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Floer Cohomology, Multiplicity and the Log Canonical Threshold

Let f be a polynomial over the complex numbers with an isolated singularity at 0. We show that the multiplicity and the log canonical threshold of f at 0 are invariants of the link of f viewed as a contact submanifold of the sphere. This is done by first constructing a spectral sequence converging to the fixed point Floer cohomology of any iterate of the Milnor monodromy map whose E^1 page is explicitly described in terms of a log resolution of f. This spectral sequence is a generalization of a formula by A'Campo. By looking at this spectral sequence, we get a purely Floer theoretic description of the multiplicity and log canonical threshold of f.


Introduction
Let f : C n+1 → C be a polynomial with an isolated singular point at 0 where n ≥ 1. Let S ǫ ⊂ C n+1 be the sphere of radius ǫ centered at 0. The link of f at 0 is the submanifold L f ≡ f −1 (0) ∩ S ǫ ⊂ S ǫ where ǫ > 0 is sufficiently small. One can ask the following question: what is the relationship between the link of f and various algebraic properties of f ? For instance Zariski in [Zar72] asked whether the multiplicity of f at 0 depends only on the embedding L f ⊂ S ǫ . Another important invariant is the log canonical 1 threshold (see [Ati70], [Mus12] or Definition 2.1). Again one can ask if this is an invariant of L f ⊂ S ǫ (see [Bud12, Section 1.6]). We will answer weaker versions of these questions. If ǫ > 0 is small enough, it turns out that L f is naturally a contact submanifold of S ǫ (See [Var82]). If g : C n+1 −→ C is another polynomial with isolated singularity at 0 then we say that f and g have embedded contactomorphic links if there is a contactomorphism Φ : S ǫ −→ S ǫ sending L f to L g . Varchenko showed in [Var82] that if there is a holomorphic change of coordinates sending f to g then they have embedded contactomorphic links. One of the goals of this paper is to prove the following theorem.
Theorem 1.1. Suppose that f, g : C n+1 −→ C are two polynomials with isolated singular points at 0 with embedded contactomorphic links. Then the multiplicity and the log canonical threshold of f and g are equal.
We will prove this theorem by finding formulas for the multiplicity and log canonical threshold in terms of a sequence Floer cohomology groups. The key technical result of this paper proving the above theorem will be a natural generalization of a formula by A'Campo The key technical result of this paper (Theorem 1.2) is a spectral sequence converging to a group whose Euler characteristic is naturally equal to the left hand side of (1.1) multiplied by (−1) n and so that the Euler characteristic of the E 1 page is naturally equal to the right had side multiplied by (−1) n . We will now explain this result.
For ǫ > 0 small enough, the Milnor fiber M f is naturally a symplectic manifold and φ can be made to be a compactly supported symplectomorphism (see Section 3). For any compactly supported symplectomorphism ψ satisfying some additional properties, one can assign a group HF * (ψ, +) called the Floer cohomology group of ψ (see [Sei01,Section 4] or Section 4 of this paper). The Euler characteristic of this group is (−1) n multiplied by the Lefschetz number of ψ (See property (HF1) in Section 4 of this paper). As a result, we have a sequence of groups HF * (φ m , +) whose Euler characteristic is (−1) n Λ(φ m ) for all m > 0. All of these groups are invariants of the link of f up to embedded contactomorphism (See Lemma 8.17). The log resolution π : Y −→ C n+1 is called a multiplicity m separating resolution if ord f (E i ) + ord f (E j ) > m for all i, j ∈ S satisfying i = j and E i ∩ E j = ∅.
Theorem 1.2. Suppose that π : Y −→ C n+1 is a multiplicity m separating resolution for some m ∈ N >0 . Let (w i ) i∈Š be positive integers so that − i∈Š w i E i is ample. Let a i be the discrepancy of E i (see Definition 2.1) and define k i ≡ Corollary 1.4. The multiplicity of f is the smallest m > 0 so that HF * (φ m , +) = 0. The log canonical threshold of f at 0 is : HF α (φ m , +) = 0 or − α 2m = 1 .
Since HF * (φ m , +) is an invariant of the link of f up to embedded contactomorphism by Lemma 8.17 we get that Theorem 1.1 follows immediately from Corollary 1.4.
For each m > 0, Denef and Loeser in [DL02] constructed natural spaces χ m,1 whose Euler characteristic is Λ(φ m ). Therefore it is natural to ask, what is the relationship between these spaces and the groups HF * (φ m , +), if any? Such a question was considered by Seidel (see [DL02,Remark 2.7]). It might be interesting to see if there is a similar spectral sequence converging to H * (χ m,1 ; Z) since these spaces admit a natural stratification induced by the strata of the log resolution π. A possible proof would exploit the spectral sequence [Pet16,Formula (3)] combined with [DL02, Lemma 2.2] (see also the calculations in the proof of [DL02, Lemma 2.5]).
1.1. Sketch of the proof of Theorem 1.2. We will now state one of the key properties of the group HF * (ψ, +) that will be used in this proof. This property is stated precisely in (HF3) Section 4 and proven in Appendix C.
Spectral Sequence Property: Suppose that the set of fixed points of ψ is a disjoint union of connected codimension 0 submanifolds B 1 , · · · , B l with boundary and corners and suppose that ψ behaves in a particular way near the boundary of B i for each i. Then there is a grading CZ(B i ) ∈ Z for each B i and there is a specific function ι : {1, · · · , l} −→ N so that there is a spectral sequence converging to HF * (ψ, +) with E 1 page equal to The spectral sequence above is an example of a Morse-Bott spectral sequence (See [Bot57, Corollary 2] and [Hut02, Section 6.4] for other similar examples). Therefore in order to prove Theorem 1.2 it would be sufficient for us to deform the monodromy symplectomorphism φ m so that the set of fixed points is a union of codimension 0 submanifolds homotopic to E o ip for each i ∈ {1, · · · , l}. The problem is that we cannot quite do this, but we can construct a new symplectomorphism with the required fixed point sets without changing HF * (φ m , +). Also, Theorem 1.2 really requires a specific ordering of the submanifolds E o ip corresponding to the sequence of positive integers (w j ) j∈S but we will ignore this detail here, as the main applications of this paper do not need such an ordering. We will now explain how to modify φ m without changing HF * (φ m , +) so that it has this fixed point property. This is done in Section 5.
We have a natural symplectic form ω Y on Y coming from the ample divisor − i∈S w i E i . This symplectic form gives us a natural Ehresmann connection on π * f away form (π * f ) −1 (0) and hence gives us a monodromy map. First of all, we deform ω Y so that it behaves well with respect to π * f (See Sections 5.1 and 5.2). The key idea is that since π * f locally looks like m i=1 z m i , we can deform ω Y so that it basically looks like the standard symplectic form in these local charts (with a few modifications). The next step is to show that the corresponding monodromy map ψ satisfies HF * (ψ m , +) = HF * (φ m , +) (See sections 5.3, 5.4 and Appendix A). Here we are using the fact that these Floer cohomology groups are invariants of the mapping tori of φ m and ψ m respectively along with an additional contact structure on these tori and some additional data. Finally we need to compute the fixed points of the monodromy map so that we can apply our spectral sequence property (See section 5.5).
1.2. Plan of the paper. In Section 2 we construct algebraic invariants of (C n+1 , f −1 (0)) which will be used to tell us the smallest non-vanishing degree of HF * (φ m , +) for each m. These invariants are constructed by looking at the multiplicities and discrepancies of the prime exceptional divisors (E i ) i∈S of a resolution.
In Section 3 we give some basic definitions of the main objects in symplectic and contact geometry that will be used in this paper. These include Liouville domains, (abstract) open books, contact mapping cylinders and gradings. In Section 4 we give a definition Floer cohomology of a symplectomorphism. We also state the three main properties (HF1) -(HF3) of the Floer cohomology group HF * (ψ, +) that will be needed for this paper. These properties will be proven in Appendix B and C.
Section 5 is the largest section of the paper. This section is used to construct a monodromy symplectomorphism nice enough so that we can use the properties from Section 4 to prove Theorem 1.2. This section heavily relies on results and notation from [MTZ14a]. Section 6 contains a proof of Theorem 1.2 and Corollary 1.3.
Appendix A is deals with gradings. It enables us to compute the quantities CZ(B i ) stated in the spectral sequence property in the sketch of the proof of Theorem 1.2 earlier. Appendix B proves that the groups HF * (φ m , +) only depend on the link L f ⊂ S ǫ as a contact submanifold. This relies heavily on results of [McL12b]. In Appendix C we prove the spectral sequence property of HF * (ψ, +) described above.
1.3. Conventions. If (M, ω) is a symplectic manifold and θ is a 1-form then its ω-dual X ω θ is the unique vector field satisfying ω(X ω θ , Y ) = θ(Y ) for all vectors Y . Sometimes we just write X θ instead of X ω θ if it is clear from the context that the symplectic form we are using is ω. For a smooth function H : M −→ R, we define X H ≡ X −dH . The time t flow of X −dH will be denoted by φ H t : M −→ M (this is called the time t Hamiltonian flow of H). Also if f : B ′ −→ B is a smooth map and π : V −→ B is a vector bundle then we will write elements of the pullback bundle f * (V ) as pairs (b ′ , v) ∈ B ′ × V satisfying f (b ′ ) = π(v). For any fiber bundle π : E −→ B and any subsets N ⊂ E, C ⊂ B, we define N | C ≡ N ∩ π −1 (C). To avoid cluttered notation, we will not distinguish between an element of a set and a subset of size 1 when the context is clear (e.g. i will quite often mean {i}). We also write Dom(f ) and Im(f ) for the domain and image of a map f . For any set I, we define N I >0 to be the set of tuples (k i ) i∈I where k i ∈ N >0 . Acknowledgments: Many thanks to Mircea Mustaţǎ for suggesting the connection with log canonical threshold. Also many thanks to Paul Seidel for answering some of my questions. This paper is supported by the NSF grant DMS-1508207.

Multiplicities and Discrepancies of Exceptional Divisors.
In this section we will introduce some of the basic tools that are needed from algebraic geometry. We will define the multiplicity and log canonical threshold of an isolated hypersurface singularity as well as some more general invariants. We will also explain how to compute these invariants in terms of certain resolutions called multiplicity m separating resolutions and show how such computational techniques do not depend on the choice of resolution.
Let f : C n+1 → C be a polynomial with an isolated singular point at 0.
Since the only singularities in this paper will be at 0 ∈ C n+1 , we will just call a log resolution at 0 of f a log resolution of (C n+1 , f −1 (0)).
The multiplicity of f along E j , denoted by ord f (E j ), is the order of π * f along E j . In other words, choose some local coordinate chart z 1 , · · · , z n centered at a generic point of E j so that E j = {z 1 = 0} and define ord f (E) ≡ k where k ∈ Z satisfies π * f = gz k 1 in this coordinate system for some holomorphic function g satisfying g(0) = 0.
The discrepancy of E j denoted by a(E j ) is calculated as follows: choose local holomorphic coordinates y 1 , · · · , y n on Y centered at a point on E j and holomorphic coordinates x 1 , · · · , x n on C n+1 centered at 0. Then a(E j ) is the order of the Jacobian determinant of f along E j expressed in these coordinates. This quantity does not depend on the choices of such holomorphic coordinates. The multiplicity of f at 0 is min{ord f (E j ) : j ∈ S − ⋆ S } and the log canonical threshold is min a(E j )+1 ord f (E j ) : j ∈ S . Throughout this paper, we will define E I to be ∩ j∈I E j for each I ⊂ S. If I is the empty set then E I is the entire manifold Y .
Definition 2.2. Let π : Y → C n+1 be a log resolution of (C n+1 , f −1 (0)) as above. For each m ∈ N >0 , we define the minimal multiplicity m discrepancy to be: Our convention here is that infimum of the empty set is ∞. Later on in Lemma 2.6 we will show that md m (π, f ) does not depend on π and hence we can define md m (f ) ≡ md m (π, f ) for some choice of log resolution π. A morphism π : Y → C n+1 is a multiplicity m separating resolution if it is a log resolution of (C n+1 , f −1 (0)) such that for any two resolution divisors E and F of π satisfying E ∩ F = ∅, the sum of the multiplicities of f along E and F is greater than m.
Multiplicity m separating resolutions make it much easier for us to compute the minimal multiplicity m discrepancy.
Proof. This follows from the fact that if j∈I k j ord f (E j ) = m and E I = ∅ for some I ⊂ S−⋆ S and (k j ) j∈I ∈ N I >0 then |I| = 1.
Lemma 2.4. If we have any log resolution then we can blow such a resolution up along strata of ∪ i∈S E i inside π −1 (0) so that it becomes a multiplicity m separating resolution.
Proof. Let π : Y −→ C n+1 be a resolution with resolution divisors (E j ) j∈S . Define Let b Y be the number of elements in the set Hence by induction we can blow up Y along subsets of the form E I until we get a log resolution π ′′ : Y ′′ −→ C of (C n+1 , f −1 (0)) so that a Y ′′ − b Y ′′ ≥ m. Since b Y ′′ ≥ 1, we get that a Y ′′ > m. Hence π ′′ is a multiplicity m separating resolution.
Lemma 2.5. Let π : Y → C n+1 be a log resolution of (C n+1 , f −1 (0)) and I ⊂ S a subset satisfying |I| ≥ 2. Letπ :Y → C n+1 be the log resolution of (C n+1 , f −1 (0)) obtained by blowing up Y along E I . Then Let E be the exceptional divisor of the blowdown mapY −→ Y . Then by looking at a local model of the blowdown map and using the chain rule we get (2.2) Suppose for someǏ ⊂ S satisfying EǏ = ∅, some k ∈ N ≥0 and (k j ) j∈Ǐ ∈ NǏ >0 we have Then by Equations (2.1) and (2.2) we have Also Equations (2.1) and (2.2) tell us that Equations (2.3) and (2.4) tell us that md m (π, f ) = md m (π, f ).
Proof. Let π : Y → C n+1 andπ :Y → C n+1 be two such resolutions. Lemma 2.5 tells us that blowing up along strata does not change the minimal multiplicity m discrepancy. Hence by Lemma 2.4, we can assume that π andπ are multiplicity m separating resolutions. Since π andπ are birational morphisms, we have that there is a birational morphism Φ : Y Y so that π =π • Φ. Let (E j ) j∈S be the resolution divisors of π and (Ě j ) j∈Š the resolution divisors ofπ. Suppose that ord f (E j ) divides m for some j ∈ S − ⋆ S . LetǏ ⊂Š be the largest subset satisfying Φ(E j ) ⊂ĚǏ . Since Φ is well defined outside a subvariety of codimension ≥ 2, we have a point p ∈ E j and holomorphic charts y 1 , · · · , y n in Y and x 1 , · · · , x n inY centered at p and Φ(p) respectively where Φ is well defined. We will also assume that E j = {y 1 = 0} andĚ k = {x α(k) = 0} for all k ∈Ǐ and for some α :Ǐ −→ {1, · · · , n}. Let J be the Jacobian of π(y 1 , · · · , y n ),J the Jacobian ofπ(x 1 , · · · , x n ) and J Φ the Jacobian of Φ(y 1 , · · · , y n ). Then By (2.5) combined with the fact thatπ is a multiplicity m separating resolution, we have

Liouville domains, Symplectomorphisms and Open Books
In this section we give basic definitions of Liouville domains and graded symplectomorphisms and open books. We will also explain the correspondence between open book decompositions and graded symplectomorphisms of Liouville domains. All of the material here is contained in [Gir02] with the exception of gradings which is contained in [Sei00]. For more details on open book decompositions see [D14].
