An introduction to Weinstein handlebodies for complements of smoothed toric divisors
Bahar Acu, Orsola Capovilla-Searle, Agnès Gadbled, Aleksandra Marinković, Emmy Murphy, Laura Starkston, Angela Wu
AAN INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OFSMOOTHED TORIC DIVISORS
BAHAR ACU, ORSOLA CAPOVILLA-SEARLE, AGN`ES GADBLED, ALEKSANDRA MARINKOVI´C,EMMY MURPHY, LAURA STARKSTON, AND ANGELA WU
Abstract.
In this article, we provide an introduction to an algorithm for constructing Weinstein handle-bodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example.This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirbycalculus. Finally, we include one complicated example at the end of the article to showcase the algorithmand the types of Weinstein Kirby diagrams it produces. Introduction
A key way to study closed symplectic manifolds is to break them down into two more easily understoodparts: a neighborhood of a divisor and a complementary
Weinstein domain . A divisor is a symplecticsubmanifold of co-dimension 2. One can allow this submanifold to have certain controlled singularities,such as normal crossing singularities or more general singularities modeled on complex hypersurfaces.Donaldson proved that every symplectic manifold has a divisor [Don96] and Giroux proved that thisdivisor can be chosen such that the complement of a regular neighborhood admits the structure of aWeinstein domain [Gir02, Gir17]. A
Weinstein domain is a symplectic manifold with convex contacttype boundary which can be broken down into symplectic handles modeled and glued as described byWeinstein [Wei91]. The symplectic topology of a Weinstein manifold is encoded in the attaching spheresof the handles. The attaching spheres can be represented by isotropic and Legendrian knots in the frontprojection. This Weinstein handlebody diagram gives a combinatorial/diagrammatic method to encode asymplectic manifold. There is a calculus of moves which relates different diagrams for equivalent Weinsteinmanifolds [Gom98, DG09].Recently, there has been increased study of symplectic divisors in symplectic manifolds, particularlyin the case when the complement is Weinstein. Some of the motivation comes from homological mirrorsymmetry, where generalizing the link between coherent sheaves and Fukaya categories to larger classesof manifolds has required one to consider a mirror pair that includes not only a space, but also a divisor[Aur07]. One way to associate a Fukaya category for a divisor pair, is to look at the wrapped Fukayacategory of the complement of the divisor. The Weinstein handle decomposition is key to understandingthe wrapped Fukaya category, due to recent results that the co-cores of the handles generate the category[CDRGG17, GPS19, GPS18]. The Floer homology of these co-cores is intrinsically tied to the LegendrianDGA of the Weinstein handlebody diagram [BEE12, Ekh19, EL17] which is combinatorially calculated byEkholm-Ng [EN15].An important class of symplectic manifolds are toric manifolds . These have been studied extensivelyas they form a large class of examples of integrable systems because of the symmetry provided by theHamiltonian action of a torus on such manifolds. According to the famous Delzant classification, allcompact symplectic toric 2 n -manifolds are uniquely (up to equivariant symplectomorphism) determined byconvex n -dimensional polytopes, which correspond to the orbit space of the action. Much of the symplecticinformation can be encoded in the combinatorics of these polytopes known as Delzant polytopes. Moreover,toric manifolds have their origin in algebraic geometry, and they come by definition with a fibration bytori given by the action, so that they have been among the first cases of interest for homological mirrorsymmetry, especially in view of the SYZ philosophy (see for instance [Abo09]). Every compact toric Mathematics Subject Classification.
Primary: 57R17. Secondary: 53D05, 53D10.
Key words and phrases.
Weinstein domains, handlebody decompositions, Kirby diagrams, toric divisiors, handleattachments. a r X i v : . [ m a t h . S G ] J a n ACU, CAPOVILLA-SEARLE, GADBLED, MARINKOVI´C, MURPHY, STARKSTON, AND WU symplectic manifold is naturally equipped with a toric divisor. This is precisely the set of all points withnon trivial stabilizer and the fixed points of the toric action are normal crossing singularities of the divisor.This can also be understood in terms of the moment map image polytope: the toric divisor is the preimageof the faces of the polytope under the moment map. The complement of a neighborhood of the divisor issymplectomorphic to a Weinstein domain whose completion is T ∗ T n . (The complement of the divisor isthe preimage of the interior of the polytope under the moment map, which is T n × P where P is a convexopen subset of R n .) Hypersurfaces with normal crossing singularities can naturally be deformed to becomeless singular at the expense of increasing the topological complexity of the divisor and its complement.A toric manifold, together with its toric divisor or any smoothing of the divisor is a log Calabi-Yau pairwhich is a convenient setting for studying mirror symmetry of a space with a divisor [GHK15, HK20].A manifold of dimension 4 will have symplectic surface divisors. Normal crossing singularities in thisdimension are just positive transverse intersections of two smooth branches, or nodes . A deformation ofthis node smooths out the surface, trading the node for an annular tube which thus joins two differentcomponents or increases the genus of the surface. For a toric 4-manifold, the complement of the (fullysingular) toric divisor looks like T ∗ T , which has a natural Weinstein structure described by a diagramdiscovered by Gompf [Gom98].The toric divisor is the preimage of the facets of the Delzant polytope ∆, and the nodes are the preimagesof the vertices of ∆. Since there is a one to one correspondence between the nodes and vertices, we willuse the same notation to denote both a node and its moment map image vertex. Each vertex V ∈ ∆, hasa corresponding ray r based at V defined as the sum of the primitive edge vectors of ∆ adjacent to V andpointing outward from V . Definition 1.
