A Log PSS morphism with applications to Lagrangian embeddings

# A Log PSS morphism with applications to Lagrangian embeddings

Sheel Ganatra and Daniel Pomerleano
###### Abstract.

Let be a smooth projective variety and an ample normal crossings divisor. From topological data associated to the pair , we construct, under assumptions on Gromov-Witten invariants, a series of distinguished classes in symplectic cohomology of the complement . Under further “topological” assumptions on the pair, these classes can be organized into a Log(arithmic) PSS morphism, from a vector space which we term the logarithmic cohomology of to symplectic cohomology. Turning to applications, we show that these methods and some knowledge of Gromov-Witten invariants can be used to produce dilations and quasi-dilations (in the sense of Seidel-Solomon [Seidel:2010uq]) in examples such as conic bundles and open parts of isotropic Grassmannians. In turn, the existence of such elements imposes strong restrictions on exact Lagrangian embeddings, especially in dimension 3. For instance, we prove that any exact Lagrangian in a complex 3-dimensional conic bundle over must be diffeomorphic to or a connect sum .

S. G.  was partially supported by the National Science Foundation through a postdoctoral fellowship — grant number DMS-1204393 — and agreement number DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
D. P. was supported by Kavli IPMU, EPSRC, and Imperial College.

## 1. Introduction

Let be a smooth projective variety, an ample strict (or simple) normal crossings divisor and its complement. The affine variety has the structure of an exact finite-type convex symplectic manifold and hence one can associate to its symplectic cohomology, [Floer:1994uq, Cieliebak:1995fk, Viterbo:1999fk], a version of classical Hamiltonian Floer cohomology defined for such manifolds. Symplectic cohomology has a rich TQFT structure and plays a central role in understanding the Fukaya categories of such [Seidel:2002ys, Abouzaid:2010kx] as well as various approaches to mirror symmetry [MR3415066].

While there are very few complete computations of symplectic cohomology in the literature, Seidel [Seidel:2002ys, Seidel:2010fk] has suggested that symplectic cohomology for as above should be computable in terms of topological data associated to the compactification and (relative) Gromov-Witten invariants of . In the case when is smooth, this proposal has been explored in some depth by Diogo and Diogo-Lisi [diogothesis, diogolisi].

The first part of this work introduces an approach to studying the symplectic cohomology in terms of the relative geometry of the compactification (when is normal crossings). In §3.2 we introduce an abelian group which captures the combinatorics and topology of the normal crossings compactification, called the log(arithmic) cohomology of

 (1.1) H∗log(M,D).

In the special case that is a smooth divisor, the log cohomology has a simple form:

 (1.2) H∗log(M,D)=H∗(M∖D)⊕tH∗(SD)[t]

where denotes the unit normal bundle to . In general, is generated by classes of the form , where lies in the cohomology of certain torus bundles over open strata of and is a multiplicity vector.

Our approach is inspired by a map introduced by Piunikhin, Salamon, and Schwarz [Piunikhin:1996aa] relating the (quantum) cohomology of and the Hamiltonian Floer cohomology of a non-degenerate Hamiltonian. Namely, we define a linear subspace of admissible classes together with a map

where is the positive or high energy part of symplectic cohomology, a canonically defined quotient of the complex defining by “low energy (cohomological) generators.”

There is a wide class of topological pairs for which the map (1.3) is particularly well behaved (e.g., take any projective variety and the union of generic hyperplane sections). For a topological pair, , and there is a canonical lifting of (1.3) to a map we term the Log PSS morphism

 (1.4) PSSlog:H∗log(M,D)→SH∗(X)

The key feature of topological pairs is that relevant moduli spaces of relative pseudo-holomorphic curves are all generically empty; for example, there should be no holomorphic spheres in which intersect a single component of in one point. See §3.3 for the precise definition of a topological pair.

###### Remark 1.1.

In a sequel to this paper [DPSGII], we will define a natural ring structure on and show that the Log PSS morphism (1.4) is a ring isomorphism for all topological pairs , leading to numerous explicit new computations of symplectic cohomology rings.

