Model projective twists and generalised lantern relations
MMODEL PROJECTIVE TWISTS AND GENERALISED LANTERNRELATIONS.
BRUNELLA CHARLOTTE TORRICELLI
Abstract.
We use Picard-Lefschetz theory to introduce a new local model for the planarprojective twists τ AP P Symp ct p T ˚ AP q , A P t R , C u . In each case, we construct an exactLefschetz fibration π : T ˚ AP Ñ C with three singular fibres, and define a compactly supportedsymplectomorphism ϕ P Symp ct p T ˚ AP q on the total space. Given two disjoint Lefschetzthimbles ∆ α , ∆ β Ă T ˚ AP , we compute the Floer cohomology groups HF p ϕ k p ∆ α q , ∆ β ; Z { Z q and verify (partially for CP ) that ϕ is indeed isotopic to (a power of) the projective twist inits local model.The constructions we present are governed by generalised lantern relations , which provide anisotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an S -fibred coisotropic submanifold of the smooth fibre. We also use these relations to generate non-exact fillings for the contact manifolds p ST ˚ CP , ξ std q , p ST ˚ RP , ξ std q , and study two classesof monotone Lagrangian submanifolds of p T ˚ CP , dλ CP q . Introduction
Dehn twists.
Given a Lagrangian sphere L – S n embedded in a symplectic manifold p M n , ω q , there is a compactly supported symplectomorphism of M called the Dehn twistalong L , and denoted by τ L P Symp ct p M q . Any Dehn twist τ L P Symp ct p M q is defined in aneighbourhood of L Ă M n by a local model Dehn twist ; the standard Dehn twist along thezero section of the cotangent bundle p T ˚ S n , dλ T ˚ S n q . The construction and well-definedness ofthese symplectomorphisms rests on the fact that the tangent bundle to a sphere has a periodicgeodesic flow, and the geodesics have a common period.In their local model given by the cotangent bundle, Dehn twists have infinite order in thesymplectic mapping class group π p Symp ct p T ˚ S n qq ([Sei00]), and in some cases it is known thata Dehn twist generates the mapping class group of the cotangent bundle–this has been verifiedfor τ S P π p Symp ct p T ˚ S qq in [Sei98b].In a global context, it is a non-trivial task to distinguish between a (power of a) Dehn twistand the identity in the symplectic mapping class group π p Symp ct p M qq , especially because thesmooth isotopy class of such a symplectomorphism is often trivial.The first non-local examples of Dehn twists symplectically non-trivial (in fact, of infinite sym-plectic order) were found in Seidel’s early investigations ([Sei98a], [Sei00]).The emergence of symplectic Picard-Lefschetz theory apported a new range of methodologiesthat are currently widely used in the field to study symplectic mapping class groups. In thiscontext, the remarkable feature that makes Dehn twists so important is that they are the classof symplectomorphisms that correctly codifies symplectic monodromy maps associated to nodaldegenerations. Technically, this means that the monodromy of a Lefschetz fibration is isotopicto a product of Dehn twists along Lagrangian spheres of the smooth fibre (by the symplecticPicard-Lefschetz formula ([Sei03, Sei08]).On the other hand, in the exact setting, any Dehn twist τ L along a Lagrangian sphere L Ă p
M, ω q can be realised as the monodromy of a Lefschetz fibration with smooth fibre p M, ω q . In this Date : August 7, 2020. a r X i v : . [ m a t h . S G ] A ug ase, the monodromy of an exact Lefschetz fibration can never be isotopic to the identity in thesymplectic mapping class group ([BGZ19], [Tor20]), so Dehn twists are an important source ofnon-trivial symplectomorphisms of exact symplectic manifolds.1.2. Projective twists.
One of the limitations of Dehn twists lies in their very definition; theirexistence is subject to the existence of embeddings of Lagrangian spheres. But the topology ofa symplectic manifold could present obstructions to the existence of such spheres (this is thecase for example for CP n and CP n ˆ CP n ). As a result, in such circumstances it is difficultto apply many of the standard techniques involving Dehn twists or Lefschetz fibrations for thestudy of the symplectic mapping class group.Seidel introduced ([Sei00]) a wider class of symplectomorphisms defined from Lagrangian sub-manifolds with periodic geodesic flow; among these are notably spheres (in which case thesymplectomorphisms are squared Dehn twists) and projective spaces. This paper focuses onthe latter class of symplectomorphisms, that we call projective twists (the appellation Dehn will be associated exclusively to Dehn twists along spheres). We use Picard-Lefschetz theory toestablish a local model for the standard projective twists τ RP , τ CP in the symplectic mappingclass group of their respective cotangent bundles.Unlike their spherical cousins, projective twists have not been in the spotlight of research insymplectic toplogy. In this case too, the definition of projective twist requires the existence ofa Lagrangian embedding of a projective space, which might impose even stronger topologicalrestrictions to the ambient manifold. Moreover, many results given by Picard-Lefschetz theorydo not apply for these maps.Nevertheless, there are a series of results that suggest that projective twists do have interestingproperties of the caliber of Dehn twists; Evans [Eva11], Harris [Har11], Mak-Wu [MW18], andthe author [Tor20].In parallel to projective twists, another class of symplectomorphisms has been introduced, thatof fibred twists; symplectomorphisms defined in the first instance by Biran-Giroux and laterin more generality by Perutz ([Per07]). Fibred twists are defined as compactly supportedsymplectomorphisms that act as Dehn twist on the fibres of a coisotropic manifold V Ñ B ,spherically fibred over a symplectic manifold p B, ω q . One important property of fibred twists,which echoes the symplectic Picard-Lefschetz formula, is that fibred twists can be realisedas monodromies of so called Morse-Bott-Lefschetz (MBL) fibrations, a class of fibrations whichadmits singularities generated by Morse-Bott degenerations ([Per07]). Other properties of thesemaps have emerged from [CDvK14], [WW16].Projective twists can be identified with S -fibred twists, in a local model in which the coisotropicsubmanifolds are given by the unit cotangent bundles ST ˚ AP n , A P t R , C , H u . The symplecticdecompositions we analyse in Section 2.3 (derived from [Bir01]), CP n – D (cid:15) T ˚ RP n Y C, C : “ z P CP n , n ÿ i “ z i “ + , (1)and CP n ˆ CP n – D (cid:15) T ˚ CP n Y Σ , Σ : “ p z, w q P CP n ˆ CP n , n ÿ i “ z i w i “ + (2)deliver a better understanding of the S -fibred coisotropics defining the fibred twists associatedto projective twists. .3. Results.
Since projective twists can be viewed as fibred twists, it should be possible togenerate global results from the MBL fibration picture. The challenge to work in the MBLsetting is that that total space of Morse-Bott-Lefschetz fibrations is in general not exact, as thesingular locus (a smooth submanifold of the singular fibre) often admits rational curves. Thisfact is not irrelevant as it can induce complications related to loss of compactness of modulispaces of pseudo-holomorphic curves.In this paper we focus on finding local models for real and complex projective twists in dimensiontwo, and we manage to do so from a Lefschetz fibration perspective which relies on generalisedlantern relations , according to which the global monodromy of an exact Lefschetz fibrationcoming from a Lefschetz pencil is always isotopic to a fibred Dehn twist in the boundary ofthe fibre. We use (generalised) lantern relations to construct a local model of planar projectivetwists from a Lefschetz fibration perspective.We employ the decompositions (1), (2) to define Lefschetz fibrations π : T ˚ AP Ñ C , A P t R , C u ,on the projective cotangent bundles, derived from a Lefschetz pencil of hyperplanes on theprojective varieties p CP , ω F S q , p CP ˆ CP , ω F S ‘ ω F S q . The generalised lantern relations ((39)and (57)) force the total (ungraded) monodromy of these fibrations to preserve every vanishingcycle. We can use this property, together with a construction of [Sei15], to build compactlysupported symplectomorphisms ϕ P Symp ct p T ˚ AP q . The construction involves lifting a Dehntwist along an annulus in the base C to a symplectomorphism Φ π of the total space, that weadjust to a compactly supported symplectomorphism ϕ P Symp ct p T ˚ AP q .We measure the Floer theoretical action of ϕ on a Lefschetz thimble by computing the Floercohomology groups HF p ϕ k p ∆ α q , ∆ β ; Z { Z q for elements ∆ α , ∆ β in a distinguished basis of Lef-schetz thimbles. In the real case, we can combine these computations with the knowledge ofthe mapping class group π p Symp ct p T ˚ RP qq ([Eva11]) to obtain: Theorem 1.
The symplectomorphism ϕ P Symp ct p D ˚ RP q is isotopic to a power of the projec-tive twist τ k RP , k P Z . In the complex case, very little is known about π p Symp ct p T ˚ CP qq ; we know for exampleabout the existence of exotic projective twists ([Tor20]), but only in higher dimensions. As aconsequence, we can only deliver a partial result based on our Floer cohomological computations. Theorem 2.
The symplectomorphism ϕ P Symp ct p D ˚ CP q is of (symplectic) infinite order. For both A P t R , C u , we then compare our results with the groups (computed in [FS05])HF p τ AP p T ˚ x q , T ˚ y ; Z { Z q for cotangent fibres T ˚ x , T ˚ y Ă T ˚ AP and a speculative comparisonwith the wrapped Floer group HW p T ˚ x , T ˚ x ; Z { Z q . We conjecture that a similar constructionapplied to a Lefschetz fibration π : D ˚ HP Ñ C would give rise to a model HP -twist (see thediscussion in Section 7).Additional results are obtained by utilising generalised lantern relations to operate so called“monodromy substitutions”. We show that these operations can produce strong fillings of unitcotangent bundles (Sections 7.3.2, 7.3.3).As another byproduct of our results we obtain information on monotone Lagrangian submani-folds L – S ˆ S Ă p T ˚ CP , dλ CP q . Theorem 3.
There are two distinguished Lagrangian isotopy classes of Lagrangian submanifolds T , T – S ˆ S Ă T ˚ CP Organisation of the paper.
Section 2 is mostly a summary of the aspects of Picard-Lefschetz theory we use in our constructions. In Section 2.3 we provide an explicit descriptionof the model projective twists seen as S -fibred twists (as in [Per07]). In Section 2.5, we explainthe generalised lantern relation , an isotopy that relates the monodromy of a Lefschetz fibration rising from a Lefschetz pencil, and a fibred twist along the normal bundle of the base locus ofthe pencil.In Section 3, we follow [Sei15] to construct a compactly supported symplectomorphism ϕ on thetotal space of an exact Lefschetz fibration π : E Ñ C . In Section 4 we use a spectral sequenceto compute Floer cohomology of two Lefschetz thimbles, one of which has been twisted by themap ϕ P Symp ct p E q .In Section 5 and 6 we explain how the constructions of the first half of the paper can beapplied to Lefschetz fibrations π : T ˚ AP Ñ C to produce local models for the projective twists τ AP P Symp ct p T ˚ AP q , A P t R , C u .Section 7 presents a short summary of additional results obtained by deploying the theoreticaldevices developed in this article.1.5. Acknowledgements.
First and foremost, I would like to express my deepest gratitude tomy supervisor Ivan Smith, whose support and guidance have been an invaluable component ofthe development and completion of this project. I am thankful to him for providing knowledge-able advice and helping me getting out of theoretical impasses along the way. Moreover, I amgrateful to him for taking the time to comment an ample number of versions of this paper, andoffering suggestions for improvement.It was Ailsa Keating who first suggested that the model RP twist could be constructed usinga Lefschetz fibration as we present here. I am very thankful to her for sharing the ideas whichprovided the momentum to engender this project, and for helpful conversations in the initialstage of the research.I am extremely thankful to Jack Smith for offering impeccable Floer theoretical advice, overnumerous conversations, which helped me articulate the technicalities of Section 4. I wouldlike to thank Jonny Evans for sharing his insights about the Morse-Bott-Lefschetz fibration ofSection 2.2 and helpful discussions.Many thanks to Denis Auroux, who helped clarify the structure of the bifibration of Section6.3 via the map (55) and Agust´ın Moreno, who provided helpful comments on Section 7.3,and mentioned [Oba20] to me. I also want to thank Jeff Hicks and Navid Nabijou for helpfuldiscussions.I am enormously indebted to the keystone of DPMMS, Vivien Gruar, whose generous help andsupport (both administrative and personal) played an essential role in fostering the optimalwork environment to pursue my research.Last but not least, I wish to express my sincere gratitude to my family and friends, whoaccompanied and guided me in these years of meandering explorations towards intellectualautonomy. 2. (Morse-Bott-) Lefschetz fibrations The aim of this section is to illustrate the existing model for projective twists, in their pre-sentation as monodromies of Morse-Bott-Lefschetz fibrations (2.3). To serve this purpose, wefirst recall the basic properties of Lefschetz and MBL fibrations (2.1, 2.2). Towards the end, wefocus the discussion on Lefschetz fibrations arising from Lefschetz pencils (2.4,2.5).2.1.
Exact Lefschetz fibrations.
We summarise the definitions and assumptions requiredwhen discussing exact Lefschetz fibrations in this paper. The notation is mainly taken from[Sei08, Section (15a)]. he version of Lefschetz fibration we consider is tailored for Liouville manifolds, a class ofsymplectic manifolds whose convexity properties facilitate the implementation of Floer theory,in the sense that pseudo-holomorphic curves are well-behaved on such manifolds. Definition 2.1.
A Liouville manifold of finite type is an exact symplectic manifold p W, ω “ dλ W q , λ W P Ω p W q such that there exists a proper function h : W Ñ r , and c ą with thefollowing property. For all x P p c , , the vector field Z (called Liouville vector field) dual to λ W (called Liouville form) satisfies dh p Z qp x q ą . W is called complete if the Liouville vectorfield is complete.A closed sublevel set M : “ h ´ pr , c sq of a Liouville manifold p W, dλ W q is a compact symplecticmanifold with contact type boundary p H : “ h ´ p q , λ W | H q , and it is called a Liouville domain.A Liouville domain p M, dλ M q admits a symplectic completion to a Liouville manifold p W, ω q : “p M Y H B M ˆr , , d p rα qq , where r is the coordinate on R and p Σ “ B M, α “ λ | H q is its contactboundary. Definition 2.2.
A Liouville domain p M, dλ M q admits an almost complex structure of contacttype near the boundary J M , i.e satisfying dh ˝ J M “ ´ λ M (where h is a function as in Definition2.1). It extends to an almost complex structure J on the completion satisfying ‚ J p BB r q “ Z , where Z is the Liouville field on the completion ‚ J is invariant under translations in the r -direction ‚ J | M “ J M .The extension of an almost complex structure of contact type near the boundary to an almostcomplex structure on the cone at infinity as above will be called cylindrical at infinity . We will only consider Liouville manifolds that are complete and of finite type, which we canidentify as the union of a Liouville domain W with a non-compact end, equipped with an almostcomplex structure cylindrical at infinity.Let p E n ` , Ω E , λ E q be an exact symplectic manifold with corners and p D, ω D q the unit discwith Liouville form λ D such that λ D | B D ą ω D “ d p λ D q . Consider a map π : E Ñ D whose only critical points, i.e points in Crit p π q : “ t x P E | D x π “ u , are non-degenerate (hencefinitely many, w.l.o.g at most one in each fibre). Let Crit v p π q be the set of critical values of π . Definition 2.3.
Let J E be an almost complex structure on p E, Ω E , λ E q and j is the standardalmost complex structure on D . An exact symplectic Lefschetz fibration on E is a p J E , j q -holomorphic map with the properties above and the following additional features. ‚ For any z R Crit p π q , the smooth fibre p M : “ π ´ p z q , ω M : “ Ω E | π ´ p z q , λ M : “ λ E | π ´ p z q q is a Liouville domain p M, dλ M q . ‚ The boundary B E has two components of codimension , the vertical boundary B v E : “ π ´ pB D q , and the horizontal boundary B h E ; the two intersect at a codimension -corner. ‚ π : E Ñ D is proper and the restriction π | B v E : B v E Ñ B D is a smooth fibre bundle. ‚ There is an open neighbourhood V Ă D ˆ M of D ˆ B M , an open neighbourhood V Ă E of B h E and a fibrewise diffeomorphism f : V Ñ V such that f ˚ p λ E q “ λ D ` λ M (3)The subspace T E v : “ ker p Dπ q Ă T E is symplectic and its symplectic complement defines acanonical connection
T E h , so that for x R Crit p π q there is a splitting T x E “ T x E v ‘ T x E h .This induces parallel transport maps, which are well defined by the triviality condition at thehorizontal boundary. Definition 2.4.
