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Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of these actions. As a by-product, we provide an algorithm that determines the g-reduced form of a blowup form. Our work is a combination of"soft"equivariant and combinatorial techniques, using the momentum map and related data, with"hard"holomorphic techniques, including Gromov-Witten invariants.


Introduction
In this paper we complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. It is in this dimension that there are abundant examples (indeed, Gompf has shown that any finitely presented group may be the fundamental group of a four-dimensional symplectic manifold [7]), and yet powerful holomorphic techniques (J-holomorphic curves and Gromov-Witten invariants) are developed enough to make classification problems tractable.
A symplectic action of a T n = (S 1 ) n on a symplectic manifold (M, ω) is Hamiltonian if it admits a moment map: an equivariant smooth map Φ : M → t * such that the components Φ ξ = Φ, ξ satisfy Hamilton's equation dΦ ξ = −ι(ξ M )ω, for all ξ ∈ t. Here ξ M is the vector field that generates the action on M of the one-parameter subgroup {exp(sξ) | s ∈ R} of T . The actions we consider are effective and the manifolds are compact and connected. By the Convexity Theorem [2,9], the image of the momentum map ∆ = Φ(M ) ⊂ t * ∼ = R n is a convex polytope. We say that two Hamiltonian T actions are equivalent if they differ by an equivariant symplectomorphism and a reparametrization of T . For an effective Hamiltonian action T n M , we are forced to have dim(T ) ≤ 1 2 dim(M ). Thus, if M is four-dimensional, the only tori that can act effectively are S 1 and T 2 .
Delzant [6] has classified symplectic manifolds equipped with a Hamiltonian action of a torus of half the dimension in terms of their momentum polytopes, up to equivariant symplectomorphism. Building on work of Audin [3] and Ahara and Hattori [1], Karshon [13] has classified symplectic four-manifolds with Hamiltonian S 1 actions in terms of decorated graphs. In particular, the only four-manifolds that admit Hamiltonian S 1 actions are blowups of S 2 -bundles over Riemann surfaces and CP 2 .  [17,18].
To complete the characterization of Hamiltonian actions on symplectic four manifolds, it remains to study the case of Hamiltonian actions on blowups of S 2 -bundles over a Riemann surface of positive genus. These do not admit Hamiltonian 2-torus actions. In what follows, we characterize Hamiltonian circle actions on them, up to (possibly non-equivariant) symplectomorphism (Theorems 2.13 and 2.14). We prove that a reduced symplectic blowup of an irrational ruled symplectic four-manifold is compatible with all the Hamiltonian circle actions: there are no "exotic actions".
We give an algorithm in pseudo-code in Appendix B, written by Tair Pnini, to count the (inequivalent) Hamiltonian circle actions on blowups of irrational ruled symplectic manifolds. We use the algorithm to derive results on the existence and number of such actions in Section 3. Along the way, we also provide an algorithm that determines the reduced form corresponding to a symplectic form on a blowup of a ruled symplectic manifold (Algorithm 6.5). The proofs combine "soft" combinatorial techniques and "hard" holomorphic techniques.
Acknowledgements. Tair Pnini is a student at the University of Haifa at Oranim and she implemented the counting algorithms for us. We are indebted to her for her work. We also thank Daniel Cristofaro-Gardiner, Yael Karshon, Dusa McDuff and Martin Pinsonnault for helpful discussions. We are grateful to the anonymous referee for remarks which improved the paper.
Holm was supported in part by the Simons Foundation under Grant #208975 and the National Science Foundation under Grants DMS-1206466 and DMS-1128155. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or Simons Foundation.