A toric manifold with a chosen subset { V , . . . , V n } of the nodes is { V , . . . , V n } -centered,if the corresponding rays r , . . . , r n all intersect at a common single point in the interior of its Delzantpolytope.We show in [ACSG +
20, Theorem 4.1] that if a toric 4-manifold is { V , . . . , V n } -centered, then thecomplement of the toric divisor smoothed at the nodes { V , . . . , V n } has a Weinstein structure which wecan explicitly describe.In this article, we explain an algorithm to produce a Weinstein handlebody diagram for the complementof any divisor obtained by smoothing the { V , . . . , V n } -nodes of a toric divisor in a { V , . . . , V n } -centeredtoric 4-manifold. We prove the Weinstein handlebody produced by this algorithm is Weinstein homotopic tothe complement of the smoothed toric divisor in our later paper [ACSG + Main Results.
There exists an algorithm to produce a Weinstein handlebody diagram for the complementof a toric divisor smoothed at the subset of nodes { V , . . . , V n } in a { V , . . . , V n } -centered toric 4-manifold.(1) Applying this algorithm to CP smoothed in one node yields the self-plumbing of T ∗ S as illus-trated in Figure 1. Moreover, the same output is obtained for the complement of a toric divisorin any toric 4-manifold smoothed in one node.Algorithm Figure 1 (2) Applying the algorithm to a monotone CP CP smoothed at all six nodes yields a Weinsteinmanifold constructed by attaching 2-handles to the 4-ball along a 5-component Legendrian link asin Figure 2. N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 3
Algorithm
Figure 2
We would like to note that the handlebody diagram in the first example above has already been observedby Casals-Murphy in [CM19], viewed as a Weinstein handlebody for the complement of an affine smoothconic in C (which is the same as the complement of a smooth conic together with a generic line in CP ,which is the once smoothed toric divisor in CP ). We obtain this diagram from a completely differentmethod and provide a systematic recipe which applies much more generally. This first example providesan accessible way to explain the steps of our more general algorithm.This paper is organized as follows. In Section 2, we give definitions and discuss the relevant preliminarybackground on Weinstein Kirby calculus. In Section 3, we provide a picture into how to see the mainstructure of the handle attachment needed to obtain the complement of the smoothed divisor by describingthe core and co-core of the handle in the smoothing local model. The remainder of the paper demonstratesthe algorithm for producing the desired handlebody diagrams. In Section 4, we produce a Weinsteinhandlebody diagram for the complement of a toric divisor smoothed in one node and apply sequences ofKirby calculus moves to simplify the diagrams. In Section 5, we present a more complicated example,coming from CP CP , with the toric divisor smoothed at all six nodes, to showcase the scope ofapplications and the corresponding Weinstein Kirby diagrams. Acknowledgments.
This project was initiated at the 2019 Research Collaboration Conference for Womenin Symplectic and Contact Geometry and Topology (WiSCon) that took place on July 22-26, 2019 atICERM. The authors would like to extend their gratitude to the WiSCon organizers and the hostinginstitution ICERM and their staff for their hospitality. The authors would like to thank Lenhard Ng foruseful conversations. BA and AM would like to thank the Oberwolfach Research Institute for Mathematicsfor hosting them during the completion of this project. AG was partially supported by Wallenberg grantno. KAW 2016-0440 and the Fondation Math´ematique Jacques Hadamard. AM is partially supportedby Ministry of Education and Science of Republic of Serbia, project ON174034. LS is supported by NSFgrant no. DMS 1904074. AW is supported by the Engineering and Physical Sciences Research Council[EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The LondonSchool of Geometry and Number Theory), University College London. OC-S is supported by NSF GraduateResearch Fellowship under grant no. DGE-1644868.2.
Weinstein handlebodies and Kirby calculus
Weinstein handle structure. A Liouville vector field Z for a symplectic manifold ( W, ω ) is a vectorfield satisfying L Z ω = ω . By Cartan’s formula for the Lie derivative and the fact that the symplectic form isclosed, this is equivalent to saying d ( ι Z ω ) = ω (here ι denotes the interior product where ι Z ω ( · ) = ω ( V, · )).In particular, when there exists a Liouville vector field, the symplectic structure is exact. The 1-form λ = ι Z ω which satisfies dλ = ω is called the Liouville form . For an introduction to these ideas, see [MS17].The primary use of Liouville vector fields is to glue symplectic manifolds along contact type boundaries.When the Liouville vector field is transverse to the boundary, it defines a contact structure on the boundaryand can be used to identify a collared neighborhood of the boundary with a piece of the symplectizationof that contact boundary.When Weinstein defined a model of a handle decomposition for symplectic manifolds [Wei91], heequipped the handle with a Liouville vector field so that the gluing of the handle attachment could beperformed using only contact information on the boundary. More specifically, the handle attachment iscompletely specified by a Legendrian attaching sphere, or an isotropic attaching sphere together with data
ACU, CAPOVILLA-SEARLE, GADBLED, MARINKOVI´C, MURPHY, STARKSTON, AND WU
Figure 3.
