the coordinates matches that of n ( A n +1 ).With this viewpoint, we rename the smooth pieces of the A n +1 -front, indexing them bynon-root vertices H i = n − P εi − ⊂ H A n +1 i ∈ n ( A n +1 ) = { , . . . , n } Likewise, we rename the smooth pieces of the of the A n +1 -Lagrangian, indexing them byvertices L = R n ⊂ L A n +1 L i = T + R n H i ⊂ L A n +1 i ∈ n ( A n +1 ) = { , . . . , n } and similarly, we rename the smooth pieces of the of the A n +1 -Legendrian, indexing them byvertices Λ i = { } × L A n +1 ,i ⊂ Λ A n +1 i ∈ v ( A n +1 ) = { , , . . . , n } Lemma 2.18.
For n ≥ , and n ∈ v ( A n +1 ) = { , , . . . , n } the unique leaf vertex, and ˚ H n ⊂ H A n +1 the interior of the corresponding smooth piece, we have H A n +1 \ ˚ H n = H A n × R inside of R n ( A n +1 ) = R n ( A n ) × R .Proof. Recall the other smooth pieces H i = n − P εi − , for i = 1 , . . . , n −
1, are independent ofthe last coordinate x n . (cid:3) General trees.
Now we consider a general signed rooted tree T = ( T, ρ, ε ).To each leaf β ∈ (cid:96) ( T ), we associate the linear signed A n +1 -rooted tree A β = ( A β , ρ, a )where A β is the full subtree of T on the vertices v ( A β ) = { α ≤ β ∈ v ( T ) } , and a is therestricted sign decoration.Consider the Euclidean space R n ( T ) . For each β ∈ (cid:96) ( T ), the inclusion n ( A β ) ⊂ n ( T )induces a natural projection π β : R n ( T ) (cid:47) (cid:47) R n ( A β ) Definition 2.19.
Let T = ( T, ρ, ε ) be a signed rooted tree.(i) The arboreal model T -front is the multi-cooriented hypersurface given by the union H T = (cid:83) β ∈ (cid:96) ( T ) π − β ( H A β ) ⊂ R n ( T ) where H A β ⊂ R n ( A β ) is the arboreal A β -front.(ii) The arboreal model T -Lagrangian is the union of the zero-section and positive conor-mal L T = R n ( T ) ∪ T + R n ( T ) H T ⊂ T ∗ R n ( T ) (iii) The arboreal model T -Legendrian is the the liftΛ T = { } × L T ⊂ J R n ( T ) Arboreal models H T , L T and Λ T corresponding to positive T are called positive. Remark . When T = A n +1 , the above definition recovers Definition 2.16 verbatim.Transporting from the case of A n +1 , we may naturally index the smooth pieces of the T -front by non-root vertices H α = π − β ( H A β ,α ) ⊂ H T α ∈ n ( T )where β ∈ (cid:96) ( T ) is any leaf with α ≤ β , and H A β ,α ⊂ H A β is the corresponding smooth piece.Likewise, we may index the smooth pieces of the T -Lagrangian by vertices L ρ = R n ( T ) ⊂ L T RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 15
Figure 2.6.
Two non A n -type fronts with different choices of signs. L α = T + R n ( T ) H α ⊂ L T α ∈ n ( T )and the smooth pieces of the T -Legendrian by verticesΛ α = { } × L α ⊂ Λ T α ∈ v ( T )Let us record a basic compatibility of the above Lagrangians and Legendrians.Fix a signed rooted tree T = ( T, ρ, ε ). Let us first consider the situation when there is asingle vertex ρ (cid:48) ∈ T adjacent to ρ . Let T (cid:48) = T \ ρ be the signed rooted tree with root ρ (cid:48) andrestricted signs.Let α , . . . , α k ∈ T (cid:48) be the vertices adjacent to ρ (cid:48) , and ε , . . . , ε k the signs of T assigned tothe respective edges from ρ (cid:48) to α , . . . , α k .Let L ∞T ⊂ S ∗ R n ( T ) be the ideal Legendrian boundary of L T ⊂ T ∗ R n ( T ) . Note that L ∞T liesin the open subspace J R n ( T (cid:48) ) (cid:39) { p ρ (cid:48) = 1 } ⊂ S ∗ R n ( T ) . Lemma 2.21.
The contactomorphism S : J R n ( T (cid:48) ) (cid:47) (cid:47) J R n ( T (cid:48) ) S ( x ρ (cid:48) , x, p ) = ( x ρ (cid:48) − (cid:80) ki =1 ε i p α i / , ˆ x, p )ˆ x α i = x α i + ε i p / , for i = 1 , . . . , k, ˆ x β = x β elsetakes the Legendrian L ∞T isomorphically to the Legendrian { } × L T (cid:48) .Thus L ∞T itself is a model arboreal Legendrian of type T (cid:48) = T \ ρ .Proof. For each leaf vertex of T , we have a linear signed type A subtree of T given by thevertices running from ρ to the leaf. By Definition 2.19, L T is the union of the correspondinglinear signed type A subcomplexes L A . Each such subcomplex is independent of the coor-dinate x β indexed by vertices β not in the subtree, hence lies in the zero locus of the dual coordinate p β . Thus transport of each L ∞A under the contactomorphism of the lemma reducesto that of Lemma 2.8. (cid:3) More generally, suppose ρ , . . . , ρ (cid:96) are the vertices adjacent to ρ . Observe that T \ ρ isa disjoint union of signed rooted subtrees T j ⊂ T \ ρ , for j = 1 , . . . , (cid:96) , with ρ j as root andrestricted signs. Let T + j = T j ∪ ρ ⊂ T be the signed rooted subtree with ρ readjoined as rootand with restricted signs. Set c j = n ( T ) \ n ( T j ).Let L ∞T ⊂ S ∗ R n ( T ) be the ideal Legendrian boundary of L T ⊂ T ∗ R n ( T ) . We similarly have L ∞T + j ⊂ S ∗ R n ( T + j ) the ideal Legendrian boundary of L T + j ⊂ T ∗ R n ( T + j ) .Since ρ j is the unique vertex adjacent to ρ within T + j , observe that L T + j is connected andin fact lies in J R n ( T j ) = { p ρ j = 1 } ⊂ S ∗ R n ( T + j ) . Moreover, observe that L ∞T is the disjoint union of the connected componentsΛ j = L ∞T + j × R c j ⊂ J R n ( T j ) × T ∗ R c j = { p ρ j = 1 } ⊂ S ∗ R n ( T ) By Lemma 2.21, L ∞T + j ⊂ J R n ( T j ) is a model aboreal Legendrian of type T j , so Λ j = L ∞T + j × R c j ⊂ J R n ( T j ) × T ∗ R c j is a stabilized model arboreal Legendrian of type T j . This proves: Lemma 2.22.
Fix a signed rooted tree T = ( T, ρ, ε ) .Let ρ , . . . , ρ k be the vertices adjacent to ρ . Let T j ⊂ T \ ρ be the signed rooted subtreewith ρ j as root and restricted signs, and T + j = T j ∪ ρ ⊂ T the signed rooted subtree with ρ readjoined as root and with restricted signs. Set c j = n ( T ) \ n ( T j ) .Then the ideal Legendrian boundary L ∞T ⊂ S ∗ R n ( T ) of the model arboreal Lagrangian L T ⊂ T ∗ R n ( T ) of type T is the disjoint union of the Legendrians Λ j = L ∞T + j × R c j ⊂ S ∗ R n ( T ) , which are stabilized model arboreal Legendrians of type T j . By Lemma 2.18, we also have the following.