A tuple p J E , j q is called compatible with π if the following holds. π is p J E , j q -holomorphic, i.e Dπ ˝ J E “ j ˝ Dπ and there is a complex structure J suchthat J E “ J in a neighbourhood of Crit p π q . ‚ The restriction J vv : “ J E | M is an almost complex structure of contact type compatiblewith the Liouville form λ M , and in a neighbourhood V of B h E as above, the almostcomplex structure is a product f ˚ p J E q “ p j, J vv q . ‚ Ω E p¨ , J E ¨q is symmetric and positive definite. This choice of almost complex structure is not generic. However, the space of compatiblealmost complex structures on the total space of an exact Lefschetz fibration is contractible([Sei03, Section 2.1]).A fibration as above can be extended to an exact fibration without boundary, by completing thevertical and the horizontal boundary. In the fibre direction, it suffices to add a cylindrical end pB M ˆ r , D (completing every fibre to a Liouville manifold). The base can be completedto C (see for example [MS10, Section 2]), but in this case require some additional properties.In that case, the condition on the neighbourhood of B h E is replaced with the correspondingcondition on an open set of E whose complement is relatively compact when restricted to eachfibre. The base of such a fibration will be the complex plane C , and vertical boundary B v E isreplaced with a set π ´ p C z D q , for a compact disc D Ă C . The properness assumption on π cannot hold anymore, and the set of critical points is assumed to be compact.Let π : E Ñ C be an exact Lefschetz fibration with smooth fibre p M, ω q , m ` w , . . . , w m Ă D R , for a disc D R Ă C of radius R . Fix a basepoint “at infinity”, z ˚ P C , such that (cid:60) p z ˚ q " R , and use it to fix a representative of thesymplectomorphism class of the smooth fibre π ´ p z ˚ q – M . Definition 2.5. (1) A vanishing path associated to a critical value w i P Crit v p π q is a prop-erly embedded path γ i : R ` Ñ C with γ ´ i p Crit v p π qq “ t u , such that outside of a compactset containing the critical values, the image of γ i is a horizontal half ray at height λ i P R : D T ą R such that @ t ą T, (cid:61) p γ i p t qq “ λ i . (4) (2) A distinguished basis of vanishing paths for π is a collection of m ` disjoint paths p γ , . . . γ m q Ă C defined as above, admitting an ordering defined by the heights, satisfying λ ă λ ă ¨ ¨ ¨ ă λ m . For all i “ , . . . , m , the fibres lim t Ñ8 π ´ p γ i p t qq will be symplecticallyidentified with the smooth fixed fibre π ´ p z ˚ q .(3) The corresponding basis of Lefschetz thimbles is the unique set of Lagrangian submani-folds p ∆ γ , . . . , ∆ γ m q Ă E which project to γ , . . . , γ m via π . Given a Lefschetz thimble L , define its height as the value h p L q : “ λ from (4) . For a pair of thimbles p L , L q set L ą L if h p L q ą h p L q holds.(4) We get an associated basis of vanishing cycles p V , . . . , V m q where for all i “ , . . . , m , V i “ B ∆ γ i “ ∆ γ i X M Ă M. Lefschetz thimbles and vanishing cycles are exact Lagrangians in the total space and fibrerespectively. ‚ A Lefschetz thimble L Ă E is an exact (i.e λ | L is exact), Lagrangian disc, which ispreserved by the flow of the vector field B z on the base C . ‚ A vanishing cycle V Ă M is an exact Lagrangian sphere which comes with an equivalenceclass in of diffeomorphisms S n Ñ V defined up to the action of O p n ` q (called a framing ). igure 1. A distinguished basis of vanishing paths p γ , . . . , γ m q Definition 2.6.
We call global monodromy a symplecticomorphism φ P Symp ct p M q obtained byparallel transport anticlockwise around a loop encircling all the critical values of the fibration,see also (21) . By the symplectic Picard-Lefschetz theorem ([Sei08, (16c)]), the global monodromy is isotopicto the product of Dehn twists along the vanishing cycles p V , . . . , V m q , φ » τ V ¨ ¨ ¨ τ V m P Symp ct p M q (5)Given the data t M, p V , . . . , V m qu , there is an exact Lefschetz fibration π : E Ñ C with fibre p M, ω q , and vanishing cycles p V , . . . , V m q Ă M , unique up to Liouville isomorphism (exactsymplectomorphism) ([Sei08, (16e)]). Hurwitz moves (the action of the braid group Br m ` asin [Sei08, (16d)]) on the set of vanishing cycles also preserves the isomorphism class of the totalspace.2.1.1. Gradings.
Let p M, ω q be a symplectic manifold satifying 2 c p T M q “
0. In this situation,it is possibe to define a notion of Z grading–for Lagrangian submanifolds, symplectomorphisms,and Floer cohomology groups.The assumption 2 c p M q “ K M : “ p Ź n T M q b´ ; a complex quadratic form that we denote by η M . Consider the squaredphase map, i.e α M : Gr p T M b q Ñ S , α M p V q “ η M pp v ^ ¨ ¨ ¨ ^ v n q q| η M pp v ^ ¨ ¨ ¨ ^ v n q q| , (6)where v , . . . , v n is a basis of V P Gr p T M q . Definition 2.7.
A graded Lagrangian is a pair p L, α L q consisting of a Lagrangian submanifold L Ă M with an associated map α L : L Ñ R such that exp p πiα L p x qq “ α M p T L x q . Definition 2.8.
A graded symplectomorphism is a pair p φ, α φ q consisting of a symplectomor-phism φ P Symp p M q and a map α φ : Gr p T M q Ñ R satisfying exp p πiα φ p V qq “ α M p Dφ p V qq α M p V q . Let π : E Ñ C be an exact Lefschetz fibration with smooth fibre M . A grading on the baseand a (choice of) grading on the total space induces a compatible grading on M as follows.Consider the bundle of relative quadratic forms K E { C “ π ˚ p K ´ C q b K E . For any regular valueof the fibration z P C , there is a (canonical) isomorphism K E { C | E z – K E z , which determines aquadratic complex volume form η E z on smooth fibres E z “ π ´ p z q (see [Sei08, Section (15c)]).This enables to upgrade symplectic parallel transport maps to graded symplectomorphisms inthe sense of Definition 2.8. In Section 2.4 we explain the grading shift associated to the globalmonodromy for the fibrations under study. .2. Morse-Bott-Lefschetz fibrations.
Lefschetz fibrations can be regarded as a specialcase of Morse-Bott-Lefschetz (MBL) fibrations, a class of fibrations which admits singularitiesmore degenerate than ordinary double points. Their monodromies are symplectomorphismscousins to Dehn twists, called fibred twists ([Per07]). We briefly introduce this class of sym-plectomorphisms, and their local models as monodromies of MBL fibrations, since projectivetwists (as formalised in [Sei00] via the geodesic flow), can be regarded as a special type offibred twists (see discussion in Section 2.3).Let D be a disc with standard almost complex structure j , and p E, Ω E q a p n ` q -manifoldwith a closed 2-form Ω E and an almost complex structure J E . A MBL fibration π : E Ñ D isdefined by a tuple p E, Ω E , π, J E , j q satisfying the same properties as a Lefschetz fibration, withthe exception that the critical locus Crit p π q Ă E can be a set of smooth sumanifolds B i of (real)codimension 2 k i ` k i ě π is onlyrequired to be non-degenerate only in the normal direction to Crit p π q . Note that in general thetwo form on the total space cannot be made exact and the singular locus might admit rationalcurves (see Section 7.3 for concrete manifestations of this phenomenon).As the name indicates, the singularities of these fibrations are modelled around Morse-Bottsingularities; every critical point in a component Q i P E crit of the singular locus has a neigh-bourhood with coordinates p x , . . . x n q such that in that neighbourhood, π p x , . . . x n q “ k i ÿ j “ x j . (7)A MBL fibration is characterised by its smooth fibre p M, ω q and its “fibred vanishing cycles”, aset of submanifolds of the smooth fibre playing a role analogous to vanishing cycles of a Lefschetzfibration. Fibred vanishing cycles are defined as sets of points in the generic smooth fibre thatare mapped into the critical locus via the limit parallel transport maps. For any component B i of the critical locus, the associated fibred vanishing cycle is a coisotropic submanifold of M withthe structure of a sphere bundle V i Ñ B i . The analogy with vanishing cycles is that the imageof V i under the limiting parallel transport into B i collapses the spherical fibres of V i ([Per07,Section 2.1]).Fibred vanishing cycles constitute a characterising piece of data of a MBL fibration, and themonodromy of a model MBL fibration with smooth fibre p M, ω q , one singularity with criticallocus B and associated fibred coisotropic V Ñ B is given by the fibred twist τ V P Symp ct p M q around V [Per07, Theorem 2.16]. On the other hand, any fibred twist can be realised asmonodromy of such a MBL fibration ([Per07, Section 2.4.1]).We briefly recall the definition of a special case of fibred twist, when the coisotropic is an S -bundle V Ñ B .Let p W, ω W q be a symplectic manifold, and V Ă W be a coisotropic submanifold admitting thestructure of an S -bundle V Ñ Q over a symplectic manifold p Q, ω Q q . Assume the structuregroup of the bundle is SO p q » S , so that there is a principal S -bundle P Ñ Q and a bundleisomorphism V – P ˆ S S , and (by the coisotropic neighbourhood theorem), a neighbourhood U of V Ă W is isomorphic to a neighbourhood of P ˆ S S in the symplectic associated bundle P ˆ S T ˚ S (as in [WW16, Theorem 2.4, Definition 2.7]). Note that this neighbourhood is justa collar neighbourhood of the contact hypersurface V . Definition 2.9. (1)
Local fibred twist.
Let τ S : T ˚ S Ñ T ˚ S the standard S -Dehntwist. Then τ loc pr x, p t, θ qsq “ r x, τ S p t, θ qs . defines a compactly supported symplectomorhism τ loc P Symp ct p P ˆ S T ˚ S q . Fibred twist along V . For ε ą , let ψ : P ˆ S D (cid:15) T ˚ S Ñ U be the symplectomorphismof neighbourhoods. Then τ V : “ " ψ ˝ τ loc ˝ ˝ ψ ´ on UId on W z U defines a compactly supported symplectomorphism in Symp ct p W q . Remark 2.10.
When p W, ω W q is a Weinstein manifold coming from a decomposition of a po-larised manifold p X, ω X , Σ q as X – W Y Σ (see Definition 2.11), where p Σ , ω Σ q is the symplectichypersurface, then there is an obvious principal S -bundle p : P Ñ Σ , (obtained as principal S -bundle associated to the line bundle determined by the hypersurface), called the prequanti-sation bundle. The contact boundary pB W, α q admits the structure of an S -fibred coisotropic B W – P ˆ S S Ñ Σ . Every fibred twist construction occurring throughout the paper will be associated to acoisotropic submanifold as in the above remark.2.3.
Models of projective twists via MBL degenerations.
In this section we explain therelation between projective twists and S -fibred twists–known to the experts but not elucidatedin the existing literature–and study S -bundles V Ñ Q as in Definition 2.9, which can be usedto construct a MBL fibration with monodromy the fibred twist τ V » τ AP , A P t R , C u . Definition 2.11. (1) A polarised symplectic manifold is a closed sympelctic manifold p X, ω X q together with a smooth and reduced symplectic hypersurface Σ Ă X whose ho-mology class r Σ s P H n ´ p X q represents the Poincar´e dual to k r ω s P H p X ; Z q for some k P N .(2) A polarised K¨ahler manifold is a tuple p X, ω X , J, Σ q comprising of a K¨ahler manifold p X, ω X , J q that is also a polarised symplectic manifold p X, ω X , Σ q . Remark 2.12.
Given a polarised manifold p X, ω X q with symplectic hypersurface p Σ , ω Σ q , thecomplement W : “ X z Σ carries the structure of a Liouville domain p W, dα q with contact form α obtained as follows. The boundary B W can be identified to a principal S -bundle p : p P, α q Ñ Σ (derived from the line bundle determined by Σ Ă X ), where α is a connection -form withcurvature dα “ ´ p ˚ ω Σ and such that the vector field definiting the principal S -action on X coincides with the Reeb vector field of α . A polarised K¨ahler manifold admits a symplectic decomposition as follows.
Theorem 2.13. [Bir01, Theorem 1.A]
Let p X, ω X , J, Σ q be a polarised K¨ahler manifold. Thereexists an isotropic CW-complex Ξ Ă X whose complement, the open dense subset p X z Ξ , ω X q ,is symplectomorphic to a standard symplectic disc bundle modeled on a disc normal bundle D ε N Σ { X , for ε ą . Throughout the paper, we will use this decomposition in two particular situations, in whichthe isotropic CW complex is a smooth Lagrangian submanifold of the given polarised K¨ahlermanifold.
Example . [Bir01, 3.1.2] Let X “ p CP n , ω F S q and Σ : “ t z ` ¨ ¨ ¨ ` z n “ u Ă CP n be thesmooth quadric. ThenΞ “ tr z : ¨ ¨ ¨ : z n s P CP n | z j P R for all 0 ď j ď n u » RP n (8)is a smooth Lagrangian in CP n . For ε ą
0, there is a disc normal bundle D ε N Σ { CP n , and byTheorem 2.13 a decomposition CP n – RP n Y D ε N Σ { CP n . (9) s Ξ » RP n Ă CP n is a smooth Lagrangian, Weinstein’s neighbourhood theorem delivers asymplectomorphism between a neighbourhood of RP n Ă p CP n , ω F S q and a neighbourhood ofthe zero section RP n Ă p T ˚ RP n , dλ T ˚ RP n q . Therefore, for some ε ą
0, there is a conformalsymplectomorphism CP n z Σ – D ε T ˚ RP n .We infer that the sphere cotangent bundle ST ˚ RP n Ă p T ˚ RP n , dλ T ˚ RP n q has the structure of S -fibred coisotropic ST ˚ RP n Ñ Σ . (10) Example . [Bir01, 3.3] Let X “ p CP n ˆ CP n , ω F S ‘ ω F S q with homogeneous coordinates pr x : ¨ ¨ ¨ : x n s , r y : ¨ ¨ ¨ : y n sq , and Σ : “ t ř ni “ x i y i “ u Ă CP n ˆ CP n . ThenΞ “ tpr x ¨ ¨ ¨ : x n s , r x : ¨ ¨ ¨ : x n sq , r x ¨ ¨ ¨ : x n s P CP n u » CP n (11)is a smooth Lagrangian in CP n ˆ CP n .As Ξ » CP n Ă CP n ˆ CP n is a smooth Lagrangian, Weinstein’s neighbourhood theorem deliversa symplectomorphism between a neighbourhood of CP n Ă p CP n ˆ CP n , ω F S ‘ ω F S q and aneighbourhood of the zero section CP n Ă p T ˚ CP n , dλ T ˚ CP n q . Then, for some ε ą
0, there is asymplectomorphism p CP n ˆ CP n qz Σ – D ε T ˚ CP n .Then the sphere cotangent bundle ST ˚ CP n Ă p T ˚ CP n , dλ T ˚ CP n q has the structure of S -fibredcoisotropic ST ˚ CP n Ñ Σ . (12)the hypersurface Σ Ă CP n ˆ CP n can be identified with the set of pairsΣ – tp X, Y q , X is a line in C n ` , Y is a 2-plane in CP n with X Ă Y u Ă CP n ˆ CP n . This admits a projection(13) Σ ÝÑ CP p X, Y q ÞÝÑ X. that realises the hypersurface Σ as the projectivised cotangent bundle Σ – P p T ˚ hol CP n q . Thismeans that the S -fibred coisotropic (12) can be identified with the bundle ST ˚ CP n Ñ P p T ˚ hol CP n q , which corresponds the S -bundle obtained as the sphere bundle of the universalline bundle associated to P p T ˚ hol CP n q .In the two-dimensional case the Lagrangian Ξ – CP is the “antidiagonal”, and Σ Ă CP ˆ CP is a Flag threefold embedded in CP ˆ CP . Write the Flag 3-fold (flag manifold of completeflags ins C ) F l : “ tp X, Y q , X Ă C is line through 0 , Y Ă CP is a plane such that X Ă Y u , (14)then we can parametrise lines X by the first CP factor, and planes Y by the second CP factorso that in coordinates we obtain the identification F l – !ř i “ x i y i “ ) “ Σ.We notice the following. Consider the Flag 3-fold as the set of pairs p X, Y q as in (14). Theprojection (13) gives F l – P p T ˚ hol CP q . Lemma 2.16.
For A P t R , C u , consider AP n with the standard Riemannian metrics g , anda circle bundle of radius r P p , q of p T ˚ AP n , dλ T ˚ AP n q , denoted by V : “ ST ˚ AP n “ pt v P T ˚ AP n ; } v } g “ r u , λ AP n q . Then V Ă T ˚ AP n is an S -fibred coisotropic submanifold, and the S -fibred twist of Definition 2.9 is isotopic to the projective twist τ AP n in Symp ct p T ˚ AP n q asdefined in [Sei00, 4.b.] .Proof. The submanifold V Ă T ˚ AP n defines a hypersurface of contact type with dλ T ˚ AP n “ α ,for a contact form α P Ω p ST ˚ AP n q . The Reeb vector field associated to α is the (co)geodesic ow, and since the latter is periodic, and all orbits have the same period, it defines an S -fibration. On the other hand, V is naturally a coisotropic submanifold of the cotangent bundle,and the characteristic (isotropic) foliation coincides with the Reeb flow–and therefore the withthe (co)geodesic flow. Hence the characteristic foliation is in fact an S -fibration p : V Ñ Q ,where the leaf space Q inherits a symplectic structure ω Q satisfying p ˚ ω Q “ dλ T ˚ AP n | V . Acareful inspection of the definitions of the two symplectomorphisms finished the proof. (cid:3) Geometry of Lefschetz pencils.