When the pair is not topological and is an admissible class, we formulate the obstruction to lifting from to in terms of a Gromov-Witten invariant associated to . When this obstruction vanishes, this provides a way to produce distinguished classes in which we will show may be applied to study the symplectic topology of .

It should be noted that the construction of the Log PSS morphism (1.4) and the proof of the chain equation are considerably more involved than for its classical analogue. We conjecture that for general , a further analysis of the moduli spaces appearing in (1.4) could produce an isomorphism between and the cohomology of a cochain model of log cohomology equipped with an extra differential accounting for non-vanishing relative Gromov-Witten moduli spaces. Such analysis would require stronger analytic results to deal with the subtle compactness and transversality questions which arise (related issues already arise in the study of relative Gromov-Witten invariants relative a normal crossings divisor). Our obstruction analysis is indeed inspired by this conjectural theory, and can be viewed as an elementary special case.

The rest of this paper explores applications of our construction to Lagrangian embeddings

A primary class of examples we consider are conic bundles over . Let be a generic Laurent polynomial in -variables and denote the zero locus of . Set to be the affine variety given by

 (1.5) X={(u,v,w)∈C2×(C∗)n−1|uv=h(w)}

For simplicity, we assume that is connected. The symplectic topology of these varieties (and the closely related conic bundles over ) is very rich and can be approached from different perspectives, see for instance [AAK, catdyn, KS]. For example, there is a standard construction of Lagrangian spheres in given by taking a suitable Lagrangian disc with boundary on the discriminant locus and “suspending” it to a Lagrangian [HIV, GM, SeidelSus].

It is natural to ask: what are the possible topologies of exact Lagrangians in these conic bundles? The suspension construction typically provides a large collection of Lagrangian spheres and Seidel [catdyn]*Chapter 11 has provided constructions of exact Lagrangian tori in certain examples. Our first application is a relatively complete classification of the diffeomorphism types of exact Lagrangian submanifolds in 3-dimensional conic bundles over .

###### Theorem 1.2.

(See Theorem 6.18) Let be a 3-dimensional conic bundle over of the form (1.5), and let be a closed, oriented, exact Lagrangian submanifold of . Then is either diffeomorphic to or (by convention, the case corresponds to ).

By combining our methods with those from [Seidel:2014aa], we also prove the following result concerning disjoinability of Lagrangian spheres:

###### Theorem 1.3.

(See Theorem 6.16) Let be a field and be an odd integer. Suppose that is a conic bundle of the form (1.5) of total dimension over and that is a collection of embedded Lagrangian spheres which are pairwise disjoinable. Then the classes span a subspace of which has rank at least .

An important special case is when , in which case the theorem immediately implies that for any embedded Lagrangian sphere , the class is non-zero and primitive (Corollary 6.17).

For our final application, we study a question posed by Smith concerning the “persistence (or rigidity) of unknottedness of Lagrangian intersections.”

###### Question 1.4 (Smith).

In a symplectic manifold , if a pair of Lagrangian 3-spheres and meet cleanly along a circle, can the isotopy class of this knot (in and ) change under (nearby) Hamiltonian isotopy?

Note that there is no smooth obstruction to changing the knot types. In [evanssmithwemyss], Evans-Smith-Wemyss study the case where is unknotted in both factors under an additional assumption on the identification of normal bundles

 η:νS1S1≅νS1S2

induced by the symplectic form on . They prove in this setting that if there is a nearby Hamiltonian isotopy of and of so that continues to meet cleanly along a knot, then this knot must be the unknot in one component and the unknot or trefoil in the other; see Proposition 6.22 for a precise statement. In §6.4, under the same assumptions on the intersection of and , we rule out the remaining trefoil case, while also giving an alternative proof of Proposition 6.22, thereby answering Question 1.4 negatively in this case. Our main result, Proposition 6.23, is somewhat stronger and rules out any pair of Lagrangian 3-spheres meeting cleanly in a non-trivial knot in a standard “plumbing” neighborhood of .