Corollary 2.23.
For β ∈ (cid:96) ( T ) a leaf vertex, and ˚ H β ⊂ H T the interior of the correspondingsmooth piece, we have H T \ ˚ H β = H T \ β × R β inside of R n ( T ) = R n ( T \ β ) × R β . Extended arboreal models.
It will be useful for us also define extended arboreal models associated with rooted, but not signed trees T = ( T, ρ ).For the unsigned rooted tree A n +1 = ( A n +1 , ρ ) we define H A n +1 := n − Γ ⊂ R n ,L A n +1 := R n ∪ T ∗ R n H A n +1 ⊂ T ∗ R n , RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 17 Λ A n +1 := 0 × L A n +1 ⊂ J R n . Similarly, for a general rooted tree T = ( T, ρ ) we define H T = (cid:83) β ∈ (cid:96) ( T ) π − β ( H A β ) ⊂ R n ( T ) where H A β ⊂ R n ( A β ) is the arboreal A β -front. Furthermore, we define L T = R n ( T ) ∪ T + R n ( T ) H T ⊂ T ∗ R n ( T ) and Λ T = { } × Λ T ⊂ J R n ( T ) Clearly, for any signed version T of the tree T we have H T ⊂ H T , L T ⊂ L T , Λ T ⊂ Λ T . Lemma 2.24.
Given a closed embedding Λ ∞T ⊂ Λ ∞ T with Λ ∞T ,α ⊂ Λ ∞ T ,α , for all α , the front π (Λ ∞T ) ⊂ H T is an embedding of H T .Proof. For each leaf vertex of T , we have a linear signed type A subtree of T given bythe vertices running from ρ to the leaf. By construction, Λ ∞T and Λ ∞ T are the union of thecorresponding type A subcomplexes L ∞A and L ∞ A . Each such subcomplex is independent ofthe coordinates x β indexed by vertices β not in the subtree. Now Lemma 2.10 confirms π ( L ∞A )is the standard embedding of H A after a change of coordinates x α indexed by vertices α inthe subtree. Moreover, the change of coordinates agrees for x α indexed by vertices α in theintersection of such subtrees. By definition, H T is the union of the H A . (cid:3) The stability theorem
In this section we define arboreal Lagrangian and Legendrian subsets and prove their sta-bility under symplectic reduction and Liouville cone operations.3.1.
Arboreal Lagrangians and Legendrians.Definition 3.1.
Arboreal Lagrangians and Legendrians are defined as follows:(a) A closed subset L ⊂ X of a 2 m -dimensional symplectic manifold ( X, ω ) is called an arboreal Lagrangian if the germ of (
X, L ) at any point λ ∈ L is symplectomorphic tothe germ of the pair ( T ∗ R n × T ∗ R m − n , L T × R m − n ) at the origin, for a signed rootedtree T with n := n ( T ) ≤ m .(b) A closed subset Λ ⊂ Y of a (2 m + 1)-dimensional contact manifold ( Y, ξ ) is calledam arboreal Legendrian if the germ of ( Y, Λ) at any point λ ∈ Λ is contactomorphicto the germ of ( J ( R n × R m − n ) = J R n × T ∗ R m − n , Λ T × R m − n ) at the origin, for asigned rooted tree T with n := n ( T ) ≤ m .(c) A closed subset H ⊂ M of an ( m + 1)-dimensional manifold M is called an arborealfront if the germ of ( M, H ) at any point m ∈ M is diffeomorphic to the germ of( R n +1 × R m − n , H T × R m − n ) at the origin, for a signed rooted tree T with n := n ( T ) ≤ m . The pair ( T , m ) is called the arboreal type of the germ of L , Λ, or H at the given point.We say L , Λ, or H is positive if it is locally modeled on positive arboreal models at all points. Remark . Later we will also allow arboreal Lagrangians to have boundary and even corners,but throughout the present discussion we restrict to the above definition for simplicity.Given an arboreal Lagrangian we call sup λ ∈ L { n ( T ( λ )) } the maximal order of L , where T ( λ ) is a the signed rooted tree describing the germ of L at the point λ . Similarly, we definethe maximal order of arboreal Legendrians and fronts.Every arboreal Lagrangian or Legendrian is naturally stratified by isotropic strata indexedby the corresponding tree type. A Lagrangian distribution η in X is called transverse toan arboreal Lagrangian L if it is transverse to all top-dimensional strata of L . Similarly aLegendrian distribution η ⊂ ξ in a contact ( Y, ξ ) is called transverse to an arboreal LegendrianΛ if it has trivial intersection with tangent planes to all top-dimensional strata of Λ.
Definition 3.3. A polarization of L or Λ is a transverse Lagrangian distribution. Remark . We emphasize the transversality to an arboreal Lagrangian means transversalityto its closed smooth pieces, and not just to open strata.Before we continue we introduce some auxiliary notions. Let V be a symplectic vectorspace and (cid:96) , (cid:96) , (cid:96) ⊂ V linear Lagrangian subspaces which are pairwise transverse. We write (cid:96) ≺ (cid:96) ≺ (cid:96) if (cid:96) corresponds to a positive definite quadratic form with respect to thepolarization ( (cid:96) , (cid:96) ) of V . Let C ⊂ V be a coisotropic subspace. For any linear Lagrangiansubspace (cid:96) ⊂ V we denote by [ L ] C the symplectic reduction of (cid:96) with respect to C .Let L be an arboreal Lagrangian whose germ at a point λ ∈ L has the type ( T =( T, ρ, ε ) , m ). Let L ρ ⊂ T λ X the tangent plane to the root Lagrangian corresponding tothe root ρ . For each vertex α connected by an edge with ρ let L α ⊂ T λ X denote the La-grangian plane tangent to the Lagrangian corresponding to the vertex a . We recall that L ρ and L α cleanly intersect along a codimension 1 subspace. Consider a coistropic subspace C α := Span( L ρ , L α ) ⊂ T λ X . Let η be a Lagrangian distribution in X transverse to L . Definethe sign(1) ε ( η, L, α ) = +1 , if [ L ρ ] C α ≺ [ L α ] C α ≺ [ η ] C α ; − , if [ L ρ ] C α ≺ [ η ] C α ≺ [ L α ] C α . Similarly, if Λ is an arboreal Legendrian in a contact manifold (
Y, ξ ), and η a Legendriandistribution transverse to Λ, then for any point λ ∈ Λ of type T = ( T, ρ, ε ) we assign a sign ε ( η, Λ , α ) for every vertex α adjacent to the root ρ as equal to ± ≺ -orderof the triple [ L ρ ] C α , [ L α ] C α , [ η ] C α in [ ξ λ ] C α .3.2. Stability of arboreal Lagrangians and Legendrians.
The following is the mainresult of this paper.
RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 19
Figure 3.1.
The notion of sign for the A singularity. Theorem 3.5.