Statement of Main Results
We now turn to the full statement of our main results. We begin by setting our notation.
For a non-negative integer k, denote by (Σ×S 2 ) k the manifold obtained from the trivial S 2 -bundle over Σ by k complex blowups at k distinct points, and by (M Σ ) k the manifold obtained from the non-trivial S 2 -bundle over Σ by k complex blowups at k distinct points. Let E 1 , . . . , E k denote the homology classes of the exceptional divisors. We fix the labeling of the exceptional divisors.
In the positive genus case, our notion of g-reduced does not coincide with the notion of reduced given in [22,24]. An explanation of the difference is in Remark 2.16.  [26,Proposition 7.21]. We give the proof in the positive genus case in [11]; the proof in the genus 0 case is in [18].
2.11. Lemma. The set of blowup forms on M is an equivalence class of symplectic forms.
The set E(M ) of exceptional classes in H 2 (M ) is the same for all the blowup forms, as discussed in Lemma 4.5. An exceptional sphere in a symplectic four-manifold (M, ω) is an embedded ω-symplectic sphere of self intersection −1. A homology class E ∈ H 2 (M ) is exceptional if it is represented by an exceptional sphere. For blowups of S 2 -bundles over a Riemann surface Σ with a positive genus, there is an explicit identification of the classes in E, by a holomorphic argument of Biran [4,Corollary 5.C].
Pinsonnault [28,Proposition 3.13] showed that every Hamiltonian circle action on a k-fold blowup of a ruled symplectic four-manifold can be obtained from an action on some ruled symplectic manifold M 0 by a sequence of k equivariant blowups of some sizes. For blowups of irrational ruled symplectic manifolds, we use the identification of E(M ) in order to derive, from the same holomorphic tools used in the non-constructive result of Pinsonnault, a constructive characterization, in which we identify the ruled manifold M 0 and the sizes of the equivariant blowups.
2.13. Theorem. Let k ≥ 1 be an integer, and let M = (Σ × S 2 ) k (respectively M = (M Σ ) k ). Assume that the genus of Σ is positive. Let ω be a blowup form on M whose cohomology class is encoded by a vector (λ F , λ B ; δ 1 , . . . , δ k ); if k ≥ 2, assume that [ω] is in g-reduced form. Then every Hamiltonian circle action is equivalent to a one that is obtained from a Hamiltonian circle action on Σ × S 2 (resp. M Σ ) with the symplectic form whose cohomology class is encoded by (λ F , λ B ) by a sequence of S 1 -equivariant symplectic blowups of sizes δ 1 ,δ 2 , δ 3 , . . ., δ k , in this order, and to b one that is obtained from a Hamiltonian circle action on M Σ (resp. Σ × S 2 ) with the symplectic form whose cohomology class is encoded by (λ F , λ B + 1 2 λ F − δ 1 ) by a sequence of S 1 -equivariant symplectic blowups of sizes λ F − δ 1 , δ 2 , . . ., δ k , in this order.
Moreover, Theorem 2.13 holds if we replace "Hamiltonian circle action" with "action of a compact Lie group G that preserves ω and induces the identity morphism on H 2 (M )." To complete the characterization we provide a description of the Hamiltonian circle actions on Σ × S 2 and on M Σ in Proposition 3.2.
Proof of Theorem 2.13. Consider an action on (M, ω) of a compact Lie group G that preserves ω and induces the identity morphism on H 2 (M ). Let J G be a G-invariant ω-tamed almost complex structure on M . The structure J G is the almost complex structure associated by the polar decomposition to the invariant Riemann metric g obtained by averaging some Riemann metric g along the action of G with respect to the Haar measure: g(u, v) := G g (σ a * u, σ a * v)da.
First assume k ≥ 2. By the identification (2.12) of E(M ) and the g-reduced assumption, the class E k is of minimal symplectic area in E(M ). Hence, by Lemma [28,Lemma 1.2], E k is represented by an embedded J G -holomorphic sphere C. To see that C is G-invariant, apply the positivity of intersections [27,Proposition 2.4.4], the fact that E · E = −1, and the assumption that the action is trivial on H 2 (M ) to get that for every a ∈ G, the J G -holomorphic sphere aC equals the J G -holomorphic sphere C.
We now let k ≥ 1. By k − 1 repeated applications of this blow down, we reduce the theorem to the following claim: There exist embedded G-invariant symplectic spheres C in E 1 and D in F − E 1 in (Σ × S 2 ) 1 (resp. (M Σ ) 1 ), with the blowup form encoded by (λ F , λ B ; δ 1 ), and blowing down equivariantly along C yields a G-action on Σ × S 2 (resp. M Σ ) with the symplectic form encoded by (λ F , λ B ), and blowing down along D yields a G-action on M Σ (resp. Σ × S 2 ) with the symplectic form encoded by (λ F , λ B + 1 2 λ F − δ 1 ). This claim follows from the existence of embedded J G -holomorphic spheres in E 1 and in F −E 1 , stated in Lemma 4.6, which is the positive genus version of the genus zero [29, Lemma 2.2], and from Lemma 4.12 and Corollary 5.3 (on blow downs).
Our second main result provides a unique representation for a blowup form on the k-fold blowup of a ruled symplectic four-manifold Σ × S 2 or M Σ .
2.14. Theorem. Let k ≥ 2 be an integer, and let M = (Σ × S 2 ) k or M = (M Σ ) k . Assume that the genus g(Σ) is positive. Given a blowup form ω on M , there exists a unique blowup form ω whose cohomology class is in g-reduced form such that (M, ω) ∼ = (M, ω ).
In fact, we prove even more than is stated in Theorem 2.14. We include an algorithm, Algorithm 6.5, which puts a blowup form into its unique g-reduced form, by a composition of permutations of the δ i s and a map we denote the Cremona transformation (see Definition 5.7). The latter map is a positive genus version of the (usual) Cremona transformation on blowups of M S 2 ; it is motivated by the combinatorial description of the transformation as interwining two ways to obtain the same Hamiltonian S 1 -manifold, as shown in Figure 2.15.
Theorem 2.14 also provides a way to determine when two symplectic forms on the same manifold, on (Σ × S 2 ) k or on (M Σ ) k , are diffeomorphic: they must have the same g-reduced form. In the case when g(Σ) = 0 and M = (M Σ ) k , the existence and uniqueness of a g-reduced form is proved in [16,Theorem 1.4].