To explain how the results about conic bundles and knottedness above connect to our discussion of Log PSS, note that the morphism gives a method for producing distinguished classes in , and algebraic relations between them111Much like the classical PSS morphism, in many cases intertwines topological/GW-type relations in with TQFT-structure relations in . This will be explored further in the sequel [DPSGII].. It is well-understood, going back to ideas of Viterbo, that the existence of solutions in satisfying certain relations often produce strong restrictions on Lagrangian embeddings [Viterbo:1996kx, Seidel:2010uq, Seidel:2014aa]. Concretely, let denote an exact Lagrangian embedding of a closed Spin manifold. The well-known Viterbo transfer map [Viterbo:1996kx, Ritter] gives a unital BV morphism from to the symplectic cohomology of a neighborhood of , , which in turn can be identified [Viterbo:1996kx, Salamon:2006ys, AbSch1, AbSch2, Abouzaid:2015ad] with the string topology BV algebra of , , a topological invariant [Chas:aa]. Thus, one can transfer distinguished elements to distinguished elements in , the existence of which in turn restricts the topology of .

As an example, a dilation [Seidel:2010uq] is an element satisfying , where denotes the BV operator. A straightforward application of Viterbo’s principle (along with a string topology computation) implies that if admits a dilation, does not contain an exact Lagrangian [Seidel:2010uq, Cor. 6.3]. It should be noted that the existence of dilations has other implications for Lagrangian embeddings as well, for example [Seidel:2014aa] has used dilations to study disjoinability questions for Lagrangian spheres. Weakening the above condition, a quasi-dilation ([catdyn]*page 194, where the definition is attributed to Seidel-Solomon) is a pair where is invertible and . The existence of quasi-dilations imposes slightly weaker, but nevertheless quite strong constraints on the topology of . We make these constraints very explicit when , using techniques from 3-manifold topology:

###### Proposition 1.5.

(Corollary 5.9) Let be an exact Lagrangian embedding of a closed oriented 3-manifold. If admits a quasi-dilation, then is finitely covered by a product or by a connected sum .

When the quasi-dilation lifts to , it is possible to refine Proposition 1.5 somewhat; see Corollary 5.9. Seidel [catdyn]* Lecture 19 has shown that quasi-dilations have similar implications for disjoinability questions as dilations. As usual, the chief difficulty in applying these results to restrict exact Lagrangian embeddings is performing partial computations of the algebra structure in symplectic cohomology to produce quasi-dilations. Using mirror symmetric considerations, Seidel [catdyn]*first paragraph of Lecture 19 has suggested the following result which we prove.

###### Theorem 1.6.

(Theorem 6.9) Let be the conic bundle appearing in (1.5). Then admits a quasi-dilation over .

In fact, our result here is more general and applies for example to standard conic fibrations over more general bases. Theorem 1.2 is proven by combining Theorem 1.6 with some additional algebraic relations which exist in the zeroth symplectic cohomology group and which enable us to rule out all of the remaining possibilities not excluded by Lemma 5.7. Following Theorem 6.18, we provide examples of exact Lagrangian embeddings for (these arise by essentially replacing the disc in the suspension construction by a suitable exact embedding of , a sphere minus -discs). Theorem 1.3 follows by combining Seidel’s results [Seidel:2014aa] with Theorem 1.6. An essential ingredient in this theorem is the specific geometry of the construction of the quasi-dilation, which is what enables us to interpret the constraints imposed by disjoinability in terms of the topology of .

We expect that our techniques apply in many other geometric situations as well. For example, we also study the case where is a Fano variety of dimension and is a smooth very ample divisor satisfying and with . Under hypotheses on the cohomology and Gromov-Witten invariants of such , we show using that admits a dilation, see Theorem 6.1 below. The theorem is closely related to a question of Seidel [catdyn]*Conjecture 18.6, see Remark 6.4. We also illustrate how this theorem may be applied by studying an interesting example where is an isotropic Grassmannian parameterizing isotropic two planes in a symplectic vector space of dimension and is a generic hyperplane section.