Let T be a signed rooted tree. Let ρ , . . . , ρ k be vertices adjacent to the root ρ and T j be subtrees with roots ρ j (where we removed the decoration of edges [ ρ j α ] ). Let φ j : T ∗ R m → J R m , m ≥ n = n ( T ) , be germs of Weinstein hypersurface embeddings withdisjoint images. Denote z j := φ j (0) , Λ j = φ j ( L T j × R m − n ( T j ) ) , j = 1 , . . . , k . Suppose that (i) π ( z j ) = 0 ; (ii) the arboreal Legendrian Λ := (cid:83) kj =1 Λ j projects transversely under the front projection J R n → R × R n ; (iii) for each edge [ ρ j α ] we have ε ( ν, Λ j , α ) = ε [ ρ j α ] .Then R m ∪ C (Λ) is an arboreal Lagrangian of type ( T , m ) or equivalently, the germ of thefront π (Λ) is diffeomorphic to H T × R m − n ( T ) . From Theorem 3.5 it follows by induction that the class of arboreal singularities as ax-iomatically defined in Section 1 coincides with the class of arboreal singularities defined interms of local models, which is equivalent to the statement of Theorem 1.2. Theorem 3.5 is acorollary of its unsigned version which is the content of the following proposition.
Proposition 3.6.
Let T be a rooted tree. Let ρ , . . . , ρ k be vertices adjacent to the root ρ and let T j be subtrees with roots ρ j . Let φ j : T ∗ R m → J R m , m ≥ n = n ( T ) , begerms of Weinstein hypersurface embeddings. Denote z j := φ j (0) , (cid:98) Λ j = φ j ( (cid:98) L T j × R m − n ( T j ) , j = 1 , . . . , k . Suppose that (i) π ( z j ) = 0 ; (ii) the extended arboreal Legendrian Λ := (cid:83) kj =1 Λ j projects transversely under the frontprojection J R n → R × R n ;Then R m ∪ C (Λ) is an extended arboreal Lagrangian of type ( T , m ) , or equivalently, the germof the front π (Λ) is diffeomorphic to H T × R m − n ( T ) . Proof of Theorem 3.5 using Proposition 3.6.
Consider the arboreal Legendrian as a closedsubcomplex of the extended model. Apply Proposition 3.6 to assume the extended frontis in canonical form. Then Lemma 2.24 implies the front of the original arboreal Legendrianis a canonical model. (cid:3)
Proposition 3.6 will be proven below in this section (see Section 3.6 ) below, but first wediscuss some corollaries of Theorem 3.5.
Corollary 3.7.
Let Λ ⊂ ∂ ∞ T ∗ M be an arboreal Legendrian. Suppose that the front projection π : Λ → M is a transverse immersion. Then L := C (Λ) ∪ M is an arboreal Lagrangian. Figure 3.2.
The zero section union the Liouville cone on a Legendrian whoseprojection to the zero section is an immersion is arboreal.
Proof.
The intersection H := M ∩ C (Λ) is the front of the Legendrian Λ. Each point a ∈ H has finitely many pre-images z , . . . , z k ∈ Λ. The germs Λ j of Λ at z j by our assumptionare images of arboreal Lagrangian models under Weinstein embeddings of their symplecticneighborhoods. Hence, by Theorem 3.5 the germ of L at z is of arboreal type. (cid:3) It is not a priori clear that even the standard Lagrangian (resp. Legendrian) arborealmodels are arboreal Lagrangians (resp. Legendrians). However, the following corollary showsthat they are.
Corollary 3.8.
Consider a model Lagrangian L T ⊂ T ∗ R n , n = n ( T ) . Then for any point λ ∈ L T the germ of L T at λ is a ( T (cid:48) , n ) -Lagrangian for a signed rooted tree T (cid:48) .Proof. We argue by induction in n . The base of the induction is trivial. Assuming the claimfor n − L T can be presented as L ρ ∪ C (Λ), where L ρ is the smooth piececorresponding to the root ρ of T and Λ is a union of model Legendrians of dimension n − ∂ ∞ T ∗ ( R n ). By the induction hypothesis Λ is an arboreal Legendrian, and hence applyingCorollary 3.7 we conclude that L T is an arboreal Lagrangian. (cid:3) RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 21
Remark . We will not need it in what follows, so only briefly comment here that it is possibleto specify precisely the type ( T (cid:48) , n ) of the germ of L T at each point λ ∈ L T . Following [N13]the underlying tree T (cid:48) is a canonically defined subquotient of T , in other words, a diagram T (cid:48) ← S → T , where S → T is a full subtree, and S → T (cid:48) contracts some edges; conversely,any such subquotient can occur. Furthermore, if we partially order T with the root ρ ∈ T asminimum, then the root ρ (cid:48) ∈ T (cid:48) is the unique minimum of the natural induced partial orderon T (cid:48) . Finally, to equip T (cid:48) with signs, we restrict the signs of T to the subtree S , then pushthem forward to T (cid:48) using that each edge of T (cid:48) is the image of a unique edge of S . Corollary 3.10.
Let L T ⊂ T ∗ R n be a model Lagrangian associated with a signed rooted tree ( T, ρ, ε ) . Let η , η be two polarizations transverse to L T . Suppose that for any vertex α of T adjacent to ρ we have ε ( η , L, α ) = ε ( η , L, α ) . Then there is a (germ at the origin of ) a symplectomorphism ψ : T ∗ R n → T ∗ R n such that ψ ( L ) = L and dψ ( η ) = η along L .Proof. There exists embeddings h , h : T ∗ R n → J R n as Weinstein hypersurfaces, such that h j ( η j ) = ν , j = 0 ,
1, where ν is the canonical Legendrian foliation of J R n by fibers of thefront projection to R n × R . Consider the arboreal Lagrangians L j := C ( h j ( L T )) ∪ ( R n × R ), j = 0 ,
1, and note that their arboreal types are described by the same signed rooted tree T obtained from T by adding a new root, connecting it by an edge to the old one, and assigningto edges [ ρα ] of T ⊂ T adjacent to the old root ρ the sign ε ( η , L, α ) = ε ( η , L, α ). ApplyingTheorem 3.5 we find the required symplectomorphism ψ . (cid:3) Corollary 3.11.
Let H ⊂ M be an arboreal front. Then for any submanifold Σ ⊂ M transverse to (all strata of ) H the intersection Σ ∩ H is an arboreal front in Σ .Proof. We can assume that H is an arboreal front germ at a point x ∈ H , and hence thegerm of ( M, H ) at x is diffeomorphic to the germ of ( R n ( T )+1 × R k , H T × R k ) for some rootedsigned arboreal tree T and k = n − n ( T ). Note that the transversality of Σ to H impliesthat codimΣ ≤ k and that the projection of p : Σ ⊂ R n ( T )+1 × R k → R n ( T )+1 to the firstfactor is a submersion, and because we are dealing with germs, it is a trivial fibration. Onthe other hand, the projection p | Σ ∩ H : Σ ∩ H → H T is the restriction of this fibration to H T ⊂ R N ( T ) . (cid:3) Parametric version.
The following is the parametric version of Theorem 3.5.
Theorem 3.12.