Definition 3.1. An exact symplectic manifold is a pair (M, θ M ) where M is a manifold and θ M is a 1-form so that ω M ≡ dθ M is symplectic. A Liouville domain is an exact symplectic manifold (M, θ M ) where M is a compact manifold with boundary and the ω M -dual X θ M of θ M points outwards along ∂M . The 1-form θ M is called the Liouville form. The contact boundary of M is the pair (∂M, α M ) where α M ≡ θ M | ∂M . Here α M is a contact form. Since X θ M points outwards along ∂M , we get that the backwards flow of X θ M exists for all time t. By considering the smooth embeddings φ ln(r M ) | ∂M where r M ∈ (0, 1], we can construct a standard collar neighborhood (0 Here r M is the coordinate given by the natural projection r M : (0, 1] × ∂M ։ (0, 1] and is called the cylindrical coordinate on M . An exact symplectomorphism φ : M −→ M is a diffeomorphism so that φ * θ M = θ M + dF φ for some smooth function F φ : M → R. Technically we want to think of this as a pair (φ, F φ ) but we will suppress F φ from the notation and just write φ. The support of such an exact symplectomorphism is the region We now need to define graded symplectomorphisms as in [Sei00]. This is needed so that we can define their Floer cohomology groups in the next section.
Definition 3.2. We define (R 2n , Ω std ) to be the standard symplectic vector space. Let Sp(2n) be the space of linear symplectomorphisms of (R 2n , Ω std ) and Sp(2n) its universal cover. Let π : E −→ V be a symplectic vector bundle with symplectic form Ω E whose fibers have dimension 2n. Sometimes we will write (E, Ω E ) or just E for such a symplectic vector bundle when the context is clear. Define the symplectic frame bundle F r(E) to be an Sp(2n) bundle whose fiber at x ∈ V is the space of linear symplectomorphisms from (R 2n , Ω std ) to (π −1 (x), Ω E | x ). A grading on a symplectic vector bundle π : (3.1) This is just a choice of reduction of the structure group of E from Sp(2n) to Sp(2n). A symplectic vector bundle with a choice of grading is called a graded symplectic vector bundle. Suppose thatπ :Ě −→V is a symplectic vector bundle and β :Ě −→ E is a bundle morphism covering a smooth map β :V −→ V so that β restricted to each fiber is a linear symplectomorphism. Let be the natural isomorphism between this bundle and the pullback bundle. ThenĚ has a natural grading β * ( F r(E)) −→V , called the grading on E pulled back by β.
The simplest example of a grading is the natural grading on the trivial bundleV ×C k −→V given by We will call such a grading the trivial grading.
Remark 3.3. In this paper we are only interested in gradings up to isotopy and so we will sometimes regard isotopic gradings as the same grading. If we have a smooth connected family of symplectic vector bundles then if one of them has a grading then all of them have a natural choice of grading up to isotopy. For more information about this see Appendix A.
Definition 3.4. Suppose that are graded symplectic vector bundles and F : E 1 −→ E 2 is a symplectic vector bundle isomorphism covering a diffeomorphism V 1 −→ V 2 . Then a grading on F is an Sp(2n) bundle isomorphism Let (X, ω X ) be a 2n dimensional symplectic manifold. A grading on (X, ω) is a choice of grading on the symplectic vector bundle T X. A symplectic manifold with a choice of grading is called a graded symplectic manifold A grading of a symplectomorpism φ between two graded symplectic manifolds (X 1 , ω X 1 ), (X 2 , ω X 2 ) is a choice of grading for the symplectic bundle isomorphism dφ : T X 1 −→ T X 2 . A graded symplectomorphism is a symplectomorphism with a choice of grading.
We also wish to have a notion of grading for contact manifolds.
Definition 3.5. Recall that a cooriented contact manifold (C, ξ C ) is a manifold C of dimension 2n − 1 with a cooriented hyperplane distribution ξ C with the property that there is a 1-form α C whose kernal is ξ C respecting this coorientation and so that α C ∧(dα C ) n−1 is a volume form on C. The 1-form α C is called a contact form compatible with ξ C . A coorientation preserving contactomorphism between two cooriented contact manifolds is a diffeomorphism preserving their respective hyperplane distributions and coorientations. A contact submanifold B ⊂ C is a submanifold so that (B, ξ B ≡ ξ C ∩ T B) is a cooriented contact manifold with the induced coorientation from ξ C .
A grading on a contact manifold (C, ξ C ) consists of a choice of contact form α C compatible with ξ C and a choice of grading on the symplectic vector bundle (ξ C , dα C | ξ C ). A cooriented contact manifold with a choice of grading is called a graded contact manifold. Since the space of contact forms compatible with ξ C is contractible, we get an induced grading on (ξ C , dα| ξ C ) for any other contact form α compatible with ξ C which is unique up to isotopy. Hence from now on we will regard this as the same grading.
A grading of a coorientation preserving contactomorphism φ between two graded contact manifolds (C 1 , ξ 1 ), (C 2 , ξ 2 ) consists of a grading on the symplectic vector bundle isomorpism dφ| ξ C 1 : ξ C 1 −→ ξ C 2 where the symplectic forms on ξ C 1 and ξ C 2 come from contact forms α C 1 and α C 2 compatible with ξ C 1 and ξ C 2 respectively satisfying α C 1 = φ * α C 2 with induced gradings. A graded contactomorphism is a coorientation preserving contactomorphism contactomorphism with a choice of grading.
In this paper, we will only deal cooriented contact manifolds and coorientation preserving contactomorphisms. So from now on, a contact manifold is a cooriented contact manifold and a contactomorphism is a coorientation preserving contactomorphism.
Definition 3.6. Suppose that (C, ξ C ) is a contact manifold and B is a contact submanifold. The normal bundle of B is a symplectic vector bundle with symplectic form defined as follows: Choose a compatible contact form α C on (C, ξ C ) and define Then T ⊥ B is a symplectic vector bundle with symplectic form dα C | T ⊥ B . The symplectic structure on N C B is the pushforward of the above symplectic form under the natural bundle isomorphism T ⊥ B → N C B.
Since the space of compatible contact forms is contractible, we have that the choice of symplectic form on the N C B is unique up to symplectic bundle isomorphism and any two choices of such isomorphism are homotopic. As a result, we will refer to this bundle as the normal bundle as we are only concerned with isomorphisms of such bundles up to homotopy.
Definition 3.7. A contact pair with normal bundle data (B ⊂ C, ξ C , Φ B ) consists of a contact manifold (C, ξ C ) where B is a codimension 2 contact submanifold along with a symplectic trivialization Φ B : N C B → B × C of its normal bundle. A contactomorphism between two such triples is a contactomorphism Ψ : C 1 → C 2 sending B 1 to B 2 so that the composition is homotopic through symplectic bundle trivializations to Φ B 1 . If there exists such a contactomorphism between the pairs as in Equation (3.2) then we say that they are contactomorphic.
A contact pair with normal bundle data (B ⊂ C, ξ C , Φ B ) is graded if there is a choice of grading on C −B. A graded contactomorphism between two graded contact pairs with normal bundle data as in Equation (3.2) consists of a contactomorphism Ψ between these contact pairs and a choice of grading of the contactomorphism Ψ| C 1 −B 1 : If a graded contactomorphism exists between two graded contact pairs with normal bundle data then we say that they are graded contactomorhic.
The main example of a contact pair with normal bundle data comes from singularity theory.
Example 3.8. Fix n > 0. Let f : C n+1 −→ C be a holomorphic function with an isolated singularity at 0 and let J 0 : T C n+1 −→ T C n+1 be the standard complex structure on C n+1 . Let S ǫ ⊂ C n+1 be the sphere of radius ǫ > 0 and let ξ Sǫ = T S ǫ ∩ J 0 T S ǫ be the standard contact structure on S ǫ . Define L f ≡ f −1 (0) ∩ S ǫ . Then a result by Varchenko in [Var82] tells us that for all sufficiently small ǫ > 0, L f ⊂ S ǫ is a contact submanifold called the link of f at 0. Also df gives us an induced map df : N Sǫ L f −→ C and hence a trivialization The contact pair with normal bundle data (L f ⊂ S ǫ , ξ Sǫ , Φ f ) is called the contact pair associated to f .
We also need a grading on this contact pair. It turns out, by the discussion in Definition 7.7, that every contact manifold with trivial first and second homology group has a unique grading up to homotopy. This means that ξ Sǫ has a grading giving us an induced grading on the contact pair associated to f . We will call this the standard grading.
We will now define open book decompositions and also graded open book decompositions. Let D ⊂ C be the unit disk with polar coordinates (r, ϑ).
Definition 3.9. An open book is a pair (C, π) where • C is a smooth manifold, • π : C − B −→ R/Z is a smooth fibration where B is a codimension 2 submanifold and • there is a tubular neighborhood B × D ⊂ C of B = B × {0} in C so that π satisfies: The submanifold B is called the binding of the open book and a page of the open book is the closure of a fiber of π which is a submanifold with boundary equal to B.
We now want open books to be compatible with contact pairs. Definition 3.10. A contact pair with normal bundle data (B ⊂ C, ξ C , Φ B ) is supported by an open book (C, π) (1) if there is a contact form α C compatible with ξ C so that dα C | π −1 (t) is a symplectic form for all t ∈ R/Z, (2) the trivialization of N C B induced by the choice of tubular neighborhood from Definition 3.9 is homotopic through orientation preserving bundle trivializations to Φ B (this does not depend on the choice of such a tubular neighborhood). We will write (C, ξ C , π) for a contact pair supported by an open book and we will call it a contact open book. Note that B and Φ B are not included in the notation as B = C − Dom(π) and Φ B is determined by the open book due to the fact that the space of orientation preserving trivializations of N C B is weakly homotopic to the space of symplectic trivializations of N C B. The contact pair (B ⊂ C, ξ C , Φ B ) is called the contact pair associated to (C, ξ C , π). If the contact pair associated to (C, ξ C , π) is graded then we call this a graded contact open book.
An isotopy between two contact open books (C 1 , ξ C 1 , π 1 ), (C 2 , ξ C 2 , π 2 ) is a contactomorphism Φ : C 1 −→ C 2 between the respective contact pairs with normal bundle data together with a smooth family of maps (π t : Dom(π 1 ) −→ R/Z) t∈[1,2] joining π 1 and π 2 • Φ so that (C 1 , ξ C 1 ,π t ) is a contact open book for all t ∈ [1, 2]. Such an isotopy is graded if we have a smooth family of graded contact open books and Φ is a graded contactomorphism.
The main example of a contact open book will come from singularity theory.
Example 3.11. Let f : C n+1 −→ C, n > 1 be a holomorphic function with an isolated singularity at 0 and let (L f ⊂ S ǫ , ξ Sǫ , Φ f ) be the contact pair associated to f as in Example 3.8. Let arg(f ) :   The mapping torus of (M, θ M , φ) is a smooth map For δ > 0 small enough, we have that the subset of the collar neighborhood of ∂M is disjoint from the support of φ. This means that there is a natural embedding: (1 − δ, 1] × ∂M × R/Z ⊂ T φ which we will call the standard collar neighborhood of ∂T φ . Note that α T φ is equal to r M α M + Cdt in the standard collar neighborhood where r M (resp. t) is the natural projection map to (1 − δ, 1] (resp. R/Z) and α M = θ M | ∂M .
If (M, θ M , φ) is a graded abstract contact open book then we get a natural grading on (T φ , ker(α T φ )) as follows: Since the kernal of dα T φ is transverse to the fibers of T φ , ξ T φ ≡ ker(α T φ ) is isotopic through hyperplane distributions H t , t ∈ [0, 1] to the vertical tangent space T ver T φ ≡ ker(Dπ φ ) of π T φ with the property that dα T φ | Ht is non-degenerate for all t ∈ [0, 1]. Therefore it is sufficient to construct a grading for the symplectic vector bundle (T ver T φ , dα T φ ). Consider the symplectic vector bundle (pr * T M, pr * (dθ M )) where is the natural projection map. We have that the symplectic vector bundle (T ver T φ , dα T φ ) is isomorphic to the symplectic vector bundle on (pr * T M/ ∼, pr * (dθ M )) where ∼ identifies pr * T M | M ×{1} with pr * T M | M ×{0} using the map Since (T M, dθ M ) has a natural grading, we get that (pr * T M, pr * (dθ M )) has an induced grading pulled back via pr. The map Dφ gives us an induced map and since φ is a graded symplectomorphism, the map above lifts to a map: Hence by using the above gluing map, we get a grading on the quotient (pr * T M/ ∼, pr * (dθ M )) and therefore a grading on (T ver T φ , dα T φ ). In turn this gives us a grading on the contact manifold (T φ , ker(α T φ )). We will call this the induced grading on T φ .
We will now construct a contact open book from an abstract contact open book. Let D(ρ) ⊂ C be the closed disk of radius ρ > 0 with polar coordinates (r, ϑ).
The above conditions ensure that is a contact form whose restriction to B φ is also a contact form and that the restriction of dα C φ to π −1 (t) is symplectic for all t ∈ R/Z. Define ξ C φ ≡ ker(α C φ ). The tubular neighborhood ∂M × D(δ/2) of B φ gives us a trivialization Φ B φ of its normal bundle and hence we get a contact pair with normal bundle data (B φ ⊂ C φ , ξ C φ , Φ B φ ) which we will called the contact pair associated to (M, θ M , φ). This contact pair with normal bundle data is supported by the open book (C φ , π C φ ). Hence is isotopic through contact manifolds to T φ , we get that the induced grading on T φ gives us a grading on (C φ − B φ , ξ C φ | C φ −B φ ). Hence we have a relative grading on (B φ ⊂ C φ , ξ C φ , Φ φ ) which we will call the induced grading. The Theorem below is a result of Giroux in [Gir02].
Theorem 3.15. The above map OBD is an bijection. Technically this monodromy map is only defined up to isotopy, and so the monodromy map is really just a choice of representative in this isotopy class. If φ is an exact symplectomorphism, then a small positive slope perturbationφ of φ is an exact symplectomorphismφ equal to the composition of φ with a C ∞ small Hamiltonian symplectomorphism of small positive slope. The action of a fixed point p is −F φ (p) where F φ is a function satisfying φ * θ M = θ M + dF φ . The action depends on a choice of F φ which has to be fixed when φ is defined although usually F φ is chosen so that it is zero near ∂M (if possible). All of the symplectomorphisms coming from isolated hypersurface singularities will have such a unique F φ . An isolated family of fixed points is a path connected compact subset B ⊂ M consisting of fixed points of φ of the same action and for which there is a neighborhood N ⊃ B where N \ B has no fixed points. Such an isolated family of fixed points is called a codimension 0 family of fixed points if in addition there is an autonomous Hamiltonian H : N → (−∞, 0] so that H −1 (0) = B is a connected codimension 0 submanifold with boundary and corners, the time t flow of X H is well defined for all t ∈ R and φ| N : N → N is equal to the time 1 flow of H. The action of an isolated family of fixed points B ⊂ M is the action of any point p ∈ B.

Fixed Point Floer Cohomology Definition
Before we define Floer cohomology, we need some definitions so that we can give it a grading. For any path of symplectic matrices (A t ) t∈[a,b] we can assign an index CZ(A t ) called its Conley-Zehnder index. The Conley-Zehnder index of certain paths of symplectic matrices A t was originally defined in [CZ84]. It was defined for a general path of symplectic matrices in [RS93] and also in [Gut13]. We will not define it here as we will only use the following properties (see [Gut13,Proposition 8  Since Sp(2n) is the universal cover of Sp(2n), its elements correspond to paths of symplectic matrices starting from the identity up to homotopy fixing their endpoints and so let β be the path corresponding to φ(id) ∈ Sp(2n). We define CZ(φ, p) ≡ CZ(β).