2.16.
Remark. Li-Li [22] and Li-Liu [24], in the context of relating the symplectic genus of an integral cohomology class to its minimal genus, define reduced forms for blowups of S 2 × Σ (with g(Σ) > 0) by different conditions. First, they work with homology classes with Z coefficients whereas we work with cohomology classes over R. More significantly, their conditions translate (by Poincaré duality) to the conditions on an integral 2-cohomology class: By contrast, we require λ F ≥ δ 1 + δ 2 , and neither require λ B ≥ δ 1 nor positivity. Note that the vector (3, 3; 2, 2) from Figure 2.15 is reduced in the sense of [22,24]. The Cremona transformation turns it into Φ(Σ min ) = 0 , Area = 2 , g ) = 0 , Area = 1 , g Figure 2.15. On the left, the same toric manifold, with the purple hexagonal moment image, can be seen in two ways as a 2-fold blowup of CP 1 × CP 1 , with different symplectic forms and blowup sizes. In this case, the Cremona transformation turns the vector (3, 3; 2, 2) (the blowup of the blue square) into (3, 2; 1, 1) (the blowup of the red parallelogram). On the right we can see the S 1 -manifold represented by the middle decorated graph as a blowup from the left (3, 3) with size 2 blowups, one each from the top and bottom; or from the right (3, 2) with size 1 blowups, one each from the top and the bottom. As before, the Cremona transformation turns the vector ), which encodes a blowup form diffeomorphic to (3, 3; 2, 2) that is g-reduced. The requirement λ F ≥ δ 1 + δ 2 is essential for the key claim that the class E k is of minimal symplectic area in our proof of Theorem 2.13.
Li and Li [22,Lemma 3.4] prove that any class in H 2 over Z of nonnegative square is equivalent to a reduced class under the action of orientation-preserving diffeomorphisms; they provide an algorithm to find the reduced class using reflections along integral homology classes that are realized by orientation-preserving diffeomorphisms. In Li-Li's terminology, the Cremona transformation and the (δ i , δ j )-transpositions we use in Algorithm 6.5 correspond to reflections along F − E 1 − E 2 and along E i − E j , respectively. However Li and Li's algorithm uses other transformations which we do not allow, e.g., −1 and the reflections along r i F + ε i E i . Their reflections represent diffoemorphisms, but possibly not symplectomorphisms. Their algorithm might send a vector encoding (the dual of) a blowup form ω to a vector that cannot encode a blowup form, or to a vector encoding a blowup form ω that is not diffeomorphic to ω. For example, Li and Li's algorithm sends the (g-reduced) vector (6, 1; 2, 1) by reflection over F − E 1 to (4, 1; 1, 0), which does not represent a blowup form. Their algorithm sends the (g-reduced) vector (12, 2; 3, 3) by reflections over F − E 1 and over F − E 2 to (10, 2; 1, 1). Those vectors represent manifestly different symplectic forms: the former admits no Hamiltonian circle actions, while the latter does admit a Hamiltonian circle action.
We conclude this section with a brief outline of the rest of the paper. In Section 3, we recall the notion of the decorated graph associated to a Hamiltonian circle action, and describe the effect of an equivariant symplectic blowup on the graph. We then derive from Theorem 2.13 results on the existence and number of Hamiltonian circle actions on a fixed symplectic four-manifold. Section 4 examines symplectic blowups of ruled symplectic four-manifolds and highlights the symplectic facts and techniques we need to prove the main theorems. We deduce from Li-Liu's characterization of symplectic forms [24] a necessary and sufficient condition for a vector to encode a cohomology class of a blowup form, and describe how these techniques can compute certain symplectic invariants like Gromov width and packing number (see Remarks 4.9 and 4.10). In Section 5, we use the correspondence between Hamiltonian circle actions and decorated graphs to deduce results on symplectic manifolds from combinatorial observations. Finally, we prove the existence portion of Theorem 2.14 in Section 6, and the uniqueness in Section 7. For each two dimensional component S of the fixed point set there is a fat vertex S labeled by two real numbers and one integer: the momentum map label Φ(S), the area label 1 2π S ω, and the genus g of the surface S. A Z k -sphere is a gradient sphere in M on which S 1 acts with isotropy Z k . For each Z k -sphere containing two fixed points p and q, the graph has an edge connecting the vertices p and q labeled by the integer k; the size 1 2π S ω of a Z k -sphere S is 1/k of the difference of the moment map values of its vertices. We note that vertical translations of the graph correspond to equivariant symplectomorphisms, and flips correspond to automorphisms of the circle.

Counting Hamiltonian circle actions on blowups of irrational ruled manifolds
Hamiltonian circle actions on irrational ruled symplectic manifolds. Karshon's results [13, Theorem 6.3 and Lemma 6.15] and [14, Table in Proof of Lemma 1] imply that Hamiltonian circle actions on a ruled symplectic four-manifolds over a closed Riemann surface of positive genus g correspond to decorated graphs that consist of two fat vertices with the same genus label g. Moreover, Karshon's results yield the following characterization of Hamiltonian circle actions on irrational ruled symplectic four-manifolds. See also [28,Corollary 3.12].
Moreover, there is a one-to-one correspondence between inequivalent Hamiltonian S 1 -actions on (M, ω) and the nonnegative integers Remark. Note that the enumeration of the inequivalent circle actions on a ruled symplectic manifold over a Riemann surface Σ of positive genus g is analogous to the enumeration of the inequivalent toric actions on a ruled symplectic manifold over S 2 as in [14].
The effect of an S 1 -equivariant blowup on the decorated graph. Recall that we can think of a symplectic blowup of size δ = r 2 /2 as cutting out an embedded ball of radius r and identifying the boundary to an exceptional sphere via the Hopf map. This carries a symplectic form that integrates on the sphere to 2πδ. For more details see [10] and [26,Section 7.1]. If the embedding of the ball is G-equivariant centered at a G-fixed point, then the G-action extends to the symplectic blowup, see details in [15]. If the action is Hamiltonian, its moment map naturally extends to the equivariant symplectic blowup.
In Figure 3.4, we describe all the possible effects of an S 1 -equivariant symplectic blowup of a ruled manifold over a surface of positive genus, up to flips. If the surface is of genus 0, an S 1 -equivariant symplectic blowup can have additional effects. For a complete description of Hamiltonian S 1 -blowups of 4-manifolds, see [13, §6]. We observe that flipping commutes with S 1 -equivariant symplectic blowup. That is, the (fat or not) vertices created in the blowup do not surpass the other pre-existing (fat or not) vertices in the same chain of edges, and the fat vertices after the blowup have positive size labels. For quantitative description, see [15,Lemma 3.3].
An algorithm to count the (non-equivalent) Hamiltonian circle actions on a k-fold blowup of an irrational ruled manifold. Let k ≥ 1. Let Σ be a Riemann surface of positive genus. When the cohomology class of a blowup form Theorem 2.13 reduces the question of counting the number of Hamiltonian circle actions on (M, ω) to a combinatorial one. It is enough to count the number of possible ways to get valid decorated graphs by k S 1 -blowups of sizes δ 1 , . . . , δ k in descending order, starting from a decorated graph with two fat vertices, the top of size λ B − n 2 λ F and the bottom of size We count the resulting decorated graphs up to vertical translations and flips.
By Corollary 5.4, it is enough to count the Hamiltonian circle actions on symplectic k-fold blowups of Σ × S 2 . The counting algorithm was implemented by Tair Pnini. Let ω λ F ,λ B ;δ1,...,δ k be a blowup form on (Σ×S 2 ) k whose cohomology class is in g-reduced form and is encoded by a vector (λ F , λ B ; δ 1 , . . . , δ k ). At step 0 the program creates the set of graphs with two fat vertices determined by λ B , λ F , corresponding to the even integers 0 ≤ n < 2 λ B λ F . At step i, for 1 ≤ i ≤ k, it creates the set of graphs (up to equivalence) which may be obtained by performing a valid blowup of size δ i on a graph in the set created at step i − 1. The output of the program is the number of graphs in the set obtained at step k. This is a "greedy" algorithm. We use data structures designed to reduce the number of tests for equivalence between pairs of graphs and to optimize the equivalence test. A detailed description of the data structures and the pseudo-code of the algorithm is given in Appendix B by Tair Pnini.
3.6. Corollary. Let k ≥ 1 and assume that we are performing k blowups of equal size: We count the possible ways to preform k blowups of size ε, each at a fat vertex, starting from one of the decorated graphs of two fat vertices listed in Proposition 3.2.
We have subtracted the equivalent actions, like those in Figure 3.9 where blowups of size λ F 2 from the top or bottom fat vertices will end up at the same height, resulting in the same decorated graphs.
• Similarly, if 2ε < λ F , the number of Hamiltonian circle actions on ((M Σ ) k , ω λ F ,λ B ;ε,...,ε ) equals the sum • In particular, if kε ≥ 2λ B then there are no Hamiltonian circle actions. Figure 3.9. This figure shows coincidentally equivalent decorated graphs arising from blowups of different graphs.
For example, for k ≥ 2 there are no Hamiltonian circle actions on (Σ × S 2 ) k with a blowup form ω 2,1; 2 k ,... 2 k ; by Lemma 4.8, such a blowup form exists. This lack of Hamiltonian actions also appears for equal blowups in the simply connected case; see for example [15].
where a denotes the smallest integer greater than or equal to a. If λ F for every even n < 2 λ B λ F , and • for every j, the sum k i=j+1 δ i < δ j , and moreover • for every j and s, we have For example, if λ B ∈ N, λ F = 1 and δ i = 1 4 i for 1 ≤ i ≤ k, the number of Hamiltonian circle actions is (λ B − 1 2 ) · (k + 1)!. (By Lemma 4.8, there exists a blowup form on (Σ × S 2 ) k whose cohomology class is encoded by this vector, for every k ≥ 1.)