A sketch of the model for the Weinstein handle of index k , with the associatedgradient-like Liouville vector field Z k .on its normal bundle. The limitation is that the index of the handle is required to be less than or equal to n in a 2 n dimensional manifold. In particular, Weinstein 4-manifolds must be built entirely from handlesof index 0, 1, and 2.The model Weinstein handle of index k in dimension 2 n for k ≤ n is a subset of R n with coordinates( x , y , · · · , x n , y n ), with the standard symplectic structure ω = (cid:80) j dx j ∧ dy j and Liouville vector field Z k = k (cid:88) j =1 (cid:0) − x j ∂ x j + 2 y j ∂ y j (cid:1) + n (cid:88) j = k +1 (cid:18) x j ∂ x j + 12 y j ∂ y j (cid:19) . As with smooth handle theory, the handles are in one to one correspondence with the critical points ofa Morse function. The Liouville vector field agrees with the gradient of such a Morse function (for somechoice of metric), in other words, the Liouville vector field is gradient-like . In the model index k handle,the Liouville vector field is the gradient (with the standard Euclidean metric) of the function φ k = k (cid:88) j =1 (cid:18) − x j + y j (cid:19) + n (cid:88) j = k +1 (cid:18) x j + 14 y j (cid:19) . The handle can be considered to be the subset of R n given by D k × D n − k where the first factorcorresponds to the coordinates ( x , · · · , x k ) and the second corresponds to the remaining coordinates( x k +1 , · · · , x n , y , · · · , y n ). The key terminology for important parts of the handle is as follows. • The core of the handle is D k × { } where x k +1 = · · · = x n = y = · · · = y n = 0. This is the stablemanifold of flow-lines of Z k which limit positively towards the zero at the origin. • The co-core of the handle is { } × D n − k where x = · · · = x k = 0. This is the unstable manifold of flow-lines of Z k which limit negatively towards the zero at the origin. • The attaching sphere is the boundary of the core, S k − × { } . This will be identified with anisotropic sphere in the boundary of the existing manifold to which the handle is attached. • The attaching region is a neighborhood of the attaching sphere S k − × D n − k . This is the entirepart of the handle which will be glued on to a piece of the boundary of the existing manifold whenthe handle is attached. Therefore the Liouville vector field Z k points inward into the handle alongthis part of the boundary (it is concave ). • The belt sphere is the boundary of the co-core { } × S n − k − . It is a co-isotropic sphere in theboundary of the manifold obtained after attaching the handle. N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 5
In general we can piece together the Liouville vector fields on the handles, and put together adjustedversions of the locally defined Morse functions to get a global Morse function on the manifold. A Weinsteinstructure is often encoded analytically as a quadruple (
W, ω, Z, φ ) where W is a smooth manifold, ω is asymplectic structure on W , Z is a Liouville vector field for ω on W , and φ is a Morse function such that Z is the gradient-like for φ . Remark . When a manifold with a Weinstein structure has contact type boundary, it is called a
Weinsteindomain . Such a domain can be extended by a cylindrical end to make the Liouville vector field completeto give a non-compact infinite volume
Weinstein manifold .2.2.
Weinstein Kirby calculus.
The data needed to encode the Weinstein domain are the attachingmaps. In dimension 4, the attaching map of a handle is completely determined by the Legendrian orisotropic attaching sphere. The attaching sphere of a 1-handle is a pair of points. Diagramatically wedraw a pair of 3-balls implicitly identified by a reflection, representing the attaching region S × D . Theattaching sphere of a 2-handle is a Legendrian embedded circle (a Legendrian knot). The 4-dimensional2-handle attachment is determined by the knot together with a framing, but in the Weinstein case, theframing is determined by the contact structure. More specifically, the contact planes along a Legendrianknot determine a framing by taking a vector field transverse to the contact planes. The contactomorphismgluing the attaching region of the 2-handle to the neighborhood of the Legendrian identifies the productframing in the 2-handle with the tb − tb denotes the contact framing (Thurston-Bennequinnumber), which is identified with an integer by looking at the difference between the contact framingand the Seifert framing (this must be appropriately interpreted when the diagram contains 1-handles–see[Gom98]).The diagram we draw should specify the Legendrian attaching knots in S along with the pairs of3-balls indicating the attachments of the 1-handles. By removing a point away from these attachments,we reduce the picture in S to a picture in R . After a contactomorphism, the contact structure on R isker( dz − ydx ) in coordinates ( x, y, z ). The front projection is the map π : R → R with π ( x, y, z ) = ( x, z ).A Legendrian curve in this contact structure is tangent to the contact planes, which happens preciselywhen the y -coordinate is equal to the slope dzdx of the front projection. Therefore, Legendrian knots can berecovered from their front projections with the requirement that the diagram has no vertical tangencies(instead it will have cusp singularities where the knot is tangent to the fibers of the projection) and thecrossings are always resolved so that the over-strand is the strand with the more negative slope (we orientthe y -axis into the page to maintain the standard orientation convention for R so the over-strand isthe strand with a more negative y -coordinate). In these front projections, the contact framing tb can becomputed combinatorially in terms of the oriented crossings and cusps of the diagram when the diagramis placed in a standard form where the pairs of 3-balls giving the attaching regions of 1-handles are relatedby a reflection across a vertical axis. Namely, tb of a Legendrian knot is the difference of the writhe ofthe knot and half the number of cusps in the front projection. For a thorough introduction to Legendrianknots, see [Etn05].The set of moves that relate Weinstein handlebody diagrams in Gompf standard form for equivalentWeinstein domains includes Legendrian Reidemeister moves (including how they interact with the 1-handles) listed in [Gom98], see Figures 4 and 5, as well as handle slides , and handle pair cancellationsand additions , shown in [DG09]. Given two k -handles, h and h , a handle slide of h over h is given byisotoping the attaching sphere of h , and pushing it through the belt sphere of h . We depict a 1-handleslide (along with intermediate Reidemeister and Gompf moves) in Figure 6 and a 2-handle slide in Figure7. A 1-handle h and a 2-handle h can be cancelled, provided that the attaching sphere of h intersectsthe belt sphere of h transversely in a single point. We call this a handle cancellation and the pair ofhandles a cancelling pair . Likewise a cancelling pair can be added to a Weinstein handlebody diagram, asdepicted in Figure 8. When multiple 2-handles intersect a single 1-handle, the simplification in Figure 9can be performed to reduce the overall complexity of a Weinstein diagram.Before approaching our goal of presenting an algorithm to construct Weinstein–Kirby diagrams forcomplements of smoothed toric divisors, we will start with the unsmoothed case, where the complementhas a Liouville completion, T ∗ T . The Legendrian handlebody we present, was originally found by Gompf[Gom98]. It follows from that article that this handlebody gives a Stein/Weinstein structure on the smooth ACU, CAPOVILLA-SEARLE, GADBLED, MARINKOVI´C, MURPHY, STARKSTON, AND WU
Figure 4.