### Related work

We note that there are several independent recent and/or ongoing works which have non-trivial conceptual overlap with the present paper. We have already mentioned the work Diogo-Lisi [diogothesis, diogolisi], who use techniques from symplectic field theory to give a formula for the abelian group in terms of topological data and relative Gromov-Witten invariants for a large class of pairs with smooth. Closer in spirit to the present work, a forthcoming paper of Borman and Sheridan [bormansheridan] constructs a particular distinguished class in for suitable pairs (see Remark 4.25 for more details about how their class fits into our framework) and uses this to develop a relationship between and , the quantum cohomology of the compactification . A modified version of their construction in the setting of Lefschetz pencils also appears in a recent preprint of Seidel [SLef].

### Acknowledgments

Paul Seidel’s work has decisively influenced many aspects of this paper; he also answered the author’s questions concerning [catdyn] and gave comments on an earlier draft. The authors had some inspiring conversations with Strom Borman, Luis Diogo, Yakov Eliashberg, Eleny Ionel, Mark McLean, Sam Lisi, and Nick Sheridan about symplectic cohomology and normal crossings compactifications at an early stage of this project. The examples following Theorem 6.18 are based upon a suggestion of Jonny Evans. We thank Jonny Evans, Ivan Smith, and Michael Wemyss for sharing with us a preliminary copy of their work [evanssmithwemyss], and Ivan Smith for conversations about how to apply our results to study Lagrangians intersecting cleanly along a circle. Alessio Corti suggested the examples and also patiently answered questions about conic bundles. Clelia Pech also helped us with these examples by calculating the matrix relating the two bases in the cohomology of the isotropic Grassmannian and giving useful clarifications concerning the (quantum) Pieri rules. We thank them all for their assistance.

### Conventions

Throughout this paper the ring will be either a field or the integers. All grading conventions in this paper are cohomological. Conventions for gradings in symplectic cohomology follow [Abouzaid:2010kx].

## 2. Symplectic cohomology

### 2.1. The definition

Let be a Liouville domain; that is, a dimensional manifold with boundary with one form such that is symplectic and the Liouville vector field (defined as the dual of ) is outward pointing along . More precisely, flowing along is defined for all times, giving rise to a canonical embedding of the negative half of the symplectization

 (2.1) j:(0,1]R×∂¯X →X (2.2) j∗(ω) =d(Rθ|∂¯X)

The (Liouville) completion of is formed by attaching (using the aforementioned identification) a cylindrical end to the boundary of :

 (2.3) X:=¯X∪∂¯X[1,∞)R×∂¯X.

The completion is also exact symplectic, equipped with one form with symplectic. Explicitly, is equal to on and on the cylindrical end, where denotes the canonical coordinate. Recall that in the above situation, is a contact manifold; it possesses a canonical Reeb vector field determined by , . The spectrum of , denoted is the set of real numbers which are lengths of Reeb orbits in . We will assume throughout that this spectrum is discrete (which follows from choosing sufficiently generic).

Observe that after choosing a compatible almost-complex structure then (negative) the first Chern class is represented by the complex line bundle . For simplicity (and in order to set up -gradings), we will assume that and moreover that we have fixed a choice of trivialization of .

###### Definition 2.1.

A (time-dependent) Hamiltonian is admissible with slope if near infinity, where is the cylindrical coordinate near (note only depends on ) and is an arbitrary constant.

For each positive real number which is not in the spectrum of , we choose an admissible Hamiltonian of slope , denoted . Each Hamiltonian determines an action functional on the free loop space of , whose value on a free loop is given by

 AH(x):=−∫S1x∗^θ+∫10H(t,x(t))dt

By definition (and partly convention), the Hamiltonian vector-field of with respect to is the unique (-dependent) vector-field on such that . Critical points of are time-1 orbits of the Hamiltonian vector-field with respect to the symplectic form . It is well-known that we may perturb the functions so that all 1-periodic orbits are non-degenerate, while keeping the functions admissible (for a nice reference directly applicable to our setting see [McLean:2012ab]*Lemma 2.2).