If we have an isolated family of fixed points B ⊂ X, then we define its Conley-Zehnder index CZ(φ, B) to be the Conley-Zehnder index of some b ∈ B. Since B is path connected, this does not depend on the choice of b ∈ B by property (CZ5) above.
In Appendix A, we also have a different way of computing the Conley-Zehnder index by looking at the mapping torus T φ of φ. This will be useful in the proof of Theorem 5.41.
Let (M, θ M , φ) be a graded abstract contact open book. We will now give a sketch of the definition of the Floer cohomology group HF * (φ, +) (see [Sei01]). Letφ be a small positive slope perturbation of φ. This can be done so thatφ is C ∞ close to φ and so that the fixed points ofφ are non-degenerate (see [SZ92,Theorem 9.1] in the case whereφ is Hamiltonian. The general case is similar [DS94, Page 586]). We can also ensure thatφ is a graded symplectomorphism due to the fact that it is isotopic to φ through symplectomorphisms.
We now choose a C ∞ generic family of cylindrical almost complex structures (J t ) t∈[0,1] satisfying φ * J 0 = J 1 . The genericity property then tells us that M(φ, J t , p − , p + ) is a compact oriented manifold of dimension 0 for all fixed points p − , p + of φ satisfying CZ(p − )−CZ(p + ) = 1 ([DS94, Theorem 3.2]). We define # ± M(φ, J t , p − , p + ) to be the signed count of elements of M(φ, J t , p − , p + ). Let CF * (φ) be the free abelian group generated by fixed points of φ and graded by the Conley-Zehnder index taken with negative sign. The differential ∂φ ,(Jt) t∈[0,1] on We will only use the following properties of these Floer cohomology groups: • ι(i) < ι(j) if the action of B i is less than the action of B j . Then there is a cohomological spectral sequence converging to HF * (φ, +) whose E 1 page is equal to (see Appendix C).

Constructing a Well Behaved Contact Open Book.
The aim of this section is to modify the Milnor monodromy map so that the set of fixed points is a union of codimension 0 families of fixed points so that we can apply (HF3) above. We will do this by constructing such a nice symplectomorphism whose mapping torus is isotopic to the mapping torus of the Milnor monodromy map. 5.1. Some Preliminary Definitions. The aim of this section is to construct a symplectic form on the resolution which behaves well with respect to the resolution divisors. To do this we need a purely symplectic notion of smooth normal crossing divisor. We will introduce some notation from [MTZ14a].
If V ⊂ X is a submanifold of a manifold X then we will use the notation for the normal bundle of V . If (V i ) i∈S is a finite collection of submanifolds of X and I ⊂ S, Also by convention we define V ∅ ≡ X. The collection (V i ) i∈S intersects transversally if for every subset I ⊂ S and every x ∈ V I , If V ⊂ X is a submanifold and X is oriented then an orientation on V induces an orientation on N X V and conversely an orientation on N X V induces an orientation on V using Equation (5.1) (if V is odd dimensional this depends on a sign convention but we will not need this since the manifolds that we will be dealing with are even dimensional). If X is oriented and (V i ) i∈S is a collection of oriented transversally intersecting submanifolds, then the submanifold V I has a natural orientation called the intersection orientation since N X V I = i∈I N X V i is oriented.
Definition 5.1. Suppose that (X, ω) is a symplectic manifold. Then (V i ) i∈S is called a symplectic crossings divisor or SC divisor if (V i ) i∈S are transversally intersecting codimension 2 symplectic submanifolds so that V I is also symplectic and so that the symplectic orientation on V I agrees with its corresponding intersecting orientation for all I ⊂ S. We will also assume that S is a finite set.
We now want to define SC divisors with particularly nice neighborhoods. We call π : (L, ρ, ∇) → V a Hermitian line bundle if π : L → V is a complex line bundle, ρ is a Hermitian metric and ∇ is a ρ-compatible Hermitian connection on L. We define ρ R and ρ i R to be the real and complex parts of the Hermitian metric ρ. We will also use the notation ρ to denote square of the norm function on L. If we view L as an oriented real vector bundle then we can recover the complex structure i ρ from the metric ρ R using the fact that for all x ∈ V and W ∈ L| x − 0, i ρ (W ) is the unique vector making W, i ρ (W ) into an oriented orthonormal basis of L| x . Hence we can define a Hermitian structure (ρ, ∇) on any oriented 2-dimensional real vector bundle L ։ V to be a pair (ρ, ∇) making (L, i ρ ) into a Hermitian line bundle.
For any such triple (L, ρ, ∇) we have an associated Hermitian connection 1-form α ρ,∇ ∈ Ω 1 (L − V ). This is the pullback of the associated principal U (1)-connection on the unit circle bundle of (L, ρ) (see [BW58] or [MTZ14a, Appendix A]). If (L i , ρ i , ∇ (i) ) i∈I is a finite collection of Hermitian line bundles over a symplectic manifold (V, ω) and pr I;i : i∈I L i → L i is the natural projection map then we define This is a symplectic form in some small neighborhood of the zero section. Given a 2dimensional symplectic vector bundle L ։ V with symplectic form Ω, an Ω-compatible Hermitian structure on L is a Hermitian structure (ρ, ∇) where the complex structure i ρ is compatible with Ω.
Suppose that Ψ :V → V is a diffeomorphism and suppose π : (L i , ρ i , ∇ (i) ) i∈I → V ,π : (Ľ i ,ρ i ,∇ (i) ) i∈I →V are two collections of Hermitian line bundles then a product Hermitian isomorphism is a vector bundle isomorphism Ψ : i∈IĽ i → i∈I L i covering Ψ and respecting the direct sum decomposition so that the induced map Ψ : is an isomorphism of Hermitian line bundles for all i ∈Ǐ.
Definition 5.2. Let V ⊂ X be a submanifold of a manifold X. A regularization is a diffeomorphism Ψ :Ň → X from a neighborhodŇ ⊂ N X V of the zero section onto its image so that Ψ(x) = x for all x ∈ V and so that the map We also need a notion of regularization compatible with the symplectic form. Because of this we first need to talk about connections induced by closed 2-forms. Recall that an Ehresmann connection on a smooth submersion π : Definition 5.3. Let π : E −→ B be a smooth submersion and let Ω be a 2-form on E whose restriction to each fiber is non-degenerate. Then the symplectic connection associated to Ω is an Ehresmann connection H ⊂ T E where H is the set of vectors which are Ω-orthogonal to the fibers of π. In other words, is the vertical tangent bundle. If (X, ω) is a symplectic manifold and V ⊂ X is a symplectic submanifold then N X V is a symplectic vector bundle on V . We write ω| N X V for the symplectic form on this vector bundle and ω| L for the restriction of ω| N X V to L where L is any subbundle L ⊂ N X V . The following definition differs in notation from [MTZ14b, Definition 2.8] but it defines the same object.
Definition 5.4. Let (X, ω) be a symplectic manifold, V a symplectic submanifold and let where ρ i is (by abuse of notation) the pullback of ρ i by the natural projection map ⊕ j∈I L j −→ L i , • ∇ (i) restricted to Dom(Ψ) ∩ L i coincides with the symplectic connection associated to ω| L i for all i ∈ I and The splitting (5.3) is called the associated splitting and the ω| L i -compatible Hermitian structure ( ρ i , ∇ (i) ) is called the associated Hermitian structure on L i . This Hermitian structure is uniquely determined by ρ i and Ψ. Also since ρ i gives us a complex structure on L i for each i ∈ I, we get a natural complex structure on N X V which we will call the complex structure associated to the ω-regularization ((ρ i ) i∈I , Ψ).
We wish to extend Definitions 5.2 and 5.4 to transverse collections of submanifolds and SC divisors respectively. To do this we need some preliminary notation. Let X be a manifold and (V i ) i∈S transversely intersecting submanifolds. We have a canonical identification for each I ′ ⊂ I where π I;I ′ : N V I−I ′ V I → V I is the natural projection map. Note that (5.5) is not an identification of bundles since the base of the left hand bundle is V I whereas the base of the right hand bundle is N V I−I ′ V I .
Definition 5.5. Let X be a manifold and (V i ) i∈S a transverse collection of submanifolds. A system of regularizations is a tuple (Ψ I ) I⊂S , where Ψ I is a regularization for V I so that Using the inclusion map ι above, define Using the equality (5.5), DΨ I;I ′ identifies the normal bundle of N V I−I ′ V I inside N X V I near 0 with the normal bundle of V I ′ near V I using the derivative of the regularization Ψ I . The map DΨ I;I ′ is also a bundle isomorphism covering the diffeomorphism The definition below tells us that Ψ I ′ and Ψ I should be equal under the identification (5.7).
Definition 5.6. Let X be a manifold and (V i ) i∈S a transverse collection of submanifolds of X. Then a regularization for (V i ) i∈S is a system of regularizations (Ψ I ) I⊂S for (V i ) i∈S so that: The following definition differs from [MTZ14b, Definition 2.11] for the same reasons that Definition 5.4 differs from [MTZ14b, Definition 2.8]. Apart from that, this definition is exactly the same. This structure also appears in [McL12a,Lemma 5.14] although the regularization maps have particular domains and it is defined in a slightly different way.
Definition 5.7. Let (X, ω) be a symplectic manifold and (V i ) i∈S an SC divisor. An ωregularization is a pair of tuples (1) (Ψ I ) I⊂S is a regularization for (V i ) i∈S as in Definition 5.6 and is a smooth map, (2) ((ρ i | Im(Ψ I ) ) i∈I , Ψ I ) is an ω-regularization for V I in X for each I ⊂ S as in Definition 5.4 and (3) the maps DΨ I;I ′ from (5.7) are product Hermitian isomorphisms for all I ′ ⊂ I ⊂ S with respect to the natural splittings , We are only interested in regularizations near (V i ) i∈S and so we want a notion of equivalence to reflect this.
Definition 5.8. Two ω-regularizations A real codimension 2 submanifold with an oriented normal bundle should be thought of as the differential geometric analogue of a smooth divisor in algebraic geometry. We wish to construct complex line bundles from such submanifolds in the same way that line bundles are constructed from Cartier divisors in algebraic geometry. The following line bundle associated to a codimension 2 submanifold V of a manifold X will depend on a choice of regularization Ψ :Ň → X of V and a complex structure i on N X V and it will come with a canonical section s V : V → O X (V ) whose zero set is V . We define: The corresponding fibration is defined in the following natural way This is a line bundle satisfying the following important canonical identities: (5.10) We will call Ψ the associated regularization. The canonical section s V : X → O X (V ) of this line bundle is defined as follows: We also define O X (0) ≡ O X ≡ X × C to be the trivial bundle. This is also O X (∅) where ∅ is the empty submanifold.

Trivializing Line Bundles.
In the previous section, we constructed a line bundle from any codimension 2 submanifold with oriented boundary. As a result, we can construct line bundles from any SC divisor. In this subsection we show that if such a line bundle is trivial and if our SC divisor admits a regularization, then our line bundle admits a trivialization which behaves well with respect to this regularization. This trivialization will be used later to construct a map from a neighborhood of our SC divisor to C with nice parallel transport maps away from the singularities. Let (X, ω) be a symplectic manifold and (V i ) i∈S an SC divisor on X admitting an ω- be the line bundle with associated regularization Ψ i and complex structure associated to the ω-regularization (ρ i , Ψ i ). Recall that these have natural sections be the canonical section of this bundle. Using the identity (5.10) we have and hence we get the following maps: Therefore we get a (not necessarily fiberwise linear) map: One can think of the above map as a section of Below is a definition of a trivialization of O( i m i V i ) with the property that locally around each point of V I , the canonical section , · · · , n} and a : R −→ R has the following graph: 1 We need to use the above function a to ensure that we have good dynamical properties (see Section 5.5). See [MTZ14a, Definition 3.8] for a related definition.
Definition 5.9. For each r > 0, we define the radius r tube of V I to be the set (5.14) over V I . Let B ⊂ X be any set. The tube radius of R along B is the largest radius r tube of V I along B that can 'fit' inside the image of Ψ I for each I ⊂ S. More precisely, it is the supremum of all r ≥ 0 with the property that T r,I ∩ (Im(Ψ I )| x ) is a compact subset of X for all x ∈ B ∩ V I and I ⊂ S. Let U ⊂ X be an open set. Now suppose that the tube radius of R along U is positive and let R > 0 be any constant smaller than the tube radius and also smaller than 1. We let . and where the norm of the linear map: 4 ,i using the identification (5.10). We say that Φ is radius R compatible with R along U if it is radius R compatible with R along U ∩ V I for each I ⊂ S. It is compatible with R along U if it is radius R compatible with R along U for some R smaller than the tube radius of R.
One should think of Equation (5.15) as saying that the trivialization Φ identifies the canonical section of O X ( i m i V i ) with the 'infinitesimal' section Π (m i ) i∈S ,I multiplied by a particular factor near V I . This particular factor that we are multiplying by is actually equal to 1 if we are very near V I . Note also that Equation (5.15) tells us that the norm of Φ 2 (s (m i ) i∈S (x)) only depends on (ρ i (x)) i∈I and if ρ i (x) ≥ 3R/4 for some i ∈ I then this norm does not depend on ρ i (x) for all x ∈ Im(Ψ I ) − ∪ i∈S−I T 3R/4,i . As a result, the above definition is consistent with the stated norm property of the linear map (5.16).
Proof. Just as in the proof of [MTZ14a, Definition 3.9] we will proceed by induction on the strata of ∪ j V j . Before we do this though we will need to construct certain natural maps that identify Π (m i ) i∈S ;I with s (m i ) i∈S near V I . Define for all i ∈ S, I ⊂ S. By Equation (5.9), we have the natural identification: and by (5.6) and (5.10) we have the identity for all i ∈ S and I ⊂ S. By using the natural projections we have a map: I by using the identity (5.17) and whose restriction to W i;I is defined by ∈ W i:I × C by using the identity (5.18). One should think of this map as a way of canonically identifying the bundle O X (V i ) near V I with its pullback along the natural projection map from D i;I to V I induced by π N X V I • Ψ −1 I . By (5.9) and (5.11), Also by (5.8) and the fact that DΨ I;I ′ is a product Hermitian isomorphism for all I ′ ⊂ I ⊂ S, We can define similar maps for the line bundle O X ( i m i V i ) in the following way Equations (5.19) and (5.20) give us the following equations: Using these equations, we will now prove our lemma by induction on the set of subsets of S. Let be a total order on the set of subsets of S with the property that if |I ′ | < |I| then I I ′ . We write I ≺ I ′ when I I ′ and I = I ′ . Since U is relatively compact, the tube radius of R along U is positive and hence we can choose any constant R > 0 smaller than this tube radius.
Suppose that there is some I * ⊂ S and a trivialization We now wish to modify the trivialization Φ ≺ so that these properties hold for all I I * . Let T r,I be the radius r tube of V I as in Equation (5.14) and define L ≺ r ≡ ∪ I≺I * T r,I . First of all, let g : be a smooth function whose value at is equal to the norm of the linear map Since the map Π V i ;I restricted to each fiber of P r I;I−i is an isometry for all i ∈ I ⊂ S, we get that g( ) by our induction hypothesis. Combining this with the fact that a R (s) = 1 for all s ≥ 3R/4, we can choose a smooth function f : ). This implies that the norm of the linear map: is 1 for all I I * . Hence Equation (5.16) holds for f Φ ≺ 2 for all I I * . We now wish to modify f Φ ≺ so that Equation (5.15) for this new trivialization holds as well. We let be a smooth trivialization. Equation (5.22) combined with the fact that Φ ≺ 2 is radius R compatible with R along U ∩ V I for all ).