The Legendrian Reidemeister moves, up to 180 degree rotation about eachaxis, where the top, middle, and bottom moves are called Reidemeister I, ReidemeisterII, and Reidemeister III, respectively.
Figure 5.
Gompf’s three additional isotopic moves, up to 180 degree rotation about eachaxis. The top, middle, and bottom moves are called Gompf move 4, Gompf move 5, andGompf move 6, respectively.manifold D ∗ T (which is the trivial bundle D × T ). More generally, Stein handlebody diagrams are givenon the smooth manifolds D ∗ Σ for any surface in [Gom98]. In [ACSG +
20, Theorem 7.1], we show thatthe Weinstein structures induced on these diagrams are Weinstein homotopic to the canonical co-tangentWeinstein structure on D ∗ Σ.For T ∗ T specifically, it is known that there is a unique Weinstein fillable contact structure on theboundary T and one can then deduce that the Gompf handlebody agrees with the canonical symplecticstructure by Wendl’s result that S ∗ T has a unique Stein/Weinstein filling up to deformation [Wen10].To see that the diagram in Figure 11 represents D ∗ T ∼ = D × T smoothly, we can start with a handledecomposition for T with one 0-handle, two 1-handles and a single 2-handle. Thickening this diagram toa 4-dimensional handlebody yields a disk bundle over T with Euler number e , agreeing with the framingcoefficient of the 2-handle attachment. One then needs to put the diagram into Gompf standard form asseen in Figure 10 by sliding the uppermost attaching ball below the attaching ball on the right so thatboth 1-handles are related by a reflection across the same vertical axis. Then we must realize the knot as aLegendrian knot by replacing vertical tangencies by cusps and making sure the crossings always have theover-strand corresponding to the more negative slope. The most obvious way to do this yields a Legendrianknot whose Thurston-Bennequin framing is 0 (see the diagram on the right of Figure 10), so this wouldcorrespond to a D bundle over T with Euler number −
1. By wrapping one strand around the lower
N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 7 (1) (2)(3)(4) (5)
Figure 6.
An example of a 1-handle slide on T ∗ T consisting of the following sequenceof isotopies: (1) Reidemeister 2, (2) Gompf move 4, (3) Gompf move 5, (4) Gompf move4, (5) Reidemeister 2. Figure 7.
An example of a 2-handle slide of the black unknot over the red unknot.
Figure 8.
An example of a 1-handle cancelling with a 2-handle.left attaching ball as in Figure 11, we obtain a smoothly isotopic picture where the new Legendrian has tb = 1, so the Euler number is 1 − D ∗ T . Since this is one Weinstein filling of S ∗ T ,and we know that such fillings are unique up to deformation, it must agree with the canonical co-tangentsymplectic structure on D ∗ T .3. The local model for our handle attachment
In this section we study the local model for the smoothing of a normal crossing singularity of a symplecticdivisor in dimension 4 and describe the local handle attachment information (core and co-core) for the2-handle attachment needed to describe the complement of the smoothing of a node. Here we give theintuitive picture behind the following theorem.
Theorem 3 (Theorem 4.1 [ACSG + . Let ( M, ω ) be a toric -manifold corresponding to Delzant polytope ∆ which is { V , . . . , V n } -centered. Let D denote the divisor obtained by smoothing the toric divisor at thenodes V , . . . , V n . Then there exist arbitrarily small neighborhoods N of D such that M \ N admits thestructure of a Weinstein domain.Furthermore, M \ N is Weinstein homotopic to the Weinstein domain obtained by attaching Weinstein -handles to the unit disk cotangent bundle of the torus D ∗ T , along the Legendrian co-normal lifts ofco-oriented curves of slope s ( V ) , . . . , s ( V n ) . Here s ( V i ) is equal to the difference of the inward normalvectors of the edges adjacent to V i in ∆ . A local model in a 4-dimensional manifold M for the normal crossing intersection of two symplecticdivisors can be given by a Darboux chart ( C , ω std ) at the intersection point, where the two divisors are ACU, CAPOVILLA-SEARLE, GADBLED, MARINKOVI´C, MURPHY, STARKSTON, AND WU
Figure 9.
An example of handle slides and cancellations when multiple 2-handles passthrough a 1-handle. Red and blue 2-handles are slid over the central green 2-handle. Thegreen 2-handle is then cancelled with the 1-handle.
Figure 10.
Diagram depicting how to move the usual picture of T ∗ T into standardform after inserting the necessary Legendrian data, i.e. replacing vertical tangencies bycusps. Figure 11.