Let denote the set of time-1 orbits of . To each orbit , there is an associated one-dimensional real vector space called the orientation line, defined via index theory as the determinant line of a local operator associated to (this goes back to [Floer:1995fk] but our treatment follows e.g., [Abouzaid:2010kx]*§C.6; note particularly that the choice of local operator is fixed by the choice of trivialization of chosen above). Over a ring , the Floer co-chain complex is, as a vector space

 (2.4) CF∗(X,Hλ):=⨁x∈X(Hλ)|ox|k

where for any real one-dimensional vector space , its -normalization

 (2.5) |V|k

is the rank one free -module generated by possible orientation of , modulo the relation that the sum of opposite orientations vanishes. To define a differential on (2.4), pick a compatible (potentially -dependent) almost complex structure which is of contact type, meaning on the conical end, it satisfies

 (2.6) ^θ∘Jt=dR.

Given a pair of orbits and , a Floer trajectory between and is formally a gradient flowline for between and using the metric on induced by , or equivalently a map , asymptotic to at satisfying a PDE called Floer’s equation:

 (2.7)

Denote by

 (2.8) ˜M(x0,x1)

the moduli space of Floer trajectories between and , or solutions to (2.7). For generic , (2.8) is a manifold of dimension

 deg(x0)−deg(x1)

where , the index of the local operator associated to , is equal to , where is the Conley-Zehnder index of (with respect to the symplectic trivialization of induced by the trivialization of ) [Cieliebak:1995fk, Floer:1995fk]. There is an induced -action on the moduli space given by translation in the -direction, which is free for non-constant solutions. Whenever , for a generic the quotient space

 (2.9) M(x0,x1):=˜M(x0,x1)/R

is a manifold of dimension .

The most delicate part of setting up such a theory for exact (non-compact) symplectic manifolds, where much of the analysis is otherwise simplified, is an a priori compactness result ensuring that solutions to the PDE cannot escape to in the target (this is vital input into the usual Gromov compactness theorems for moduli spaces, which apply to maps into a compact target). Technically, one can ensure a priori compactness by carefully choosing the behavior of and near so that a sort of maximum principle holds for solutions to Floer’s equation (at least outside a compact set); see e.g., [Seidel:2010fk]. The maximum principle ensures that any solution to Floer’s equation remains in some compact subset of the target , which depends only on the asymptotics of (and possibly , , ).

Given that input, a version of Gromov compactness ensures that for generic choices, whenever , the moduli space (2.9) is compact of dimension 0. Moreover, orientation theory associates, to every rigid element an isomorphism of orientation lines and hence an induced map . Using this, one defines the component of the differential

 (2.10) (∂CF)x1,x0=∑u∈M(x0,x1)μu

whenever (and 0 otherwise). When applied to 2 dimensional moduli spaces (which are 1 dimensional after quotienting by ), Gromov compactness and gluing analysis imply that:

###### Lemma 2.2.

Given such that for generic , admits a compactification with

 ∂¯¯¯¯¯¯¯M(x0,x1):=⨆y∈X(Hλ),deg(y)=deg(x1)+1M(x0,y)×M(y,x1).

By a standard argument this implies that and hence that the cohomology of is well-defined. We will denote by

 (2.11) HF∗(X,Hλ):=H∗(CF∗(X,Hλ),∂CF);

standard techniques involving continuation maps shows that this group only depends on . Furthermore, whenever there are continuation maps [Seidel:2010fk]

 (2.12) cλ1,λ2:HF∗(X,Hλ1)→HF∗(X,Hλ2)

defined as follows: Let be a map from which agrees with near and near and which is monotonic, meaning (at least away from a compact set in ) . Let be a compatible dependent almost complex structure agreeing with a choice of used to define near and a choice of used to define near . Then, if is an orbit of and is an orbit of , the component of the map (2.12) is defined on the chain level by counting maps satisfying Floer’s equation for and :

 ∂su+Js,t(∂tu−XHs,t)=0

which in addition satisfy requisite asymptotics:

 (2.13) ⎧⎨⎩lims→−∞u(s,−)=x2lims→+∞u(s,−)=x1.

(A standard transversality, compactness, and gluing argument ensures that the chain level map is in fact a chain map, as long as a maximum principle holds for elements of the moduli space. This maximum principle is where one requires .).