Combining this with the fact that T R,I * deformation retracts on to V I * ∪ (T R,I * ∩ L ≺ R ), we can construct a smooth trivialization Equations (5.21), (5.24) and (5.25) tell us that Φ is radius R compatible with R along Because the norm of the linear map (5.23) is 1 for all I I * , we have by Equations (5.24) and (5.25) that the norm of the linear map is equal to 1 for all I I * . Hence we are done by induction.

Links of Divisors and Open Books.
In this subsection, we first give a purely symplectic definition of a divisor which looks like the resolution divisors of a log resolution of an isolated hypersurface singularity. We then construct the 'link' of this resolution divisor, which corresponds to the embedded link of our isolated singularity. Finally we construct an open book decomposition of this resolution divisor corresponding the Milnor open book of our hypersurface singularity.
We have the following definition from [McL12a].
Definition 5.11. Let (X, ω) be a symplectic manifold of dimension 2n and let θ ∈ Ω 1 (X −K) satisfy dθ = ω| X−K for some compact K ⊂ X. Suppose that K admits an open neighborhood U which deformation retracts onto K. Let ρ : X −→ [0, 1] be a smooth function equal to 0 along K and equal to 1 outside a compact subset of U . Then the dual is an SC divisor and each V i is connected and compact. Then a Mayor-Vietoris argument tells us that H 2n−2 (∪ i∈S V i ; R) is freely generated by the fundamental classes [V i ] and hence there are unique real numbers (w i ) i∈S so that The wrapping number does not depend on the choice of neighborhood U or bump function ρ. We can calculate the wrapping number w(θ, V j ) in the following way (See the discussion after Definition 5.4 in [McL16]): Let D ⊂ X be a small symplectically embedded disk with polar coordinates (r, ϑ) which intersects V i positively once at 0 ∈ D and does not intersect V j for all j ∈ S − i. Then the wrapping number w i is the unique number so that (w i /2π)dϑ is cohomologous to The computation of the wrapping number using the disk D above enables us to define wrapping numbers in the case when each V i is properly embedded but not necessarily compact. We will need this broader definition of wrapping number in the proof of Lemma 5.23 below. The next definition is supposed to be a way of describing a log resolution of a pair (C n+1 , f −1 (0)) (as in Definition 2.1) in a symplectic way. The key motivating example for such a definition is given in Example 5.14 below.
Definition 5.12. A resolution divisor is a pair (X, (V i ) i∈S ) where (V i ) i∈S are transversally intersecting properly embedded codimension 2 submanifolds of a manifold X so that there is a unique element ⋆ S ∈ S with the property that V ⋆ S is non-compact and V i is compact where (m i ) i∈S are positive integers satisfying: (1) (X, (V i ) i∈S ) is a resolution divisor as above, (4) m ⋆ S = 1 and the wrapping number The form ω is called the symplectic form associated to (O X ( i∈S m i V i ), Φ, θ). A grading on this model resolution is a grading on (X − ∪ i∈S−⋆ S V i , ω).
Definition 5.13. Two model resolutions These model resolutions are graded isotopic if they are isotopic as above with the additional property that the model resolutions Y t all admit gradings that smoothly depend on t ∈ [0, 1] and where the grading on Y 0 coincides with the grading on Y and the grading on Y 1 coincides with the grading on Y under the identification Ψ.
We will now give an example of a model resolution.
Example 5.14. Let f : C n+1 −→ C be a polynomial with an isolated singularity at 0 and let U ⊂ C n+1 be an open set containing 0 so that f | U −0 is regular. Let π : Y −→ C n+1 be a log resolution of (C n+1 , f −1 (0)) obtained by a sequence of blowups along smooth loci and define X ≡ π −1 (U ) (such a resolution exists by [Hir64]). Let (E j ) j∈S be the resolution divisors of this resolution and let E ⋆ S ⊂ X be the proper transform of f −1 (0) ∩ U . Because such a resolution is obtained by a sequence of blowups along smooth loci, there are positive integers (w j ) j∈S−⋆ S so that A ≡ − i∈S−⋆ S w i E i is ample on X. By the divisor line bundle correspondence, let L −→ X be the corresponding ample line bundle with a meromorphic section satisfying (s) = A. Choose a Hermitian metric · on L whose curvature form is a positive 1-1 form. Define θ = −d c ln( s ). The 2-form dθ extends uniquely to a Kähler form ω on X. The wrapping number of θ around E j is w j for all j ∈ S − ⋆ S .
We also let m i ∈ N >0 be the multiplicity of f along E i for all i ∈ S. Since i∈S m i E i is the divisor defined by f • π| X we get, by the divisor line bundle correspondence, that the Such a resolution also has a grading as follows: is a biholomorphism we get a canonical holomorphic trivialization as a unitary vector bundle coming from the trivialization on T C n+1 . The grading on TX is equal to the trivial grading onX × C n+1 pulled back by Φ. We will call this the standard grading.
We now wish to associate a pair of contact manifolds with normal bundle data to a model resolution. In the case of Example 5.14, this is the link of our singularity (as defined in the introduction) with some additional data.
Definition 5.16. Let Ψ,Ψ be regularizations of a submanifold V of a manifold X. Then a smooth family of regularizations Ψ t of V connects Ψ andΨ if Ψ is germ equivalent to Ψ 0 anď Ψ is germ equivalent to Ψ 1 .
Lemma 5.17. Let (X, (V i ) i∈S ) be a resolution divisor. For any two tuples of regularizations Proof. Choose a metric making V ⋆ S into a totally geodesic submanifold. Define T r X ⊂ T X to be the set of tangent vectors of length at most r. Fix i ∈ S. Choose a relatively compact neighborhood W i of V i in X and let δ > 0 small enough so that the exponential map restricted to T δ w X is a diffeomorphism onto its image for all w in . Let X be a smooth manifold and let (V i ) i∈Š be transversely intersecting compact codimension 2 submanifolds of X. A smooth function • a real number b i > 0 and • a smooth function q i : Dom(Ψ i ) −→ [0, 1] equal to the square of some norm on N X V i near V i , equal to 1 outside a compact subset of Dom(Ψ i ) and non-zero on Dom(Ψ i )−V i for each i ∈ S and a smooth function τ : We say that f is strongly compatible with (V i ) i∈S if in addition τ = 0.
Lemma 5.19. Let (X, (V i ) i∈S ) be a resolution divisor and let equal to the square of a norm near V i , equal to 1 outside a compact set and non-zero outside First of all, we can smoothly deform f and g through smooth functions compatible with (V i ) i∈S by changing q i ,q i , b i ,b i and τ andτ so that q i =q i , b i =b i and τ =τ = 0. Hence we can assume that Lemma 5.17 tells us that there is a smooth family of regulations (Ψ t i ) i∈S−⋆ S , t ∈ [0, 1] compatible with V ⋆ S connecting (Ψ i ) i∈S−⋆ S and (Ψ i ) i∈S−⋆ S and hence we get a smooth family of functions compatible with (V i ) i∈S (after possibly shrinking the region on which q i is not equal to 1 so that it fits inside ∩ t∈[0,1] Dom(Ψ t )). This is a smooth family of functions compatible with (V i ) i∈S joining f and g.
Lemma 5.20. Let (X, ω) be a symplectic manifold with a choice of grading and C ⊂ X a contact hypersurface with a contact form α compatible with the contact structure satisfying dα = ω| C . Then (C, ker(α)) has a natural induced choice of grading.
We will call such a grading the induced grading on C.
Proof. Let X α be a smooth section of the bundle T X| C −→ C equal to the ω-dual of α.
Since dα = ω| C and α is a contact form, we get that X α is transverse to C. Let R be the Reeb vector field of α and define ξ C ≡ ker(α). Let H ⊂ T X| C be the 2-dimensional symplectic vector subbundle spanned by X α and R. Then we have the following direct sum decomposition of symplectic vector bundles Since X α , R is a symplectic basis for H at each point of C, we have a natural symplectic trivialization of (H, ω| H ) sending X α , R to the standard symplectic basis vectors on C. Hence we have a natural isomorphism: Now choose an almost complex structure J on M compatible with ω so that its restriction to T M | C is equal to J C ⊕ i with respect to the splitting (5.26) where J C is an almost complex structure on ξ C compatible with dα| ξ C and i is the standard complex structure on C. Now we will use the natural correspondence between gradings and trivializations of the canonical bundle as stated in Appendix A. Let Φ : κ J −→ X ×C be the choice of trivialization of the canonical bundle of (T M, J) associated to the grading on (X, ω) (See Definition 7.7 in Appendix A). Since J| C = J C ⊕ i under the splitting (5.26), we get a natural trivialization Φ C : κ J C −→ C × C induced from the trivialization Φ| C . The induced grading on (C, ξ C ) is then the grading associated to the trivialization Φ C as in Definition 7.7.
We wish to use functions compatible with a resolution divisor to construct it's 'link'. The following proposition tells us how to at least start doing this.
Then there is a smooth function g : X − K −→ R and an open neighborhood U of K so that df (X ω θ+dg )| U > 0 and df ⋆ (X ω⋆ θ⋆+dg⋆ )| U ∩V⋆ S > 0 where ω is the symplectic form associated to our model resolution, The proof of this proposition is almost exactly the same as the proof of [McL16,Proposition 5.8]. The only difference is that we have to take into account the additional submanifold V ⋆ S . For the sake of completeness we have produced the proof below. We also have a parameterized version of the proposition above.
Then there is a smooth family of functions The proof of this proposition is almost exactly the same as the proof of Proposition 5.21 except that all variables are now parameterized by t. Therefore for notational simplicity we will just prove Proposition 5.21. Before we prove this we need a few preliminary technical lemmas. The following lemma is very similar to [McL16,Lemma 5.12]. This lemma should be thought of as a local version of Proposition 5.21.
Lemma 5.23. Let (O X ( i∈S m i V i ), Φ, θ) be a model resolution and fix a metric · on X. Define K ≡ ∪ i∈S−⋆ S V i . Fix I ⊂ S and let ω be the symplectic form associated to our model resolution. Let U ⊂ K be an open set with the property that U ∩ V I ′ is contained inside a contractible Darboux chart of V I ′ for all I ′ ⊂ S and so that U ∩ V i = ∅ for all i ∈ S − I.
Then there is a smooth function g : X − K −→ R so that for any function f : Proof. Define I ≡ I−⋆ S and let n be the dimension of X divided by 2. Since U ∩V i is contained inside a contractible Darboux chart we have, by a Moser argument, symplectic coordinates x i 1 , y i 1 , · · · , x i n , y i n defined on some neighborhood We can also choose W i so that W i ∩ V j = ∅ for all j ∈ S − I and so that W i is a compact contractible codimension 0 submanifold of X with boundary so that ∂W i , (V j ) j∈S are transversely intersecting. Define W ′ i ≡ W i −V i and let P i : W ′ i −→ W ′ i be the cover corresponding to the subgroup of π 1 (W ′ i ) generated by loops wrapping around V i near V i for each i ∈ I. In other words, the cover corresponding to the image of . Therefore there is a natural symplectic embedding of ι i : W ′ i ֒→ C n+1 so that the standard symplectic coordinates x 1 , y 1 , · · · , x n , y n in C n restricted to W ′ i are ρ i , ϑ i , x i 2 , y i 2 , · · · , x i n , y i n respectively. Let W ′ i ⊂ C n+1 be the closure of P −1 i (W ′ i ) inside C n+1 and let V ′ I−i be the closure of P −1 i (V I−i ) inside C n+1 . Then W ′ i is a codimension 0 submanifold of C n+1 with boundary and corners and V ′ I−i is a codimension 2(|I| − 1) submanifold of W ′ i with boundary and corners where one part of the boundary iš Let w i > 0 be the wrapping number of θ around V i . Let H ⊂ T C n+1 |V I−i be a 2 dimensional symplectic subbundle overV I−i containing ker DP i |V (a) g i (x 1 , y 1 , · · · , x n , y n ) = g i (x 1 , y 1 + 2π, x 2 , y 2 , · · · , x n , y n ) + w i , Define W ′ ≡ ∩ i∈ I W i . Let θ 1 ∈ Ω 1 (W ′ ) be any 1-form of bounded norm satisfying dθ 1 = ω| W ′ . Define By (a), we get that the wrapping number of Θ around V i ∩ W ′ is w i for all i ∈ I. Hence (after shrinking (W i ) i∈ I so that W ′ deformation retracts on to W ′ ∩ K) there is a function g : Let f : X − K −→ R be compatible with (V i ) i∈S and let (Ψ i ) i∈S−⋆ S be its associated regularizations. Then by definition for some c 1 , c 2 > 0 and θ 1 is bounded, part (3) of our lemma holds. Since τ and θ 1 are bounded and since Equation (5.27) holds, it is sufficient for us to prove the following statements: ( that is also isotopic to the identity through such diffeomorphisms and pulling back s i to log(ρ i ), we have that inequalities (1) and (2) above can be deduced from: ) → 0 as we approachV I−i for all j ∈ I − i and (3) X d g i is tangent to V ⋆ S alongV I−i for all i ∈ I. These properties follow from (a)-(c) above.
Lemma 5.24. Let (O X ( i∈S m i V i ), Φ, θ) be a model resolution and fix a metric · on X. Define K ≡ ∪ i∈S−⋆ S V i and let f : X − K −→ R be a smooth function compatible with (V i ) i∈S . Then there is a smooth function h : X − K −→ R and constants a 1 , a 2 > 0 so that a 1 df < θ + dh < a 2 df near K.
Proof. We will use part (3) of Lemma 5.23 and an induction argument to do this. Choose open sets U 1 , · · · , U m in X along with subsets I 1 , · · · I m ⊂ S so that {1, · · · , m} and • so that U j ∩ V k = ∅ for all j ∈ S − I j and all j ∈ {1, · · · , m}.
Proof of Proposition 5.21. By Lemma 5.24 we can add an exact 1-form to θ so that inside a neighborhood N of K for some constants b 1 , b 2 > 0. By Lemma 5.23 we can find open sets W 1 , · · · , W m of X covering K, smooth functions g i : X − K −→ R, i = 1, · · · , m, and a constant c > 0 so that (3) c df < θ + dg i <č df inside W i for some constants c,č > 0. Now choose smooth functions ρ i , i = 1, · · · , m so that m i=1 ρ i | K = 1 and ρ i = 0 outside a compact subset of W i for each i = 1, · · · , m. We define We define g ⋆ ≡ g| V⋆ S −K . The inequality (5.30) combined with property (3) above tells us that |g i | < C|f | +Č for some C,Č > 0 near W i ∩ K. This means that df (X ω θ+dg ) > c θ + dg df near K, df (X ω θ⋆+dg⋆ ) > c θ ⋆ + dg ⋆ df near V ⋆ S ∩ K and a 1 df < θ + dg < a 2 df near K for some constants a 1 , a 2 > 0.
Hence by Proposition 5.21 we have that df (X θ+dg ) > 0 and df ⋆ (X θ⋆+dg⋆ ) > 0 in an open neighborhood U of K for some smooth function g : Finally the trivialization Φ B of the normal bundle of B in C is induced from the trivialization Φ since the normal bundle of B is naturally isomorphic as an oriented vector bundle to has a grading then this gives us an induced grading on the link as in the proof of Lemma 5.20 since C − B is a contact hypersurface of (X − ∪ i∈S−⋆ S V i , ω) where ω is the symplectic form associated to our model resolution. We will call this the induced grading on (B ⊂ C, ξ C , Φ B ).