Stein structure on D bundle over T with framing coefficient e ( T ∗ T ) ≤ = (cid:8) ( z , z ) ∈ C | z = 0 (cid:9) and Σ = (cid:8) ( z , z ) ∈ C | z = 0 (cid:9) . Smoothing thisnormal crossing means that locally one substitutes the union of these divisors by the smooth surfaceΣ = (cid:8) ( z , z ) ∈ C | z · z = ε (cid:9) N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 9 for some ε >
0. Topologically, the complement of the smoothed surface differs from the complement ofthe normal crossing divisors by one 2-handle attachment for each smoothed normal crossing intersection.Under the centered hypothesis (Definition 1), Theorem 3 says that when considering the symplectic andLiouville structures, the difference between the complements also corresponds to a collection of
Weinstein < ε (cid:48) < ε ,the disk D defined as the image of the map φ : [0 , × [0 , π ] → C ( r, θ ) (cid:55)→ (cid:18) rε (cid:48) e iθ rε (cid:48) e − iθ (cid:19) The size of ε (cid:48) is determined by the size of the neighborhood of the smoothed divisor which is deleted. As ε (cid:48) → ε , the open disk of radius ε provides the analog of the co-core in the non-compact complement of thedivisor itself.This disk is Lagrangian as for r (cid:54) = 0, the derivative of φd ( r,θ ) φ = (cid:18) ε (cid:48) e iθ irε (cid:48) e iθ ε (cid:48) e − iθ − irε (cid:48) e − iθ (cid:19) is an isomorphism so that the tangent space to D at the point φ ( r, θ ) is spanned by the vectors u = (cid:18) ε (cid:48) e iθ ε (cid:48) e − iθ (cid:19) and u = (cid:18) irε (cid:48) e iθ − irε (cid:48) e − iθ (cid:19) and one can check that ω std ( u , u ) = (cid:61) m ( (cid:104) u , u (cid:105) ) = 0where (cid:104)· , ·(cid:105) is the standard Hermitian product in C (equivalently one can check that φ ∗ ( i ( dz ∧ d ¯ z + dz ∧ d ¯ z )) = 0). The point for r = 0 corresponds to the origin of C . At this point, the two followingcurves c and c in D parametrized for t ∈ ( − ,
1) by: c ( t ) = (cid:18) tε (cid:48) tε (cid:48) (cid:19) and c ( t ) = (cid:18) itε (cid:48) − itε (cid:48) (cid:19) give the two independent vectors in the tangent space at the origin of the disk D : u = (cid:18) ε (cid:48) ε (cid:48) (cid:19) and u = (cid:18) iε (cid:48) − iε (cid:48) (cid:19) One can note that we have again ω std ( u , u ) = 0.The boundary of this disk is the image by φ of { } × [0 , π ] , that is, the circle B = (cid:26) (cid:18) ε (cid:48) e iθ ε (cid:48) e − iθ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ ∈ [0 , π ] (cid:27) . One can choose ε (cid:48) < ε such that the circle B lies on the boundary of the small neighborhood of thesmoothed Σ (since z z = ε (cid:48) e iθ ε (cid:48) e − iθ = ε (cid:48) goes to ε if ε (cid:48) approaches ε ). It limits to the origin when ε (cid:48) goes to 0, so that B is the belt sphere of the handle and D is indeed its co-core.The core is characterized (uniquely up to Lagrangian isotopy) as a Lagrangian disk with unknottedboundary in the smooth 2-handle which intersects the co-core transversally at one point and which avoidsthe boundary of the Weinstein manifold, so in our model, it should avoid the smoothed Σ. Let D be the disk defined as the image of the map ψ : [0 , × [0 , π ] → C ( r, θ ) (cid:55)→ (cid:18) rεe iθ rεe − i ( θ + π ) (cid:19) = (cid:18) rεe iθ − rεe − iθ (cid:19) This disk is also Lagrangian as, similarly as before, the tangent space to D at the point ψ ( r, θ ) for r (cid:54) = 0 is spanned by the vectors u = (cid:18) εe iθ − εe − iθ (cid:19) and u = (cid:18) irεe iθ irεe − iθ (cid:19) and one can check that ω std ( u , u ) = (cid:61) m ( (cid:104) u , u (cid:105) ) = 0 . Similarly, at the origin, the two following curves c and c in D parametrized for t ∈ ( − ,
1) by c ( t ) = (cid:18) tε − tε (cid:19) and c ( t ) = (cid:18) itεitε (cid:19) give the two independent vectors in the tangent space at the origin of the disk D : u = (cid:18) ε − ε (cid:19) and u = (cid:18) iεiε (cid:19) Note again that ω std ( u , u ) = 0. This disk does not intersect the smoothed Σ as z z = rεe iθ ( − rεe − iθ ) = − r ε (cid:54) = ε . Moreover, D and D intersect at the origin and this intersection is transverse as one can check that thefamily ( u , u , u , u ) spans C as a real vector space. This shows that D is the core of the handle andthe attaching sphere is the image by ψ of { } × [0 , π ], that is, A = (cid:26) (cid:18) εe iθ εe − i ( θ + π ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ ∈ [0 , π ] (cid:27) . Identifying these transverse Lagrangian disks with the core and co-core in Weinstein’s model for a 4-dimensional 2-handle, allows us to place a Weinstein structure (a Liouville vector field and correspondingMorse function) on this new piece which lies in the complement of the regular neighborhood of the smooth-ing. If the polytope is centered with respect to all the nodes we are smoothing, the Weinstein structures onthese pieces glue together consistently with a Weinstein structure on the complement of the unsmoothedtoric divisor (a Weinstein domain which completes to T ∗ T ). It is shown in [ACSG +
20, Section 4.5] howthese fit together to give an explicit global Weinstein structure.In the toric description of the toric manifolds and divisors we consider, the Hamiltonian torus action inthe local Darboux model corresponds to the torus action on C given in coordinates: ( e iθ , e iθ ) ∗ ( z , z ) =( e iθ z , e iθ z ). In particular, through the symplectomorphism between the complement of the normalcrossing divisor we consider and T ∗ T , the orbit of a point corresponds to the torus T and the quotientspace under the Hamiltonian action corresponds to a cotangent fiber. The attaching sphere in the modelcorresponds in this symplectic identification to a lift of a circle of slope (1 , −
1) in the base T to thecotangent bundle T ∗ T (lift corresponding to the point ( ε , ε ) in the quotient space ( R > ) ). The standardmodel neighborhood corresponds to the standard cone R ≥ × R ≥ , and the corresponding attaching slopeis (1 , −
1) = (1 , − (0 , SL (2 , Z ) transformation, the attaching sphere for the 2-handle correspondingto the smoothing of a normal crossing singularity at a chosen vertex V will be the co-normal lift of acircle of slope s ( V ), where s ( V ) is the difference of the inward normal vectors of the edges adjacent to V ,(see [ACSG +
20, Lemma 4.2]).