Let T be a signed rooted tree. Let ρ , . . . , ρ k be vertices adjacent to theroot ρ and T j be subtrees with roots ρ j (where we removed the decoration of edges [ ρ j α ] ).Let φ yj : T ∗ R m → J R m , m ≥ n = n ( T ) , be families of germs of Weinstein hypersurfaceembeddings with disjoint images, parametrized by a manifold Y . Denote z yj := φ yj (0) , Λ jy = φ yj ( L T j × R m − n ( T j ) ) , j = 1 , . . . , k . Suppose that Figure 3.3.
Illustration that Σ ∩ H is an arboreal front in Σ.(i) π ( z yj ) = 0 ; (ii) the arboreal Legendrian Λ y := (cid:83) kj =1 Λ jy projects transversely under the front projection J R n → R × R n ; (iii) for each edge [ ρ j α ] we have ε ( ν, Λ jy , α ) = ε [ ρ j α ] .Then there exists a family of diffeomorphisms φ y between H T × R m − n ( T ) and the front π (Λ y ) .If K ⊂ Y is a closed subset and the φ yj are the standard embeddings of the local model for y ∈ O p ( K ) , then we may further assume φ y = Id for y ∈ O p ( K ) . The parametric version of Proposition 3.6 is formulated similarly. As a consequence ofTheorem 3.12 we get the following result, which was formulated as Theorem 1.3 in Section 1.
Corollary 3.13.
Fix a signed rooted tree T = ( T, ρ, ε ) , set n = | n ( T ) | and consider thearboreal T -front H T ⊂ R n . Let D ( R n , H T ) be the group of germs at of diffeomorphisms of R n preserving H T as a front, i.e. as a subset along with its coorientation.Then the fibers of the natural map D ( R n , H T ) → Aut( T ) are weakly contractible.Proof. We deduce Corollary 3.13 from Theorem 3.12. We will argue for T = A n +1 when H A n +1 = n − Γ; the case of general T is similar.Since Aut( A n +1 ) is trivial, we seek to show D ( R n , n − Γ) is weakly contractible. Note any ϕ ∈ D ( R n , n − Γ) preserves 0, and moreover, preserves the canonical flag in T R n given by thetangents to the intersections (cid:84) i
RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 23
Corollary 3.14.
Let L T ⊂ T ∗ R n be a model Lagrangian associated with a signed rootedtree ( T, ρ, ε ) . Let η y , η y be two families of polarizations transverse to L T parametrized by amanifold Y . Suppose that for any vertex α of T adjacent to ρ we have ε ( η y , L, α ) = ε ( η y , L, α ) . Then there is a family of (germ at the origin of ) symplectomorphisms ψ y : T ∗ R n → T ∗ R n such that ψ y ( L ) = L and dψ y ( η y ) = η y along L . Moreover, if η y = η y for y ∈ O p ( K ) for K ⊂ Y a closed subset, then we can take ψ y = Id for y ∈ O p ( K ) . The proof is just like in the non-parametric case, but applying Theorem 3.12 instead ofTheorem 3.5.3.4.
Tangency loci.
Before proving Proposition 3.6 and its parametric analogue we need toanalyze more closely the geometry of hypersurfaces forming arboreal fronts.
Definition 3.15.
Given smooth hypersurfaces X , X ⊂ R n +1 , we denote by T ( X , X ) ⊂ R n +1 their tangency locus , i.e. the subset of points x ∈ X ∩ X such that T x X = T x X . Remark . Given smooth Legendrians L , L ⊂ J R n whose fronts X = π ( L ) , X = π ( L ) ⊂ R n +1 are smooth hypersurfaces, note that T ( X , X ) = π ( L ∩ L ).For 0 ≤ j < i ≤ n , recall the nontation h i,j := h i − j ( x j +1 , . . . , x i )so in particular h i, = h i ( x , . . . , x i ) and h i,i − = h ( x i ) = x i . Set T i,j = { h i,j = 0 } ⊂ R n +1 Note h i,j is independent of x , . . . , x j , and we have T i,j = R j +1 × n − j − Γ i − j − Lemma 3.17.
For ≤ j < i ≤ n , the tangency locus T ( n Γ i , n Γ j ) ⊂ R n +1 is the intersectionof either n Γ i or n Γ j with the union { h i,j = 0 } ∪ j − (cid:83) k =0 { h i,k = h j,k = 0 } = T i,j ∪ j − (cid:83) k =0 ( T i,k ∩ T j,k ) Proof.
Since n Γ i , n Γ j are the graphs of h i , h j , the projection of T ( n Γ i , n Γ j ) to the domain R n is cut out by h i = h j dh i = dh j Note h i = h i, = x − h i, , h j = h j, = x − h j, . By examining the dx -component of dh i = dh j , we see it implies h i = h j . Thus the projection of T ( n Γ i , n Γ j ) is cut out by thesingle equation dh i = dh j which in turn implies h i = h j .To satisfy dh i = dh j , so in particular h i = h j , there are two possibilities: (i) h i = h j = 0; or(ii) h i = h j (cid:54) = 0. In case (i), we find the subset { h i, = h j, = 0 } appearing in the union of the assertion of the lemma. In case (ii), we observe dh i = dh j is then equivalent to dh i, = dh j, which in turn implies h i, = h j, .Now we repeat the argument. To satisfy dh i, = dh j, , so in particular h i, = h j, , thereare two possibilities: (i) h i, = h j, = 0; or (ii) h i, = h j, (cid:54) = 0. In case (i), we find the subset { h i, = h j, = 0 } appearing in the union of the assertion of the lemma. In case (ii), we observe dh i, = dh j, is then equivalent to dh i, = dh j, which in turn implies h i, = h j, .Iterating this argument, we obtain the subset (cid:83) j − k =0 { h i,k = h j,k = 0 } , and arrive at thefinal equation dh i,j = 0 . By examining the dx j +1 -term, we see dh i,j = 0 holds if and only if h i,j = 0, which gives the remaining subset of the assertion of the lemma. (cid:3) Remark . The only evident redundancy in the description of the lemma is T i,j − ∩ T j,j − ⊂ T i,j since h i,j − = x j − h i,j , h j,j − = x j , so their vanishing implies the vanishing of h i,j .We will be particularly interested in the locus T i,j ⊂ T ( n Γ i , n Γ j ) and formalize its structurein the following definition. Definition 3.19.
Given smooth hypersurfaces X , X ⊂ R n +1 , we denote by τ ◦ ( X , X ) ⊂ T ( X , X ) the subset of points x ∈ X ∩ X where in some local coordinates we have X = { x = 0 } , X = { x = x } . We write τ ( X , X ) ⊂ T ( X , X ) for the closure of τ ◦ ( X , X ),and refer to it as the primary tangency of X , X . Remark . Given smooth Legendrians L , L ⊂ J R n whose fronts X = π ( L ) , X = π ( L ) ⊂ R n +1 are smooth hypersurfaces, note that τ ◦ ( X , X ) is the front projection ofwhere L , L intersect cleanly in codimension one.We have the following consequence of Lemma 3.17. Corollary 3.21.
For ≤ j < i ≤ n , the primary tangency τ ( n Γ i , n Γ j ) ⊂ R n +1 is theintersection of either n Γ i or n Γ j with T i,j . Before continuing, let us record the following for future use.