The link does not depend on the choice of neighborhood U , constant C or function f by the following Lemma: be the natural projection map for all I ⊂ S. We say that θ is compatible with R if the restriction of (Ψ I ) * θ − i∈I pr * I;i ρ I;i + w i 2π α I;i (5.33) to each fiber of π N X V I | Ψ −1 I (X−K) is 0 for every I ⊂ S. Lemma 5.28. Let (O X ( i∈S m i V i ), Φ, θ) be a model resolution so that (V i ) i∈S admits an ωregularization R. Define K ≡ ∪ i∈S−⋆ S V i . Then there is a smooth function g : X − K −→ R so that θ + dg is compatible withŘ for some regularizationŘ which is germ equivalent to R.
Proof. This is done by induction on the strata of ∪ i V i . We will use the notation from Definition 5.27 above. Let be a total order on the set of subsets of S with the property that if |I ′ | < |I| then I I ′ . We write I ≺ I ′ when I I ′ and I = I ′ . Suppose for some I * ⊂ S, we have constructed open sets U ≺ I inside Dom(Ψ I ) containing V I for all I ≺ I * and a smooth function g ≺ : X − K −→ R with the property that vanishes along each fiber of π N X V I | U ≺ I ∩Dom(Ψ I ) for all I ≺ I * . We now want these properties to hold for all I I * . For all I I * , let U I ⊂ Dom(Ψ I ) be open sets containing V I whose closure is compact so that the closure of U I is contained in U ≺ I when I ≺ I * and so that π N X V I | U I has contractible fibers for all I I * . Since the wrapping number of θ around V i is w i for all i ∈ I and since (Ψ I ) * (dθ) = ω (ρ I;i ,∇ (I;i) ) i∈I | Dom(Ψ I ) , we get that the restriction of to each fiber of π N X V I | Dom(Ψ I * ) is exact. Also by our induction hypothesis, the restriction of A to the fibers of π N X V I * | (Ψ I * ) −1 (Ψ I (U ≺ I )) is 0 for all I I * . This means that there is a smooth function g = : X − K −→ R so that g = restricted to a small neighborhood of the closure of Ψ I (U I ) is 0 for all I I * and so that A + (Ψ I * ) * dg = restricted to each fiber of π N X V I * | U I * ∩Dom(Ψ I * ) is 0. Define g ≡ g ≺ + g = . Then be a model resolution with associated symplectic form ω and let U ⊂ X be an open set. A regularization of M of radius R along U for some R < 1 is an ω-regularization R of (V i ) i∈S as in Equation (5.31) so that (1) the line bundle O X ( i∈S m i V i ) is also defined using the regularization maps (Ψ i ) i∈S from R, (2) Φ is radius R compatible with R along U as in Definition 5.9 and (3) θ is compatible with R. A regularization of M along U is a regularization of M of radius R along U for some R < 1 smaller than the tube radius of R along U .
We wish to show that every model resolution is isotopic to one admitting a regularization as above. Before we do this we need a preliminary lemma.
Lemma 5.30. Let X be a smooth manifold with a smooth family of cohomologous symplectic forms (ω t ) t∈[0,1] and (V i ) i∈S a compact SC divisor on X with respect to ω t for all t ∈ [0, 1]. Define K ≡ ∪ i∈S V i and let θ be a 1-form on X − K satisfying dθ = ω 0 | X−K . Then there exists a smooth family of 1-forms (θ t ) t∈[0,1] on X − K so that dθ t = ω t | X−K for all t ∈ [0, 1] and so that the wrapping number of θ t around V j does not depend on t ∈ [0, 1] for all j ∈ S.
Proof. Since ω t − ω 0 is exact for all t ∈ [0, 1], there is (by exploiting the Hodge decomposition theorem for differential forms) a smooth family of 1-forms (β t ) t∈[0,1] on X so that dβ t = ω t − ω 0 . Let U ⊂ X be a neighborhood of K whose closure is a compact manifold with boundary which deformation retracts onto K and ρ : X −→ [0, 1] a smooth function equal to 0 near K and equal to 1 outside a compact subset of U . Defineβ t ≡ θ + β t ∈ Ω 1 (X − K) for all t ∈ [0, 1]. Since ρβ t = 0 near K, we can think of this as a smooth 1-form on X by defining it to be 0 along K. Let c t ≡ [ω t − d(ρβ t )] ∈ H 2 (X, X − U ; R) for all t ∈ [0, 1] and define c ≡ [ω 0 − d(ρθ)] ∈ H 2 (X, X − U ; R). Since H 2 (X, X − U ; R) = H 2 (X, X − K; R) we have the long exact sequence: is independent of t which proves our lemma. Lemma 5.30 then tells us that there is a smooth family of 1-forms (θ t ) t∈[0,1] on X − ∪ i∈S−⋆ S V i so that θ 0 = θ and dθ t = ω| X−∪ i∈S−⋆ S V i for all t ∈ [0, 1] and so that the wrapping number of θ t around V i is independent of t for each i ∈ S − ⋆ S . We can assume that O X ( i∈S m i V i ) is defined using the regularizations (Ψ i ) i∈S as changing the regularization needed to define a line bundle as in Equation (5.9) creates an isomorphic line bundle. Now we isotope Φ through trivializations to a trivialization Φ so that Φ is compatible with R along U by Lemma 5.10. By Lemma 5.28 we have, after replacing R with a germ equivalent regularization, that θ ≡ θ 1 +dg is compatible with R for some g ∈ C ∞ (X − ∪ i∈S−⋆ S V i ). Hence which admits a regularization along U .
Lemma 5.32. Let (O X ( i∈S m i V i ), Φ, θ) be a model resolution admitting a regularization along U for some relatively compact open set U ⊂ X as in Definition 5.29. Let Φ 2 be the composition of Φ with the natural projection X × C ։ C. Define Since U is relatively compact, it is sufficient for us to show that for every x ∈ U ∩ (∪ i∈S V i ), there is a small open set U x ⊂ X containing x so that π Φ | Ux∩π −1 Φ (C−0) has symplectic fibers and so that θ restricted to π −1 Φ ({|z| = ǫ ′ }) ∩ U x is a contact form. We will first show that the fibers are symplectic. Suppose that I ⊂ S is the largest set satisfying x ∈ V I . Let a R be the function defined in Definition 5.9 and let Π (m i ) i∈S ,I (v) be as in Equation (5.13). Near x, we have that a R (ρ i ) = ρ i . Therefore π Φ (y) = Φ 2 (Π (m i ) i∈S ,I (Ψ −1 I (y))) for all y ∈ X near to x. Since Ψ I is a regularization, it is sufficient for us to show that the fibers of Φ 2 • Π (m i ) i∈S ,I restricted to a small neighborhood of x inside N X V I are symplectic with respect to where pr I;i is the natural projection map from Equation (5.32). Let W x ⊂ V I be a small open neighborhood of x that is contractible and choose unitary trivializations where β ∈ Ω 2 (N X V I | Wx ) is a closed 2-form whose restriction to the fibers of π N X V I | Wx is zero and whose restriction to the zero section is also zero. This means that near x we have that We can choose our trivializations T i so that Φ 2 • Π (m i ) i∈S ,I is equal to i∈I z m i i inside π N X V I | Wx . Since the fibers of i∈I z m i i are symplectic with respect toω near W x ∩ V I and since ω is equal toω at the point x, there is a small neighborhood U x ⊂ N X V I containing x so that the fibers of i∈I z i | Ux are symplectic with respect to ω (ρ I;i ,∇ (I;i) ) i∈I | Wx . Hence the fibers of π Φ | Ux are symplectic where U x ≡ Ψ I ( U x ).
We now wish to show that θ restricted to Y r ≡ π −1 Φ ({|z| = r}) ∩ U x is a contact form for all r > 0. Since ω restricted to the fibers of π Φ | Ux are symplectic, it is sufficient for us to show that θ restricted to the kernel of ω| Yr is non-zero at every point for all r > 0. By Equation (5.34) and the fact that Φ 2 • Π (m i ) i∈S,I is equal to i∈I z m i i inside N X V I | Wx , we get that the kernel of Ψ * I ω| Ψ −1 I (Yr) is tangent to the fibers of π N X V I inside Dom(Ψ I ). Therefore, since the restriction of the 1-form (5.33) to the fibers of π N X V I inside Dom(Ψ I ) is zero, Ψ * I θ restricted to the kernel of Ψ * I ω| Ux is non-zero so long as U x ⊂ Dom(Ψ I ). Hence θ restricted to Y r is a contact form for all r > 0 so long as U x ⊂ Im(Ψ I ).
along a relatively compact open set U ⊂ X as in Definition 5.29 where U contains K ≡ ∪ i∈S−⋆ S V i . Let T r,I be the radius r tube of V I as in Equation (5.14). Let Φ 2 : X −→ C be the composition of Φ with the natural projection map X × C −→ C. Choose ǫ > 0 small enough so that and so that the fibers (Φ 2 • s (m i ) i∈S ) −1 (z) ∩ U are symplectic for z ∈ D ǫ − 0 by Lemma 5.32. LetŤ be a smoothing of the compact manifold with corners ∪ i∈S−⋆ S T R,i so that

38)
X θ points outwards along ∂Ť ∩ T R,⋆ S and so thatŤ ⊂ ∪ i∈S−⋆ S T R,i . We also assume that the smoothing is small enough so that ∪ i∈S−⋆ S T 3R/4,i is contained in the interior ofŤ . Define ). This is a Liouville domain for ǫ > 0 small enough since X θ is tangent to V ⋆ S − K and X θ is transverse to ∂Ť and pointing outwards.
• the 1-form (5.33) restricted to each fiber of π N X V I | Ψ −1 I (X−K) is 0 for every I ⊂ S, • Equation (5.38) holds • Φ is radius R compatible with R along U and • ∪ i∈S−⋆ S T 3R/4,i is contained in the interior ofŤ , we get that the monodromy map φ : M −→ M of π Φ around the loop [0, 1] −→ ∂D(ǫ), t −→ ǫe 2πit with respect to the symplectic connection associated to ω exists and has compact support. In addition, since ω| π −1 Φ (∂D(ǫ)) = dθ| π −1 Φ (∂D(ǫ)) , φ is an exact symplectomorphism with compact support. We call (M, θ M , φ) the abstract contact open book associated to Now suppose that our model resolution has a choice of grading. Since π −1 Φ (∂D(ǫ)) is a contact submanifold of (X − K, ω) with contact form given by restricting θ by Lemma 5.32 after possibly shrinking ǫ, we get an induced grading on this contact submanifold by Lemma 5.20. Since the contact distribution is isotopic to Q ≡ ker(Dπ Φ | π −1 Φ (∂(ǫ)) ) through hyperplane distributions Q t , t ∈ [0, 1] where ω| Qt is non-degenerate for all t, we get a grading ι : F r(Q) × Sp(2n) Sp(2n) ∼ = F r(Q) on Q and hence on (M, dθ M ). Since the parallel transport maps of π Φ along ∂D(ǫ) have lifts to F r(Q), φ has an induced grading and hence (M, θ M , φ) is a graded abstract contact open book. We will call this the induced grading on (M, θ M , φ).
Proof. By Gray's stability theorem, there is a smooth family of contactomorphismsΦ t : C −→ C starting from the identity map so thatΦ t is a contactomorphism between (C, ξ 0 ) and (C, ξ t ). Therefore by pulling everything back byΦ t , we can assume that ξ t = ξ 0 for all t ∈ [0, 1]. Also by Gray's stability theorem, there is a smooth family of embeddings Again by Gray's stability theorem, there is a neighborhood N of B and a smooth family of contact embeddings ι t : (N, ξ 0 | N ) −→ (C, ξ C ) whose restriction to B 0 is ι t and where ι 0 | N : N −→ N is the identity map. Let H t : ι t (N ) −→ R be a smooth family of functions generating the contact isotopy ι t . By definition this means that there is a contact form α compatible with ξ 0 so that where R is the Reeb vector field of α (see [MS98,Lemma 3.49]). Choose a smooth family of functions K t , t ∈ [0, 1] equal to H t near ι t (B t ). Then K t generates a smooth family of contactomorphisms Ψ t satisfying the properties we want.
of radius R < 1 along an open set U ⊂ X containing K ≡ ∪ i∈S−⋆ S V i as in Definition 5.29. Since m ⋆ S = 1, we let a closed hypersurface transverse to X θ and V ⋆ S and define B ≡ C ∩ V ⋆ S . Let Φ B be a trivialization of the normal bundle of the contact submanifold B ⊂ C induced by Φ ⋆ | B .
Then the contact pair (B ⊂ C, ker(θ| C ), Φ B ) is contactomorphic to the link of the model resolution (O X ( i∈S m i V i ), Φ, θ). If our model resolution is graded, then both of these contact pairs have induced gradings by Lemma 5.20 and the above contactomorphism becomes a graded contactomorphism with respect to these gradings.
Then f is compatible with (V i ) i∈S−⋆ S as in Definition 5.18. Let c ≪ −1 and defině The normal bundle ofB insideČ has a natural trivialization ΦB induced by the trivialization Φ ⋆ . Since df (X θ ) > 0 near K and c ≪ −1 , we get that (B ⊂Č, ξČ, ΦB) is the link of our model resolution (O X ( i∈S m i V i ), Φ, θ) by Definition 5.25.
Since df (X θ ) > 0 inside ∪ i∈S−⋆ S T R,i − K and ∪ i V i is connected, we can choose a smooth family of hypersurfaces (C t ) t∈[0,1] joining C andČ so that C t is transverse to X θ and V ⋆ S for all t ∈ [0, 1]. Define B t ≡ C t ∩ V ⋆ S and ξ t ≡ ker(θ| Ct ). Also let Φ Bt be the trivialization of the normal bundle of B t inside C t induced by Φ ⋆ so that Φ B 0 = Φ B and Φ B 1 = ΦB. Then (B t ⊂ C t , ξ t , Φ Bt ) is a smooth family of contact pairs joining (B ⊂ C, ker(θ| C ), Φ B ) and (B ⊂ C, ξČ , ΦB). Therefore they are isomorphic by Lemma 5.34. Also if (O X ( i∈S m i V i ), Φ, θ) is graded then they are graded isomorphic since all of our contact pairs have induced gradings from our model resolution by Lemma 5.20. Proof. In this proof we will use the same notation as in Definition 5.33. We will introduce it again here for the sake of clarity. By Lemma 5.31 we can isotope our model resolution so that it admits a regularization R ≡ ((ρ i ) i∈I , (Ψ I ) I⊂S ) of radius R along U for some relatively compact open U containing K ≡ ∪ i∈S−⋆ S V i . By Lemma 5.26, the link does not change after this isotopy. Let T r,I be the radius r tube of V I as in Equation (5.14). LetŤ be a smoothing of the manifold with corners ∪ i∈S−⋆ S T R,i as in Definition 5.33. In other words,Ť satisfies Equation (5.38), X θ points outwards along ∂Ť andŤ ⊂ ∪ i∈S−⋆ S T R,i . Also we require thatŤ contains ∪ i∈S−⋆ S T 3R/4,i . Define Define be the natural projection map and (r, ϑ) polar coordinates on C. Let W : L R ∩ V ⋆ S −→ [0, 1] be a smooth function equal to 0 inside L 4R/5 ∩ V ⋆ S and equal to 1 inside (L R − L 5R/6 ) ∩ V ⋆ S and define W : We now define θ t ∈ Ω 1 (((T R,⋆ S ∩ L R ) ∪ L 4R/5 ) − K) for t ∈ [0, 1] to be θ inside L 4R/5 − K and equal to For R 1 > 0 small enough with respect to R, we get that dθ t is a symplectic form inside L ≡ L 4R/5 ∪ (L R ∩ T R 1 ,⋆s ) and dθ t restricted to π −1 Φ (x) ∩ L is a symplectic form for all x ∈ C − 0 and t ∈ [0, 1].