N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 11 The algorithm through an example
Here we will show how to obtain a Weinstein handle diagram for the complement of a toric divisor withexactly one node smoothed. Note that for any Delzant polytope, and any single vertex V , the polytopeis vacuously { V } -centered. To fix a specific example, we could start with the toric 4-manifold CP whosetoric divisor is a collection of three CP ’s intersecting at three points, and smooth this divisor at one ofthe intersection points. More explicitly in homogeneous coordinates [ z : z : z ] on CP , the toric divisoris given by the union of the three lines L = { z = 0 } , L = { z = 0 } and L = { z = 0 } . Let ussmooth the intersection of L with L at [1 : 0 : 0]. In the affine coordinate chart where z = 1, withcoordinates ( z , z ), this aligns exactly with our local model in Section 3. After smoothing, the lines L and L are joined to form a conic Q , which intersects the remaining line L at two points. By Theorem 3,the complement of this smoothed divisor is obtained by attaching a single 2-handle to D ∗ T along theLegendrian lift of a curve in T of slope (1 , −
111 12222
Figure 12.
Left: The Legendrian unknot K in the boundary of the 4-dimensional 0-handle of the Gompf diagram, indicating the intersection of this boundary with the La-grangian torus (0-section of D ∗ T ). The black portions coincide with segments of theattaching circle of the 2-handle, and the blue portions give the attaching arcs of the2-dimensional 1-handles of the torus. Right: The corresponding decomposition of theLagrangian torus.In order to translate Legendrian attaching circles in S ∗ T described as the co-normal lift of a curve in T into Legendrian curves drawn in the Gompf diagram (Figure 11), we need to understand how these twopictures get identified. As mentioned in Section 2.2, the Gompf handle diagram is obtained by startingwith a smooth handle decomposition of T with a single 0-handle, two 1-handles, and one 2-handle. Thisdiagram is thickened by two dimensions to obtain T × D , and then the attaching curve of the 2-handle isisotoped around until it agrees with a Legendrian front diagram with induced framing tb − D ∗ T which has critical locus along the 0-section. In [ACSG +
20, Theorem 7.1], we provethat these two structures are Weinstein homotopic and identify the image of the Lagrangian torus givingthe zero-section of D ∗ T in the Gompf diagram. The handle decomposition on the 4-manifold inducesthe corresponding handle decomposition on the Lagrangian torus by intersection. In particular, we see aLegendrian (un)knot K in the boundary of the 4-dimensional 0-handle, which partially coincides with theattaching sphere of the 2-handle, and partially corresponds with attaching arcs for the 1-handles of theLegendrian torus. See Figure 12.Now consider the Legendrian co-normal lift to S ∗ T of the circle in T which is the boundary of the 2-dimensional 0-handle, with the inward co-orientation. This is a Legendrian push-off of K in ∂B = ∂D ∗ D ,because a small positive Reeb flow applied to K = ∂D yields the co-normal lift of a concentric circle closeto ∂D . We will perform an isotopy to the curves in our torus corresponding to attaching spheres of theadditional 2-handles so that these curves agree with parallel copies of such circles except where they enterthe 1-handles. Circles which are further inward will be pushed off more in the positive Reeb direction. Notethat every Legendrian circle has a standard neighborhood which is contactomorphic to a neighborhoodof the zero section in J ( S ) with the contact form dz − ydx where x is the coordinate on S . We willtranslate the diagram on the torus to a front projection diagram of J ( S ), where we think of the S
12 ACU, CAPOVILLA-SEARLE, GADBLED, MARINKOVI´C, MURPHY, STARKSTON, AND WU
Figure 13.
The circle with (1 , T with the dotted arrow indicating the co-orientation of the curve, and an isotoped version which runs parallel to the boundary ofthe 0-handle of T except where it passes through a 1-handle. Figure 14.
The curve in J ( S ) which is identified with the curve (1 ,
0) on T as inFigure 13.as the Legendrian boundary of the 0-handle of the T ∂ z direction in J ( S ), circles which are further inwards in the torus (pushed further by the Reeb flow)will correspond to curves which are pushed upwards more in the J ( S ) diagram. See an example of thisprocedure in Figures 13 and 14. Finally, once we have our diagram in the 1-jet space, we can satellitethe diagram onto the image of S in the Gompf diagram. The images of the co-normal lifts of curvescontained in a neighborhood U of the curve γ that coincides with the attaching circle of the 2-handle andthe attaching arcs of the 2 dimensional 1 handles of the torus are illustrated in Figure 15. As indicatedby the shading gradient, curves which lie further inward towards the center of the square correspond tocurves at greater z -height values in the jet-space. In turn, curves at a higher z -height in the jet space willcorrespond to higher Reeb pushoffs of the complicated looking Legendrian unknot of Figure 15(c).Initially, let us apply this procedure in the simple example where we are attaching a 2-handle along theco-normal lift of a circle in the torus with slope (1 , ,
0) to the upper side of the square, so it lies close to theboundary, and then cut the rectangle at the bottom left vertex to map to J ( S ), obtaining Figure 14.Satelliting this onto the Legendrian unknot in Figure 12, we obtain Figure 16.In general, when satelliting, we need to be somewhat careful with the behavior of the curves nearthe 1-handle. If the curves pass above the attaching region of a 1-handle without entering the 1-handle,they will follow an upward Reeb push-off of the cusps that appear inside the 1-handle attaching balls inFigure 12. Note that we will typically push these cusps out of the attaching regions of the 1-handles bya Legendrian isotopy. If the curves pass through the 1-handle in the torus, they will pass through thecorresponding 1-handle in the 4-dimensional handlebody. See Figure 17 for the conventions in a morecomplicated example.Although our initial explicit example asked us to attach a handle along a curve of slope (1 , − ,
0) slope. Figure 18 showsthe series of 1 handle slides, Reidemeister and Gompf moves that take the Legendrian lift of the resultingdiagram obtained by attaching along a (1 , −
1) curve to the diagram corresponding to attaching along a(1 ,
0) curve.