Lemma 3.22.
Fix ≤ k < j ≤ n − .We have τ ( τ (Γ n , n Γ k ) , τ ( n Γ j , n Γ k )) = τ ( n Γ n , n Γ j ) ∩ τ ( n Γ j , n Γ k ) where the primary tangency of τ (Γ n , n Γ k ) , τ ( n Γ j , n Γ k ) of the left hand side is calculated in n Γ k (cid:39) R n .Proof. By the preceding corollary, the left hand side is the intersection n Γ k ∩ τ ( T n,k , T j,k ).Note n Γ k ∩ T j,k = τ ( n Γ j , n Γ k ) = n Γ j ∩ T j,k . Hence n Γ k ∩ τ ( T n,k , T j,k ) = n Γ j ∩ τ ( T n,k , T j,k )since y ∈ n Γ k ∩ τ ( T n,k , T j,k ) ⇐⇒ y ∈ n Γ k ∩ T j,k , y ∈ τ ( T n,k , T j,k ) ⇐⇒ y ∈ n Γ j ∩ T j,k , y ∈ τ ( T n,k , T j,k ) ⇐⇒ y ∈ n Γ j ∩ τ ( T n,k , T j,k ). RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 25
Figure 3.4.
Verification of the conclusion of Lemma 3.22 for n =2, in this case both the right and left hand sides of the equality τ ( τ ( Γ , Γ ) , τ ( Γ , Γ )) = τ ( Γ , Γ ) ∩ τ ( Γ , Γ ) consist of the origin.Next, recall T n,k = R k +1 × n − k − Γ n − k − T j,k = R k +1 × n − k − Γ j − k − Hence by the preceding corollary, we have τ ( T n,k , T j,k ) = T j,k ∩ { h n,j = 0 } Thus the left hand side is given by n Γ j ∩ T j,k ∩ T n,j .On the other hand, by the preceding corollary, the right hand side is also given by n Γ j ∩ T n,j ∩ T j,k . (cid:3) More on distinguished quadrants.
Corollary 3.23.
For ≤ j < i ≤ n , we have n Γ εi ∩ n Γ εj = T ( n Γ εi , n Γ εj ) = τ ( n Γ εi , n Γ εj ) and they coincide with the closed boundary face of n Γ εi cut out by h i,j = 0 . Proof.
For j = 0, we have n Γ ε = n Γ = { x = 0 } . From the definitions, we have n Γ εi ∩ n Γ = T ( n Γ εi , n Γ ) = τ ( n Γ εi , n Γ )which is cut out of n P εi by h i, = h i = 0 . For j >
0, the assertions follow from Lemma 2.4 by induction on n . (cid:3) Remark . Note for any 0 ≤ j < i ≤ n , we have τ ( n Γ i , n Γ j ) = (cid:83) ε τ ( n Γ εi , n Γ εj ) To see this, consider x ∈ τ ( n Γ i , n Γ j ), so that h i,j ( x ) = 0 by Corollary 3.21. Choose ε so that x ∈ n Γ εi . Then by Corollary 3.23, we have x ∈ τ ( n Γ εi , n Γ εj ).For i = 0, let n L ε = R n ⊂ T ∗ R n denote the zero-section. For i = 1 , . . . , n , consider theconormal bundles n L εi = T ∗ n − Γ εi − R n ⊂ T ∗ R n and their union n L ε = (cid:83) ni =0 n L εi Similarly, for i = 0 , . . . , n , consider the smooth Legendrian n Λ εi ⊂ J R n that maps diffeomorphically to n Γ εi ⊂ R n +1 under the front projection π : J R n → R n +1 , andtheir union n Λ ε = (cid:83) ni =0 n Λ εi Note the contactomorphism of Lemma 2.1 takes n Λ εi ⊂ J R n isomorphically to { } × n L εi ⊂{ } × T ∗ R n , and thus n Λ ε ⊂ J R n isomorphically to { } × n L ε ⊂ { } × T ∗ R n .We have the following topological consequence of Lemma 2.4. Corollary 3.25.
As a union of smooth manifolds with corners, n Γ ε ⊂ R n +1 is given by thegluing n Γ ε = ( n − Γ ε (cid:48) × R ≥ ) (cid:96) ( n − Γ ε (cid:48) ×{ } ) ( R n × { } ) where ε (cid:48) = ( ε ε , ε , . . . , ε n ) . The front projection takes n L ε ⊂ J R n homeomorphically to n Γ ε ⊂ R n +1 . Before continuing, let us record the following for future use.
Corollary 3.26.
For < j < i ≤ n , the closure of the codimension one clean intersection of n L εi , n L j is precisely n L εi ∩ n L εj .Proof. The closure of the codimension one clean intersection of n L εi , n Λ j is conic and projectsto the primary tangency of n − Γ εi − , n − Γ j − . By Corollary 3.21, the primary tangency of n − Γ i − , n − Γ j − is cut out by h i − ,j − = 0. By Corollary 3.23, this is precisely the tangency T ( n − Γ εi − , n − Γ j − ) and hence lifts precisely to the conic intersection n L εi ∩ n L εj . (cid:3) The case of A n +1 -tree. The following Theorem 3.27 will play a key role in provingProposition 3.6.
Theorem 3.27.
Let ϕ : T ∗ R n → J R n be an embedding as a Weinstein hypersurface. Assumethat the image of n L under ϕ is transverse to the fibers of the projection J R n → R n . Let Υ = π ( ϕ ( n L )) ⊂ R × R n be (the germ of ) the front at the central point.Then there exists a diffeomorphism R × R n → R × R n taking Υ to the germ at the originof n Γ ⊂ R × R n . RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 27
The proof of Theorem 3.27 will proceed by induction on the dimension n . At each stage,we will prove the fully parametric version: Theorem 3.28.
Let ϕ y : T ∗ R n → J R n be a family of Weinstein hypersurface embeddingsparametrized by a manifold Y . Assume that the image of n L under ϕ y is transverse to thefibers of the projection J R n → R n . Let Υ y = π ( ϕ y ( n L )) ⊂ R × R n be (the germs of ) thefronts at the central points.Then there exists a family of diffeomorphisms ψ y : R × R n → R × R n taking Υ y to the germat the origin of n Γ ⊂ R × R n . If ϕ y = Id for y ∈ O p ( K ) , where K ⊂ Y is a closed subset,then we may assume ψ y = Id for y ∈ O p ( K ) . As usual the case of general pairs (
Y, K ) follows from the case Y = D k and K = S k − . Base case n = 0 . The k -parametric version states: the germ of any graphical hypersurfaceΥ ⊂ R × R k is diffeomorphic to the germ of the zero-graph Γ × R k = { } × R k . This canbe achieved by an isotopy generated by a time-dependent vector field of the form h t ∂ x . Thisvector field is zero at infinity if Υ is standard at infinity. Case n = 1 . The next case of the induction n = 1 is elementary but slightly different fromthe others, so it is more convenient to treat separately.With the setup of the theorem, consider the front Υ = π ( Λ) ⊂ R , and assume withoutloss of generality that the origin is the central point. By induction, we may assume, the fronttakes the form Υ = Γ ∪ Υ ⊂ R where Γ = { x = 0 } . Near the origin, the intersectionΓ ∩ Υ and tangency locus T (Γ , Υ ) coincide and consist of the origin alone. Moreover, byconstruction, the origin is a simple tangency, and so Υ = { x = αx } with α (0) (cid:54) = 0. Now itis elementary to find a time-dependent vector field of the form h t x ∂ x , hence vanishing on Γ ,generating an isotopy taking Υ to either Γ = { x = x } or − Γ = { x = − x } . In the formercase, we are done; in the latter case, we may apply the diffeomorphism ( x , x ) (cid:55)→ ( − x , x )to arrive at the configuration Γ ∪ Γ . Finally, it is evident the prior constructions can beperformed parametrically, with the vector field zero at infinity if Υ is standard at infinity. Inductive step.