Blowups of ruled symplectic four-manifolds
Let Σ be a compact connected Riemann surface. Fix an integer As a smooth manifold, M is obtained by k iterations of a connected sum with CP 2 , CP 2 equipped with the opposite orientation. In general, the connected sum construction N = N #CP 2 depends on a number of choices, but if N is another connected sum obtained from different choices, then there exists a diffeomorphism from N to N that respects the splitting H 2 ( N ) = H 2 (N ) ⊕ ZE induced by the constructions. This follows from standard arguments in differential topology (see e.g., [5,Chapter 10]). In particular, if M is obtained from Σ × S 2 or M Σ by blowing up at k distinct points and M is obtained by blowing up at a permutation of the same points, then the two manifolds are the same, up to relabeling the exceptional classes E 1 , . . . , E k . 4.1. Lemma. Fix a vector (λ F , λ B ; δ 1 , . . . , δ k ) and a permutation σ ∈ S k . Suppose that there exists a blowup form ω on M whose cohomology class is encoded by this vector. Then there exists a blowup form ω whose cohomology class is encoded by the vector (λ F , λ B ; δ σ(1) , . . . , δ σ(k) ) and (M, ω), and (M, ω ) are symplectomorphic. A (parametrized) J-holomorphic sphere is a map f : (CP 1 , j) → (M, J) that satisfies the Cauchy-Riemann equations df • j = J • df at every p ∈ CP 1 . An embedding is a one-to-one immersion which is a homeomorphism with its image. An embedded J-holomorphic sphere C ⊂ M is the image of a Jholomorphic embedding f : CP 1 → M . If J is ω-tamed then such a C is an embedded ω-symplectic sphere. We will refer to the genus zero Gromov Witten invariant (with point constraints) GW : For the precise definition, see [27]. Fixing a symplectic form ω, if GW(A) = 0, then for generic ω-tamed almost complex structure J there exists a J-holomorphic sphere in the class A. Because Note also that B n · B −n = 0.
(4) The genus zero Gromov-Witten invariant GW(F ) with respect to any blowup form is not zero.
The last item follows from [25, §4] and the fact that Gromov-Witten invariants are consistent under the natural inclusions  The following lemma is a consequence of (2.12) and Facts 4.3 that we will need in the proof of the uniqueness theorem.
• When k = 0, the class F is the unique class in H 2 (M ; Z) satisfying (1) Its symplectic area with respect to a symplectic ruling of the S 2 -bundle is positive; (2) Its self intersection number is zero; and • In the case k = 0, a class A ∈ H 2 (M ; Z) is written as A = pB + qF for p, q ∈ Z. Assume that A satisfies the three conditions. Since A · A = 0, we get, using yield that 2q = 1 contradicting the fact that q is an integer. We conclude that p = 0 hence 2q = c 1 (T M )(A) = 2, i.e., q = 1.
If A satisfies the first condition, then, by (2.12) and item(2) in Facts 4.3, for all On the other hand, by (2.12) and Facts 4.3, the class F satisfies the conditions.
Necessary and sufficient conditions for a vector to encode a blowup form. We now turn to the question of when a vector (λ F , λ B ; δ 1 , . . . , δ k ) ∈ R 2+k encodes the cohomology class of a blowup form. • the numbers λ F , λ B ; δ 1 , . . . , δ k are positive; • λ F > δ i for all i; and • the volume inequality λ F λ B − 1 2 (δ 2 1 + . . . + δ 2 k ) > 0 holds. Proof. If the vector (λ F , λ B ; δ 1 , . . . , δ k ) encodes the cohomology class of a blowup form ω, then the symplectic areas of the embedded ω-symplectic exceptional spheres representing the classes in E(M ) = {E 1 , . . . , E k , F −E 1 , . . . , F −E k } and the symplectic volume of M are necessarily positive. This establishes the fact that δ i > 0, λ F > δ i and the volume inequality. The definition of a blowup form requires λ F > 0, including for the case k = 0. Finally becuase the symplectic volume and λ F are positive, λ B > 0 as well.
The fact that the listed conditions on the vector (λ F , λ B ; δ 1 , . . . , δ k ) are sufficient to guarantee the existence of a blowup form follows from Li and Liu's characterization of symplectic forms with a standard canonical class on blowups of ruled symplectic manifolds [24,Theorem 3]. They prove that for any symplectic 4-manifold with b + = 1 (which includes all blowups of ruled surfaces), the symplectic cone of symplectic forms for which the first Chern class is the same as that of a blowup form (i.e., encoded as in item (3)