In this paper we will construct exact Lefschetz fibra-tions arising from pencils of hyperplane sections on a given algebraic variety. This subsectionis a short revision on the techniques involved.Let X be an p n ` q -dimensional smooth projective variety, and let L Ñ X be an ample linebundle. Let s , s be linearly independent sections in H p L q . ThenΣ r λ : µ s : “ t λs ` µs “ u r λ : µ sP CP Ă X (15)defines a family of (projective) hypersurfaces. Assume Σ : “ Σ r s “ s ´ p q is smooth, andlet Σ be another smooth fibre in the family. There is a rational function s s : X (cid:57)(cid:57)(cid:75) CP , notdefined on the base locus B : “ č r λ : µ sP CP Σ r λ : µ s “ t s “ s “ u . (16)The family is called a (algebraic) Lefschetz pencil if(1) The base locus B is a smooth submanifold of X (of codimension 4).(2) The map p X : X z B Ñ CP admits a (finite) set of non-degenerate critical pointsCrit p p X q (and is a submersion away from Crit p p X q . Denote the set of critical valuesby Crit v p p X q .An algebraic Lefschetz pencil satisfies (deduced by the above description)(3) In a neighbourhood of p P B there are local coordinates p z , . . . , z n q such that B “ t z “ z “ u and p X p z , . . . , z n q “ z z .(4) In a neighbourhood of p P Crit p p X q there are local coordinates p z , . . . , z n q such that p X p z , . . . , z n q “ z ` ¨ ¨ ¨ ` z n .The smooth fibres F of p X are affine varieties F – Σ z B Ă Σ. A natural way of turning aLefschetz pencil into a Lefschetz fibration is to perform a blow-up of X at the base locus B ; r X : “ Bl B X “ tp z, y q P CP ˆ X : s p y q “ zs p y qu . (17)Then the projection map r p : r X Ñ CP defines a Lefschetz fibration over CP with closed fibres,each of which contains a copy of the base locus embedded as a smooth hypersurface. Theexceptional divisor of the blow up is given by the projectivisation of the normal bundle to B , E “ P p N B { X q .Recall that there is an isomorphism r X z E – X z B . The geometry of the blown-up space yields awell known formula relating the Euler characteristics of the various components of a Lefschetzpencil. Lemma 2.17.
Let X Ă CP N smooth variety of complex dimension n ` . Let p Σ r λ : µ s q r λ : µ sP CP be a Lefschetz pencil on X with base locus B , generic fibre Σ and p r ` q critical points, thenthe Euler characteristics of X , Σ and B are related as follows: χ p X q “ χ p Σ q ´ χ p B q ` p´ q n ` p r ` q . (18) (cid:3) nother strategy to define a Lefschetz fibration from the projection map of a Lefschetz pencilis to remove the smooth generic fibre Σ “ s ´ p q from the family (15). In the process, thebase locus is removed, and the restriction p X | X z Σ gives rise to a well-defined map p : X z Σ ÝÑ C (19)with affine fibres p F, ω F q .Pencils as described above exist on any projective variety (see [Voi03, Lemma 2.10]). Donaldson([Don96, Don99]) proved the existence of a Lefschetz pencil on any closed polarised symplecticmanifold p X, ω q , so the discussion above could be reformulated in that more general setting.The map p can be completed into an exact Lefschetz fibration π : E Ñ C with smooth fibresLiouville manifolds p M, ω q exact symplectomorphic to a completion of p F, ω F q (see [Sei08, Sec-tion (19b)] and [Sei01, Section (3A)] for this adaptation) which shares the same symplecticinvariants of p .2.5. Generalised lantern relations.
The symplectic Picard-Lefschetz formula expresses theglobal monodromy of a Lefschetz fibration as the product of Dehn twists along the Lagrangianvanishing cycles of the singularities. If the given Lefschetz fibration is induced by a (algebraic orsymplectic) Lefschetz pencil, then the global monodromy is isotopic to a fibred twist as follows.
Lemma 2.18. [see [Aur03, Section 2.4] , [Gom04, Section 3] ] Let π : E Ñ C be an exact Lef-schetz fibration obtained from a Lefschetz pencil on a smooth projective variety X (or a sym-plectic polarised manifold) with base locus B . Assume π has r ` critical fibres with associatedLagrangian vanishing spheres p V , . . . , V r q , and let p M, ω q be the smooth exact fibre. Then thereis an isotopy in π p Symp ct p M qq : τ V ¨ ¨ ¨ τ V r » τ V (20) where the map τ V is a fibred twist along the S -fibred coisotropic submanifold V : “ B M Ñ B ,obtained as in Remark 2.12. We call the expression (20) a generalised lantern relation . Consider the fibration r p : r X Ñ CP with closed fibre Σ, obtained by (symplectically) blowingup the pencil at the base locus B . Given a set of generators p (cid:96) , . . . , (cid:96) r q of π p CP z Crit v p ˜ π qq ,the monodromy representation induced by parellel transport, ψ : π p CP z Crit v p r p q , ˚q Ñ Symp p Σ q{ Ham p Σ q (21)gives a relation ψ p (cid:96) q ˝ ¨ ¨ ¨ ˝ ψ p (cid:96) r q “ Id in the symplectic mapping class group of the (closed)fibre. The Picard-Lefschetz formula then implies that, for any basis of vanishing paths, theproduct of Dehn twists in the vanishing spheres associated to the singularities is isotopic to theidentity: φ : “ τ V ¨ ¨ ¨ τ V r » Id P Symp p Σ q . (22)The Dehn twists are supported in a neighbourhood of the vanishing cycles, so the mon-odromy representation preserves the base locus pointwise and the isotopy above can be liftedto Symp ct p Σ , B q , the group of symplectomorphisms of Σ fixing B pointwise.On the other hand, the global monodromy φ twists the fibres of the unit sphere bundle SN B { Σ of a disc normal bundle D ε N B { Σ , ε ą
0, and it is not possible to lift the isotopy (22) toSymp ct p Σ z B q . Instead, we obtain φ » τ V P Symp ct p Σ z B q (23)where τ V is the fibred twist along the coisotropic hypersurface V : “ SN B { Σ .Recall the closed smooth fibre is polarised, and admits a decomposition into an affine part F (whose completion is exact symplectomorphic to p M, ω q ) and the base locus, so the coisotropic V can be identified with the boundary of the affine fibre of the fibration p : X z Σ Ñ C . Moreover, admits an identification with P ˆ S S Ñ B , for the prequantisation bundle P Ñ B as inRemark 2.12.The monodromy of the fibration π : E Ñ C is isotopic to the product φ , and hence also to thefibred twist τ V . Remark 2.19.
Lemma 2.18 assigns to an exact Lefschetz fibration π : E Ñ C obtained froma Lefschetz pencil, a fibred twist τ V P Symp ct p M q , where p M, ω q is the smooth exact fibre of π .On the other hand, by the discussion of Section 2.2, there is a MBL fibration with fibre (exact)symplectomorphic to p M, ω q and global monodromy φ » τ V . We discuss further implications inSection 7.3. Grading of the monodromy.
Let π : E Ñ C be an exact Lefschetz fibration. If 2 c p T E q “
0, the symplectic parallel transport maps can be endowed with a grading. There is family ofpencils, to which we will restrict to in our applications, that induce Lefschetz fibrations withthis property. Consider a Lefschetz pencil defined by an ample line bundle L Ñ X over an p n ` q -dimensional projective variety X , satifying Assumption A . K X – L b´ d , d ą K X is the canonical bundle of X , and d P Z .Under this assumption, the global monodromy can be considered as a graded symplectomor-phism, and we obtain a graded version of Lemma 2.18.As before, let s , s be linear independent sections of H p L q generating the pencil. The imageof a section of L b´ d (which locally has the form s b´ d ) under (24) is a p n ` q -form, which hasa zero ( d ă
0) or a pole ( d ą
0) along Σ “ s ´ p q . The image of s b´ d gives a trivialisationof K X , the bundle of quadratic forms, over the complement W “ X z Σ , where it restricts to aquadratic form η X . The latter has an associated relative quadratic form η X { dz (here dz is thevolume form on C ). Lemma 2.20. [Sei08, Section (19b)]
Let π : E Ñ C be an exact Lefschetz fibration with smoothfibre p M, ω q . Assume π is obtained from a Lefschetz pencil satisfying Assumption 24. The globalmonodromy φ P Symp ct p M q acts as a shift on every graded closed exact Lagrangian V Ă M .In other words, it acts by the identity equipped with a constant grading, and according to thediscussion above we have φ p V q » V r ´ d s . (25) Proof.
Let X be the projective variety on which the pencil is defined. The graded monodromyaround infinity acts on closed Lagrangians by a shift, which depends on the pole order of η X { dz .Under the assumption (A), the relative quadratic form η X { dz does not extend smoothly overinfinity– where the section s has a pole.By (24), η X has a pole of order 2 d at infinity, while a standard argument shows that dz has apole of order 4. Therefore, η X { dz has a pole of order 2 d ´ η : “ z ´ d p η X { dz q is holomorphic. In particular, the phase function of η is preserved under parallel transportaround a large circle, which means that η X { dz “ z d ´ η undergoes a phase shift of 2 d ´ V i isshifted as φ p V i q » V i r´p d ´ qs “ V i r ´ d s . (cid:3) Remark 2.21.
Note that the a full rotation in the base increases the phase of dz by , so thegrading of an arc winding in the base will by altered by ´ (again, by the sign change of [Sei08] ). . A compactly supported symplectomorphism of the total space
Adapting a construction from [Sei15], we build a compactly supported symplectomorphimon the total space of a Lefschetz fibration π : E Ñ C , under the assumption that the globalmonodromy of π fixes the isotopy class of every vanishing cycle (Assumption B).Let π : E Ñ C be an exact Lefschetz fibration with smooth fibre a Liouville domain p M, ω q (ora Liouville manifold after completion), m ` w , . . . , w m Ă D R , where D R Ă C is a disc of radius R ą
0. Fix a base point z ˚ P C , (cid:60) p z ˚ q " R , and use it to fix a representativeof the symplectomorphism class of the smooth fibre π ´ p z ˚ q – M .Fix a basis of vanishing paths p γ , . . . γ m q and the corresponding set of Lagrangian vanishingcycles p V , . . . , V m q in M following the conventions of Section 2.1. Assumption B . For every i “ , . . . , m , the total (ungraded) monodromy of the fibration pre-serves every vanishing cycle, i.e there is φ P Symp ct p M q and an isotopy φ » τ V ¨ ¨ ¨ τ V m inSymp ct p M q such that @ i “ , . . . , m, φ p V i q “ V i . Let D R ´ ε Ă D R ´ ε Ă C be two discs centered at the origin, of radii R ´ ε and R ´ ε respectively,containing all the critical values w , . . . , w m . Via Assumption B, the map φ P Symp ct p M q hassupport disjoint from the vanishing cycles, so it can be extended to an element of Symp p E q acting fibrewise as φ (on all fibres, including the critical ones).We first consider an anticlockwise rotational vector field on the base, supported on D R z D R ´ ε ,defined as follows. Let ψ : R ` Ñ R ` be a smooth function such that(26) ψ p r q “ r ď R ´ εψ p r q “ r ą R ´ ε Define a Hamiltonian function H : C Ñ r , , H p z q “ ψ p| z |q and let p b t q be its associatedHamiltonian flow. The 2 π -flow p b π q defines a Dehn twist in the annulus A ; this map acts asthe identity on D R ´ ε , and as an anticlockwise 2 π rotation on C z int p D R q , and interpolatesbetween the two on A . For every t , p b t q can be lifted via parallel transport to a family ofsymplectomorphisms p Φ t q of the total space supported on π ´ p A q . The time 2 π -(lifted)flow,that we denote by Φ π , covers the base twist b π , and it is fibre preserving over C z int p A q ,namely over D R ´ ε and over C z int p D R q . In particular: ‚ Φ π acts fibrewise as the identity over D R ´ ε . ‚ There is an isotopy over C z int p D R q , such that fibrewise Φ π | M » φ .From the discussion above, the map φ ´ P Symp ct p M q can be extended to a symplectomorphism˜ φ ´ P Symp p E q , and we consider the composition ˜ ϕ : “ ˜ φ ´ ˝ Φ π . The symplectomorphism˜ ϕ P Symp p E q defines another lift of b π , so it is supported over the compact region D R . Hencethe support of ˜ ϕ is disjoint from the “vertical boundary” π ´ p C z D R q .However, ˜ ϕ is not compactly supported away from the horizontal boundary: the problem beingthe non-trivial action of ˜ ϕ on π ´ p A q X B h E . In what follows we adjust the map by finding anisotopy ˜ ϕ » ϕ to a compactly supported symplectomorphim ϕ P Symp ct p E q .The horizontal boundary B h E can be trivialised as in Definition 2.3: an open neighbourhoood V Ă E of B h E admits an isomorphism (compatible with the Liouville forms and the almost omplex structures) with an open neighbourhood of the trivial bundle C ˆ B M of the form U h – C ˆ M out Ă C ˆ M , where M out Ă M is an open neighbourhood of B M .The restriction of r ϕ to that region can be written r ϕ | U h – τ C ˆ id , where τ C is the Dehn twist ina circle in C . The latter admits a symplectic isotopy with the identity, so there is a symplecticisotopy supported in the non-compact region D ˆ M out , so that τ C ˆ Id » Id C ˆ Id . Figure 2.
The action of Φ π (left) and that of r ϕ Floer cohomology computations
This section focuses on the computation of Floer cohomology groups HF ˚ p ϕ k p ∆ α q , ∆ β ; Z { Z q ,where ∆ α , ∆ β Ă E are disjoint Lefschetz thimbles of a Lefschetz fibration π : E Ñ C admit-ting a map ϕ P Symp ct p E q as defined in the previous section. In Section 4.2, we adapt aspectral sequence from [MS10]. The Lagrangians thimbles we consider intersect cleanly, andthe spectral sequence encodes the data from [Po´z94] for Floer cohomology of clean intersectingLagrangians. The underlying idea is to approximate (or, in the best scenario, fully compute)the Floer cohomology HF p ϕ k p ∆ α q , ∆ β ; Z { Z q by computing the singular cohomology of theconnected components C i of the intersection ϕ k p ∆ α q X ∆ β “ Ť i “ ,...,k ´ C i , i.e k ´ à i “ H ˚ p C i ; Z { Z q ñ HF p ϕ k p ∆ α q , ∆ β ; Z { Z q . (27)4.1. Floer cohomology conventions.
The Lagrangian submanifolds occuring in the com-putations are either vanishing cycles or vanishing thimbles of π : E Ñ C with smooth fibre p M, ω q : ‚ A pair p L , L q of exact Lagrangian thimbles of p E, Ω E q . ‚ A pair p V , V q of closed exact compact Lagrangians of p M, ω q .The ground field for all the Floer cohomology rings is assumed to be Z { Z ; this avoids involvingspin structures.In our applications, both types of Lagrangians admit a Z -grading. Recall that given a pair p L , L q of exact closed Lagrangian submanifolds of M or of exact cylindrical Lagrangian sub-manifolds of E , the grading shifts are compatible with Floer cohomology in the following sense([Sei00, Section 2f])HF ˚ p L r (cid:96) s , L q “ HF ˚´ (cid:96) p L , L q “ HF ˚ p L , L r´ (cid:96) sq , (cid:96) P Z . (28) .1.1. Total space.
Let π : E Ñ C be an exact symplectic fibration (as in Definition 2.3) with acylindrical almost complex structure J E making π p J E , j C q -holomorphic (Definition 2.4).For non-compact Lagrangians such as vanishing thimbles, we set the following conventions. Fix c ą R and K ą
0. Consider two Lagrangian thimbles L , L Ă E projecting to vanishing paths γ , γ : R ` Ñ C with h p L i q “ λ i . For a point in the base z P C , let x : “ (cid:60) p z q , y : “ (cid:61) p z q . Definition 4.1.
For (cid:15) ą λ ´ λ , define a map H (cid:15) P C p C , R q satisfying (29) H (cid:15) p x q “ x ă cH (cid:15) p x q “ (cid:15) x ą c ` K. Define the Hamiltonian vector field Y (cid:15) “ H (cid:15) p x qB y , and call its time-1 flow χ (cid:15) . An “admissiblebending” is a lift r χ (cid:15) of χ (cid:15) to the total space. If L is a Lagrangian thimble, then r χ (cid:15) p L q is aanother vanishing thimble isotopic to L and π p r χ (cid:15) p L qq “ χ (cid:15) p γ q . Definition 4.2.