The inductive step takes the following form. Suppose the fully parametricassertion has been established for dimension n −
1. Starting from n Λ ⊂ T ∗ R n , remove the lastsmooth piece to obtain n Λ (cid:48) = n Λ \ n Λ n , and consider the corresponding front Υ (cid:48) = π ( n Λ (cid:48) ).Note that n Λ (cid:48) = n − Λ × R ⊂ T ∗ ( R n − × R ), and so by an inductive application of the1-parametric version of the theorem, we may assumeΥ (cid:48) = n − Γ × R Set Υ n = π ( n Λ n ). We will find a diffeomorphism R n +1 → R n +1 that preserves Υ (cid:48) (as asubset, not pointwise), and takes Υ n to n Γ n . Moreover, it will be evident the diffeomorphismcan be constructed in parametric form, including the relative parametric form. This willcomplete the inductive step and prove the theorem. Two propositions.
The proof of the inductive step is based on the following 2 proposi-tions.
Proposition 3.29.
Fix n ≥ .With the setup of Theorem 3.27, suppose Υ = (cid:83) n − i =0 n Γ i ∪ Υ n where Υ n = π ( n Λ n ) . Supposein addition Υ n has primary tangency loci satisfying τ (Υ n , n Γ i ) ⊃ τ ( n Γ n , n Γ i ) i = 0 , . . . , n − Then Υ n = { x = αh n } where α = 1 + β n − (cid:81) j =1 h n,j = 1 + βh n, · · · h n,n − Moreover, the same holds in parametric form.Proof.
We have Υ n = { x = g } for some g . Since τ (Υ n , n Γ ) ⊃ τ ( n Γ n , n Γ ) = { h n = 0 } ,we must have g is divisible by h n , hence g = αh n , for some α . Next, for any j (cid:54) = 0 , n , byLemma 3.17, τ ( n Γ n , n Γ j ) is cut out by h n,j = 0. Since τ (Υ n , n Γ j ) ⊃ τ ( n Γ n , n Γ j ), and h n (cid:54) = 0along a dense subset of { h n,j = 0 } , taking the ratio g/h n shows we must have α = 1 + h n,j β j ,for some β j . Repeating this argument, and using the transversality of the level-sets of thecollection h n,j , we arrive at the assertion. (cid:3) Proposition 3.30.
Fix n ≥ .With the setup of Theorem 3.27, suppose Υ = (cid:83) n − i =0 n Γ i ∪ Υ n where Υ n = π ( n Λ n ) . Supposein addition Υ n = { x = αh n } where α = 1 + β n − (cid:81) j =1 h n,j = 1 + βh n, · · · h n,n − Consider the family Υ tn = { x = (1 − t + tα ) h n } so that Υ n = n Γ n , Υ n = Υ n .Then there exist functions h t : R n +1 → R such that the vector fields h t v n − = h t (cid:80) n − i =0 x i i ∂ x i = h t x ∂ x + h t x ∂ x + · · · + n − h t x n − ∂ x n − generate an isotopy ϕ t : R n +1 → R n +1 such that ϕ t (Υ n, ) = Υ n,t .In addition, the functions h t , hence vector fields h t v n − , are divisible by the product (cid:81) n − j =1 h n,j . Moreover, all of the above holds in parametric form.
The following lemmas are needed for the proof of Proposition 3.30.
Lemma 3.31.
For all ≤ i ≤ n , the vector field v i = (cid:80) nj =0 x j j ∂ x j = x ∂ x + x ∂ x + · · · + i x i ∂ x i preserves each n Γ j ⊂ R n +1 , for j = 0 , . . . , i . RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 29
Proof.
Since n Γ j ⊂ R n +1 is independent of x j +1 , . . . , x n , it suffices to prove the case i = j = n .Recall n Γ n is the zero-locus of f = x − h n . We will show v ( h n ) = h n and so v ( f ) = f .Recall h n = h n, = x − h n, , and in general h n,j = x j +1 − h n,j +1 with h n,n − = x n . Thus v n ( h n,n − ) = n h n,n − , and by induction, v ( h n,j ) = j +1 h n,j , so in particular v ( h n, ) = v ( h n ) = h n . (cid:3) Remark . In the context of the inductive step outlined above, we will use Lemma 3.31 inparticular the vector field v n − = (cid:80) n − i =0 x i i ∂ x i = x ∂ x + x ∂ x + · · · + n − x n − ∂ x n − to move Υ n to n Γ n . The lemma confirms we will preserve Υ (cid:48) = n − Γ × R = (cid:83) n − i =0 n Γ i . Lemma 3.33.
For any ≤ j < i ≤ n , and ≤ k ≤ i , we have ∂h i ∂x k = − ( − k k − (cid:89) j =0 h i,j = − ( − k h i, h i, · · · h i,k − Proof.
Recall h i = h i, and the inductive formulas h i,j = x j +1 − h i,j +1 with h i,i − = x i . Thuswe have ∂h i,j ∂x j +1 = 2 h i,j ∂h i,j ∂x k = − h i,j ∂h i,j +1 ∂x k k > j + 1and the assertion follows. (cid:3) Proof of Proposition 3.30.