N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 13 (a) The neigh-borhood U of theco-normal lift of acurve γ ⊂ T (b) Identification of U with neighbor-hood of the zero section of J ( S ).(c) The front projection of U in the Gompf diagramfor D ∗ T Figure 15.
Transferring curves from T to J ( S ) and then to the Gompf diagram. Figure 16.
The Legendrian handle diagram of the complement of the toric divisorsmoothed in one node. That is, T ∗ T ∪ Λ (1 , . Proposition 4.
For any toric -manifold, let Σ be the result of smoothing the toric divisor at a singlenode. Then the complement of a small regular neighborhood of Σ is a Weinstein domain which is Weinsteinhomotopic to the domain represented by the handle diagram of Figure 16.Proof. First note that for a single node, the centered hypothesis is automatically satisfied. By Theorem 3such a complement is a Weinstein domain obtained by attaching a single 2-handle to D ∗ T along theLegendrian curve given by a co-oriented co-normal lift of a curve of slope ( a, b ) in the torus. We verifythat the resulting Weinstein domain does not depend on the choice of the slope up to Weinstein homotopy.We can see that T ∗ T ∪ Λ ( a,b ) is Weinstein homotopic to T ∗ T ∪ Λ (1 , , by performing 1-handle slides on T ∗ T ∪ Λ ( a,b ) similar to Figure 18, to take a Λ ( a,b ) curve to Λ ( a,b ± a ) or to Λ ( a ± b,b ) . Using the Euclideanalgorithm, with an appropriate choice of 1-handle slides, one can start with T ∗ T ∪ Λ ( a,b ) , for any pair a, b ∈ Figure 17.
Mapping the red, blue and purple curves from J ( S ) to T ∗ T . The relativeheights of these curves with respect to the black curve are preserved. The purple curvefollows a Reeb push off of the cusping curve in the 1-handle’s attaching region (see Figure12) while the red and blue curves pass through the 1-handle. Figure 18.
In columns from left to right, a series of Reidemeister and Gompf moves and1-handle slides that take the Legendrian lift of a (1 , −
1) curve to the Legendrian lift ofa (1 ,
0) curve. (1) Reidemeister III, (2) Reidemeister II and I, (3) Reidemeister II, (4)Reidemeister II, (5) slide green 1-handle over orange 1-handle, (6) Gompf move 5, (7)Gompf move 4, and (8) Reidemeister II. Z that are relatively prime, and end with T ∗ T ∪ Λ (1 , . That the Weinstein domains are symplectomorphiccan also be proved using toric arguments (see [ACSG +
20, Proposition 5.5]). (cid:3)
N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 15 (1)(2)(3)(4)(5)(6)(7)(8)(9)
Figure 19.
A series of Reidemeister moves and handle slides simplifying the Weinsteinhandle diagram of the complement of the toric divisor smoothed in one node. (1) Rei-demeister III, (2) Reidemeister II and I, (3) 2-handle slide, (4) Handle cancellation, (5)Reidemeister III, (6) Reidemeister I and II, (7) Reidemeister III, I and II, (8) Gompf move6, (9) Reidemeister II.
Figure 20.
The diagram of the complement of the toric divisor smoothed in one nodeafter simplifications, in particular after applying a single Reidemeister I to the last diagramin Figure 19.The diagram where we attach along the (1 ,
0) curve is preferable to the one where we attach along slope(1 , −
1) (or a more complicated ( a, b ) curve) because it is easier to simplify. Starting with Figure 16, wecan perform Reidemeister moves, Gompf moves, handle cancellations and handle slides. We choose such asimplifying sequence in Figure 19 to obtain the diagram illustrated in Figure 20.5.
A more complicated example: smoothing a toric divisor in CP CP Consider CP blown up three times with the same size of the blow ups. The corresponding Delzantpolytope is a hexagon illustrated in Figure 21. Observe that the sizes of the blow-ups have been chosenprecisely to make the polytope centered with respect to all of its vertices. The inward normals of the sixcorners are: (1 , , (0 , − , ( − , − , ( − , , (0 , , (1 , T ∗ T with 2-handles attached alongΛ (1 , , Λ (0 , − , Λ ( − , − , Λ ( − , , Λ (0 , , Λ (1 , as shown in Figure 22. Figure 21.
The Delzant polytope of monotone CP CP .Figure 22 shows the curves Γ = (1 , , (0 , − , ( − , − , ( − , , (0 , , (1 ,
1) on T . As in the previoussection, we isotope the curves giving a Legendrian isotopy of their co-normal lifts, in order to have themrun parallel to the boundary of the 0-handle in T except possibly at 1-handles. This allows us to identifythem in J ( S ). We isotope each curve so that at the end, the co-orientation points inward. Since thereare multiple curves, we choose an isotopy which minimizes the number of crossings.Once these curves agree (except where they enter the 1-handles) with parallel copies of suitable concen-tric circles which are positive Reeb push-offs of boundary of the 0-handle of the T J ( S ). This identifies all our curves in T with curves in J ( S ) as in Figure 23. We thensatellite the image of the curves in J ( S ) onto the image of S in the Gompf diagram of T ∗ T , using theconventions described in the previous section and illustrated in Figures 15 and 17 to maintain the relativepositions of the curves. The result is Figure 24. N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 17
Figure 22.