The Floer cochain (with Z { Z coefficients) of two Lagrangian thimbles L , L Ă E , is defined as CF p L , L q : “ CF p χ (cid:15) p L q , L q , where the admissible bending is chosen such that h p r χ (cid:15) p L qq ą h p L q .The Floer differential is defined as the count of solutions to the Cauchy-Riemann equationwith boundary conditions on p χ ε p L q , L q and with respect to an almost complex structure as inDefinition 2.3. Given intersection points q ´ , q ` P C p L , L q , let M p q ´ , q ` ; J E q be the moduli space of (un-parametrised) solutions u : R ˆ r , s Ñ E to the (perturbed) Floer equation with Lagrangianand asymptotic conditions u p s, q Ă L , u p s, q Ă L . lim s Ñ˘8 u p s, t q “ q ˘ . and for q P C p L , L q , we set B q “ ř p P C p L , L q M p p, q ; J E qx p y , for an almost complex structure J E making π a p J E , j C q -holomorphic projection. The choice is not generic, therefore a smallperturbation should be applied, and that preserves the arguments below.The moduli spaces of curves as above are compact. Recall the thimbles are necessarily disjointfrom the horizontal boundary B h E (as every vanishing cycle is), and so is every intersection pointin C p L , L q . Near B h E , the almost complex structure become product-like, and a maximumprincipe in the fibre implies the Floer solutions cannot reach the horizontal boundary.For compactness in the base direction, we proceed as follows. By exactness, the action functionalgives a common upper bound for the energy of the curves in these moduli spaces, so Gromovcompactness applies (and there is no bubbling, again by exactness) and J E -holomorphic curvescannot escape the domains bounded by the pair p L , L q .As π is p J E , j C q -holomorphic, the open mapping theorem forces J E -holomorphic curves boundedby p L , L q to project to holomorphic curves bounded by p γ p R ` q , γ p R ` qq , and no non-trivialstrip in the total space can project to an open domain in the base. Remark 4.3.
By definition, for a distinguished basis of thimbles p ∆ , . . . , ∆ m q , we have HF p ∆ j , ∆ k q “ for all j ą k . Fibre.
For Lagrangian vanishing cycles, the situation is fairly standard. For a pair p V , V q Ă p M, ω q of closed exact Lagrangian submanifolds of the fibre, the Floer cohomol-ogy groups HF p V , V ; Z { Z q are well defined (see for example [Sei08, Section 8]). We adopt thesame orientations for strips as above. .2. A spectral sequence.
For grading reasons, we assume that the exact Lefschetz fibration π : E Ñ C comes from a Lefschetz pencil satisfying Assumption (A) (this automatically implies(B) via the lantern relation). Assume π has regular fibre p M, ω q , m ` v “ t w , . . . , w m u Ă C . Let z ˚ P C be a base-point with (cid:60) p z ˚ q " π ´ p˚q – M . Fix a distinguished basis of vanishing paths p γ , . . . , γ m q for the critical values, associated basis of Lagrangian vanishing thimbles p ∆ , . . . , ∆ m q . Call p V , . . . , V m q Ă M the resulting basis of vanishing cycles in the fixed fibre. Choose two elementsof the basis ∆ α , ∆ β , α, β P t , . . . , m u with the property that h p ∆ α q ą h p ∆ β q .For the entire section, fix the isotopy class of the global monodromy φ » τ V ¨ ¨ ¨ τ V m with φ p V i q “ V i , i “ , . . . , m .The properties of the pencil ensure the existence of a symplectomorphism ϕ P Symp ct p E q asconstructed in the previous section.4.2.1. Arcs in the base.
Let R ą
0, and D R ´ ε Ă D R Ă C two discs in the base (of radii R ´ ε and R respectively, for ε ą
0) containing the critical values. Let b π be a Dehn twist in theannulus A “ D R z D R ´ ε as defined in Section 3. Note that the image ϕ p ∆ γ q of the vanishingthimble associated to a vanishing path γ is isotopic to the vanishing thimble ∆ b π p γ q associatedto the twisted path b π p γ q .For k P Z , set γ kα : “ b k π p γ α q and ∆ kα : “ ∆ γ kα . The vanishing paths are chosen such thatevery component of the intersection γ kα X γ c is transverse. Call the set of intersection points I : “ t z , . . . , z k ´ u , where z is the innermost and z k ´ the outermost intersection point. Figure 3.
The intersection pattern of the pair of paths p γ kα , γ β q Consider the holomorphic strips bounded by the two arcs, and connecting the intersection pointsof I . We can assign a grading to these points as follows. Fix deg p z q “
0. This determines thegrading of every z (cid:96) , (cid:96) “ , . . . , k ´ p z (cid:96) q “ (cid:96) ¨ p´ q . (30)For any z P I , we have ´ p k ´ q ă deg p z q ă he set I of intersection points admits an order relation: z ´ , z ` P I satisfy z ` ą z ´ if there isa sequence v , . . . , v r : R ˆ r , s Ñ C of holomorphic curves with bounded energy, of the form v i p R ˆ t uq Ă γ kα p R ` q , v i p R ˆ t uq Ă γ β p R ` q , lim s Ñ`8 v i p s, ¨q “ lim s Ñ´8 v i ` p s, ¨q , lim s Ñ`8 v p s, ¨q “ z ` , lim s Ñ´8 v r p s, ¨q “ z ´ , In our case z k ´ ą ¨ ¨ ¨ ą z and the the base-grading of (30) is compatible with this order. Infact, for z ´ , z ` P I , deg p z ` q ą deg p z ´ q ðñ z ` ą z ´ .4.2.2. Lagrangian vanishing cycles in the fibre.
For every intersection point z P I , denote theassociated fibre by E z : “ π ´ p z q and consider the Lagrangians V z,α : “ E z X ∆ kα , V z,β : “ E z X ∆ β . (31)The Lagrangians V z,α , V z,β Ă E z are closed exact (and disjoint from the horizontal boundary), sothe Floer cohomology groups HF p V z,α , V z,β ; Z { Z q are well-defined (see Section 4.1) and admita Z -grading (see below). Note that for (cid:96) “ , . . . , k ´ V z (cid:96) ,α – ϕ i p V z ,α q » V z ,α r (cid:96) ¨ s ϕ s , where s ϕ is the degree shift we compute below.4.2.3. Lagrangian thimbles in the total space.
Each component of the intersection locus ∆ kα X ∆ β lying over z i P I is determined by the intersection of the pair p V z i ,α , V z i ,β q Ă π ´ p z i q . Whennecessary, we consider a compactly supported fibrewise Hamiltonian perturbation of one of thethimbles to make this intersection transverse. Then the cochain complex CF p ∆ kα , ∆ β ; Z { Z q isgenerated by intersection points q P ∆ kα X ∆ β , and its Floer cohomology is well-defined (Section4.1).Under the assumption 2 c p E q “
0, all Lagrangian submanifolds of the total space can beequipped with a grading, which induces a grading of the smooth fibres as in Section 2.1.1.For our computations, it will be enough to consider a relative grading, obtained by fixing thegrading of the Lagrangians over the first intersection point z P I , and tracking the (grading)shifts on the successive intersection loci caused by the action of ϕ .The global monodromy is a graded symplectomorphism of the fibre, which operates a degreeshift given by 4 ´ d (Lemma 2.20). On the other hand, ϕ is a graded symplectomorphism ofthe total space, and its grading shift is given by s ϕ : “ p ´ d q ´ “ ´ d, (32)where the ´ (cid:96) “ , . . . k ´ ˚ p V z (cid:96) ,α , V z (cid:96) ,β ; Z { Z q – HF ˚ p V z ,α r (cid:96) ¨ p ´ d qs , V z ,β ; Z { Z q – HF ˚´ (cid:96) ¨p ´ d q p V z ,α , V z ,β ; Z { Z q . (33)4.2.4. Spectral sequence for Floer cohomology.
Proposition 4.4.
There is a cohomological spectral sequence with bigraded differentials d r : E p,qr Ñ E p ` r,q ´ r ` r , converging to HF ˚ p ∆ kα , ∆ β ; Z { Z q . The starting page is generatedby (34) E pq “ " ‘ z P I p HF p ` q p V z,α , V z,β ; Z { Z q ´ p k ´ q ď p ď
00 otherwise where I p : “ t z P I, deg p z q ě p u Ă I . roof. From now onwards, we will omit the coefficient field Z { Z in the notation. The spectralsequence is obtained as a special instance of of [MS10, Proposition 4.1], whose proof is adaptedbelow. Let p J t q t Pr , s be a family of almost complex structures on E such that each tuple p J t , j q is compatible with the fibration π in the sense of Definition 2.4. Then, for a pair ofintersection points p q ´ , q ` q P ∆ kα X ∆ β we can define a relation q ` ą q ´ if there is a sequenceof J t -holomorphic curves of bounded energy with u i p R ˆ t uq Ă ∆ kα , u i p R ˆ t uq Ă ∆ β , lim s Ñ`8 u i p s, ¨q “ lim s Ñ´8 u i ` p s, ¨q , lim s Ñ`8 u p s, ¨q “ q ` , lim s Ñ´8 u r p s, ¨q “ q ´ . If q ` ą q ´ , then q ˘ must either lie in the same fibre or their projection satisfy π p q ` q ą π p q ´ q in the ordering defined for intersection points on the base. This means that q ` ą q ´ impliesdeg p π p q ` qq ě deg p π p q ´ qq .As noted in Section 2.1, the choice of almost complex structures is not generic, but by applyinga small perturbation to the family p J t q , these curves meet the usual regularity and compactnessrequirements ([Sei03, Section 2.2]).Let I p : “ t z P I, deg p z q ě p u Ă I , where the intersection points are graded as in 4.2.1;this determines ´ p k ´ q ď p ď
0. The complex CF p ∆ kα , ∆ β q admits a filtration F ˚ , whereeach term F p is generated by the intersection points in π ´ p I p q . By definition of the ordering,the Floer differential preserves F p so that there is an induced differential on F p { F p ` . Thisgives rise to a spectral sequence, whose first page is given by the cohomology H ˚ p F p { F p ` q “ HF p V z p ,α , V z p ,β q . The last equality holds since regular elements in the restriction to a regularfibre M p V p,α , V p,β q Ă M p ∆ kα , ∆ β q also define regular elements in the original moduli space M p ∆ kα , ∆ β q (see original proof of [MS10] for details). (cid:3) The Floer complex CF p ϕ k p ∆ a q , ∆ c q . Recall that ∆ α and ∆ β are disjoint and h p ∆ α q ą h p ∆ β q so HF p ∆ α , ∆ β q “ HF p ∆ α , ∆ β q “ Lemma 4.5.
The first power of ϕ satisfies HF ˚ p ϕ p ∆ α q , ∆ β q “ HF ˚ p ∆ α , ∆ β q – HF ˚ p V z ,α , V z ,β q . (35) Proof.
If we consider the first power of the map ϕ , the vanishing paths associated to the twothimbles intersect in a single point z . Then there can be no “horizontal” Floer differential.Namely, all generators are contained in π ´ p z q , so any pseudo-holomorphic curve connect-ing these generators must be confined in that fibre, as π is p J E , j C q -holomorphic (and theopen mapping theorem applies). The spectral sequence then collapses at the first page, andHF ˚ p ϕ p ∆ α q , ∆ β q – HF ˚ p V z ,α , V z ,β q . (cid:3) Now let k ą
0, and I “ t z , . . . , z k ´ u be the set of intersection points in the base, with theirsubsets I p “ t z P I, deg p z q ě p u . Then the nontrivial p p, q q -entries of the first page of thespectral sequence (34) are given by E pq “ ‘ z P I p HF p ` q p V z,α , V z,β q – ‘ p HF p ` q p V z ,α r p p d ´ qs , V z ,β q – ‘ p HF p ` q ´ p p d ´ q p V z ,α , V z ,β q (36) Lemma 4.6.
The inclusion CF ˚ p V z ,α , V z ,β q ã Ñ CF ˚ p ϕ k p ∆ α q , ∆ β q is a cochain homomorphismpreserving the Floer product structure.Proof. The element F is generated by the intersection points in π ´ p z q , which are generatorsof CF p V z ,α , V z ,β q . igure 4. Top: the initial vanishing paths p γ α , γ β q , middle: the first power p γ α , γ β q , bottom: the second power p γ α , γ β q .As z is the innermost point, there can be no pseudo-holomorphic curve in the moduli space M p ∆ kα , ∆ β q associated to the Floer differential of z , unless its image is fully contained in thefibre π ´ p z q (this follows by the conventions for Floer strips that we have adopted and theopen mapping theorem).The generators of CF p V z ,α , V z ,β q are closed under the Floer differential of the complexCF ˚ p ∆ kα , ∆ β q , and cochains of CF ˚ p V z ,α , V z ,β q must be cochains in CF ˚ p ∆ kα , ∆ β q . The Floerproduct is also defined by a count of pseudo-holomorphic curves, so the same reasoning appliesto show that these curves must remain in the same fibre. (cid:3) RP twist From a pencil of quadrics on CP , we define a Lefschetz fibration π : T ˚ RP Ñ C , with generalfibre M a 2-sphere with four punctures and three singular fibres (Section 5.1). The smooth fibre M can be thought of as a plumbing of circles along two points; in this picture, the vanishingcycles are the two Lagrangian cores and their surgery.Using the construction of Section 3, we can define a non-trivial, compactly supported symplec-tomorphism ϕ P Symp ct p T ˚ RP q of the total space, and prove that ϕ is isotopic to a nontrivialpower of the standard planar real projective twist τ RP P Symp ct p T ˚ RP q . .1. Lefschetz fibration.
We begin by describing a Lefschetz fibration on T ˚ RP arising fromfrom a Lefschetz pencil of quadrics on CP (this is a well-known case study in symplectic toplogy,see for example [Aur03, Section 3.1]).Consider a Lefschetz pencil C r λ : µ s : “ t λs ` µs “ u r λ : µ sP CP on CP , generated by sections of O CP p q denoted by s and s . In a generic family of conics, the curves intersect in a commonset of four points, the base locus B “ t s “ “ s u . Generically, there are three singularcurves in such a family, any of which has one singularity which is at most an ordinary doublepoint. Each singular conic is composed by two lines in CP containing two of the base pointseach; three such configurations can arise (the number of critical fibres is given by the formula(18)). The three associated vanishing cycles are the classes of circles which collapse to theintersection of the two lines in each singular conic. Hence, each vanishing cycle in a smoothfibre is represented by a loop encircling two points of the base locus (see Figure 5). Figure 5.
A smooth fibre of the pencil, and the three configurations of vanishingcycles.Let r z : z : z s be homogeneous coordinates on CP . There is a rational map CP (cid:57)(cid:57)(cid:75) CP , z “ r z : z : z s ÞÑ r s p z q : s p z qs sending each point to the hyperplane in CP containing it. By removing one of the smoothfibres, for example Σ : “ s ´ p q , we obtain a well defined map p : “ s s : CP z C ÝÑ CP zt8u – C . (37)This defines a Lefschetz fibration whose fibres are 2-spheres with four punctures (the base locus B has been removed in the process) and base C . Lemma 5.1.
The total space W : “ CP z C is exact symplectomorphic to a disc cotangentbundle p D ε T ˚ RP , dλ RP n q , ε ą .Proof. Follows from Example 2.14. (cid:3)
The fibration (37) can be adjusted to become an exact Lefschetz fibration E Ñ C as in Section2.1, whose (fibrewise) completion is Liouville isomorphic, after rounding off the corners, to aLefschetz fibration π : T ˚ RP ÝÑ C (38)with smooth fibres M isomorphic to four punctured 2-spheres. The vanishing cycles V , V , V Ă M are exact Lagrangian circles which partition the four boundary components in three possibleconfigurations of pairs. There is only one Hamiltonian class for each vanishing cycle (each omotopy class has only one exact Lagrangian representative, by Stokes) and since Diff p S q » O p q , there is a unique choice of framing. Remark 5.2. In [Joh11, Section 3.1] , the Lefschetz fibration induced by the Lefschetz pencil ofSection 5.1 is proved to be isomorphic to the complexification of a Morse function f : RP Ñ R with three real critical values w , w , w of indices , , (for a choice of Riemannian metric g for which the pair p f, g q is Morse-Smale). An appropriate choice of pencil delivers a Lefschetzfibration π : D ˚ RP Ñ C such that Crit p π q Ă RP , π p RP q Ă R and π | RP “ f . In this setting,one can ensure that there is a choice of thimbles intersecting the real locus RP in the total spacetransversally at a single point. Then, as π “ f C , the gradient flow of f can be identified witha multiple of the parallel transport flow of π , and a suitable choice of thimbles intersecting thecentral Lagrangian RP transversely can be made globally isotopic to cotangent fibres of T ˚ RP . Monodromy.
In this section we study the monodromy properties of π : T ˚ RP Ñ C . LetCrit v “ t w , w , w u be the set of critical values of π . We fix a base-point z ˚ P C , the smoothfibre π ´ p z ˚ q – M (a four punctured sphere) and basis of vanishing paths p γ , γ , γ q associatedbasis of Lagrangian thimbles p ∆ , ∆ , ∆ q and vanishing cycles p V , V , V q Ă M . Lemma 5.3.