Suppose Υ = (cid:83) n − i =0 n Γ i ∪ Υ n where Υ n is the graph of h β = (1 + β n − (cid:81) j =1 h n,j ) h n = (1 + βh n, · · · h n,n − ) h n Our aim is to find a normalizing isotopy, generated by a time-dependent vector field v t ,taking the graph Υ n = { x = h β } to the standard graph n Γ n = { x = h n } , i.e. to the graphwhere β = 0, while preserving (cid:83) n − i =0 n Γ i . Thus for any infinitesimal deformation in the classof functions h β , we seek a vector field v realizing the deformation and preserving the functions h , . . . , h n − , i.e. we seek to solve the system˙ h i = 0 , i = 0 , . . . , n − h β = γ n − (cid:89) j =0 h n,j = γh n, · · · h n,n − (2)where ˙ h β denotes the derivative of h β with respect to v , and γ is any given smooth function.Let Λ β ⊂ T ∗ R n +1 denote the conormal to the graph of h β . Any vector field v = (cid:80) nj =0 v j ∂/∂ x j on R n +1 extends to a Hamiltonian vector field v H on T ∗ R n +1 with Hamil-tonian H = (cid:80) nj =0 p j v j . We will find v deforming the graph of h β by finding H so that v H deforms the conormal to the graph Λ β . In general, for a function f : R n → R , with graph Γ f = { x = f } ⊂ R n +1 , denote theconormal to the graph by T ∗ Γ f ⊂ T ∗ R n +1 . With respect to the contact form p dx + . . . p n dx n − x dp , the conormal T ∗ Γ f is given by the generating function F ( x , . . . , x n ) = − p f ( x , . . . , x n ) , i.e. it is cut out by the equations p i = − p ∂f∂x i , i = 1 , . . . , nx = f ( x , . . . , x n )Hence given a Hamiltonian H = (cid:80) nj =0 p j v j , its restriction to the conormal T ∗ Γ f is given by H | T ∗ Γ f = p v | x = f − p n (cid:88) j =1 ∂f∂x j v j | x = f and so further restricting to p = 1, we find the Hamilton-Jacobi equation H | T ∗ Γ f ∩{ p =1 } = v | x = f − n (cid:88) i =1 ∂f∂x i v i | x = f = v | x = f − ˙ f Let us apply the above to h β and h i , for i = 0 , . . . , n −
1. It allows us to transform system (2)into the system v ( x , . . . , x n , h i ) − n (cid:88) j =1 ∂h i ∂x j v j = 0 , i = 0 , . . . , n − v ( x , . . . , x n , h β ) − n (cid:88) j =1 ∂h β ∂x j v j = γ n − (cid:89) j =0 h n,j (3)Note we can reformulate Lemma 3.31 from this viewpoint: when β = γ = 0, given anyfunction h = h ( x , . . . , x n ), the functions v = x h, v = x h, v = x h, . . . , v n = x n n h (4)satisfy system (3).Now let us choose v , v , . . . v n − as in (4) but set v n = 0. This will satisfy the first n equations of system (3), independently of β, γ . From hereon, we will restrict to this class ofvector fields and focus on the last equation of system (3).Let us first set β = 0, so that h β = h n , and solve system (3) in this case. Using Lemma 3.33,we can then rewrite the left-hand side of the last equation of system (3) in the form v ( x , . . . , x n , h n ) − n − (cid:88) j =1 ∂h n ∂x j v j = h h n − n − (cid:88) j =1 ∂h n ∂x j x j j = h h n + n − (cid:88) j =1 ( − j x j j − (cid:89) k =0 h n,k RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 31
Using h n = h n, , h n,k − x k +1 = − h n,k − , we can inductively simplify the term in parentheses h n + n − (cid:88) j =1 ( − j x j j − (cid:89) k =0 h n,k = h n ( h n − x + n − (cid:88) j =2 ( − j x j j − (cid:89) k =1 h n,k )= h n ( − h n, + n − (cid:88) j =2 ( − j x j j − (cid:89) k =1 h n,k )= h n h n, ( − h n, + x + n − (cid:88) j =3 ( − j x j j − (cid:89) k =2 h n,k ) · · · = ( − n − h n h n, h n, · · · h n,n − = ( − n − n − (cid:89) j =0 h n,j Thus for β = 0, the last equation of system (3) reduces to( − n − h n − (cid:89) j =0 h n,j = γ n − (cid:89) j =0 h n,j and hence can be solved by h = ( − n − γ n − (cid:89) j =0 h n,j Now for general β , we will similarly calculate the left-hand side of the last equation ofsystem (3). To simplify the formulas, set F = n − (cid:81) j =0 h n,j θ = βF Thus we have h β = (1 + θ ) h n , and our prior calculation showed when β = 0, the last equationof system (3) took the form ( − n − hF = γF so was solved by h = ( − n − γF .For general β , after factoring out the function h to be solved for, the left-hand side of thelast equation of system (3) takes the form( − n − (1 + θ ) F − h n n − (cid:88) j =1 j ∂θ∂x j x j Thus the equation itself takes the form(5) (( − n − (1 + θ ) F − h n n − (cid:88) j =1 j ∂θ∂x j x j ) h = γF
22 DANIEL ´ALVAREZ-GAVELA, YAKOV ELIASHBERG, AND DAVID NADLER
Since θ = βF , we have ∂θ∂x j = F ∂β∂q j + β ∂F ∂q j = F ∂β∂q j + 2 F β ∂F∂q j and hence ∂θ∂x j is divisible by F . Thus we can divide equation (5) by F , and after renaming γ , write equation (5) in the form (1 + O ( x )) h = γF where O ( x ) vanishes at the orign. We conclude we can solve the equation by h = (1 + O ( x )) − γF .This completes the proof of Proposition 3.30. (cid:3) Proof of Theorem 3.27.
In this section, we use Propositions 3.29 and Proposition 3.30to complete the inductive step outlined in 3.5, and thus, complete the proof of Theorem 3.27.Let us assume n ≥ (cid:48) ∪ Υ n where Υ (cid:48) = (cid:83) n − i =0 n Γ i , Υ n = π ( n Λ n ). We will implement the followingstrategy. Suppose for some 0 < k ≤ n −
1, we have moved Υ n , while preserving Υ (cid:48) , so thatwe have the relation of primary tangencies τ (Υ n , n Γ j ) ⊃ τ ( n Γ n , n Γ j ) j > k Then using Proposition 3.29 and Proposition 3.30, or alternatively, the cases n = 0 , k = n − , n −
2, we will move Υ n , while preserving Υ (cid:48) , so that we have therelation of primary tangencies τ (Υ n , n Γ j ) ⊃ τ ( n Γ n , n Γ j ) j ≥ k Proceeding in this way, we will arrive at k = 0, where all primary tangencies have beennormalized. Then a final application of Proposition 3.29 and Proposition 3.30 will completethe proof.To pursue this argument, we need the following control over primary tangencies. Lemma 3.34.
Fix ≤ k < j ≤ n − .We have τ ( τ (Υ n , n Γ k ) , τ ( n Γ j , n Γ k )) ⊃ τ (Υ n , n Γ j ) ∩ τ ( n Γ j , n Γ k ) Moreover, when k = n − , the tangency of τ (Υ n , n Γ n − ) and τ ( n Γ n − , n Γ n − ) is nonde-generate.Proof. We will assume k > k = 0 as an exercise.Fix a point y ∈ τ (Υ n , n Γ j ) ∩ τ ( n Γ j , n Γ k )In particular y ∈ Υ n and so y = π (˜ y ) for some ˜ y ∈ n Λ n . Recall n Λ n = (cid:83) ε n Λ εn and so ˜ y ∈ n Λ εn ,for some ε . RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 33
Figure 3.5.
The strategy of the proof: inductively normalize tangencies.Note y ∈ τ (Υ n , n Γ j ) implies ˜ y is in the closure of the clean codimension one intersection of n Λ n , n Λ j .By Corollary 3.26, this locus intersects n Λ εn precisely along n Λ εn ∩ n Λ εj and so ˜ y ∈ n Λ εj .Similarly, note y ∈ τ ( n Γ j , n Γ k ) implies ˜ y is in the closure of the clean codimension oneintersection of n Λ j , n Λ k . By Corollary 3.26, this locus intersects n Λ εj precisely along n Λ εj ∩ n Λ εk and so ˜ y ∈ n Λ εk .Thus altogether ˜ y ∈ n Λ εn ∩ n Λ εj ∩ n Λ εk = ( n Λ εn ∩ n Λ εk ) ∩ ( n Λ εj ∩ n Λ εk ).By Corollary 3.26, the intersections n Λ εn ∩ n Λ εk and n Λ εj ∩ n Λ εk are closures of clean codimen-sion one intersections, hence their projections lie in the primary tangencies τ (Υ n , n Γ k ) and τ ( n Γ j , n Γ k ). Moreover, n Λ εn ∩ n Λ εk and n Λ εj ∩ n Λ εk intersect along their primary tangency. Since π restricted to n Λ k has no critical points, the projection of this primary tangency is again aprimary tangency. Hence y ∈ τ ( τ (Υ n , n Γ k ) , τ ( n Γ j , n Γ k )), proving the asserted containment.We leave the nondegeneracy of the case k = n − (cid:3) Now we are ready to inductively normalize the primary tangencies.