The curves (1 , , (0 , − , ( − , − , ( − , , (0 , , (1 ,
1) on T before and af-ter isotoping them to agree with concentric circles co-oriented inward (except where theypass through the 1-handles). Figure 23.
The curves in J ( S ) corresponding to the curves(1 , , (0 , − , ( − , − , ( − , , (0 , , (1 , Figure 24.
A Legendrian handle diagram obtained by simplifying the Legendrian han-dlbody diagram of the complement of the toric divisor of CP CP smoothed in sixnodes.Performing a sequence of handle slides and Legendrian isotopy yields a simpler Weinstein handle di-agram shown in Figure 25. The Weinstein handle diagram allows us to easily compute the homology of CP CP \ ν ( (cid:102) Σ ), where (cid:102) Σ is the smoothed toric divisor and ν ( (cid:102) Σ ) is the neighborhood of (cid:102) Σ . Thehandle structure determines the cellular/Morse homology chain complex, which allows us to determinethe homology groups. A presentation for the fundamental group can also be computed from the handledecomposition where the 1-handles correspond to generators and 2-handles provide relations. Figure 25.
A Legendrian link obtained by simplifying the Legendrian handle diagramof the complement of the toric divisor of CP CP smoothed in six nodes.In this example we obtain: π ( CP CP \ ν ( (cid:102) Σ ); Z ) = 0 ,H ( CP CP \ ν ( (cid:102) Σ ); Z ) = Z ,H ( CP CP \ ν ( (cid:102) Σ ); Z ) = 0 ,H ( CP CP \ ν ( (cid:102) Σ ); Z ) = Z ⊕ with intersection form − − − − − − − −
10 1 0 − − ,H i ( CP CP \ ν ( (cid:102) Σ ); Z ) = 0 for i = 3 , CP CP . The complement of the smoothed divisor given by T ∗ T with 2-handlesattached along Λ (1 , , Λ ( − , − , Λ (0 , is pictured in Figure 26. Performing a sequence of handle slides andLegendrian isotopy on Figure 26 yields a simpler Weinstein handle diagram shown in Figure 27 consistingof a Legendrian (2,4) torus knot.In this picture we can see two Lagrangian spheres in distinct homology classes in the complement ofthis smoothed divisor. Again, the homology of CP CP \ ν ( (cid:102) Σ ), where (cid:102) Σ is the smoothed toric divisorand ν ( (cid:102) Σ ) is the neighborhood of (cid:102) Σ can be computed from the Weinstein handle diagram. In particular, π ( CP CP \ ν ( (cid:102) Σ ); Z ) = 0 ,H ( CP CP \ ν ( (cid:102) Σ ); Z ) = Z ,H ( CP CP \ ν ( (cid:102) Σ ); Z ) = 0 ,H ( CP CP \ ν ( (cid:102) Σ ); Z ) = Z ⊕ Z with intersection form (cid:20) − − (cid:21) ,H i ( CP CP \ ν ( (cid:102) Σ ); Z ) = 0 for i = 3 , ∈ S , A (Λ) (see [EN19] for a goodsurvey on how to calculate the Legendrian DGA). In this case is generated by a, b, x, y, z, w and markedpoints t and t , as shown in Figure 28. The generators have gradings | a | = | b | = 1 , | x | = | y | = | z | = | w | = 0 N INTRODUCTION TO WEINSTEIN HANDLEBODIES FOR COMPLEMENTS OF SMOOTHED TORIC DIVISORS 19
Figure 26.
A Legendrian handle diagram obtained by simplifying the Legendrian han-dlbody diagram of the complement of the toric divisor of CP CP smoothed in threenodes. Figure 27.
A Legendrian (2,4)-torus link obtained from Figure 26 by a sequence ofhandleslides and Legendrian isotopy. x y z wab •• t t Figure 28.
A diagram of the complement of the toric divisor of CP CP smoothedin three nodes in the Lagrangian projection with labelled Reeb chords and marked pointsgenerating the Chekanov-Eliashberg DGA.with differential ∂x = ∂y = ∂z = ∂w = 0 ∂a = xyzw + xw + xy + zw + 1 + t − ∂ab = wzyx + yx + wz + wx + 1 + t . The DGA of Λ is an interesting invariant to consider because the Wrapped Fukaya chain complexof CP CP \ ν ( (cid:102) Σ ) is A ∞ -quasi isomorphic to the DGA A (Λ) [BEE12, Ekh19, EL17]. Note that A (Λ) has no negatively graded Reeb chords and therefore the degree-0 Legendrian contact homology of A (Λ), denoted by LCH (Λ), is finitely generated. Furthermore, it is expected that LCH (Λ) is Morita equivalent to a commutative ring R such that ˆ X = Spec ( R ) where ˆ X is the mirror of CP CP \ ν ( (cid:102) Σ ).See [CM19, EL19] for other examples of such a phenomenon. References [Abo09] M. Abouzaid,
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Department of Mathematics, Northwestern University, Evanston, IL, U.S.A.
Email address : [email protected] (O. Capovilla-Searle) Department of Mathematics, Duke University, Durham, NC, U.S.A.
Email address : [email protected] (A. Gadbled) Universit´e Paris-Saclay, CNRS, Laboratoire de Math´ematiques d’Orsay, 91405, Orsay, France
Email address : [email protected] (A. Marinkovi´c) Matematiˇcki Fakultet, Belgrade, Serbia
Email address : [email protected] (E. Murphy) Department of Mathematics, Northwestern University, Evanston, IL, U.S.A.
Email address : e [email protected] (L. Starkston) Department of Mathematics, UC Davis, Davis, CA, U.S.A.
Email address : [email protected] (A. Wu) Department of Mathematics, University College London, London, United Kingdom
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