of Facts 4.3) is described by
By "exceptional class" Li and Liu refer to a homology class E that is represented by a smoothly embedded sphere with self intersection −1. The facts that • a class E is "exceptional" with c 1 (T M )(E) = 1 if and only if E ∈ E(M ); and • every symplectic form with first Chern class as that of a blowup form is a blowup form follow from results that are given in [24, Lemma 3.5 Part 2], and [23, Theorem A]; see also the explanation in [16,Section 6]. Thus, any cohomology class encoded by a vector (λ F , λ B ; δ 1 , . . . , δ k ) satisfying the conditions of the lemma is in C 0 and so is the cohomology class of some blowup form ω. We will call C 0 the standard symplectic cone.

4.9.
Remark. Note that Lemma 4.8 may be used to determine the Gromov width of a symplectic 4-manifold M = (Σ × S 2 ) k or M = (M Σ ) k with g(Σ) > 0 and k ≥ 0. To do so, we must determine for which positive numbers α there is a symplectic embedding of a ball of capacity α into M . This is equivalent to determining if there is a symplectic blowup of M of size α. We start with a vector (λ F , λ B ; δ 1 , . . . , δ k ) ∈ R k+2 >0 that encodes a blowup form on M . Lemma 4.8 tells us that there exists a blowup of M of size α so long as 0 < α < λ F and the volume inequality is satisfied. The Gromov width of M is the sup of all such α. Hence, the Gromov width will be determined by λ F and the volume of M . For such M with a blowup form encoded by the vector (λ F , λ B ; δ 1 , . . . , δ k ), it is straight forward to show that In the special case when k = 0 and M = S 2 × Σ, this is Biran's result [4, Corollary 5.C].
Symplectic blow down. By Weinstein's tubular neighborhood theorem [31], a neighborhood of an exceptional sphere is symplectomorphic to a neighborhood of the exceptional divisor in a standard blowup of C 2 . We can then blow down along C and get a symplectic manifold whose symplectic blowup is naturally isomorphic to (M, ω).

Hamiltonian circle actions on ruled symplectic four-manifolds with one or two blowups
In this section we use the correspondence between Hamiltonian circle actions and decorated graphs to deduce results on symplectic manifolds from combinatorial observations. We first consider Hamiltonian S 1 -actions on a symplectic manifold obtained from a ruled symplectic manifold by a single blowup.
(1) There is a Hamiltonian S 1 -action on (Σ × S 2 ) 1 with the blowup form encoded by (s, r; ε), for which there exists an equivariant sphere C in the class E 1 and an equivariant sphere D in the class F − E 1 , and blowing down along C yields a Hamiltonian S 1 -action on Σ × S 2 with the compatible symplectic form encoded by (s, r), and blowing down along D yields a Hamiltonian S 1 -action on M Σ with the compatible symplectic form encoded by (s, r + 1 2 s − ε).   (λ F , λ B ; δ). For every embedded ω-symplectic sphere in the class F −E 1 , blowing down along it yields a symplectic manifold that is symplectomorphic to M Σ (resp. Σ×S 2 ) with the symplectic ruling encoded by (λ F , λ B + 1 2 λ F − δ).
We now continue our study by looking at the effect of two blowups on a decorated graph. In Figures 5.5 and 5.6 we describe two different Hamiltonian blowups of different ruled symplectic manifolds that yield the same symplectic manifold with Hamiltonian S 1 -action. Assume that 0 < ε 1 , ε 2 < min{r, s}.  On the right, we have a decorated graph for MΣ with λB = r + s 2 and λF = s. From the right to the center, we take two blow-ups of MΣ of sizes ε1 from the bottom and ε2 from the top.
Consequently, we get an analogue of the Cremona transformation, adjusted for the positive genus case.
Proof. First assume k = 2. In In particular, there exists a blowup form on M 2 such that the pullback by ψ is a blowup form. By Lemma 2.11, it follows that for every blowup form, its pullback by ψ is a blowup form. Because of how ψ acts on homology, if ω is a symplectic form that is encoded by the vector (λ F , λ B ; δ 1 , δ 2 ), then ψ * ω is a symplectic form that is encoded by the vector (λ F , λ B ; δ 1 , δ 2 ). The case k ≥ 3 then follows by the uniqueness of symplectic blowups of given sizes.

Existence of g-reduced form
In this section we prove the existence part of Theorem 2.14. That is, given a k-fold blowup of a ruled symplectic manifold over a surface of positive genus, M = (Σ × S 2 ) k or M = (M Σ ) k , we show that there exists a blowup form on M whose cohomology class is g-reduced. We achieve this by providing an algorithm that starts with a vector encoding the cohomology class of a blowup form on a k-fold blowup of an irrational ruled manifold, and returns a symplectomorphic blowup form which is g-reduced.
Let k ≥ 2. It is straight forward to check that the Cremona move satisfies the following properties.