Define symplectic cohomology to be the colimit of this directed system

 (2.14) SH∗(X):=lim−→λHF∗(X,Hλ)

It is not hard to prove that this definition is independent of the various choices made along the way [Seidel:2010fk]. There are canonical morphisms, for each

 (2.15) cλ1,∞:HF∗(X,Hλ)→SH∗(X).

In particular, for , ; hence there is a canonical map [Viterbo:1999fk]. The cone of (a chain-level version of) this map is often called high energy (or positive) symplectic cohomology and denoted . (Recall from the instroduction that the most canonical formulation of the Log PSS map in the presence of holomorphic spheres is in terms of )

Let us review a convenient alternate construction of high energy symplectic cohomology , which makes use of a slightly more restrictive choice of Hamiltonians (but manages to avoid using chain-level colimit constructions). Consider Hamiltonians for which:

• on

• Over the collar region satisfy , with and

• For some near 1, we have that for .

After taking a suitable small perturbation, which for simplicity we may assume is time independent in the interior, action considerations show that the (orientation lines associated to) orbits in the interior generate a subcomplex . Set

 (2.16) CF∗+(X,Hλ):=CF∗(X,Hλ)CF∗−(X,Hλ)

This construction passes to direct limits, giving rise to

 (2.17) SH∗+(X):=lim−→λHF∗+(X,Hλ)

It is not difficult to see that there is a long exact sequence

 (2.18) ⋯→H∗(X)→SH∗(X)→SH∗+(X)→H∗+1(X)→⋯

Finally, we observe that for such Hamiltonians, the integrated maximum principle of [Abouzaid:2010ly]*Lemma 7.2 implies that all Floer solutions connecting such orbits actually lie in the region where ; in particular all curves we study lie in the compact region .

### 2.2. Algebraic structures

Among its many other TQFT structures, symplectic cohomology is a BV-algebra; in particular it has a pair of pants product and a BV operator which we now define.

Recall that a negative cylindrical end, resp. a positive cylindrical end near a puncture of a Riemann surface consist of a holomorphic map

 (2.19) ϵ−:(−∞,0]×S1→Σ

resp. a holomorphic map

 (2.20) ϵ+:[0,∞)×S1→Σ

asymptotic to . Let be a Riemann surface equipped with suitable cylindrical ends . Suppose to each cylindrical end we have associated a time dependent Hamiltonian . Let be a 1-form on with values in smooth functions on which, along the cylindrical ends, satisfies:

 ϵ∗i(K)=Hidt

whenever is large. To such a , we may associate a Hamiltonian vector field-valued 1-form , characterized by the property that for any tangent vector , is the Hamiltonian vector-field of the Hamiltonian function (on the ends, ).

In order to define Floer-theoretic operations, we fix the following additional data on :

• a (surface-dependent) family of admissible , meaning should be of contact type. Further, when restricted to cylindrical ends should depend only on .

• a subclosed 1-form , which restricts to , for , when restricted to the cylindrical ends.

• A perturbation one-form restricting to on the ends such that outside of a compact set on ,

 (2.21) XK=XHλ⊗β.

The most general form of Floer’s equation that we will be studying in this paper is:

 (2.22) {u:Σ→X,(du−XK)0,1=0.

To such data we can associate the geometric energy

 (2.23) Egeo(u):=12∫Σ||du−XK||

as well as the topological energy

 (2.24) Etop(u)=∫Σu∗ω−d(u∗K)

Let us assume that our perturbation datum is of the form (2.21) and . Then,

 (2.25) Egeo(u)=Etop(u)+∫ΣΩK

where . The positivity of and sub-closedness of therefore implies we have an inequality

 (2.26) Egeo(u)≤Etop(u).

In particular, if , it follows that everywhere. Under our assumptions, (2.21) and hold outside of a compact set, so we obtain a bound on the geometric energy of a solution to Floer’s equation in terms of the topological energy and a constant depending only on and .