Let p d , d , d , d q Ă M be simple closed curves with d i X V j “ H ( i “ , , , and j “ , , ), such that each d i encircles a distinct boundary component of the fibre. Theglobal monodromy φ P Symp ct p T ˚ RP q of π satisfies φ : “ τ V τ V τ V » ź i “ τ d i . (39) In particular, φ commutes with each individual twist, and preserves each vanishing cycle.Proof. The expression (39) is the lantern relation (see for example [FM11, Proposition 5.1]),a special instance of the generalised lantern relations of Section 2.4, according to which theglobal monodromy is isotopic to the fibred twist in the circle bundle of the normal bundle to B Ă CP . In this low-dimensional case, SN B { CP is the union of the four boundary circles. Theisotopy (39) also implies that any cyclic permutation of τ V τ V τ V defines the same element ofthe mapping class group. (cid:3) The lantern relation forces the (ungraded) monodromy to act trivially on vanishing cycles.However, if we consider φ as a graded symplectomorphism, it acts on vanishing cycles by a shiftdetermined as in Lemma 2.20. The line bundle L “ O CP p q associated to the conic C α , andthe anticanonical bundle K CP “ O p´ q satisfy Assumption (A), with O p´ q – O p q b´ . (40)On graded Lagrangian vanishing cycles V i Ă M , we therefore have the shift φ p V i q » V i r ´ s “ V i r s .5.3. The real projective twist.
Following the instructions of Section 3, we constructa compactly supported symplectomorphism on the total space of the Lefschetz fibration π : T ˚ RP Ñ C . Using the Floer theoretical computations for Lefschetz thimbles of Section 4.2,we prove that ϕ is isotopic to a power of the projective twist τ k RP P π p Symp ct p T ˚ RP qq , k P Z .According to the conventions of Section 4.2, for j, k P t , , u , j ‰ k any two thimbles ∆ j , ∆ k are disjoint and satisfy h p ∆ j q ą h p ∆ k q if j ą k . Any element in the basis of vanishing cycles p V , V , V q Ă M satisfies Condition (B) by Lemma 5.3, so Section 3 yields a well-definedcompactly supported symplectomorphism ϕ P Symp ct p T ˚ RP q on the total space of theLefschetz fibration π : T ˚ RP Ñ C . V V d d d d Figure 6.
The boundary circles around the punctures are disjoint and there-fore the twists in the composition Π i “ τ d i commute (depiction of the fibre withboundary components, before completion).Let γ k “ b k π p γ q and ∆ k : “ ∆ γ k – ϕ k p ∆ q and I : “ γ k p R ` q X γ p R ` q “ t z , . . . , z k ´ u . Overeach intersection point z P I , the thimbles ∆ k and ∆ intersect in their vanishing cycles, whichare Lagrangian circles meeting at two points (see Figure 6) V z, X V z, : “ t q ´ , q ` u Ă π ´ p z q (41)For each z P I , the Lagrangians p V z, , V z, q trace four punctured strips between q ´ and q ` , so thepairs p q ´ , q ` q generate the Floer cohomology HF ˚ p V z, , V z, ; Z { Z q – Z { Z ‘ Z { Z . Togetherwith Proposition 4.5, we obtain: Proposition 5.4.
The symplectomorphism ϕ P π p Symp ct p T ˚ RP qq is a non-trivial element ofthe symplectic mapping class group. (cid:3) Endow the arcs in the base, the Lagrangian vanishing cycles in the smooth fibres and theLagrangian thimbles and with a Z -grading as in Section 2.1.1. Set deg p z q “ p z (cid:96) q “ ´ (cid:96) for (cid:96) “ , . . . , k ´ p q ´ , q ` q be the pair of generators of CF ˚ p V z , , V z , q . The Floer product structure onCF p V z , , V z , q can be identified with the product structure computed in [Sei01, (3.2)]. SinceCF ˚ p V z , , V z , q ã Ñ CF ˚ p ϕ k p ∆ q , ∆ q is an inclusion of cochain complexes (by Lemma 4.6), thesame product relations hold in CF ˚ p ϕ k p ∆ q , ∆ q . In particular, this product structure informson the grading of the generators p q ´ , q ` q , which is zero in both cases. Therefore, we fixHF ˚ p V z , , V z , ; Z { Z q – Z { Z ‘ Z { Z for ˚ “ . (42)With these gradings, the non-trivial entries of the spectral sequence (36), generated by the Floercohomology groups of the vanishing cycles in the intersection fibres, are given by (see (36)) E pq “ ‘ p HF p ` q p V z , r p p d ´ qs , V z , q – ‘ p HF p ` q p V z , , V z , q . (43) Z { Z Z { Z Z { Z p ´ ´ ´ ´ Equation 43.
The shape of the spectral sequence for k “ d : E p p,q q Ñ E p p ` ,q ´ q , so it is not possibleto directly compute the Floer cohomology groups HF p ϕ k p ∆ q , ∆ q . However, using [Eva11], wecan infer the following. Corollary 5.5.
The symplectomorphism ϕ P π p Symp ct p T ˚ RP qq is isotopic to a power of theprojective twists τ k RP , k P Z .Proof. By Proposition 5.4, ϕ is a non-trivial compactly supported symplectomorphism of T ˚ RP . The symplectic mapping class group Symp ct p T ˚ RP q is known to be generated by τ RP ([Eva11]) so ϕ is sotopic to a power τ k RP , k P Z . (cid:3) We expect the rank of HF p ϕ k p ∆ q , ∆ q to increase linearly with k (see next section); if that wasindeed the case, ϕ would be forced to be isotopic to the projective twist, or its inverse.5.3.1. Expected results.
Let N k : “ t , . . . , k ´ u . The Floer cohomology group are expectedto be of the shape(44) HF ˚ p ϕ k p ∆ q , ∆ ; Z { Z q “ " Z { Z ‘ Z { Z ˚ “ ´ N k . Note that if all the differentials of the spectral sequence (43) vanish, (44) is the limit to whichthe sequence converges to. The nature of the above prediction rests in a series of claims, someof which are still in conjectureal state in the literature. We sketch the reasoning below.Consider the symplectomorphism of the total space Φ π P Symp p T ˚ RP q (called “global mon-odromy” in [Sei09, Section 3]) we defined in Section 3.Assume the total space has been smoothed to a Liouville domain with contact bound-ary ST ˚ RP and ∆ , ∆ suitably deformed into Lagrangians with Legendrian boundary in ST ˚ RP to become objects of the wrapped Fukaya category W p T ˚ RP ; Z { Z q . Then the di-rect limit lim ÝÑ k HF ˚ p Φ k π p ∆ q , ∆ ; Z { Z q is expected to compute the wrapped Floer cohomologyHW ˚ p ∆ , ∆ ; Z { Z q (see [MS10, Remark 3.1], and see [BEE12, Appendix] for the formulationin terms of symplectic cohomology).Moreover, if we idenitify the thimbles with cotangent fibres T ˚ q , T ˚ q P T ˚ CP (after completionof the total space to a Liouville manifold), we obtainlim ÝÑ k HF ˚ p Φ k π p ∆ q , ∆ q “ HW ˚ p ∆ , ∆ q – HW ˚ p T ˚ q , T ˚ q q – HW ˚ p T ˚ q , T ˚ q q (45) here the last isomorphism is induced by invariance under Hamiltonian isotopies of wrappedFloer cohomology ([AS10, Section 3.1]).Let Ω RP be the based loop space of RP . By [AS06], there is an isomorphismHW ˚ p T ˚ q , T ˚ q ; Z { Z q – H ´˚ p Ω RP q ; which, combined with (45) yields the conjectural relationlim ÝÑ k HF ˚ p Φ k π p ∆ q , ∆ q – H ´˚ p Ω RP ; Z { Z q (46) Lemma 5.6.
The homology of the based loop space Ω RP is given by H ˚ p Ω RP ; k q – k r x s ‘ k r x s (47) x is a generator of degree , and k the ground field.Proof. Consider the fibration Z { Z Ñ S Ñ RP , and let P RP the space of based paths on RP .Pulling back the path-loop fibration Ω RP Ñ P RP Ñ RP n to S Ñ RP yield another fibrationΩ RP Ñ Z { Z Ñ S , see for example [May99, Section 5] (alternatively, this can be deducedfrom the homotopy lifting property). Iterating this process we obtain Ω S Ñ Ω RP Ñ Z { Z .In this situation, despite having a disconnected base Z { Z (and Ω RP has two components),the fibration does have homotopy equivalent fibres, an explicit homotopy equivalence being thefollowing. Let γ be a fixed based loop in RP representing the non-trivial homology class. Then,concatenation with γ defines a map from a component of Ω RP to the other; concatenating twicegives a map from each component to itself. Therefore, H ˚ p Ω RP q – H ˚ p Ω S q ‘ H ˚ p Ω S q , and H ˚ p Ω S q – k r x s , for x in degree 1. (The homology of based loop spaces of spheres is known([FHT12, p.235]), and can be obtained by applying the Serre spectral sequence). (cid:3) Remark 5.7.
The expected result (44) matches the computation of [FS05, Theorem 1.3] . CP -twist We find a Lefschetz fibration π : T ˚ CP Ñ C with smooth fibre a clean plumbing of 3-spheres T ˚ S S T ˚ S and three singular fibres (see Appendix B for the definition of clean Lagrangianplumbing). The three vanishing cycles appear as the two core components of the plumbing, andthe surgery thereof.Following the approach of Section 5, we build a symplectomorphism ϕ P Symp ct p T ˚ CP q on thetotal space of π : T ˚ CP Ñ C .We obtain strong evidence that the construction can be used as a local model for the projectivetwist τ CP (Proposition 6.12).6.1. Lefschetz fibration on T ˚ CP . Let p CP ˆ CP , ω F S ‘ ω F S q with homogeneous coordi-nates p x, y q : “ pr x : x : x s , r y : y : y sq on the first and second factor respectively. Fix ahypersurface Σ : “ ÿ i “ x i y i “ + Ă CP ˆ CP , (48)obtained as the image of the embedding of the Flag variety F l ã Ñ CP ˆ CP . Let s , s besections of O CP ˆ CP p , q . Consider a pencilΣ r λ : µ s : “ t λs p x, y q ` µs p x, y q “ u r λ : µ sP CP Ă CP ˆ CP . (49)A generic fibre of such a pencil is the three-fold Flag variety F l Ă C . Let Σ : “ s ´ p q – F l the smooth fibre “at infinity”. Then for a disc bundle D ε T ˚ CP , ε ą
0, there is a symplecto-morphism CP ˆ CP z Σ – D ε T ˚ CP (see Example 2.15). emma 6.1. The base locus B is diffeomorphic to the 3-point blow up of CP , B – CP CP ,equipped with its monotone symplectic form.Proof. The base locus is a symplectic manifold of dimension 4, and since it is obtained as theintersection of hyperplane sections of CP ˆ CP of bidegree p , q , its Chern class is a positiveclass by the adjunction formula. Hence B is a monotone Fano (or Del Pezzo) surface, i.e anon-singular projective algebraic surface with ample anticanonical divisor. Thus B is eitherisomorphic to CP ˆ CP or to the blow up CP r CP at r points in general position, for0 ď r ď CP ˆ CP in CP via the Segre embedding, and by Lemma 2.17, we know χ p B q “ χ p F l q ´ χ p CP ˆ CP q ` . (50)Now χ p CP ˆ CP q “ χ p CP q “ F l is known to have Euler charac-teristic χ p F l q “ “
6, so that χ p B q “
6, and hence B – CP CP . (cid:3) Given a generic pencil of p , q divisors (49), consider the map p “ s p x, y q s p x, y q : CP ˆ CP z Σ ÝÑ C . (51)Adjust the fibration p to an exact Lefschetz fibration E Ñ C that completes to a Lefschetzfibration Liouville isomorphic to π : T ˚ CP ÝÑ C . (52)In the next subsection we show that the fibre of π is (exact) symplectomorphic to a plumbingof 3-spheres along a circle. Remark 6.2.
Let f : CP Ñ r , s be a Morse function on CP with three critical points ofindices , , . Let g be a Riemannian metric such that p f, g q is a Morse-Smale pair. Johns( [Joh08, Sections 3, 4.1,4.2] ) constructs a Lefschetz fibration F : E Ñ D with three prescribedvanishing -spheres, fibres given by the plumbing of two of the vanishing cycles, and satisfying ‚ CP embeds as a Lagrangian submanifold in E . ‚ Crit p F q “ Crit p f q Ă CP and F | CP “ f : CP Ñ R . ‚ After smoothing the corners, the total space is conformally exact symplectomorphic to DT ˚ CP . Topology of the fibre: A MBL fibration on the affine Flag 3-fold.
In this section weenhance our understanding of the topology of the fibres of the Lefschetz fibration π : D ˚ CP Ñ C . The following construction is due to Jonny Evans. Let F l be the Flag 3-fold embedded in CP ˆ CP as in Example 2.15.Consider a pencil of divisors; Y r λ : µ s : “ t λ p x y ´ x y q ` µx y “ u r λ : µ sP CP Ă F l .The general hyperplane section of the Flag 3-fold (i.e a generic section of the pencil Y r λ : µ s ) is acopy of the del Pezzo surface CP CP . The base locus of this pencil is the set Θ : “ t x y “ x y “ x y “ u , composed of six lines L ijk “ t x i “ x j “ y k “ u , L ijk “ t y i “ y j “ x k u where ijk is a permutation of 012. For all i , let z i “ x i y i . Remove a smooth fibre Y and definea Morse-Bott Lefschetz fibration p : F l z Y Ñ C , pr x : x : x s , r y : y : y sq ÞÑ x y x y ´ x y . (53) he general fibre is isomorphic to p C ˚ q and there are three singular fibres over t , ˘ u isomor-phic to C ˚ ˆ p C _ C q ; each critical fibre has a singular locus of which is a copy of C ˚ . Lemma 6.3.
This construction admits three -dimensional matching spheres of the total space,which pairwise intersect cleanly along a circle.Proof. There is a Hamiltonian T -action on CP ˆ CP given by p θ, φ q ¨ pr x : x : x s , r y : y : y sq “ pr e iθ x : x : e iφ x s , r e ´ iθ y : y : e ´ iφ y sq . (54)This action preserves the fibres of p and exhibits three circles given by φ “ , θ “ φ “ θ , each of which collapses in one of the singular fibres. Each pair of critical values canbe connected by an arc, and parallel transport along such arc collapses the fibre C ˚ ˆ C ˚ into C ˚ ˆ p C _ C q at each endpoint, where one of the three circles mentioned above collapses. Thismeans that over each arc, there is a matching 3-sphere obtained as union of two Morse-Bottthimbles S ˆ D Y S ˆ S D ˆ S along S ˆ S Ă C ˚ ˆ C ˚ in a smooth fibre. Each pair ofmatching spheres meet in a critical fibre, cleanly along a circle: these intersection loci are thethree components of the critical locus. (cid:3) Lemma 6.4 (Evans) . The symplectic completion of the affine Flag 3-fold M : “ F l z Y issymplectomorphic to the Lagrangian plumbing T ˚ S S T ˚ S of three-spheres (which intersectcleanly along the circles of Lemma 6.3). See Appendix B for the definition of clean Lagrangianplumbing.Proof. The lemma is proved by defining a pluri-subharmonic function
F l z Y Ñ R with skeletonformed by the pair of Lagrangians 3-spheres over r´ , s and r , s , meeting cleanly in a circleover 0. (cid:3) Remark 6.5.
The plumbing in the Lemma does not coincide with the construction in [Tor20] .The difference lies in the choice of trivialisation of the normal bundle of S inside the twospheres, which in this case also determines the gluing of the two core components of the plumbing.In this case, the framing is such that the surgery of the two spheres along S is again sphere. Topology of the fibre: a bifibration.
Consider p , q -divisors on p CP ˆ CP , ω F S ‘ ω F S q given by a linear combination of monomials x y , x y , x y . This gives a rational map (in thiscase, a net ) CP ˆ CP (cid:57)(cid:57)(cid:75) CP (55) pr x : x : x s , r y : y : y sq ÞÝÑ r z “ x y : z “ x y : z “ x y s . (56)The base locus of this map is the set Θ as in Section 6.2, so consider the well-defined restriction q : CP ˆ CP z Θ Ñ CP . The generic smooth fibre of q is isomorphic to p C ˚ q ; this is obtained after quotienting p C ˚ q by the C ˚ -action C ˚ ¨ p x i , y i q “ p ζx i , ζ ´ y i q for all i P t , , u ζ P C ˚ .Over the coordinate lines t z i “ x i y i “ uztr s , r s , r su of CP , the fibrationis singular with singular fibres C ˚ ˆ p C _ C q . There are three such degenerations, one for eachfactor of p C ˚ q .For every generic line (cid:96) : “ t b z ` b z ` b z “ u , ( b i ‰
0) in the base CP , the restriction ofthe fibration over that line, q | (cid:96) , defines an open subset of the Flag 3-fold in CP ˆ CP (exactsymplectomorphic to the plumbing M ), that we denote by M (cid:96) . This yields a p C ˚ q -fibrationadmitting three singular fibres, over the points in which (cid:96) intersects the coordinate axes. Byconstruction, such a restriction is isomorphic to a Morse-Bott-Lefschetz fibration with threecritical fibres, of the type of p : F l z Y Ñ C encountered in Section 6.2. deformation of (cid:96) in a family (cid:96) r α : β s of lines in CP corresponds to a pencil of flag 3-folds in CP ˆ CP , and the restriction of q over (cid:96) r α : β s , r α : β s P CP defines a Lefschetz fibration π asin Section 6.1.In this family, consider a path δ p t q joining a generic line δ p q : “ (cid:96) gen to a line δ p q “ (cid:96) crit passing through either of r s , r s , r s ; in the total space this traces afamily of Morse-Bott-Lefschetz fibrations of the type of p , each with a configuration of threesingular fibres. As the path reaches (cid:96) crit “ δ p q , two of the critical loci of this configurationwill come together over the singular restriction q | (cid:96) crit and this produces a matching sphere forthe fibration, as described in Lemma 6.3. Namely, for any w P tr s , r s , r su , q ´ p w q “ p C _ C q ˆ p C _ C q : as two of the singular points of q come together, two of theoriginal factors in p C ˚ q collapse, and after quotienting by the C ˚ -action, the degenerations arerecognisable as ordinary double points. As the restriction q | (cid:96) crit defines a singular fibre for afibration of the type of π , the matching spheres of p (Lemma 6.3) coincide with the vanishingcycles of π . Remark 6.6.