Lemma 3.35.
Fix ≤ k < n − .Suppose τ (Υ n , n Γ j ) = τ ( n Γ n , n Γ j ) j > k Then there exists a diffeomorphism ψ : R n +1 → R n +1 preserving Υ (cid:48) = (cid:83) n − i =0 n Γ i such that τ ( ψ (Υ n ) , n Γ j ) = τ ( n Γ n , n Γ j ) j ≥ k Moreover, when k (cid:54) = n − , the diffeomorphism is an isotopy.Proof. We will assume k < n −
3. We leave the elementary cases k = n − , n − n = 0 , R n +1 → R n to identify n Γ k = R n .On the one hand, we have τ ( n Γ j , n Γ k ) = R k × n − k − Γ j − k − k < j < n On the other hand, by Lemma 3.34 and assumption, we have τ ( τ (Υ n , n Γ k ) , τ ( n Γ j , n Γ k )) = τ (Υ n , n Γ j ) ∩ n Γ k = τ ( n Γ n , n Γ j ) ∩ n Γ k k < j < n Hence within n Γ k = R n , the loci τ (Υ n , n Γ k ) and τ ( n Γ n , n Γ k ) have the same tangencies with τ ( n Γ j , n Γ k ) = R k × n − k − Γ j − k − k < j < n Thus Proposition 3.29 and Proposition 3.30 provide a time-dependent vector field of theform v t = h t n − (cid:80) i = k +1 12 i x i ∂ x i generating an isotopy ϕ : R n − k → R n − k satisfying ϕ ( τ (Υ n , n Γ k )) = τ ( n Γ n , n Γ k )In addition, the function h t , hence vector field v t , is divisible by the product (cid:81) n − j = k +1 h n,j , andthus ϕ preserves its zero-locus.Let us complete v t to the vector field V t = h t n − (cid:80) i =0 12 i x i ∂ x i and consider the isotopy ψ : R n +1 → R n +1 generated by V t .Then ψ satisfies ψ ( τ (Υ n , n Γ k )) = τ ( n Γ n , n Γ k )It also preserves n Γ i , for 0 ≤ i ≤ n −
1, as well as τ (Υ n , n Γ j ) = τ ( n Γ n , n Γ j ), for j > k . Inaddition, it preserves τ ( n Γ j , n Γ k ) = R k × n − k − Γ j − k − k < j < n since this is the zero-locus of h n,j . (cid:3) Finally, let us use the lemma to complete the inductive step of the proof of Theorem 3.27as outlined above. Suppose for some 0 < k ≤ n −
1, we have moved Υ n , while preserving Υ (cid:48) , RBOREALIZATION I: STABILITY OF ARBOREAL MODELS 35 so that we have the sought-after primary tangencies τ (Υ n , n Γ j ) = τ ( n Γ n , n Γ j ) j > k Then using Lemma 3.35, we can move Υ n , while preserving Υ (cid:48) , so that we have the sought-after primary tangencies τ (Υ n , n Γ j ) = τ ( n Γ n , n Γ j ) j ≥ k Proceeding in this way, we arrive at k = 0, where all primary tangencies have been normalized.Now a final application of Proposition 3.29 and Proposition 3.30 move Υ n to n Γ n , whilepreserving Υ (cid:48) , and thus complete the proof of Theorem 3.27.3.6. Conclusion of the proof.
We are now ready to prove Proposition 3.6. As a consequencewe establish Theorem 3.5, and by the analogous parametric argument this also establishes theparametric version Theorem 3.12.
Proof of Proposition 3.6.
Take any point λ in the front H := π (Λ) and let π − ( λ ) = { λ , . . . , λ k } . Let Λ , . . . , Λ k be germs of Λ at these points of arboreal types ( T j , n ), n ( T j ) = n j . We need to show that the germ of the front H at λ is diffeomorphic to thegerm of a model front H T , where T is a signed rooted tree obtained from (cid:70) T j by adding theroot ρ and adjoining it to the roots ρ j of the trees T j by edges [ ρρ j ]. The signs of all edges ofthe trees T j are preserved, while previously unsigned edges ρ j α get a sign ε ( ν, L, α ), see (1).We proceed by induction on the number of vertices in the signed rooted tree T = ( T, ρ, ε ).The base case of a ( A , m )-front H ⊂ R m is the same geometry as appearing in 3.5: anygraphical hypersurface H ⊂ R × R m − is isotopic to the germ of the zero-graph { } × R m − .For the inductive step, fix a rooted tree T = ( T, ρ, ε ), and as usual set n = | n ( T ) | .Consider a ( T , m )-front H ⊂ R m , with by necessity m ≥ n .Fix a leaf vertex β ∈ (cid:96) ( T ), which always exists as long as T (cid:54) = A . Consider the smallersigned rooted tree T (cid:48) = T \ β , and the corresponding ( T (cid:48) , m )-front H (cid:48) = H \ ˚ H [ β ] ⊂ R m ,where ˚ H [ β ] ⊂ H is the interior of the smooth piece indexed by β . By induction, we mayassume H (cid:48) = H T (cid:48) × R m − n +1 ⊂ R m Thus it remains to normalize the smooth piece H [ β ].Let A β = ( A β , ρ, ε β ) be the linear signed rooted subtree of T = ( T, ρ, ε ) with vertices v ( A β ) = { α ∈ v ( T ) | α ≤ β } . Set d = v ( T ) \ v ( A β ) = n ( T ) \ n ( A β ) to be the complementaryvertices.Consider the ( A β , m )-front K ⊂ H given by the union K = (cid:83) α ∈ n ( A β ) K [ α ] of the smoothpieces of H ⊂ R m indexed by α ∈ n ( A β ). Note for A (cid:48) β = A β ∩ T (cid:48) , and K (cid:48) = K ∩ H (cid:48) , wealready have K (cid:48) = H A (cid:48) β × R m − n +1+ d ⊂ R m and seek to normalize the smooth piece K [ β ] = H [ β ]. Now we can apply Theorem 3.27 to normalize K [ β ] viewed as the final smooth piece of K . More specifically, we can apply Theorem 3.27 to normalize K [ β ] while preserving K (cid:48) andviewing the complementary directions R m − n +1+ d as parameters, see Figure 3.6. This ensureswe preserve H (cid:48) and hence do not disturb its already arranged normalization.This concludes the proof of Proposition 3.6. (cid:3) Figure 3.6.
Treating the complementary directions as parameters.
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139
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