To define the pair of pants product,

 (2.27) (−⋅−):HF∗(Hλ)⊗HF∗(Hλ)→HF∗(H2λ)

we specialize to the case where is the pair of pants, viewed as a sphere minus three points. Labeling the punctures of by , and , we equip with positive cylindrical ends around and and a negative end around . We count solutions to Equation (2.22) such that

 (2.28) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩u:Σ→X,lims→−∞u(s,−)=x0lims→∞u(s,−)=x1lims→∞u(s,−)=x2

We will need to consider certain parameterized operations on symplectic cohomology as well. In the simplest case, consider the cylinder . We will consider operations parameterized by a value . Set

 H(r)t(x)=H(t−r,x),J(r)t,x=Jt−r

Choose a pair such that along the cylindrical ends

 (2.29) {(Js,t,r,Hs,t,r)=(J(r),H(r))(Js,t,r,Hs,t,r)=(J,H).

We will denote the rotated Hamiltonian orbits of by . Note that these orbits are bijection with those of . Under this correspondence, Floer trajectories of with asymptotics , are naturally in bijection with trajectories of with asymptotics , ; i.e., there is a canonical isomorphism between the Floer complexes of and . For later use, we denote this former moduli space, the solutions to Floer’s equation for with asymptotics , , by . For generic , the BV operator defines an operation

 Δ:HF∗(X,Hλ)→HF∗−1(X,Hλ)

by counting solutions to the parameterized Floer equation:

 (2.30)

Furthermore this operation is compatible with the continuation maps . The BV operator and the product together give the structure of a unital BV-algebra.

The final general property of symplectic cohomology that we need concerns its functoriality. Namely, let be a sub-Liouville domain. We have a Viterbo functoriality map

 (2.31) j!:SH∗(X)→SH∗(W)

which respects the BV structures on both sides:

###### Lemma 2.3.

The Viterbo functoriality map is a morphism of unital BV algebras (and in particular preserves the BV operator and the product).

## 3. Complements of normal crossings divisors

### 3.1. Nice symplectic and almost-complex structures

###### Definition 3.1.

A log-smooth compactification of a smooth complex -dimensional affine variety is a pair with a smooth, projective -dimensional variety and a divisor satisfying

 (3.1) X=M∖D; (3.2) The divisor D is normal crossings in the strict sense, e.g., D:=D1∪⋯∪Di∪⋯∪Dk where Di are smooth components of D; and (3.3) There is an ample line bundle L on M together with a section s∈H0(L) whose divisor of zeroes is ∑iκiDi with κi>0.

For all pairs in this paper, we will assume that the canonical bundle is supported on

 (3.4) ∧nCT∗M≅O(∑i−aiDi),

and choose a holomorphic volume form on which is non-vanishing on and has poles of order along as in (3.4).

Given a subset define

 (3.5) DI:=∩i∈IDi;

we refer to the associated open parts of the stratification induced by as

 (3.6) ˚DI=DI∖∪j∉IDj

By convention, we set and . Denote by the natural inclusion map.

###### Definition 3.2.

Let be a symplectic manifold equipped with a one-form such that . Let denote the -dual of . We say that is a finite-type convex symplectic manifold if there exists an exhausting function together with a such that over all of

See [McLean:2012ab]*§A for a comprehensive survey of these structures including the notion of deformation of these structures that we will use. Any affine variety has a canonical (up to deformation equivalence) structure of a finite-type convex (exact) symplectic manifold constructed as follows:

###### Example 3.1 (Stein symplectic structure).

Pick a holomorphic embedding and equip with

• the one-form (in terms of polar coordinates on ); and

• the exhausting function .

Up to deformation equivalence, the resulting convex symplectic structure is independent of choice of .

There is a different construction of a (deformation equivalent) convex symplectic structure which is more natural from the point of view of normal crossings compactifications.

###### Example 3.2 (Logarithmic symplectic structure).

Let If be a log-smooth compactification of as above, and a section of a line bundle cutting out as in Definition 3.1, we can equip with

• the one-form .

• the exhausting function is given by ,

where is any choice of positive Hermitian metric on the line bundle (once more, the result is independent up to deformation equivalence of ).