Lemma.
(1) The Cremona move preserves the standard symplectic cone (2) If the vector v = (λ F , λ B ; δ 1 , . . . , δ k ) is obtained from v = (λ F , λ B ; δ 1 , . . . , δ k ) by the Cremona move and v is different from v, then δ i ≤ δ i for all i, and for at least one i we have δ i < δ i . Sketch of Proof. Let v = (λ F , λ B ; δ 1 , . . . , δ k ) ∈ C 0 and v = (λ F , λ B ; δ 1 , . . . , δ k ) the vector obtained from v by the Cremona move. In proving part (1), we note that both the "volume" λ F λ B − 1 2 k i=1 δ i 2 and λ F are preserved by the Cremona move. Hence, since the "volume" of v and λ F = λ F are positive and all other terms in the "volume" are nonpositive, we get that λ B > 0 as well. Also, by Lemma 6.2 and since v ∈ C 0 , for all i, the numbers δ i and λ F − δ i are positive as well. Part (2) and part (3) are immediate from the definition of the Cremona move.
6.4. Notation. Let g > 0. A vector in R 2+k (k ≥ 2) with all positive entries encodes a g-reduced cohomology class if it satisfies equation (2.2); we will call such a vector g-reduced.
Assume k ≥ 2 and g > 0. We will now use the Cremona move to obtain a g-reduced vector.
Proof. We prove this by contradiction. Suppose that this process does not terminate for some vector v in the standard symplectic cone. Then we get an infinite sequence v (n) , for any n ∈ N, of vectors in the standard symplectic cone in R 2+k . That is, v (1) = sort(v) and v (n+1) = sort cremona v (n) . By part (2) of Lemma 6.3, for some 1 ≤ i 0 ≤ k, the sequence δ has an accumulation point, contradicting Lemma 6.2. 6.7. Remark. Lemma 6.6 implies that Algorithm 6.5 terminates. By item (3) of Lemma 6.3 it produces a g-reduced vector. Moreover, by Corollary 4.1 and Corollary 5.8, if the input vector encodes the cohomology class of a blowup form ω on M = (Σ × S 2 ) k or M = (M Σ ) k , then the output vector encodes the cohomology class of a blowup form on M that is diffeomorphic to ω. This completes the proof of the existence portion of Theorem 2.14.

Uniqueness of the g-reduced form
Our goal in this section is to prove the uniqueness part of Theorem 2.14. Let M = (Σ × S 2 ) k or M = (M Σ ) k ; assume that g(Σ) > 0. For any blowup form ω the set of exceptional classes of minimal area in (M, ω) only depends on the cohomology class [ω]. We denote this set by where v ∈ R 2+k is the vector that encodes the cohomology class [ω]. If k ≥ 2 and [ω] is in g-reduced form, then by (2.12), E k ∈ E v min . We proceed to identify all the elements of E v min . 7.1. Notation. Let v = (λ F , λ B ; δ 1 , . . . , δ k ) be the vector encoding the cohomology class of a blowup form ω on M = (Σ × S 2 ) k or M = (M Σ ) k . If k ≥ 2, assume that [ω] is in g-reduced form. In particular, δ 1 ≥ . . . ≥ δ k , and the numbers λ F , λ B , δ 1 , . . . , δ k are positive.
We denote by j v the smallest nonnegative integer j such that δ j+1 = . . . = δ k . Thus, either j v = 0 and δ 1 = . . . = δ k , or 1 ≤ j v ≤ k − 1 and As an immediate corollary of the identification of the exceptional classes in (2.12), and the definition of a g-reduced form, we have the following proposition. Let (M, ω) and v be as in Notation 7.1. Assume that g(Σ) > 0.