Any smooth fibre q ´ p z q , z “ r z : z : z s with z i ‰ compactifies to q ´ p z qY Θ “ CP CP . Note that this coincides with the base locus of the pencil in Lemma 6.1. For theclosure of the singular fibres, see [Tyu11, Section 2] . Let Br “ x a, b | aba “ bab y the braid group on three strands. This group can be identified withthe fundamental group of 3-points configurations π p Conf p C qq .There is a surjective group homomorphism Br Ñ S into the symmetric group, with kernelgenerated by a , b , ab a ´ . This subgroup is called the pure braid group, denoted P Br : “x a , b , ab a ´ y Ă Br . Lemma 6.7.
There is a natural representation ρ : P Br Ñ π p Symp ct p T ˚ S S T ˚ S qq .Proof. This is a consequence of the local description above. Consider a line (cid:96) Ă CP in thebase space of q , and the affine Flag 3-fold M (cid:96) Ă CP ˆ CP z Θ with associated Morse-Bott-Lefschetz fibration M (cid:96) Ñ C coming from the restriction q | (cid:96) . As above, consider a deformationof (cid:96) in a family of lines (cid:96) r α : β s parametrised by r α : β s P CP . Parallel transport along a loopin (cid:96) r α : β s , based at (cid:96) and encircling one of the three critical values of the family q | (cid:96) r α : β s definesa symplectomorphism of M (cid:96) corresponding to the Dehn twist along a sphere in M (cid:96) (either oneof the cores of the plumbing, or the surgery thereof). In the base of the MBL fibration, thisoperation amounts to the action of an element of the pure braid group P Br . This showsthat there is a representation ρ : P Br Ñ π p Symp ct p T ˚ S S T ˚ S qq that sends each of thegenerators of P Br to one of the Dehn twists along the vanishing cycles of the original Lefschetzfibration π . (cid:3) Monodromy and generalised lantern relations.
Let π : T ˚ CP Ñ C be the Lefschetzfibration discussed in the previous subsections, with Crit v “ t w , w , w u . Let z ˚ P C be a base-point, which fixes the smooth fibre π ´ p z ˚ q – M isomorphic to a plumbing M : “ T ˚ S S T ˚ S .Choose a distinguished basis of vanishing paths p γ , γ , γ q and associated vanishing thimbles p ∆ , ∆ , ∆ q (so any two of them, ∆ j , ∆ k are disjoint and satisfy h p ∆ j q ą h p ∆ k q if j ą k ).Let p V , V , V q Ă M be the basis of vanishing cycles associated to the three singular points,from which we define the Dehn twists τ V , τ V , τ V P Symp ct p M q . The global monodromy of thefibration π is isotopic to τ V τ V τ V . Lemma 6.8.
The global monodromy satisfies the generalised lantern relation τ V τ V τ V » τ B P π p Symp ct p M qq (57) where τ B is the fibred twist in the unit normal bundle to base locus B “ CP CP Ă CP ˆ CP . b ab a ´ Proof.
The discussion of Section 2.4 applies, for the pencil defined in Section 6.1. (cid:3)
Recall that the monodromy can be made into a graded symplectomorphism, and that for thepencil we consider, the canonical bundle K CP ˆ CP – O CP ˆ CP p´ , ´ q satisfies K CP ˆ CP “ L b´ where L : “ O CP ˆ CP p , q . So if V Ă M is a graded vanishing cycle, we have φ p V q » V r´ s . Corollary 6.9.
The global monodromy commutes with every single twist τ V i for i “ , , .Proof. This follows from Lemma 6.8. Note that the same result can also be obtained by consid-ering the representation ρ : P Br Ñ π p Symp ct p T ˚ S S T ˚ S qq from Lemma 6.7, which sendsthe generators a , b , ab a ´ of the pure braid group to the Dehn twists τ V i along the 3-spheres V i , i “ , , (cid:3) A symplectomorphism ϕ P Symp ct p T ˚ CP q of infinite order. We define an elementof Symp ct p T ˚ CP q , following the instructions of Section 3. Using the techniques of Section 4.4,we compute the Floer cohomology of two vanishing thimbles, one of which acted upon by apower ϕ k . We find that the rank of the Floer cohomology is given by 2 k , which implies that ϕ has infinite order in Symp ct p T ˚ CP q . Comparing our outcome with results of [FS05] andpredictions from wrapped Floer cohomology computations, we make the following conjecture. Conjecture 6.10.
The symplectomorphism ϕ is isotopic to the CP -twist. Floer cohomology computations.
Let π : T ˚ CP Ñ C be the (completed) exact Lefschetzfibration (52) with smooth fibre isomorphic to the plumbing of spheres p M “ T ˚ S S T ˚ S , ω q and three vanishing 3-spheres p V , V , V q Ă M . By Lemma 6.8, the ungraded monodromypreserves the vanishing cycles, and the graded version shifts their grading as φ p V i q » V i r´ s .Then Corollary 6.9 grants the well-definedness of the construction of Section 3, that we use tobuild a compactly supported symplectomorphism ϕ P Symp ct p T ˚ CP q of the total space of theLefschetz fibration π : T ˚ CP Ñ C .To compute the Floer cohomology group HF p ϕ k p ∆ q , ∆ ; Z { Z q , we utilise the spectral sequenceof Proposition 4.4, and much of the discussion for the RP case (Section 5.3) applies.Over each intersection point z P I , the intersection of the thimbles is determined by the in-tersection of their associated vanishing 3-spheres p V z, , V z, q . The latter intersect cleanly in a ircle V z, X V z, – S . In a neighbourhood of the intersection locus, one can apply a pertur-bation by a Morse function with two critical points (one minimum and one maximum), so thatafter perturbing, the two vanishing cycles intersect in two points t q ´ , q ` u Ă π ´ p z q , and thelatter generate the Floer complex CF p V z, , V z, q in each fibre π ´ p z q . The generators t q ´ , q ` u represent non-trivial cocycles in CF p V z j , , V z j , q , since the clean intersection V z, X V z, has onecomponent, and by exactness, the results of [Po´z94, Theorem 3.4.11] apply (see also[Sei98a,Theorem 3.1]). It follows that the Floer cohomology of the vanishing cycles is the standardcohomology (up to a grading shift (cid:96) P Z discussed below)(58) @ z P I, HF ˚` (cid:96) p V z, , V z, ; Z { Z q – H ˚ p S ; Z { Z q – Z { Z ‘ Z { Z . By Lemma 4.5, we know that HF ˚ p ϕ p ∆ q , ∆ ; Z { Z q – Z { Z ‘ Z { Z generated by t q ´ , q ` u .On the other hand, HF p ϕ p ∆ q , ∆ ; Z { Z q “ Proposition 6.11.
The symplectomorphism ϕ P Symp ct p T ˚ CP q is a non-trivial element ofthe symplectic mapping class group. (cid:3) Let γ k “ b k π p γ q and ∆ k : “ ∆ γ k – ϕ k p ∆ q and I : “ γ k p R ` q X γ p R ` q “ t z , . . . , z k ´ u . Equipthe arcs in the base, the Lagrangian vanishing cycles in the smooth fibres and the Lagrangianthimbles and with a Z -grading as in Section 2.1.1. Set deg p z q “ p z (cid:96) q “ ´ (cid:96) for z (cid:96) P I . The elements t q ´ , q ` u Ă π ´ p z q of any generating pair of the complex CF ˚ p V z, , V z, ; Z { Z q must be shifted in their degrees asdeg p q ` q “ deg p q ´ q ˘
1. Namely, this shift is established by the Morse perturbation appliedabove (see [Sei98a, (4.4)]). Set deg p q ´ q “ ´
1, deg p q ` q “
0. This fixesHF ˚ p V z , , V z , ; Z { Z q – Z { Z , ˚ “ ´ , ˚ p V z, , V z, ; Z { Z q , forevery z P I . Inserting d “ ´ p k ´ q ď p ď E pq “ à z P I p HF p ` q p V z, , V z, q – à p HF p ` q ´ p p d ´ q p V z , , V z , q – à p HF q ´ p p V z , , V z , q . (59)The position of the generators in the first page indicate that the spectral sequence does notadmit any non-trivial differential. Therefore, the sequence collapses at the first page, and wehave HF p ∆ k , ∆ q – ‘ k ´ j “ HF p V z j , , V z j , q k . Theorem 6.12.
Let N k : “ t , . . . , k ´ u . The graded Floer cohomology groups are given by (60) HF ˚ p ϕ k p ∆ q , ∆ ; Z { Z q “ " Z { Z ˚ “ ´ N k , N k ´ . In particular, omitting the gradings, HF p ϕ k p ∆ q , ∆ ; Z { Z q – p Z { Z ‘ Z { Z q k . Corollary 6.13.
The symplectomorphism ϕ has infinite order in π p Symp ct p T ˚ CP qq . Remark 6.14.
The symplectic mapping class group π p Symp ct p T ˚ CP qq is not known, norwhether it is solely generated by the standard projective twist τ CP . For example, potential gen-erators could exotic CP -twists, not Hamiltonianly isotopic to τ CP . However, exotic projectivetwists have been observed only in dimensions n ě ( [Tor20] ), so we cannot draw furtherconclusions. Comparisons.
Let T ˚ q , T ˚ q Ă T ˚ CP be two cotangent fibres at q , q P CP . We canapply the observations of Section 44 to getlim ÝÑ k HF ˚ p Φ k π p ∆ q , ∆ ; Z { Z q – HW ˚ p ∆ , ∆ ; Z { Z q –– HW ˚ p T ˚ q , T ˚ q ; Z { Z q – HW ˚ p T ˚ q , T ˚ q ; Z { Z q . (61) ´ ´ ´ ´ Z { Z
00 0 0 0 Z { Z ´
10 0 Z { Z ´
20 0 Z { Z ´ Z { Z ´ Z { Z ´ q Equation 59.
The shape of the spectral sequence for k “ CP is not a Spin manifold, but since we work over a field of coefficient two, theisomorphism HW ˚ p T ˚ q , T ˚ q ; Z { Z q – H ´˚ p Ω CP ; Z { Z q from [AS06] still holds. We thereforeobtain the expected isomorphismlim ÝÑ k HF ˚ p Φ k π p ∆ q , ∆ ; Z { Z q – H ´˚ p Ω CP ; Z { Z q . The homology of the loop space Ω CP can be computed by considering the fibration Ω S Ñ Ω CP Ñ S obtained by “looping twice” the standard fibration S Ñ S Ñ CP and (as wedid in Seciton 5.3.1) and applying the Serre spectral sequence (see [FHT12, Chapter 16 (a)]). Lemma 6.15.
For a ground field k of characteristic , H ˚ p Ω CP q – k r x s ‘ k r y s{ y , where x has degree ´ , and y has degree ´ . (cid:3) Remark 6.16.
The Floer cohomology of a cotangent fibre T ˚ q of T ˚ CP twisted by a projectivetwist and another (untwisted) cotangent fibre T ˚ q Ă T ˚ CP was computed in [FS05, Theorem2.13] . Let q , q P CP be two distinct points and T ˚ q , T ˚ q P T ˚ CP their associated cotangentfibres. Then, according to [FS05] , HF p τ k CP p T ˚ q q , T ˚ q ; Z { Z q – p Z { Z ‘ Z { Z q k . (62) 7. Implementations
In this last section, we discuss a set of further developments (some more conjectural than others)catalized by our investigations. .1. The HP -twist. We can try to apply the theoretical core of this paper to illustrate aplausible local model for the HP -twist. To do this, an appropriate Lefschetz fibration withtotal space the cotangent bundle T ˚ HP is necessary.By [Joh08, Theorem 1] (see Rmk 6.2), there is an exact Lefschetz fibration π : T ˚ HP Ñ C with three singular fibres. The smooth fibre of such a fibration is a clean plumbing of spheres T ˚ S S T ˚ S along S Ă S , such that the 3-vanishing cycles of the fibration are the two7-spheres of the core of the plumbing and their surgery ([Joh08, Section 4.9]).The corresponding algebro-geometric picture (in which the fibration is induced by a Lefschetzpencil), should arise from a pencil in the space Gr p , q , the Grassmannian of 2-dimensionallinear subspaces of a 6-dimensional vector space (there is a Pl¨ucker embedding Gr p , q ã Ñ CP ). There is a polarised K¨ahler decomposition of Gr p , q as follows. Lemma 7.1. see e.g [Voe16, Lemma 4.4.1]
The space HP embeds as an open subvariety of Gr p , q , with closed complement the symplectic Grassmannian SpGr p , q (the Grassmannianof -dimensional symplectic subspaces of a -dimensional vector space). Assuming the existence of such a fibration, we can build an element ϕ P Symp ct p T ˚ HP q from theconstruction of Section 3, and run the same Floer theoretical computations with two Lefschetzthimbles ∆ , ∆ Ă T ˚ HP . Then, assuming the vanishing of all differentials of the spectralsequence (34) forced by degree reasons, the Floer cohomology groups would be given byHF p ϕ p ∆ q , ∆ q – k à i “ HF p V p i , , V p i , q – k à i “ H ˚ p S ; Z { Z q – p Z { Z ‘ Z { Z q k . (63)We could then compare this outcome with the local projective twist τ HP P π p Symp ct p T ˚ HP qq ,which behaves as follows. Lemma 7.2. [FS05]
Let x, y P HP be distinct points and T x , T y P T ˚ HP their associatedcotangent fibres. Then HF i p τ k HP p T x q , T y ; Z { Z q – p Z { Z ‘ Z { Z q k . (64)7.2. The RP - twist. In the same vein, it should be possible to extend our construction toprovide an adequate local model for the RP -twist.Consider a pencil of quadrics in CP , i.e sections of L “ O CP p q . Any smooth quadric in CP isisomorphic to CP ˆ CP , via the Segre embedding, and two quadrics in CP intersect genericallyin a 2-torus. Namely, by O CP p q| CP ˆ CP “ O CP ˆ CP p , q , the base locus has bidegree p , q ,and the genus formula g “ p d ´ qp d ´ q yields g “ B a divisor of bidegree p , q , and 4 singularfibres. The number of singular fibres follows from χ p CP q “ χ p CP ˆ CP q ´ χ p B q ´ r. and χ p B q “ D ε T ˚ RP , ε ą
0, there is a symplectomorphism CP zp CP ˆ CP q – D ε T ˚ RP . The fibration CP zp CP ˆ CP q Ñ CP derived from the pencil thereforeinduces (as in the previous situations) a Lefschetz fibration π : T ˚ RP Ñ C . (65)The smooth fibre of π is exact symplectomorphic to the completion of M : “ CP ˆ CP z B ,there are four singular fibres with associated Lagrangian vanishing spheres V , V , V , V Ă M . he global monodromy of the fibration π is a product of the twists along the 4 vanishingthree-spheres and satisfies a generalised lantern relation τ V τ V τ V τ V » τ V (66)where τ V is a fibred twist along ST ˚ RP Ñ B .The construction of Section 3 can be applied to the fibration π : T ˚ RP Ñ C above, to obtain acompactly supported symplectomorphism ϕ P Symp ct p T ˚ RP q . We conjecture that ϕ is isotopicto the projective twist τ RP P π p T ˚ RP q .7.3. Monodromy substitution and symplectic fillings of unit cotangent bundles.