The deformation equivalence of these two structures is proven in [Seidel:2010fk]. Here we recall a further deformation of the convex symplectic structure on , due to McLean [McLean:2012ab] (see [Seidel:2010fk] for the case ), with “nice” properties at infinity with respect to a given compactification . To begin, after a deformation we assume that the smooth components of intersect orthogonally in :

###### Theorem 3.3 ([McLean:2012ab]*Lemma 5.4, 5.15).

There exists a deformation of the divisors such that they intersect orthogonally with respect to the symplectic structure on . This does not change the symplectomorphism type of the complement , or the deformation class of its convex symplectic structure.

###### Definition 3.4 ([McLean:2012ab]*Lemma 5.14).

A convex symplectic structure constructed from a log-smooth compactification is nice if there exist tubular neighborhoods of with symplectic projection maps

 πi:Ui→Di

such that on a -fold intersection of tubular neighborhoods

 UI:=∩i∈IUi=Ui1∩⋯∩Ui|I|,

iterated projection times

 πi1∘πi2∘⋯∘πi|I|:UI→DI

is a symplectic fibration with structure group and with fibers symplectomorphic to a product of standard symplectic discs of some radius and such that restricts to

 ∑i∈I(12r2i−κi)dφi,

where are standard polar coordinates for . Moreover, each for is fiber-preserving, sending

 ∏j∈IDϵ→∏j∈I|j≠iDϵ.

Observe that the coordinates induce smooth functions

 (3.7) r2i:Ui→R

on the neighborhoods . On the intersections , these functions give rise to commuting Hamiltonian actions. Denote by the union

 UD:=∪iUi
###### Definition 3.5.

Define

 SI

to be the unit bundle around each , e.g. the set of points in where for some small and . We will use the notation for the one dimensional torus bundle over . Let

 ˚SI

denote the restriction of the bundle to .

Let denote a subset of . There is a codimension stratum of corresponding to the restricted torus bundle . Near these strata, we have radial coordinates . We let denote set of points in where . We let denote the natural closure of the open manifold . This is a manifold with corners whose boundary is the region

 ∂¯¯¯¯SI=∪j({rj=ϵ∖∪k∉jrk<ϵ}).

We adopt the convention that and

 ¯S∅:=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯M∖UD

Let denote the normal bundles to the stratum and denote the restriction of these bundles to . These bundles have been equipped with canonical structures and there is a canonical identifications of the associated torus bundles with and respectively.

###### Theorem 3.6 ([McLean:2012ab]*Lemma 5.20).

There exists a convex symplectic structure deformation equivalent to which is nice.

Henceforth, we replace by the corresponding nice structure. In this setting, there is a nice Liouville domain , whose boundary is a smoothing of the hypersurface with corners, . Below, we describe an explicit model for .

First, choose a smooth function satisfying:

1. There exists such that iff

2. The derivative of is strictly negative when .

3. near .

4. There is a unique with and .

(Compare [McLean2]*proof of Theorem 5.16). Define

 (3.8) F:UD →R (3.9) F(x) =k∑i=1q(r2i),

where we implicitly smoothly extend to be 0 outside of the region where is defined.

###### Lemma 3.7.

For sufficiently small, the Liouville vector field associated to is strictly outward pointing along .

###### Proof.

This computation is very close to the that in [McLean:2012ab]*Lemma 5.17 (also, compare [McLean2]*Theorem 5.16 for a related calculation in the case of concave boundaries); we include a sketch for convenience. First, note that if is small, then and ; the latter condition implies that the Liouville vector field is defined at all points of . We need to show that at any , , where denotes the Hamiltonian vector field of . Let denote the Hamiltonian vector field of , so . The condition that be nice implies that the symplectic orthogonal to the tangent space to any fiber of the projection is contained in a level set of , in particlar applied to any such vector is zero. Hence, in (where is non-zero), is tangent to every fiber of , and

 (Xq(r2i))|Dϵ=Xq(r2i)|Dϵ=−12q′(r2i)∂∂ϕi.

Applying , we have that

 (3.10) θ(Xq(r2i))=(12r2i−κi)dϕi(Xq(r2i))=12q′(r2i)(κi−12r2i)

(this extends smoothly by zero outside ). Note that with equality exactly when