This is a smooth hypersurface in X since Φ is radius R compatible with R along U and since ǫ > 0 can be made small enough so that ǫ < 3R/4. We can also ensure that ǫ > 0 is small enough so that π −1 Φ (D ǫ ) ⊂ L. This ensures that dθ t is a symplectic form near π −1 Φ (D ǫ ) and that dθ t restricted to the fibers of π Φ | π −1 Φ (∂Dǫ) is a symplectic form for all t. Define B ≡ C ∩ V ⋆ S . This is also equal toŤ ∩ V ⋆ S = κ −1 (0). For R small enough, we have that (C, ker(θ t )| C ) is a smooth family of contact submanifolds of X. The trivialization Φ ⋆ gives us a trivialization Φ B,t of the normal bundle of the contact submanifold B inside (C, ker(θ t )| C ) since C is transverse to V ⋆ S . Hence we get a smooth family of contact pairs which are all contactomorphic by Lemma 5.34. Also by Lemma 5.35, the contact pair P 0 is contactomorphic to the link of (O X ( i∈S m i V i ), Φ, θ) for R small enough and hence P 1 is contactomorphic to the link of (O X ( i∈S m i V i ), Φ, θ). Therefore to complete this Lemma, it is sufficient to show that the contact pair P 1 is contactomorphic to the contact pair associated to OBD(M, θ M , φ). In fact since (M, θ t | M ) is a smooth family of Liouville domains and since the monodromy map of π Φ around the path (5.39) with respect to the fiberwise symplectic 2-form dθ t | π −1 Φ (∂D(ǫ) is equal to φ for all t, it is sufficient for us to show that the contact pair P 1 is contactomorphic to the contact pair associated to OBD(M, θ 1 | M , φ). Note that there is a resemblance between the construction of C and the construction of OBD(M, θ 1 | M , φ) from Definition 3.14. We will now make this precise.
The contact pair P 1 can be constructed as follows: sending (x, s) to the parallel transport of x ∈ M along ∂D(ǫ) in the counter clockwise direction from ǫ ∈ ∂D(ǫ) to ǫe is ∈ ∂D(ǫ) with respect to the 2-form dθ 1 | π −1 Φ (∂D(ǫ)) . Hence we will assume that T φ = π −1 Φ (∂D ǫ ) under the identification Φ and that V is naturally a subset of T φ . Since φ has compact support inside M , we have the standard collar neighborhood as in Definition 3.13 (here δ > 0 is the same small constant defined above, which might have to be made smaller). We can choose κ so that e κ | T φ is the natural projection to (1 − δ, 1] in the neighborhood (5.42). This means that θ 1 restricted to the region (5.42) is equal to e κ α M + πǫ 2 dt where t parameterizes R/Z and where α M = θ 1 | ∂M by Equation (5.40).
Using the diffeomorphism Φ and definition (5.41) of C, we have that C is naturally diffeomorphic toČ Because θ 1 restricted to T R,⋆ S ∩ (L R − L 5R/6 ) is equal to P * ⋆ S (θ| V⋆ S −K ) + 1 2 ρ ⋆ S (Ψ −1 ⋆ S ) * Φ * ⋆ (dϑ) by Equation (5.40) and because P * ⋆ S (θ| V⋆ S −K )| M ∩ κ −1 (−δ,0] = e κ α M inside the cylinder (1 − δ, 1] × ∂M ⊂ M , we have that the contact form θ 1 | C insideČ under the above identification is equal to Notice that this description of P 1 resembles the construction of the open book associated to the abstract contact open book (M, θ 1 | M , φ) as in Definition 3.14. All we need to do is deform the above construction until it is actually equal to OBD(M, θ 1 | M , φ).

Dynamics of abstract contact open books associated to model resolutions.
In this subsection we show that the fixed points of a positive slope perturbation of the symplectomorphism associated to the graded abstract contact open book associated to a model resolution form a union of specific codimension 0 families of fixed points. We also compute the indices of these fixed points.
Definition 5.37. Let (O X ( i∈S m i V i ), Φ, θ) be a graded model resolution with associated symplectic form ω on X where n + 1 = 1 2 dim(X). Let J be an ω-compatible almost complex structure on X. By Definition 7.6, the grading onX ≡ X − ∪ i∈S−⋆ S V i corresponds to a trivialization Φ : κ J |X −→X × C of the canonical bundle. Let U be a small neighborhood of ∪ i∈S−⋆ S V i which deformation retracts on to V i . Choose a smooth section s of κ J which is transverse to 0 and so that Φ•s|X −U is a non-zero constant section of (X−U )×C. By a Mayor-Vietoris argument, the homology group H 2n (∪ i∈S−⋆ S V i ; Z) = H 2n (U ; Z) is freely generated by the fundamental classes In the case of Example 5.14, the discrepancy and multiplicity of E i as defined in Definition 5.37 is identical to the discrepancy and multiplicity of f along E i as in Definition 2.1. Similarly we have a notion of multiplicity m separating resolution as in Definition 2.2 for model resolutions which coincide in the case of Example 5.14: . Then the natural m i -fold covering of V o i is the m i -fold covering of V o i given by a disjoint union of covers diffeomorphic to the cover corresponding to the normal subgroup The proof of this lemma also gives us a geometric interpretation of V o i . Proof. After an isotopy, we can assume that our model resolution admits a regularization of radius R along U for some relatively compact open U containing ∪ j∈S−⋆ S V i . Let T r,i be the radius r tube of V i as in Equation (5.14) for some r < 3R/4. We can assume that the open neighborhood U i from Definition 5.39 is equal to T r,i − ∪ j∈S−i V j . We have that our map Q i is equal to for ǫ > 0 small enough is a fibration whose fibers are smooth manifolds with corners. Combining this with the fact that π 2 (D(ǫ) * ) = 0 we get that the map is an isomorphism by a fibration long exact sequence argument. Therefore the natural map π 1 (Q −1 i (ǫ)) −→ π 1 (U i ) has image G i . Also for 0 < ǫ ≪ r ≪ 1, the map Proof of Theorem 5.41. In this proof we will use the same notation as in Definition 5.33. We will introduce it again here for the sake of clarity. After an isotopy, we can assume that our model resolution admits a regularization We now wish to show that φ satisfies properties (1)-(3) listed in the statement of this theorem. To do this, we need to recall the construction of (M, θ M , φ). Let T r,I be the radius r tube of V I as in Equation (5.14) and let T o r,I be the interior of T r,I . LetŤ be a smoothing of the manifold with corners ∪ i∈S−⋆ S T R,i as in Definition 5.33. In other words,Ť satisfies Equation (5.38), X θ points outwards along ∂Ť ,Ť ⊂ ∪ i∈S−⋆ S T R,i and ∪ i∈S−⋆ S T 3R/4,i is contained in the interior ofŤ . Define is the composition of Φ with the natural projection map X × C −→ C. Then we can assume that (M, θ M ) ≡ (π −1 Φ (ǫ), θ| π −1 Φ (ǫ) ) for some small ǫ > 0. We will assume that ǫ > 0 is small enough so that M ⊂ ∪ i∈S T R/4,i and the fibers of π Φ | π −1 Φ (D(ǫ)) are symplectic by Lemma 5.32. Let ω be the associated symplectic form of our model resolution. Define φ : M −→ M to be the monodromy map around the loop [0, 1] −→ ∂D(ǫ), t −→ e 2πit with respect to the symplectic connection associated to ω.
First of all, we will compute the fixed points of the map φ m . To do this, we will show that they correspond to certain periodic orbits of the flow of a Hamiltonian onŤ . Define It is sufficient for us to find the periodic orbits of X H starting inside M which map under π Φ to loops in C * which wrap around 0 exactly m times in the anti-clockwise direction. This is because there is a 1 − 1 correspondence between fixed points of φ m and such orbits. This correspondence sends a fixed point p of φ m to the unique flowline of X H starting and ending at p whose image under π Φ wraps around 0 exactly m times in the anti-clockwise direction.
DefineŤ R,I ≡ T o 3R 4 ,I − ∪ i∈S−I T o 3R 4 ,i for each I ⊂ S. Since M ⊂ ∪ I⊂SŤR,I , it is sufficient for us to calculate the fixed points of φ m inside M ∩Ť R,I for each I ⊂ S. Therefore we will now compute the periodic orbits of X H starting inside M ∩Ť R,I for all I ⊂ S which project to loops in C * wrapping m times around 0 in an anti-clockwise direction. Let a R : [0, ∞) → [0, ∞) be the smooth function defined in Definition 5.9. In other words, a R satisfies: ( ) be the natural projection map. InsideŤ R,I we have that since the bundle trivialization Φ is radius R compatible with our regularization R along T R,I . Hence This means that all the periodic orbits of X H starting insideŤ R,I are contained inside the fibers of p I since the vector fields X ρ i are tangent to these fibers. Also since b R (ρ i (x)) > 0 and b ′ R (ρ i (x)) > 0 for all x ∈Ť R,I and all i ∈ I, we have that any disk contained inside a fiber of p I bounding any such orbit must intersect V i positively for all i ∈ I. This implies that the projection of this orbit to C * wraps around 0 more than m times if |I| > 1 since our model resolution is a multiplicity m separating resolution. This means that if the set of periodic orbits of X H starting inside M ∩Ť R,I whose image in C * wraps m times around 0 is non-empty then |I| = 1. Hence all fixed points of φ m are contained inside ∪ i∈S M ∩Ť R,i . Similar reasoning ensures that i ∈ S m ∪ {⋆ S } and that the set of fixed points of φ m inside and f j ≡ 1 2π arg(π Φ )| B j for j = 1, 2, we have that B i is a codimension 0 family of fixed points of φ m for all i ∈ S m . Since B i is homotopic to π −1 Φ (ǫ) ∩ T R,i which in turn is homotopic to the fiber Q −1 i (ǫ) constructed in the proof of Lemma 5.40 we have We now need to construct a small positive slope perturbationφ of φ m without creating any extra fixed points so that B ⋆ S disappears and so that φ =φ near ∪ i∈Sm B i . Since B ⋆ S is a codimension 0 family of fixed points of φ, there is a neighborhood N ⋆ S of B ⋆ S and a Hamiltonian H ⋆ S : N ⋆ S −→ (−∞, 0] so that φ m is the time 1-flow of H ⋆ S inside N ⋆ S and so that B ⋆ S = H −1 ⋆ S (0). Choose δ ⋆ > 0 small enough so that H ⋆ S has no q-periodic orbits inside H −1 ⋆ S (−δ ⋆ , 0) for all q ∈ [0, 2]. Since the vector field (5.45) is tangent to the fibers of p I inside T R,I ∩ T 3R/4,⋆ S for all I ⊂ S and since Ψ I is a regularization, we have that H ⋆ S must be a function of the variables (ρ i ) i∈I inside T R,I ∩ T 3R/4,⋆ S only. This implies that we can construct a smooth functionb : where r M is the radial coordinate on M and •b has no critical points. This implies that the time 1-flow ofb has no fixed points inside N ⋆ S and is equal to H ⋆ S outside a compact subset of N ⋆ S . Defineφ to be equal to φ m outside N ⋆ S and the time 1-flow ofb inside N ⋆ S . This is a positive slope perturbation of φ m so that the set of fixed points of φ is ∪ i∈S−⋆ S B i andφ = φ m in a neighborhood of these fixed points.
Next we need to compute the action of B i for each i ∈ S − ⋆ S . Let p ∈ B i and let γ : R/Z →Ť be the unique loop starting at p ∈ M which is symplectically orthogonal to the fibers of π Φ and satisfying π Φ • γ(t) = e 2iπmt for all t ∈ R. Then the action of p is equal to − We now need to compute the Conley-Zehnder index of B i for each i ∈ S m . Fix i ∈ S m and let p ∈ B i ⊂ M . Let γ : R/Z →Ť be the unique loop starting at p ∈ M which is symplectically orthogonal to the fibers of π Φ and satisfying π Φ • γ(t) = e 2iπmt for all t ∈ R. Let J be an ω-compatible almost complex structure on X so that π Φ becomes J-holomorphic.
be the vertical tangent bundle. Let (T verŤ ) ⊥ ⊂ T (Ť − K) be the set of vectors which are ω-orthogonal to the vertical tangent bundle. This is a J-holomorphic subbundle of T (Ť − K). Let τ C * : T C * −→ C * × C be the holomorphic trivialization which sends ∂ ∂ϑ to the constant section 1 and let τ C * ,2 : T C * −→ C be the composition of τ C * with the natural projection map C * × C −→ C. We then have a trivialization Let κ J,φ be the canonical bundle of T verŤ . Then we have a canonical isomorphism Since (X − K, ω) is a graded symplectic manifold, we get a natural choice of trivialization τ : κ J | X−K −→ (X − K) × C by Definition 7.7. The trivializations τ and (τ ⊥ ) * give us a trivialization τ ver : κ J,φ −→ (Ť − K) × C of κ J,φ by the identity (5.46). Let s φ be a section of κ J,φ so that it is equal to the constant section 1 with respect to our trivialization τ ver . Let s be a section of κ J so that s −1 (0) is transverse to 0 and contained inside a small neighborhood N of ∪ i∈S−⋆ S V i which deformation retracts on to ∪ i∈S−⋆ S V i and so that τ • s| X−N is the constant section 1. Then by definition [s −1 (0)] is a homology class homologous to i∈S a i [V i ]. Now defineŤ ǫ ≡ π −1 Φ (∂D(ǫ)). By construction, τ ver |Ť ǫ is homotopic to the induced trivialization from Lemma 5.20 (after identifying the contact hyperplane distribution inŤ ǫ with T verŤ using an isotopy between these symplectic subbundles).
We can choose J so that near the image γ, the Hamiltonian flow φ for all x ∈Ť ǫ ∩ N γ given by the highest wedge power of the J-holomorphic bundle map : κ J,φ −→ C be the composition of τ and τ ver respectively with the natural projection map to C. The winding number of the map we have that the winding number of w τ is equal to k i (−1 − a i ). Hence by Lemma 7.8 and the fact that the winding number of w φ is the winding number of w τ , we have that the Conley-Zehnder index of the fixed point p of φ m is minus 2 times the winding number of w τ and so CZ(φ m , B i ) = 2k i (a i + 1).
Here is a technical lemma that was used in the proof of Theorem 5.41 above.
If we smoothly deform f 2 inside B 2 through fibrations whose fibers are always transverse to the line field given by ker(ω| B 2 ) then the symplectic form on the fibers and the monodromy map does not change. This is because such a deformation can be realized by a flow along a vector field tangent to the line field ker(ω| B 2 ). In particular, we can assume that in some small (5.47) Let pr 1 : V × AČ −δ,Č+δ −→ V be the natural projection map. Define V 2 ≡ f −1 2 (0) ∩Ǔ and ω V 2 ≡ ω| V 2 . We can assume thatǓ is small enough so that pr 1 | V 2 : V 2 −→ V is a diffeomorphism onto its image. This map is also a symplectic embedding by Equation (5.47). Define Since ϑ) insideǓ and pr 1 | V 2 is a symplectic embedding, we get that the vector field −X ∂ ∂ϑ is tangent toǓ ∩ B 2 and lies in the kernel of ω| B 2 ∩Ǔ . Then for all m > 0, φ m 2 is equal to the time 1 flow of −mH near B 1 ∩ f −1 2 (0) inside the symplectic manifold (V 2 , ω V 2 ). Hence B 1 ∩ f −1 2 (0) is a codimension 0 family of fixed points of φ 2 .