Proposition.
• In the case of one blowup, k = 1, we have the following possibilities.
(1) Suppose that • For k ≥ 2 blowups, we have the following possibilities.
(1) Suppose that For the uniqueness of a g-reduced form, we will also need the following observations on symplectomorphic blowups of ruled manifolds.
(2) The isomorphism ϕ * sends the set E v min to the set E v min . If k ≥ 2 then (7.4) δ k = δ k .
If k = 1 then Proof. Item (1) and the first part of (2) follow immediately from the fact that ϕ carries ω to ω . The second part of (2) follows from (2.12) and the "g-reduced" assumption. Item (3) follows from Lemma 4.7. Item (4) follows from the fact that the symplectomorphism preserves symplectic volume, because and a similar formula with ω .
We now have all the ingredients to prove that the g-reduced form is unique.
Proof of the uniqueness part of Theorem 2.14. Recall that we have a manifold M that is a k-fold blowup, either (M Σ ) k or (S 2 × Σ) k , where Σ has positive genus. Let ω and ω be blowup forms on M whose cohomology classes are encoded by the vectors v = (λ F , λ B ; δ 1 , . . . , δ k ) and v = (λ F , λ B ; δ 1 , . . . , δ k ), and, if k ≥ 2, are both in g-reduced form. Suppose that (M, ω) and (M, ω ) are symplectomorphic. We aim to show that v = v . The case enumeration refers to the case enumeration in Proposition 7.2.
Suppose that k ≥ 1 and that the size of E v min equals 2k. By item (2) of Lemma 7.3, the set E v min also has 2k elements. If k = 1, by Proposition 7.2, both v and v exhibit case (3). If k ≥ 2, by Proposition 7.2, both v and v exhibit case (1b). In either case, we have that v = (λ F , λ B ; λ F 2 , . . . , λ F 2 ) and v = (λ F , λ B ; . From (7.4) we deduce that λ F = λ F , hence by item (4) of Lemma 7.3, (and the fact that λ F > 0) also λ B = λ B , and thus v = v .
Suppose that the size of E v min equals k. By item (2) of Lemma 7.3, the set E v min also has k elements. If k = 1 then Proposition 7.2 says that the vector v also exhibits one of the cases (1) or (2), and the vector v exhibits one of the cases (1) or (2). When k ≥ 2, Proposition 7.2 implies that the vector v exhibits one of the cases (1a) with j v = 0 or (2b); and the vector v also exhibits one of the cases (1a) or (2b). For the purpose of what follows, if k = 1, we will use (1a) to refer to (1), and (2b) to refer to (2).
The classes in E v min can be represented simultaneously by k pairwise disjoint embedded ω-symplectic spheres. Indeed, by Lemma 4.5 there exists an ω-tamed almost complex structure J such that each of the exceptional classes in E v min is represented by an embedded J-holomorphic sphere. Since in cases (1a) and (2b), for each two classes in this set the intersection number is zero, we conclude, by positivity of intersections of J-holomorphic spheres [27,Proposition 2.4.4], that these J-holomorphic spheres are pairwise disjoint. By item (2) (2b), v = (λ F , λ B ; λ F − δ, . . . , δ) and v = (λ F , λ B ; λ F − δ , . . . , δ ). In each of these cases, we conclude, by (7.4), that δ = δ . Then, for k ≥ 0, by Item (4) of Lemma 7.3 (and the fact that Suppose that k ≥ 2 and that the size of E v min is smaller or equal to k − 1. By item (2) of Lemma 7.3, the set E v min has size at most k − 1. Then by Proposition 7.2 each of the vectors v and v either exhibits case (1a) or case (2a). So Since ω and ω are blowup forms, the pairwise disjoint exceptional classes E jv+1 , . . . , E k can be represented simultaneously by pairwise disjoint embedded ω-symplectic or ω -symplectic spheres. By Lemma 4.12, blowing down along these spheres in (M, ω) and in (M, ω ) gives manifolds with a blowup form that is encoded by the vectorv = (λ F , λ B ; δ 1 , . . . , δ j ) and the vectorv = (λ F , λ B ; δ 1 , . . . , δ j ). By "uniqueness of blowdown" (Lemma 4.11) the resulting manifolds are symplectomorphic. Because the cohomology classes encoded by vectorsv andv are in g-reduced form, we may proceed by induction. We note that the case k = 0 is included in the case where the size of E v min equals k.
Since Algorithm 6.5 and Lemma 6.6 prove that a g-reduced form always exists, see Remark 6.7, the proof of Theorem 2.14, the existence and uniqueness of a g-reduced form, is now complete.
Appendix A. Proof of Lemma 4.6 In the proof we will use the following corollary of the positivity of intersections of J-holomorphic curves, see [18,Proposition 2.3].
A.1. Lemma. Let (M, ω) be a symplectic four-manifold. Let A and B be homology classes in H 2 (M ) that are linearly independent (over R). Suppose that GW(B) = 0, that c 1 (T M )(A) ≥ 1, and that there exists an almost complex structure J 0 ∈ J τ (M, ω) such that the class A is represented by a J 0 -holomorphic sphere. Then the intersection number A · B is nonnegative.
We now give a proof of Lemma 4.6.
Proof of Lemma 4.6. Assume that g = g(Σ) > 0. If M = (Σ × S 2 ) 1 then (by (2.12)) there are two cases: Fix J ∈ J τ (M, ω). To start, suppose that A is a homology class that is represented by a simple J-holomorphic sphere. Write with p, q, r integers. We claim that the coefficient p ofB is nonnegative and that if this coefficient is zero then A is one of the classes F ,Ê, and F −Ê. Indeed, the adjunction inequality [27, Cor. E.1.7] says that In case (a), A · A = 2pq − r 2 and c 1 (T M )(A) = (2 − 2g)p + 2q − r, so we get In case (b), A · A = p 2 + 2pq − r 2 and c 1 (T M )(A) = (3 − 2g)p + 2q − r, so we get Because A is represented by a J-holomorphic sphere for an ω-tamed J, the symplectic area of A is positive, i.e., Also note that, because r is an integer, r(r − 1) ≥ 0 with equality only if r = 0 or r = 1. By Lemma [28, Lemma 1.2], the minimal exceptional classÊ is represented by a simple J-holomorphic sphere, hence, and by the assumption on A and the positivity of intersections of J-holomorphic curves in almost complex four-manifolds [27,Proposition 2.4.4], we get that either A =Ê, i.e. p = 0, q = 1, r = 1, or r = A ·Ê ≥ 0. Assume the latter case.
By Lemma 4.5, we have GW(F −Ê) = 0, hence by Gromov's compactness theorem, we can write F −Ê = C 1 + . . . + C N where each C i is represented by a simple J holomorphic sphere. We would like to show that N = 1; the simple J holomorphic sphere in the class F −Ê = C 1 would then be embedded by the adjunction formula [27, Cor. E. 1.7]. Write each C i as a combination ofB, F , andÊ. By the previous claim, all the coefficients ofB are nonnegative. Because they sum to zero, they must all be equal to zero. Applying the previous claim again, every C i is either F , orÊ, or F −Ê. In particular, all the coefficients of F are nonnegative integers. Because they sum to 1, the coefficient of F in all but one of the C i -s are equal to zero, and in one of the C i -s, say in C 1 , the coefficient of F is equal to 1, so C 1 is either F or F −Ê. Because the coefficients ofÊ sum to −1, at least one of the C i -s is F −Ê. We conclude that C 1 = F −Ê, and N = 1. Introduction. This is a "greedy" algorithm. It starts by creating the set of Graphs determined by λ F , λ B . Then, at each stage BlowupGraph creates the set of Graphs (up to equivalence) which may be obtained by performing a blowup in the given size on a Graph from the set received in the previous step. The output of the program is the number of Graphs in the set at the final stage. In order to optimize the program, we use data structures designed to reduce the number of tests for equivalence between pairs of Graphs and to optimize the equivalence test.
Data Structure. For each Graph we arbitrarily define one of the two fat vertices to be the bottom fat vertex and the other to be the top fat vertex. In addition, we keep the distance between these two vertices and define it as the Height of the Graph, λ F in this case. We relate to different paths between the two fat vertices as Chains. A Chain will be represented by a doubly linked list of rational positive numbers, consists of the serial representation of the path, built from vertices and edges that appear one by one alternately. Each vertex is encoded by its distance from the bottom fat vertex and each edge is encoded by its label. The list representing each Chain starts with the lowest non-fat vertex in the Chain and ends with the highest non-fat vertex in the Chain (since edges between a fat vertex and a non-fat vertex are all labeled 1, we do not need to store them). For each list we will keep its first and its last nodes. In addition, every Chain in the Graph will receive a unique serial ID number, so that in a Graph with n Chains the IDs would be from 0 to (n − 1). The Chains of a Graph are kept in an array so that each Chain is put in its ID number cell of the array.
We define two different sorting methods on Chains; SortByStart and SortByEnd. Sorting by start is a lexicogaphic sort in which each node of the list is considered a letter. Sorting by end is defined by the same principle, except that the lists is read backwards and the value encoding each vertex is its distance from the top fat vertex (equals the Height of the Graph minus its distance from the bottom). Notice that sorting the Chains of a Graph by end will give the same result as sorting the Chains of the flipped Graph by start. For each Graph we will keep two additional arrays, keeping the IDs of the Chains. Both arrays keep the IDs in order, according to the corresponding Chains sort; one according to the "sort by start" and the other according to the "sort by end". See Figure B Another structure we use is referred to as "Graphs Tree". It is a balanced binary search tree, storing a set of non-equivalent Graphs. Each node in the tree contains two fields -a key field and a field containing a linked list of non-equivalent Graphs of that key. The key of a Graph is encoding the areas of the two fat vertices in the Graph -the larger area first, and the number of the Chains in the Graph. Notice that two Graphs may be equivalent only if they have the same key, meaning that they are stored in the same Graphs Tree node. The operation of adding a Graph to the Graphs Tree would be done only if there is no equivalent Graph already kept in the Graphs Tree.
Algorithms. The main algorithm counts the number of non-equivalent Graphs that can be produced as blowups of sizes δ 1 , . . . , δ n from the Graphs determined by λ B and λ F . It calls on the remaining algorithms defined in this section.
• Output: The number of non-equivalent Graphs that can be created by performing the blowups sequentially, starting from the Graphs that are determined by λ B and λ F . 1: Create a Graphs Tree structure gt (as described in the data structures section) of the Graphs defined by λ B , λ F using Graphs(λ F , λ B ; ). 2: Create a Graphs Tree structure t of all of the non-equivalent Graphs obtained from performing the blowups of sizes δ 1 , δ 2 , . . . , δ n sequentially, in every possible way, on each of the Graphs in gt, using BlowupSet(gt, δ 1 , δ 2 , . . . , δ n ). 3: Return the number of Graphs in t.