Let π : E Ñ D be an exact Lefschetz fibration with fibre a Liouville domain p M, ω “ dλ q andmonodromy φ P Symp ct p M q . After rounding off the corners, the boundary B E admits a contactstructure compatible with an open book decomposition p M, λ, φ q with page M and monodromy φ .Assume π is induced by a Lefschetz pencil on a polarised symplectic manifold p X, ω X q withclosed fibre a polarised symplectic submanifold p Σ , ω Σ q and base locus a symplectic manifold p B, ω B q Ă p Σ , ω Σ q . The complement of the smooth divisor X z Σ is a Liouville domain p W, dα q ([Gir17]) exact symplectomorphic to the total space E , and Σ z B is exact symplectomorphic tothe smooth exact fibre p M, ω q .In this section we discuss an operation (called monodromy substitution ) which exploits thegeneralised lantern relation (20) to derive a Morse-Bott-Lefschetz fibration π : E Ñ D withfibre symplectomorphic to p M, ω q , boundary contactomorphic to B E but whose total space E is topologically distinguished from E .In the context of study, the contact boundary B E is a unit cotangent bundle, and such anoperation serves to produce an alternative filling of B E which differs from the standard disccotangent bundle.The monodromy φ P Symp ct p M q of the fibration π is a product of positive powers of Dehntwists, which, by the generalised lantern relation, is isotopic to a fibred Dehn twist φ » τ V along the coisotropic V “ B M . By the general results of Section 2.2, there is a MBL fibration π : E Ñ D with fibre (exact) symplectomorphic to p M, ω q and monodromy τ V P Symp ct p M q .The total space E of the MBL fibration is symplectomorphic to a disc bundle D ε N ˚ Σ { X , for ε ą L Ñ Σ with c p L q “ ´r B s obtained as the restriction L “ L ˚ | Σ , where L Ñ X is the line bundle satisfying c p L q “ r Σ s .The boundary of the total space of such a fibration is then modelled on the unit sphere bundleof L .The boundary of a MBL fibration π : E Ñ D with smooth fibre p M, ω “ dλ q and monodromy φ admits a contact structure compatible to the open book p M, φ, λ q ([Oba20, A.2]). Then thecontact boundaries B E and B E are both contactomorphic to the open book p M, λ, φ q .Therefore, the contact manifold defined by the open book p M, λ, φ q has two “natural” fillings:one given by the total space of the exact Lefschetz fibration E , and the other by the totalspace of the MBL fibration E . The former is an exact filling, while the latter is a strong filling([Oba20, Proposition A.3]) which in general contains rational curves. Theorem 7.3. [Oba20, Corollary A.4]
Let p M, ω “ dλ q be a Liouville domain, φ P Symp ct p M q .Suppose there is a collection of spherically fibred coisotropic submanifolds p V , . . . , V k q Ă M suchthat φ » τ V . . . τ V k in π p Symp ct p M q . Then, a contact structure compatible with the open book p M, λ, φ q is strongly fillable. (cid:3) he choice of identification of the monodromy, either to a product of twists, or to a fibred Dehntwist via the generalised lantern relation, gives rise to two (possibly distinguished) symplecticfillings, and passing from one identification to the other is called a monodromy substitution (see[EG10, EMVHM11]). Definition 7.4. [Oba20, 4.3]
Let π i : E i Ñ C , i “ , be two (Morse-Bott-) Lefschetz fibrationswith fibre a symplectic manifold p M, ω q and let p V i, , . . . , V i,k i q the collection of (coisotropic)vanishing cycles of π i for some distinguished basis. The two fibrations are related by a mon-odromy substitution if D ď j ď min t k , k u such that V ,j “ V ,j for any j ă j and there is anisotopy in π p Symp ct p M qq , τ V ,j ˝ ¨ ¨ ¨ ˝ τ V ,k » τ V ,j ˝ ¨ ¨ ¨ ˝ τ V ,k (then the global monodromiesare also isotopic). A monodromy substitution can be performed in a general context, and when applicable itindicates the presence of multiple fillings, potentially non-equivalent (see [Oba20, Example3.13, Proposition 4.2]). We will see applications in Sections 7.3.1, 7.3.2, 7.3.3.To examine the topology of the new filling, we need to consider the decomposition of thepolarised symplectic manifold p X, ω X q that generates the Lefschetz pencil/fibration.In the applications of this paper, we restrict to the special case in which p X, ω X q is a polarisedK¨ahler manifold to use the results of [Bir01] (summarised in Section 2.3).Moreover, in every example under study we are in a fortuitous setting in which the decomposition(2.13) has the shape X – Ξ Y D ε N Σ { X , (67)for a smooth Lagrangian Ξ Ă X . The Liouville domain p W, dα q is then exact symplectomorphicto a disc cotangent bundle p D ˚ ε Ξ , dλ Ξ q , and the boundary of the Lefschetz fibration is contac-tomorphic to the sphere cotangent bundle p ST ˚ Ξ , ξ std q , with the canonical contact structureobtained from the Liouville form ξ std : “ ker p λ Ξ | ST ˚ Ξ q .Then the monodromy substitution replaces the standard filling of p ST ˚ Ξ , ξ can q with the bundle D ˚ ε N Σ { X , dual to the normal bundle to the hypersurface Σ Ă X .7.3.1. Known example: two Stein fillings of ST ˚ RP . Consider the Lefschetz fibration π : T ˚ RP Ñ C of Section 5. After smoothing the corners, the boundary of the total spaceis the unit cotangent bundle p ST ˚ RP , ξ std q , which, equipped with the standard contact struc-ture, is contactomorphic to the lens space p L p , q , ξ S q , equipped with the structure inducedby the standard contact structure on S . By Theorem 7.3, the lantern relation τ V τ V τ V » τ d τ d τ d τ d (68)indicates that the lens space p ST ˚ RP , ξ std q admits two fillings. On one hand, the total space ofthe Lefschetz fibration π yields the standard filling, i.e a disc cotangent bundle of RP . The otherfilling is presented as the total space of a MBL (which in this case is also Lefschetz) fibration,with fibre isomorphic to a 4-punctured sphere, and monodromy given by the product of Dehntwists around the boundary circles of the fibre (the RHS of (68)). By the decomposition (9),the total space of the latter is given by a disc normal bundle to the quadric Σ Ă CP D ε N ˚ Σ { CP , ε ą
0, which in this case is isomorphic to O CP p´ q .Note that H ˚ p D ˚ RP ; Q q – H ˚ p B ; Q q , where B is the 4-ball. Therefore, as shown in [EG10],a monodromy substitution via the isotopy (68) can be geometrically interpreted as a rationalblowdown, in which a neighbourhood of a sphere with self intersection ´ p ST ˚ RP , ξ std q »p L p , q , ξ S q found in [McD90]. .3.2. Fillings of ST ˚ RP . We can apply the principles of Section 7.3 as done in the case ofthe unit cotangent bundle ST ˚ RP in 7.3.1, but using the Lefschetz fibration π : T ˚ RP Ñ C constructed in Section 7.2 instead. This serves to identify a “natural” choice of alternativesymplectic filling of ST ˚ RP . Theorem 7.5.
Let π : T ˚ RP Ñ C be the Lefschetz fibration (65) . The monodromy substitutionperformed using the relation (66) yields a strong symplectic fillling of the contact manifold p ST ˚ RP , ξ std q , that is not a Stein filling.Proof. As in the other examples, we use the lantern relation to perform a monodromy sub-stitution: between the Lefschetz fibration (restricted over the disc) π | D : T ˚ RP | D Ñ D withmonodromy τ V τ V τ V τ V and fibre M – p CP ˆ CP qz B , and a MBL fibration π : E Ñ D withfibre M and monodromy the fibred tiwst along ST ˚ RP Ñ B .The total space E is a strong filling of ST ˚ RP obtained as the normal bundle to CP ˆ CP Ă CP with reverse orientation, E – O CP ˆ CP p´ , ´ q . (69)The topology of E reveals that this filling is distinguished from the standard cotangent bundle: H p E ; Q q is of rank 2, while H p D ˚ RP ; Q q has rank zero. Moreover, the filling (69) is notexact as the zero section is embedded as a symplectic submanifold. (cid:3) Fillings of ST ˚ CP . In the spirit of the previous two examples, we can use the generalisedlantern relation (57) to obtain the following.
Proposition 7.6.
Consider the Lefschetz fibration π : T ˚ CP Ñ C as in (52) . The monodromysubstitution performed using the relation (57) yields a strong symplectic filling of the contactmanifold p ST ˚ CP , ξ std q which is not exact.Proof. Consider the restriction π : T ˚ CP | D Ñ D of the Lefschetz fibration (52) with fibreexact symplectomorphic to M – DT ˚ S DT ˚ S , B “ CP CP and monodromy φ . Afterrounding off the corners, π has contact boundary contactomorphic to p ST ˚ CP , ξ std q . Thelantern relation (57) can be used to operate a monodromy substitution (Section 7.3) to get aMBL fibration π : E Ñ D with fibre M and monodromy τ V , for V “ SN ˚ B { F l Ñ B . The totalspace E is a strong filling (Theorem 7.3) given by the dual bundle E “ D ε N F l { CP ˆ CP – O CP ˆ CP p´ , ´ q| F l (70)in that replaces the Lagrangian CP with O CP ˆ CP p´ , ´ q , the normal bundle over the Flagthreefold F l Ă CP ˆ CP with reverse orientation. This symplectic filling is not exact (theFlag 3-fold is embedded as a symplectic submanifold). (cid:3) Remark 7.7.
Recall that the Flag -fold is isomorphic to the projectivisation P p T hol CP n q . Ifwe consider the hypersurface singularity model of Appendix A, it is tempting to believe that thestrong filling of Theorem 7.6 corresponds to the holomorphic cotangent bundle that appears inthe small resolution of the singular central fibre r χ ´ p q – T ˚ hol CP n . This is however not the case.If we consider the holomorphic cotangent bundle with its standard K¨ahler structure ω K , i.e theone which renders the zero section S – CP Ă T ˚ hol CP as a symplectic (K¨ahler) submanifold,we have ω K | S ą . The boundary of a neighbourhood of the zero section is contactomorphic tothe unit cotangent bundle ST ˚ CP , and ω K cannot be of contact type when restricted to thisboundary. Namely, the Gysin sequence applied to the unit cotangent bundle ST ˚ CP Ñ CP yields H p ST ˚ CP q – H p CP q . Together with the above consideration implies that the cless r ω K s cannot vanish on ST ˚ CP . n the other hand, the correct K¨ahler form for the filling (70) is encoded in the central fibre (93) the blow up space Bl Z . Note that a similar phenomenon happens in the well-studied caseof the -fold node (see [STY02] ). Monotone Lagrangian submanifolds of T ˚ CP . As a last note we employ thefibrations we have constructed to shed light on monotone Lagrangian submanifolds of p T ˚ CP , λ T ˚ CP q . Inspired by the Clifford and Chekanov tori in T ˚ S , we study Lagrangian sub-manifolds obtained via parallel transport of the vanishing cycles of the fibration π : T ˚ CP Ñ C .The Lefschetz fibrations π : T ˚ AP Ñ C , A P t R , C u are natural cousins of the well-studiedLefschetz fibration on p T ˚ S , dλ T ˚ S q with smooth fibre p T ˚ S , dλ T ˚ S q , two critical fibres, onevanishing cycle V (the zero section of the smooth fibre) and a matching sphere correspondingto the zero section. This fibration arisises from a pencil of conics on p CP ˆ CP , ω F S ‘ ω F S q (whose base locus is a pair of points), by removing the conic at infinity, the holomorphicdiagonal CP – ∆ Ă CP ˆ CP .In the past two decades, exotic classes of monotone tori in p CP ˆ CP , ω F S ‘ ω F S q and p CP , ω F S q have been discovered. In this context, a torus is called exotic if it is not Hamil-tonian isotopic to the Clifford torus. The most prominent of such tori is the Chekanov torus([Che96], [CS]).By a result of Gadbled ([Gad13]), both in CP ˆ CP and CP , the Chekanov torus can beidentified with a torus obtain by applying Biran’s circle bundle construction ([Bir06]).Consider the decomposition of polarised K¨ahler manifolds (see Section 2.3) CP – D ε T ˚ RP Y C (71) CP ˆ CP – D ε T ˚ S Y ∆(72)where C Ă CP is the quadric at infinity and ∆ Ă CP ˆ CP is the holomorphic diagonal.Since the Chekanov tori are constructed as circle bundles over C and ∆ respectively, theymust survive in the complement of these divisors, giving rise to tori in T ˚ S and T ˚ RP .It is known that the Chekanov torus in T ˚ S admits a presentation as Lagrangian submanifoldof the total space of the Lefschetz fibration T ˚ S Ñ C above. In this picture, the torus emergesfrom flowing the vanishing cycle V by parallel transport over a loop that does not encirle anyof the critical values. The Chekanov torus is fibred by V over this loop. Question . Is there a “natural” analogue of the Chekanov torus in T ˚ CP ?Recall the decomposition CP ˆ CP – D ˚ CP Y Σ (73)where the divisor at infinity is the Flag 3-fold. We could attempt to answer Question 1 by usingBiran’s circle bundle construction over a three-sphere contained in Σ , however, unlike for theequator in S – ∆ Ă CP ˆ CP , in this case there is apriori no a preferred choice of sphere.Given a choice of S Ă Σ , the construction yields a trivial bundle, hence a Lagrangian S ˆ S Ă T ˚ CP . This is because any circle bundle is trivial iff its the Euler class vanishes, and for acircle bundle over S the Euler class would be an element of H p S ; Z q “ π : T ˚ CP Ñ C , as we do in the next section. .5. Lagrangian in the total space of the standard fibration on C . We begin by study-ing monotone Lagrangian submanifolds of C obtained by parallel transport in the total spaceof the “standard fibration” C Ñ C . Consider the standard fibration q : C ÝÑ C z ÞÝÑ z ` z ` z ` z (74)with smooth fibres q ´ p z q “ tp x, y q P C , | x | ´ | y | “ , x x, y y “ u – T ˚ S for z ‰ T ˚ S collapses to a point. The monodromyaround a loop in the base t ÞÑ e πit is then given by the Dehn twist along this vanishing cycle,i.e τ S P Symp ct p T ˚ S q . For z P C ˚ , let V z – S Ă q ´ p z q be the zero section of the fibre over z (a representative of the vanishing cycle).We construct two (distinct) families of Lagrangian S ˆ S Ă C as follows. Let σ : r , s Ñ C be a loop in the base and let T σ : “ ď z P Im p σ q V z Ă C . (75)This is a Lagrangian submanifold of the total space that can be obtained by flowing the vanishingcycle by parallel transport over the loop σ . If 0 P Im p σ q then T σ is an immersed four-sphere(with a nodal singualrity at 0). If, on the other hand, 0 R Im p σ q , we call T σ – S ˆ S of type1 if σ encloses the origin, and of type 2 otherwise. Figure 7.
Type 1 (left) and Type 2 (right) LagrangiansFor simplicity, consider the exact Lefschetz fibration f : E Ñ C obtained by cutting the thefibres into Liouville domains exact symplectomorphic to a disc cotangent bundles D r T ˚ S , r ą
0, and call the new fibration f : E Ñ C (see [Sei03, Section 1.2] for the original exact localmodel). Lemma 7.8.
Let R Im p σ q . The lagrangian T σ is monotone.Proof. Recall that a Lagrangian L Ă p
X, ω q is monotone if the the area homomorphism ω : π p X, L q Ñ R and the Maslov (index) homomorphism µ : π p X, L q Ñ Z are proportional, i.e @ u P π p X, L q , ω p u q “ cµ p u q . (76)By the homotopy long exact sequence for pairs π p C , T σ q – π p T σ q – Z , for T σ – S ˆ S Ă C .The two homomorphisms have to be proportional. (cid:3) The characterisation of pseudo-holomorphic discs we present below revolves around the use of asection count invariant associated to Lefschetz fibrations as defined in [Sei03, Section 2.1]. Let p J, j q be a pair of an almost complex structure J on C and an almost complex structure j on C , compatible with π (as in Definition 2.4).Let D Ă C be the disc in the base bounded by the loop σ , i.e such that B D “ Im p σ q , and considerpseudo-holomorphic sections of the restriction f | D , which are curves s : D Ñ E satisfying(1) @ z P D : f p s p z qq “ z . s pB D q “ s p σ q Ă T σ (i.e the section s has a boundary condition defined by T σ )(3) Ds p z q ` J p s q ˝ Ds p z q ˝ j “ M E { D : “ t s : D Ñ E satisfies p q , p q , p qu be the moduli space of p J, j q -holomorphic sec-tions satisfying the above. A Fredholm analysis can be adapted to this situation to show that M E { D is a smooth manifold (see [Sei03, Lemma 2.5], [Sei08, p.237]). The choice of almostcomplex structures forces J -holomorphic discs in the total space to project to j -holomorphicdiscs in the base, so by the open mapping theorem, the moduli space M E { D corresponds to themoduli space of J -holomorphic discs u : p D, B D q Ñ p E, T σ q . Moreover, M E { D can be identifiedwith the unit cotangent bundle ST ˚ S (see [Sei03, Lemma 2.16]). Lemma 7.9.
Assume the loop σ : S Ñ C bounding D Ă C is such that P D . Then theLagrangian T σ has minimal Maslov index .Proof. Let N T σ be the minimal Maslov number associated to T σ Ă C . The moduli space M E { D of J -holomorphic discs s : p D, B D q Ñ p X, T σ q with boundary in the homotopy class ofthe generating loop of π p L q has expected dimension dim p M E { D q “ ` N T σ ´
3. By theobservation above, we know dim p M E { D q “
5, so that N T σ “ (cid:3) Lemma 7.10.
Assume the loop σ : S Ñ C bounding D Ă C is chosen such that R D . Thenthe Lagrangian T σ has minimal Maslov index .Proof. Since 0 R D , the restriction f | D : E | D Ñ D can be trivialised so that E | D “ f ´ p D q – D r T ˚ S ˆ D . The only non-trivial discs are generated by the base loop σ onto which T σ projectsto (there is no contribution from the smooth fibre since T σ | z : “ T σ X f ´ p z q Ă f ´ p z q – T ˚ S are both simply connected). Any disc generated this way has Maslov index 2. (cid:3) Corollary 7.11.