Proof of Theorem 1.2. Let L ≡ (L f ⊂ S ǫ , ξ Sǫ , Φ f ) be the contact pair associated to f with the standard grading as in Example 3.8. Let (O X ( i∈S m i E i ), Φ, θ) be a graded model resolution coming from the log resolution π : Y −→ C n+1 as in Example 5.14. The wrapping number of θ around E j is w j for all j ∈ S − ⋆ S . By using the function |z| 2 on C n+1 combined with Lemma 5.34, one can show that the link of this model resolution is contactomorphic to L. Hence Theorem 1.2 follows from Theorem 5.41, (HF2) and (HF3).
Lemma 6.1. Suppose we have a cohomological spectral sequence converging to a Z-graded abelian group G * with E 1 page (E 1 p,q ) p∈Z,q∈Z . Define m ≡ sup{p + q : E p,q 1 = 0} and k p ≡ sup{p + q : q ∈ Z, E p,q 1 = 0} ∀ p ∈ Z. Suppose that m is finite and k p = m − 1 for all p ∈ Z. Then G m = 0 and G k = 0 for all k > m.
We will show that each element of E pm,m−pm j can never kill or be killed by the spectral sequence differential for each j ≥ 1. Since k pm−j = m − 1 for all j, we get that k pm−j < m − 1 for all j ≥ 1. Therefore E pm−j,m−pm+j−1 j = 0 for all j ≥ 1. Hence the differential: is zero for all j. Also since (p m + j) + (m − p m − j + 1) = m + 1 > m, we get that E pm+j,m−pm−j+1 j = 0 for all j. Therefore the differential: is zero for all j . Hence G m = 0. Also G k = 0 for all k > m since E p,q 1 = 0 for all p, q ∈ Z satisfying p + q = k.
Proof of Corollary 1.3. The numbers µ m do not depend on the choice of log resolution for all m > 0 by Lemmas 2.3 and 2.6. Hence Corollary 1.3 follows immediately from Theorem 1.2 combined with Lemmas 2.4 and 6.1.

Appendix A: Gradings and Canonical Bundles
In this section we will develop tools so that we can construct gradings (See Definition 3.2) and relate them to other kinds of topological information. In this paper we will only need to study gradings up to isotopy which will be defined now. We will first give a definition of a grading for any principal G bundle and then relate it to gradings of (E, Ω). Throughout this section, G will be a Lie group, G its universal cover and p : W −→ B will be a principal G bundle. Also π : E −→ V , will be a symplectic vector bundle with symplectic form Ω whose fibers have dimension 2n.
Definition 7.1. A grading of p consists of a principal G bundle p : W −→ B along with a G bundle isomorphism ι : W × G G ∼ = W. Note that a grading of (E, Ω) is equivalent to a grading of the principal Sp(2n) bundle F r(E). Let ι j : W j × G G ∼ = W, j = 0, 1 be gradings of p. An isotopy between these two gradings consists of a G-bundle isomorphism together with a smooth family of G bundle isomorphisms: An isotopy between two gradings of (E, Ω) is an isotopy between the corresponding gradings on the principal Sp(2n) bundle F r(E). We can define isotopies of gradings of symplectic manifolds and contact manifolds in a similar way.
Definition 7.2. Let ι : W × G G ∼ = W be a grading of p. The associated covering map of this grading is the natural map: The following lemma gives a topological characterization of gradings. For simplicity we will assume that the base B is connected. Let ⋆ ∈ W be a choice of basepoint.
Lemma 7.3. Let N W be the set of normal subgroups A ⊳ π 1 (W, ⋆) so that p * : π 1 (W, ⋆) −→ π 1 (B, p(⋆)) restricted to N W is an isomorphism. Let Gr W be the set of isotopy classes of gradings of W . Then the map Q W : Gr W −→ N W sending a grading to is an isomorphism where P is the associated covering map of this grading.
Proof. We will first show that the map Q W is well defined by showing that the image (7.1) is a normal subgroup. Let ι : W × G G ∼ = W be a grading of p. The covering map P of such a grading is isomorphic (using the map ι) to the natural map The deck transformations of this map are equal to ker( G −→ G) and these act transitively.
Hence the image (7.1) is a normal subgroup. Combining this with the fact that the fibers of the natural fibration W −→ B are simply connected, the image (7.1) is contained in N W and hence the map Q W is well defined.
We will now construct an inverse to Q W . Let N ⊳ π 1 (W, ⋆) be an element of N W . Let P : W −→ W be a covering map with a choice of basepoint ⋆ ∈ W mapping to ⋆ so that the map has image equal to N . Let F be a fiber of p. Let ι F : F ֒→ W be the natural inclusion map. Since P * | P −1 * (N ) is injective, we get that (ι F ) −1 * (P −1 * (N )) = {id} and hence P | P −1 (F ) : P −1 (F ) −→ F is the universal covering map. This implies that each fiber has a natural G action and hence We define Q −1 W (N ) to be the above grading. This is an inverse to Q W .
We have the following immediate corollary of Lemma 7.3.
Corollary 7.4. Suppose that p j : W j −→ B is a principal G j bundle for some Lie group G j for j = 1, 2. Let Φ : W 1 −→ W 2 be a map of fiber bundles so that the induced map on the fibers is a fundamental group isomorphism. Then the map Φ * : N W 1 −→ N W 2 induces a natural bijection between isotopy classes of gradings on W 1 and isotopy classes of gradings on W 2 .
Here the sets N W 1 and N W 2 in this corollary are defined as in Lemma 7.3.
Lemma 7.5. Let π K : K −→ B be a principal U (1) bundle. Then there is a natural 1 − 1 correspondence between homotopy classes of trivializations of π K and isotopy classes of gradings of π K .
Proof. There is a 1 − 1 correspondence between trivializations Φ : K −→ B × U (1) of K up to homotopy and sections of π K up to homotopy given by the map sending Φ to the section whose image is Φ −1 (1). Hence all we need to do is construct a natural 1 − 1 correspondence between sections up to homotopy and isotopy classes of gradings. Let S be the set of sections up to homotopy and Gr K the set of isotopy classes of gradings. By Lemma 7.3, it is sufficient for us to construct a bijection between S and N K . We define Ψ : S −→ N K , Ψ(s) = Im(s * : π 1 (B, ⋆) −→ π 1 (K, s(⋆))).
The inverse of this map is constructed as follows: we start with a normal subgroup N ∈ N K . This gives us a grading ι : K × R U (1) ∼ = K by Lemma 7.3 since U (1) = R. Since the fibers of K are contractible, there is a smooth section s : B −→ K by [Pal66, Theorem 9] combined with the Steenrod approximation theorem [Ste99, Section 6.7, Main Theorem]. The composition s ≡ P K • s where P K : K −→ K is the associated covering map is then a smooth section of K. We then define Ψ −1 (N ) ≡ s. This is the inverse of Ψ.
We will now focus on the principal Sp(2n) bundle F r(E).
Definition 7.6. Let J be a complex structure on E compatible with Ω. The frame bundle F r(E, Ω, J) of the unitary bundle (E, Ω, J) is the principal U (n)-bundle whose fiber over v ∈ V is the space of unitary bases e 1 , · · · , e n of (E, Ω, J)| v . The anti-canonical bundle κ * J of (E, J) is the highest exterior power of the complex vector bundle (E, J). The associated U (1)-bundle κ * Ω,J ⊂ κ * J of κ * J has a fiber at v ∈ V equal to the subset of elements e 1 ∧ · · · ∧ e n where e 1 , · · · , e n is a unitary basis for (E, Ω, J)| v . Therefore we have a natural map det Ω,J : F r(E, Ω, J) −→ κ Ω,J .
The canonical bundle κ J of (E, J) is the dual of κ * J (or equivalently the anti-canonical bundle of the dual bundle of (E, J)). In a similar way, we can define the (anti-)canonical bundle of a symplectic manifold with a choice of compatible almost complex structure, or of a contact manifold (C, ξ C ) with a choice of compatible contact form α and a dα| ξ C -compatible almost complex structure J on ξ C .
Definition 7.7. Let J be an Ω-compatible complex structure on E. Let ι J : F r(E, Ω, J) −→ F r(E, Ω) be the natural inclusion map. By [MS04, Proposition 2.22, 2.23], the natural maps det Ω,J and ι J above are bundle maps whose restriction each fiber is a fundamental group isomorphism. Hence by Corollary 7.4 and Lemma 7.5 there is a natural 1 − 1 correspondence between isotopy classes of gradings of (E, Ω) and homotopy classes of trivializations of κ * Ω,J . Combining this with the fact that there is a natural 1 − 1 correspondence homotopy classes of trivializations of κ * Ω,J and homotopy classes of trivializations of κ * J and hence of trivializations of the canonical bundle κ J , we get a natural 1 − 1 correspondence: Given a trivialization Φ of κ J we will call the grading Gr(Φ) the grading associated to Φ. Given a grading g of (E, Ω), we will call Gr −1 (g) the trivialization associated to this grading.
The above discussion enables us to compute the Conley-Zehnder index of a fixed point of a graded symplectomorphism in some nice cases. Let φ : M −→ M be a graded exact symplectomorphism of a Liouville domain (M, θ M ). Let V be the unique vector field on the mapping torus T φ from Definition 3.13 given by the lift of the vector field d dt on R/Z satisfying ι V dα T φ = 0. Let φ V t be the time t flow of V . Let x be a fixed point of φ. Suppose that there is a compatible complex structure J on the vertical tangent bundle ( Since φ is graded, we see by Definition 3.13 that there is a natural grading on the vertical tangent bundle. Therefore by Definition 7.7, there is a natural trivialization Φ : κ J −→ T φ ×C of κ J associated to this grading. Let Φ 2 : κ J −→ C be the composition of Φ with the natural projection map to C. T φ (0) be a fixed point of φ and suppose that Dφ| x : T x M −→ T x M is the identity map. Then CZ(φ, x) is equal to minus 2 times the winding number of the map: Proof. Let γ : R/mZ −→ T φ be the m-periodic orbit of V whose initial point is x. Then there is a unique (up to homotopy) unitary trivialization T of γ * T ver T φ so that Φ is equal to the highest wedge power of T . Because of the correspondence (7.2), we can also ensure that T maps the grading on γ * T ver T φ (given by pulling back the grading on T ver T φ via γ) to the trivial grading on (R/mZ) × C n+1 (maybe after changing the grading to an isotopic one). Under this trivialization, the flow of V corresponds to a smooth family of Hermitian matrices (A t ) t∈[0,1] and the degree of w is −1 times the winding number of t −→ det C (A t ). Using the correspondence (7.2) and the trivialization T , such a family of matrices corresponds to a point in the universal cover F r(T M )| x of F r(T M )| x . Hence the Conley-Zehnder index of (A t ) t∈[0,1] is equal to CZ(φ, x). Since A t are unitary matrices, we get that CZ((A t ) t∈[0,1] ) is equal to twice the winding number of t −→ det C (A t ). Hence CZ(φ, x) is −2 times the winding number of w.

Appendix B: Contactomorphisms of Mapping Tori and Floer Cohomology
The aim of this section is to show that Property (HF2) holds. Here is a statement of this property: Suppose that (M 1 , θ M 1 , φ 1 ), (M 2 , θ M 2 , φ 2 ) are graded abstract contact open books so that the graded contact pairs associated to them are graded contactomorphic. Then HF * (φ 0 , +) = HF * (φ 1 , +).
Note that the definition above has many similarities with the definition of the mapping torus from Definition 3.13. Also if we define the mapping torus π Tφ : Tφ −→ R/Z, α Tφ of our positive slope perturbationφ in exactly the same way as in Definition 3.13, then Wφ = R × Tφ, θφ = (s − κ)dt + κ C αφ for some C > 0 and πφ = id R × π Tφ . The following calculation will be useful later on. If we have a Hamiltonian H equal to π * φ K for some K : R × (R/Z) −→ R then X H is equal to the horizontal lift of X ds∧dt K with respect to the symplectic connection associated to dθφ.
Definition 8.4. Let (Wφ, πφ, θφ) be a mapping cylinder. An almost complex structure J on Wφ is strictly compatible with (Wφ, πφ, θφ) if (1) J is compatible with dθφ, (2) πφ : Wφ −→ R × R/Z is (J, j)-holomorphic (I.e. Dπφ • J = j • Dπφ) where j is the complex structure sending ∂ ∂s to ∂ ∂t where (s, t) parameterizes R × R/Z, (3) the restriction of J to the cylindrical end Cφ is a product j ⊕ J M where J M is a fixed cylindrical almost complex structure inside the cylindrical end (1 − δφ, 1] × ∂M and (4) J is invariant under translations in the s coordinate. We will call J M the associated cylindrical almost complex structure on M . An almost complex structure J on Wφ is compatible with (Wφ, πφ, θφ) if there is an almost complex structureJ compatible with (Wφ, πφ, θφ) and a compact subset K in the interior of Wφ so that J| Wφ−K = J| Wφ−K .
Definition 8.5. Recall that a 1-periodic orbit of a time dependent Hamiltonian H t : Wφ −→ R is a smooth map γ : R/Z −→ M satisfying d dt γ = X dθφ Ht . Since there is a natural 1-1 correspondence between 1-periodic orbits and fixed points of the corresponding Hamiltonian symplectomorphism φ Ht 1 , we will call γ the 1-periodic orbit associated to the fixed point γ(0). A Hamiltonian is autonomous if it does not depend on time. An S 1 -family of 1-periodic orbits of H is a family (γ t ) t∈R/Z of 1-periodic orbits where γ t (ť) = γ 0 (ť + t) for all t,ť ∈ R/Z. Let h : R −→ R be a smooth function which is bounded from below with positive derivative satisfying h ′′ (s) = 0 whenever s is sufficiently positive and h ′ (s) < 1 for s sufficiently negative. The value of h ′ (s) for large enough s is called the slope of h. A Hamiltonian is strictly compatible with (Wφ, πφ, θφ) if it is equal to π * φ h(s) everywhere. A Hamiltonian is compatible with (Wφ, πφ, θφ) if it is equal to π * φ h(s) outside a compact subset of the interior of Wφ. The slope of such a Hamiltonian is defined to be the slope of h.
All the 1-periodic orbits of h(s) on the symplectic manifold (R × R/Z, ds ∧ dt) wrapping around R × R/Z once come in S 1 families in the region h ′ (s) = 1 and for the unique s satisfying h ′ (s) = 1 we have 1-periodic orbits for all q ∈ [0, 1). Also the 1-periodic orbits of π * φ h(s) project to 1-periodic orbits of h(s). A pair (H, J) is (strictly) compatible with (Wφ, πφ, θφ) if H is a Hamiltonian (strictly) compatible with (Wφ, πφ, θφ) and J is an almost complex structure (strictly) compatible with (Wφ, πφ, θφ). A smooth family (H s , J s ) s∈R of pairs compatible with (Wφ, πφ, θφ) has non-increasing slope if the slope of H s is greater than or equal to the slope of Hš for all s ≤š.
We say that (H σ , J σ ) σ∈R satisfies the maximum principle if there is a sequence of relatively compact open sets (V i ) i∈N whose union is Wφ so that (H σ , J σ ) σ∈R satisfies the maximum principle with respect to V i for all i ∈ N.