2:
Create a Graph with the following data: bottom fat vertex area = λ B + iλ F top fat vertex area = λ B − iλ F Height = λ F Chains = none 3: Add the Graph created to the Graphs Tree structure gt 4: Return gt Algorithm B.4 -BlowupSet (gt, δ 1 , δ 2 , . . . , δ n ) • Input: A Graphs Tree structure gt and n positive real numbers δ 1 , δ 2 , . . . , δ n in decreasing order. • Output: Graphs Tree structure t, storing the non-equivalent Graphs that can be created by performing the blowups sequentially, starting from the Graphs in gt. Define t as a new empty Graphs Tree 3: for each Graph g in gt 4: Create all the Graphs that can be obtained by performing a blowup of size δ i on g (perform the blowup in every possible way in g), using BlowupGraph(g, δ i ).

5:
Add each Graph that have been created in the previous step to the Graphs Tree t using AddGraph(g, t) (an algorithm that ensures that a Graph would be added if and only if there is no equivalent Graph in the Graphs Tree).

6:
Set gt = t 7: Return gt Algorithm B.5 -BlowupGraph(g, δ) • Input: A Graph g and a positive number δ • Output: A list ist of all the Graphs can be obtained by performing a blowup of size δ on the Graph g 1: if a blowup can be performed at the bottom fat vertex (that is, if δ < bottomFatVertexArea) then 2: Create a deep copy g of Graph g. Let n be the number of Chains in g, make the size of the arrays in g (Chains, ChainsByStart, ChainByEnd) be n + 1.

3:
Subtract δ from the area of the bottom fat vertex.

4:
Add a new Chain with one node δ to g . Let n be the number of Chains in g, the ID of the new Chain will be n and a pointer to the new Chain will be placed in Chains[n].

5:
Use binary search and the sorting methods defined in the data structure section to find the right place for the ID of the new Chain in the ChainsByStart and ChainsByEnd arrays and place it there (move other IDs to the right if needed to "make place").

6:
Add g to the list. 7: if a blowup can be performed on the top fat vertex (δ < topFatVertexArea) then 8: Do the same as when a blowup can be performed at the bottom fat vertex, except subtract δ from the top fat vertex area and the new Chain will have the node height −δ 9: for every Chain chain 10: for every node v that represents a vertex in chain 11: if a monotone blowup can be made on the vertex then 12: Create a new Chain replacing to chain with a blowup on v, with the same ID.

13:
Create a deep copy g of the Graph g and replace the copy of chain with the new Chain that have been created.

14:
Use binary search and the sorting methods described above, to find the right place for the ID of the new Chain in the ChainsByStart and ChainsByEnd arrays and place it there (move other IDs to the right if needed to "make place").

15:
Add g to list. 16: Return list