The two Lagrangians are distinguished by their Maslov index, hence they areneither Lagrangian isotopic, nor Hamiltonian isotopic. (cid:3)
Figure 8.
Deformation form a Clifford type Lagrangian (top left) to a Chekanovtype Lagrangian (bottom). Any such deformation is monotone, and there canbe no wall crossing (which would normally occur in the top right stage).7.6.
Monotone Lagrangian S ˆ S Ă T ˚ CP . Let π : T ˚ CP Ñ C the Lefschetz fibration(52), with smooth fibre M – T ˚ S S T ˚ S , critical values Crit v p π q “ t w , w , w u and van-ishing cycles V , V , V . For i “ , ,
4, set V z,i to be the representative of V i in the fibre π ´ p z q .For each critical value w i P Crit v p π q there is a pair of Lagrangians S ˆ S Ă T ˚ CP as follows. efinition 7.12. For i P t , , u , fix w i P Crit v p π q . Let σ : S Ñ C be a loop in the base ofthe fibration π : T ˚ CP Ñ C , bounding a disc D with B D “ Im p σ q , Crit v p π q X Im p σ q “ H and w j R D if j ‰ i .There is a monotone Lagrangian T σ,w i : “ ď z P Im p σ q V z,i Ă T ˚ CP (77) obtained by flowing the vanishing cycle V i under parallel transport around the loop σ .We say T σ,w i is(1) of Type if w i P D ,(2) of Type if w i R D . Lemma 7.13.
Fix w i P Crit v p π q . Let σ , σ : S Ñ C be two loops as in Definition 7.12, andassume that the Lagrangian T σ ,w i is of type , and T σ ,w i is of type , both associated to thesame critical value. Then the Lagrangians T σ and T σ are not Lagrangian isotopic.Proof. Consider the homotopy long exact sequence for the pair p X, L q “ p T ˚ CP , S ˆ S q .Exactness of X and monotonicity of L imply that the Maslov homomorphism µ : π p X, L q Ñ Z descends to a map µ : π p L q Ñ Z , which represents an element of H p L ; Z q . Therefore, we canuse the considerations of the previous section to prove the claim. (cid:3) ppendix A. CP n -singularities We illustrate another model in which the complex projective twist arises as monodromy of anisolated hypersurface singularity. This is taken from [Man07, Sections 2.1 and 4.3].The “local model” of this construction is given by the following subspace of sl n ` p C q : Z : “ t A P sl n ` p C q| D t P C s.t dim p ker p A ´ tId qq ě n u . (78)Elements of Z are traceless p n ` q ˆ p n ` q´ matrices that have an eigenspace of dimension atleast n . Endow Z with the restriction of the standard symplectic form Ω std : “ ω C p n ` q . Themap χ : Z Ñ C , A ÞÑ t (79)that sends a matrix in Z to its eigenvalue with highest multiplicity (at least n ) is a fibrationwith smooth fibres Z t “ χ ´ p t q , t P C ˚ , and a central fibre Z “ χ ´ p q admitting a singularityat the zero matrix. Remark A.1.
Let C ˚ act on C n ` by ζ ¨ p x, y q “ p ζ ¨ x, ζ ´ ¨ y q , p x, y q “ p x , x , . . . , x n , y , . . . y n q P C n ` . (80) Then Z can be identified with the GIT quotient Z – C n ` { C ˚ ( [Man07, Lemma 4.4] ), wherethe quotient map is given by f : C n ` Ñ Z, f p x, y q “ x T y ´ n p x ¨ y q Id. (81)
This is also isomorphic to the symplectic quotient H ´ p q{ S , where H : C n ` Ñ R , H p v, w q “| v | ´| w | is the associated Hamiltonian (moment map), see [Man07, proof of Lemma 4.8] . Fromthis perspective, we see dim p Z q “ n ` . Lemma A.2. [Man07, Lemma 4.5, 4.8]
Any smooth fibre Z t , t P C ˚ contains a vanishingLagrangian CP n .Sketch of the proof. Given t P C ˚ , define Y t : “ diag p t, t, . . . , t, ´ nt q P Z t . Conjugating Y t with an element in U p n ` q yields another matrix in Z t , so the conjugacy class U t : “ t U Y t U ´ | U P U p n ` qu (82)is contained in the fibre Z t . By standard Lie theory, this orbit is isomorphic to U t – U p n ` q{p U p n q ˆ U p qq – CP n . Namely, all the matrices in U t have one eigenvalue of multiplicity n and one simple eigenvalue and elements in U p n q ˆ U p q Ă U p n ` q stabilise such matrices underthe adjoint action.The subspace U t is Lagrangian, as the following reasoning shows. Let γ : r , s Ñ C , γ p s q “p ´ s q t be a path connecting t to 0 in the base of the fibration (79), and consider the paralleltransport map (well defined away from Z ) P γ | r ,w s : Z γ p q Ñ Z γ p w q . Then U t – ! A P Z t , P γ | r ,w s p A q is defined near A for all w ă , lim w Ñ P γ | r ,w s p A q Ñ ) (83)which implies that U t is a vanishing cycle; its image is transported to the singular point under P γ . The isomorphism (83) is proven by showing that the vector field associated to P γ at A P Z t is proportional to the tangent map D χ p A q – see [Man07, Lemma 4.5 and Lemma 4.8] for thedetails. (cid:3) emark A.3. Note that the map χ ˝ f : C n ` Ñ C is a reparametrisation of the standardmodel z “ p x, y q ÞÑ ř n ` i “ z i , so there are parallel transport maps whose vanishing cycles are n ` spheres. These cycles are invariant under the Hamiltonian S action, so Z – H ´ p q{ S implies that the vanishing cycles (83) associated to the parallel transport in Z can be seen asthe quotients of the vanishing spheres of the standard model. A.0.1.
Desingularising by small resolution.
We explain how to resolve the singularity of Z ina way that is compatible with the whole family χ ; by performing(1) A small resolution, to obtain a space r χ : r Z Ñ C that has smooth fibres r χ ´ p t q “ T ˚ CP n (with Lagrangian zero fibres) and central fibre r χ ´ p q “ T ˚ hol CP n (holomorphic zerosection)(2) A blow up in the zero section of the central fibre (the holomorphic CP n ).In [Man07, Sections 2.1-2.2], the author shows that χ has a simultaneous resolution that fits inthe following diagram r Z r χ (cid:15) (cid:15) p (cid:47) (cid:47) Z χ (cid:15) (cid:15) C (cid:47) (cid:47) C (84)The resolution r Z is constructed as follows. Let F p Z q be the flag variety associated to thepartition p n, q ; to any element in A P Z with eigenvalues λ i and eigenspaces L i i “ , F “ p Ă V “ L Ă V “ L ‘ L “ C n ` q P F p Z q . where p resolves fibrewise the singularities of χ .The space(85) r Z : “ tp X, F q , X P Z, F P F p Z q , XV i Ď V i , X acts on V i { V i ´ with weight λ i u projects to Z through the forgetful map p : r Z ÝÑ Z p A, F q ÞÑ A. Proposition A.4. [Man07, Proposition 2.2]
The diagram (84) is a simultaneous resolution of χ , and r χ is a differentiable fibre bundle. See [Man07] (the proof of this result is beyond the scope of the discussion).
Lemma A.5.
For t P C ˚ , there is an isomorphism r χ ´ p t q – χ ´ p t q . On the other hand, thecentral fibre r χ ´ p q is isomorphic to T ˚ hol CP n .Proof. For any A P Z with eigenvalues different from zero (not contained in the central fibre χ ´ p q ), the preimage p ´ p A q consists of pairs p A, F q for flags F P F p Z q preserved by A . Thereis only one such flag, which means that for any t P C ˚ , we have r χ ´ p t q – χ ´ p t q . The central fibre χ ´ p q is made of nilpotent matrices in Z ( λ i “ i “ , n .This means that r χ ´ p q consists of pairs p A, F q P r Z where any matrix A is either the zeromatrix or a matrix of rank 1, similar to a block diagonal matrix with one Jordan-block of size2 and n ´ A , the associated flag F P F p Z q can be any element in the flag variety.In our case the flag variety is isomorphic to the Grassmannian of n ´ dimensional subspaces of C n ` , which corresponds to CP n . s a complex projective variety, the flag variety F p Z q – CP n is a holomorphic subvariety ofthe central fibre r χ ´ p q .Then, the projection(86) π : r χ ´ p q ÝÑ F p Z q – CP n p A, F q ÞÑ F is the holomorphic cotangent bundle T ˚ hol CP n .Namely, over any F “ p Ă V Ă V “ C n ` q P F p Z q π ´ p F q “ tp A, F q P r χ ´ p q , A P End p C n ` q : AV i Ď V i ´ u – Hom p V { V , V q . It is then known that
Hom p V { V , V q is isomorphic to a fibre T ˚ F CP n of the cotangent bundle. (cid:3) The picture that one should keep in mind is the following. The “vanishing” CP n ’s arising asLagrangian zero sections of the smooth fibres χ ´ p t q – r χ ´ p t q limit to the holomorphic CP n inthe resolved central fibre r χ ´ p q – T ˚ hol CP n .We then perform a blow up of r Z at CP n Ă r χ ´ p q , that we denote by ˆ Z : “ Bl CP n r Z ˆ Z ˆ χ (cid:15) (cid:15) ˆ p (cid:47) (cid:47) r Z r χ (cid:15) (cid:15) p (cid:47) (cid:47) Z χ (cid:15) (cid:15) C (cid:47) (cid:47) C (cid:47) (cid:47) C (87)The submanifold CP n Ă r Z is a smooth, so the exceptional divisor of ˆ Z is given by the pro-jectivisation of its normal bundle ν CP n { r Z – T ˚ hol CP n ‘ O where O is the trivial bundle (thecomponent of the normal bundle in the base direction). On the other hand, the restriction ofthe exceptional divisor to the central fibre is given by E – P p T ˚ hol CP n q (88)( E is the exceptional divisor of the blow up of the central fibre at the holomorphic zero sec-tion, Bl CP n r χ ´ p q ). In the resolved central fibre, we therefore have a S -bundle ST ˚ CP n Ñ P p T ˚ hol CP n q (see for example [BT13, Chapter IV]).The boundary of a tubular neighbourhood of the Lagrangian zero section of a smooth fibre iscontactomorphic to ST ˚ CP n , and since the transformations we have performed are local at thesingularity 0 P Z , so is the boundary of a neighbourhood of E Ă ˆ χ ´ p q .A.0.2. Desingularising via blow up.
Another way of resolving the singularity 0 P Z is to directlyblow up Z in this singular point. To perform this (singular) blow up, we need a set of linearequations describing Z Ă C p n ` q ; we do this using linear algebra facts. Lemma A.6.
Let z “ p z q ij be a matrix of any dimension. The rank of z is at most if andonly if all the ˆ minors of z vanish. (cid:3) Recall Z : “ t A P sl n ` p C q| D t P C s.t dim p ker p A ´ tId qq ě n u Ă C p n ` q . For A P Z , the matrix z “ A ´ t ¨ Id has rank at most one (for any allowed eigenvalue t of A )this implies that all the p ˆ q minors of such matrices must vanish, and the space Z can bedefined by a set of linear homogenous equations in degree two. Recall that non-zero eigenvaluescome in pairs t “ ´ t , and rank p A ´ t i ¨ Id q “
1, while if t “
0, rank p A q “ o blow up Z at the zero matrix, we first the blow up the ambient space C p n ` q at 0 P C p n ` q ;denote that by Bl C p n ` q . The exceptional divisor of this blow up is a copy of CP p n ` q ´ .Then, the exceptional divisor of the blow up Z “ Bl Z is obtained by restricting E to the set oflinear equations given by the vanishing of the minors for the eigenvalue t “
0; this correspondsexactly to the image of CP n ˆ CP n into CP p n ` q ´ under the Segre embedding. Definition A.7.
Recall the Segre embedding is defined as
Σ : CP m ˆ CP n Ñ CP p m ` qp n ` q´ (89) pr x : x : ¨ ¨ ¨ : x m s , r y : y : ¨ : y n sq ÞÑ r x y : x y : ¨ ¨ ¨ : x i y j : ¨ ¨ ¨ : x m y n s By identifying the coordinates of CP p m ` qp n ` q´ with the entries z ij “ x i y j of an p n ` q ˆp m ` q -matrix z , we can write the embedding as (90) ¨˚˝ z ¨ ¨ ¨ z n ... ... z m ¨ ¨ ¨ z mn ˛‹‚ “ ¨˚˝ x ... x m ˛‹‚ b ` y ¨ ¨ ¨ y n ˘ The description above indicates that the image of the Segre embedding admits a presentation (90) as a matrix obtained by the Kronecker product of two vectors. In particular, this impliesthat the image Σ p CP m ˆ CP n q Ă CP p m ` qp n ` q´ can be thought of as the set of p m ` q ˆ p n ` q matrices with rank at most . Lemma A.8.
The exceptional divisor of the blow up Z “ Bl Z is CP n ˆ CP n .Proof. The exceptional divisor of this blow up is the set of matrices with the property that all2 ˆ CP p n ` q ´ . This is precisely the image ofthe Segre embedding of Σ p CP n ˆ CP n q Ă CP p n ` q ´ . (cid:3) The normal bundle to the exceptional divisor E “ CP n ˆ CP n is the restriction Bl p C p n ` q q| E – O CP n p´ q b O CP n p´ q , and we have a diagram O CP n p´ q b O CP n p´ q (cid:15) (cid:15) O CP n p´ q b C n ` (cid:15) (cid:15) CP n ˆ CP n p m,n (cid:47) (cid:47) CP n (91)The blow up Z “ Bl Z is the endspace ˆ Z obtained after performing 1) and 2) in the previoussubsection, and blowing down one of the CP n factors of CP n ˆ CP n Ă Z yields a small resolutionof Z . There are two such resolutions, one of which corresponds to r Z , and another one that wecall Z . Denote by b i the blow-up maps associated to pr i . Zs.res b s.res r Z b Z “ Bl Z b Z In the diagram, the explanations above translate in the relation b “ s.res i ˝ b i , i “ ,
2. Thedashed arc should be the analogue of a Mukai flop (see [ADM19]).Therefore, the central fibre of Bl Z can be seen as the reducible variety O CP n ˆ CP n p´ , ´ q| P p T ˚ hol CP n q Y P p T ˚ hol CP n q CP n ˆ CP n . (93) Appendix B. Clean Lagrangian plumbings
We construct the clean plumbing of two Riemannian manifolds Q , Q along a submanifold B Ă Q i , i “ , B, Q , Q closed smooth manifolds,for each i “ , B ã Ñ Q i and an isomorphism (cid:37) : N B { Q Ñ N ˚ B { Q . Pick aRiemannian metric on B , an inner product and a connection on N B { Q – N ˚ B { Q (which inducesan inner product and connection on N B { Q ). This data induces a metric on the total spaces N B { Q i , and a nieghbourhood U i of B Ă Q i can be identified with a disc subbundle D (cid:15) N B { Q i ofradius (cid:15) ą
0. With this identification we write x P U i as x “ p a, b q for b P B , a P D (cid:15) p N B { Q i q b (the fibre over b ).For each x “ p a, b q P U i , the connection gives a decomposition of the fibres T ˚ x Q i – T ˚ b B ‘p N ˚ B { Q i q b . We get an identification of a neighbourhood of B Ă T ˚ Q i as D (cid:15) N B { Q i ‘ D (cid:15) T ˚ B ‘ D (cid:15) N ˚ B { Q i . (94)Let V i be a bounded neighbourhood of Q i Ă T ˚ Q i which in (94) coincides with D (cid:15) T ˚ B ‘ N ˚ B { Q i over U i – D (cid:15) N B { Q i . Definition B.1. (1) As a smooth manifold, the clean plumbing M : “ T ˚ Q B T ˚ Q of Q , Q along B is defined by gluing V to V along D (cid:15) N B { Q ‘ D (cid:15) T ˚ B ‘ D (cid:15) N ˚ B { Q Ă V identified with D (cid:15) N B { Q ‘ D (cid:15) T ˚ B ‘ D (cid:15) N ˚ B { Q via (cid:37) , ´p (cid:37) ˚ q and id T ˚ B .(2) The plumbing construction inherits an exact symplectic structure, since the identificationmaps of p q preserve the canonical structures on T ˚ Q i . Let Z i be the standard radialLiouville vector field on V i . We define a Liouville vector field Z on the plumbing byletting Z “ ρ Z ` ρ Z , for smooth functions ρ : M Ñ r , s supported on V i . Thenthis endows the plumbing M with an exact symplectic structure. The submanifolds Q i Ă M are embedded as exact Lagrangians, and M retracts to the union Q Y B Q . Remark B.2.
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Brunella Torricelli, Centre for Mathematical Sciences, University of Cambridge, CB3 0WB, UK
E-mail address : [email protected]@cam.ac.uk