Lemma 8.7. Let (M, θ M , φ) be an abstract contact open book and let W ≡ (Wφ, πφ, θφ) be a mapping cylinder of some positive slope perturbation of φ. Let (K σ ≡ π * φ (k σ (s))) σ∈R be a smooth family of Hamiltonians strictly compatible with W so that dkσ dσ ≤ 0, dk ′ σ dσ ≤ 0 and dk ′′ σ dσ ≤ 0. Let Y be an almost complex structure strictly compatible with W . Let δ, S > 0 and let rφ be the vertical cylindrical coordinate of W . Let (H σ , J σ ) σ∈R be a smooth family of pairs compatible with W which are equal to (K σ , Y ) near the boundary of the set V δ,S ≡ π −1 φ ((−S, S) × (R/Z)) ⊂ Wφ and also in the region {rφ ≥ 1 − δ}. Then V δ,S satisfies the maximum principal with respect to (H σ , J σ ) σ∈R .
Definition 8.9. For any Mapping cylinder W ≡ (Wφ, πφ, θφ), define βφ ⊂ H 1 (Wφ) to be the set of homology classes represented by loops which project under πφ to loops homotopic to For each a, b ∈ [−∞, ∞] where a < b, any non-degenerate Hamiltonian (H t ) t∈[0,1] and smooth family of almost complex structures (J t ) t∈[0,1] compatible with W we can define HF * [a,b],βφ (φ Ht 1 ) in the same way as Floer cohomology of φ Ht 1 except that we only consider fixed points of action in [a, b] and whose associated 1-periodic orbits represent an element of βφ. We also define HF * [−∞,∞],βφ (φ Ht 1 ) ≡ HF * βφ (φ Ht 1 ). A non-degenerate autonomous Hamiltonian is an autonomous Hamiltonian H : Wφ −→ R so that for every fixed point p of the time 1 flow φ H 1 , the eigenspace of Dφ H 1 (p) associated with the eigenvalue 1 is one dimensional (this eigenspace is the tangent line to the 1-periodic orbit associated to p). Now suppose that the mapping cylinder (Wφ, πφ, θφ) is graded with grading ι : F r(T Wφ) × Sp(2n) Sp(2n) ∼ = F r(T Wφ).
Since any Hamiltonian H t is isotopic to the identity map id Wφ , we get that φ Ht 1 is naturally graded since the identity map on F r(T Wφ) makes id Wφ into a graded symplectomorphism. We will call this the standard grading and from now on we will assume that every graded Hamiltonian symplectomorphism has the standard grading. A standard perturbation of a nondegenerate autonomous Hamiltonian H where φ H 1 is graded is a time dependent Hamiltonian (H t ) t∈[0,1] which is C ∞ close to H and equal to H outside a compact set where • every 1-periodic orbit γ of (H t ) t∈[0,1] is a 1-periodic orbit of H and • for every S 1 family of 1-periodic orbits γ of H there are exactly two 1-periodic orbits γ − and γ + in this family which are also 1-periodic orbits of H t . These orbits satisfy CZ(φ Ht 1 , γ ± ) = CZ(φ H 1 , γ) ± 1/2. Such a perturbation exists by [CFH95, Proposition 2.2].
In order to prove our theorem we need another group called S 1 -equivariant Hamiltonian Floer cohomology. See [Vit99] and [BO12] for a definition. We will not define this here but we will just state some of the properties that we need. We write these groups as This means that for each γ ∈ S H , there are two 1-periodic orbits γ − , γ + of (H t ) t∈[0,1] which are also orbits in γ satisfying CZ(φ Ht 1 , γ ± ) = CZ(φ H 1 , γ) ± 1/2. Let S Ht be the set of such orbits γ ± .
Then the chain complex CF * S 1 ,[a,b],βφ (H) defining HF * S 1 ,[a,b],βφ (H, J) is a free Z[u]module generated by S Ht and graded by the Conley-Zehnder index taken with negative sign and where the degree of u is −2. Let C * [a,b],βφ (H) ⊂ C * S 1 ,[a,b],βφ (H) be the Z-submodule generated by elements of S Ht . Then the differential ∂ on CF * Here ∂ 0 is equal to the differential defining HF * [a,b],βφ (H, J). Also ∂(u i γ − ) = u i−1 γ + plus 1-periodic orbits of higher action for all i ≥ 1. The differential is Z-linear but not necessarily Z[u]-linear. Definition 8.10. We define where the direct limit is taken over all pairs (H, J) compatible with (Wφ, πφ, θφ) using the partial ordering ≤ on H. Let be a partial order on a set S. A cofinal family is a subset S ′ ⊂ S so that for all s ∈ S, there exists an s ′ ∈ S ′ so that s s ′ . In the definition above, it is sufficient to compute SH * S 1 ,βφ (Wφ, πφ, θφ) by taking the direct limit over some cofinal family of pairs (H, J) as above.
Proof. Let (Ȟ,J ) be a pair which is strictly compatible with our mapping cylinder and so that the slope ofȞ is equal to the slope of H. Then there is a constant c > 0 so thať H + c > H and H + c >Ȟ. Consider the continuation maps By (S1HF2), the composition of any two such maps is an isomorphism and hence the continuation map α is an isomorphism. Therefore it is sufficient for us to assume that (H, J) is strictly compatible with our mapping cylinder. We can also assume that H = π * φ h(s) where h ′′ (s) ≥ 0.
Choose S > 0 large enough so that π −1 φ ((−S, S) × R/Z) contains all the 1-periodic orbits of H representing a class in βφ. Let h σ : R −→ R, σ ∈ [0, ∞) be a smooth family of functions so that • h σ (s) = h(s) for all s ∈ (−S, S) and h 0 (s) = h(s) for all s ∈ R, σ (s) = 0 for all large enough s and • the slope of h σ tends to infinity as σ tends to infinity. By (S1HF3) combined with Lemma 8.7, the natural continuation map is an isomorphism for all σ ≥ 0. Also by (S1HF2), the natural continuation map is an isomorphism for all σ ≥ 0. Since (π * φ h σ (s)+ σ, J) is a cofinal family of pairs with respect to the ordering ≤, we get our result by (S1HF2). where the direct limit is taken over pairs (H, J) compatible with our mapping cylinder satisfying H| π −1 φ ((−∞,q]×R/Z) < 0. Let A be a constant smaller than the action of all the 1-periodic orbits ofφ. We say that h : R −→ R, i ∈ N is a compatible function with respect to q if • h ′ , h ′′ ≥ 0, h is bounded below, • if h ′ (x) = 1 then h(x) < x + κA and • h i | (−∞,q] < 0. Then SH * S 1 ,βφ (Wφ, πφ, θφ) = lim − → h HF * S 1 ,βφ (π * φ h(s), J) where the direct limit is taken over compatible functions with respect to q ordered by ≤ and where J is an almost complex structure compatible with our mapping cylinder. Since π * φ h(s) has no 1-periodic orbits representing βφ of positive action where h is any compatible function with respect to q we have by (S1HF3), where the direct limit is taken over pairs (H, J) compatible with our mapping cylinder satisfying H| π −1 φ ((−∞,q]×R/Z) < 0 and (H, J)| π −1 φ ([q+1,∞)×R/Z) = (Ȟ + C H , J)| π −1 φ ([q+1,∞)×R/Z) for some constant C H ∈ R.
Proof. Let (H i , J i ) i∈N be a cofinal family of pairs with respect to the directed system mentioned in the statement of this lemma. Such a countable family exists since sup H| π −1 φ ((−∞,q]×(R/Z)) < 0 for any (H, J) in the directed system above. We have that (H i , J i ) is compatible with our mapping cylinder and H i | (π −1 φ ((−∞,q]×R/Z)) < 0 and (H i , J i )| π −1 φ ([q+1,∞)×R/Z) = (Ȟ + C H i , J i )| π −1 φ ([q+1,∞)×R/Z)) for some constant C H i ∈ R. We can assume that H i < H i+1 and hence C H i < C H i+1 for all i ∈ N. After passing to a subsequence, we can assume that C H i > i for all i ∈ N.
is an isomorphism for all i ≥ 0. Then If these groups are graded then the above isomorphism respects this grading.
Proof. DefineB ≡ ∂(uB − ) and define ∂B : uB − −→B, ∂B(x) = ∂(x). Since the map (8.5) is an isomorphism for all i ≥ 0, we get that ∂ 1 | uB − : uB − −→ (B − ⊕ B + )/(B − ⊕ 0) is an isomorphism. Hence B − ⊕ B + ∼ = A − ⊕ uB − ⊕B and the differential with respect to this splitting is the matrix  Computing the homology of this chain complex using the above matrix gives us our result since ∂ B ′ is an isomorphism.
Proof. Let (H = π * φ h(s), J) be a pair strictly compatible with (Wφ, πφ, θφ) where H has slope 1.5, h ′′ ≥ 0 and where h ′ | (−∞,0) < 1. Let h t : R × R/Z −→ R, t ∈ [0, 1] be a standard perturbation of h(s) viewed as a Hamiltonian on the symplectic manifold (R × R/Z, ds ∧ dt). Then h t has exactly two 1-periodic orbits γ − , γ + . Also H t ≡ π * φ h t is a standard perturbation of H and the 1-periodic orbits of H t project to γ − or γ + . A compactness argument ([BEH + 03]) tells us that (H t , J) satisfies the maximum principle so long as H t is sufficiently C ∞ close to H and hence we can define HF * βφ (φ Ht 1 ) in the usual way. We can also define HF * S 1 ,βφ (H, J) using the standard perturbation H t by the same compactness argument.
Proof of Property (HF2). This follows from Lemmas 8.2, 8.3, 8.14 and 8.16. Now the only issue is that if we have two polynomials with embedded contactomorphic links. Then we need to show that the associated contact pairs are isomorphic. In other words, we need to show that the normal bundles coincide up to homotopy. This is contained in the proof of the following lemma.
Proof. Let (L f ⊂ S ǫ , ξ Sǫ , Φ f ) and (L g ⊂ S ǫ , ξ Sǫ , Φ g ) be the contact pairs associated to f and g respectively as in Example 3.8. Let Ψ : S ǫ −→ S ǫ be the contactomorphism sending L f to L g . We need to show that Ψ is in fact a contactomorphism of graded contact pairs by (HF2). Since H 1 (S ǫ ; Q) = 0, we get that Ψ is a graded contactomorphism by Equation (7.2) in Definition 7.7. Therefore we just need to show that the composition This is true since the trivialization Φ f (and similarly Φ g ) is uniquely determined by the following topological property: Let Ψ f :Ň Sǫ L f −→ S ǫ be a regularization of L f as in Let (J t ) t∈[0,1] be a smooth family of almost complex structures cylindrical near ∂M . Then there is an neighborhood N B ⊂ M of B so that for any sufficiently small C ∞ perturbationφ ′ , any Floer trajectory of (φ ′ , (J t ) t∈[0,1] ) connecting non-degenerate fixed points p,p ∈ N B ofφ ′ is contained inside N B .
Proof. We choose a relatively compact open neighborhood N B of B so that any fixed point ofφ inside N B is actually contained inside B. LetŇ B ⊂ M be an open neighborhood of B whose closure is contained in N B .
Let (φ k ) k∈N be a sequence of symplectomorphisms of M which C ∞ converge toφ. Suppose (for a contradiction) that φ k has a fixed point p k ∈ N B −Ň B for all k. Then after passing to a subsequence, we have that p i converges to some p ∈ N B −Ň B . Since p is a fixed point of φ, we get that p ∈ B which is impossible. Therefore φ k has no fixed points inside N B −Ň B for all sufficiently large k.
Now suppose that p k ,p k are fixed points of φ k and suppose that we have a sequence of Floer trajectories u k : R × [0, 1] −→ M of (φ ′ , (J t ) t∈[0,1] ) joining p k andp k . Define W ≡ R × [0, 1] × M with a symplectic form ω W ≡ ds ∧ dt + dθ M where s, t are the standard coordinates on R × [0, 1]. Let i be the standard complex structure on R × [0, 1] ⊂ C where (s, t) is identified with s + it. Define J W | (s,t,x) ≡ i| (s,t) ⊕ J t | (s,t) . Define u W k : R × [0, 1] −→ W, u W k (s, t) ≡ (s, t, u k (s, t)) for all k ∈ N. This is a sequence of J W -holomorphic maps. Now suppose (for a contradiction) that the image of u k is not contained inside N B for all k. Then after passing ot a subsequence, there is a sequence of points (s k , t k ) ∈ R × [0, 1] so that u k (s k , t k ) ∈ N B −Ň B and u k (s k , t k ) converges to some point q ∈ N B −Ň B . After reparameterizing the domain by translations in the s direction, we can assume that s k = 0 for all k. Also after passing to a subsequence we can assume that t k −→ť ∈ [0, 1] for someť. Define w k ≡ u W k | [−1,1]×[0,1] . Then by the main result in [Fis11], we get that w k C 0 converges to a continuous map v : [−1, 1] × [0, 1] −→ W which is smooth and J W -holomorphic on a dense open subset of its domain.
Let π M : W −→ M be the natural projection map. Since p k andp k converge to points in B, their difference in action converges to zero which implies that [−1,1]×R/Z v * dθ M = 0.
As a result of the above lemma, we have the following definition: Definition 9.3. Let B ⊂ M be an isolated family of fixed points of some positive slope perturbationφ of φ and let N B be a neighborhood of B as in Lemma 9.2. Letφ ′ be a C ∞ small perturbation so that all the fixed points ofφ ′ inside N B are non-degenerate. Since all Floer trajectories of (φ ′ , (J t ) t∈[0,1] ) are contained inside N B , we can define the Floer cohomology group HF * (φ, B) to be defined in the usual way where we only consider fixed points inside N B . Such a group is called the local Floer cohomology of B. Again it does not depend on the choice of perturbationφ ′ or (J t ) t∈[0,1] although we will not need this fact here.
Note that if B is the only set of fixed points ofφ of action in the interval [a, b], then the above definition coincides with the definition of HF * (φ, B) from Definition 9.1. More generally, if B is a union of isolated families of fixed points B 1 , · · · , B l all of the same action then HF * (φ, B) = l i=1 HF * (φ, B i ). : N B i −→ N B i be the time 1 flow of H B for each i. After possibly shrinking each neighborhood N B i , we can assume that N B 1 , · · · , N B l are all disjoint. Let (J t ) t∈[0,1] be a generic smooth family almost complex structures cylindrical near ∂M . By Lemma 9.2, any sufficiently small C ∞ perturbationφ ′ ofφ has the property that any Floer trajectory connecting fixed points inside ∪ l i=1 N B i is actually contained inside N B j for some j. Therefore Hence by [SZ92,Theorem 7.1] combined with (CZ4), we have that Equation (9.2) holds.
Proof of Property (HF3). Letφ ′ be a C ∞ small perturbation ofφ and let (J t ) t∈[0,1] be a C ∞ generic smooth family of almost complex structures cylindrical near ∂M . Let α i be the action of B i for each i ∈ {1, · · · , l}. For each p ∈ N, choose β p ∈ R so that α i = β p for all i ∈ {1, · · · , l} and so that α i > β p if and only if ι(i) ≥ p. Let F p be the subgroup of the chain complex CF * (φ ′ ) generated by fixed points of action greater than β p . Then (F p ) i∈p is a filtration on this chain complex. By Lemma 9. for all p = 1, · · · , l and H * (F p /F p−1 ) = 0 if p ∈ N − {1, · · · , l}. Therefore the spectral sequence associated to the filtration (F p ) p∈N is (9.1).