Hamiltonian circle actions with minimal isolated fixed points
aa r X i v : . [ m a t h . S G ] J a n HAMILTONIAN CIRCLE ACTIONS WITH ISOLATEDFIXED POINTS
HUI LI
Abstract.
Let the circle act in a Hamiltonian fashion on a compactsymplectic manifold (
M, ω ) of dimension 2 n . Then the S -action has atleast n + 1 fixed points. We consider the cases when the fixed point setconsists of precisely n + 1 and n + 2 isolated points. We find equivalentconditions on the first Chern class of M and the largest weight of the S -action at the fixed points. For two interesting cases, we show that thelargest weight can completely determine the integral cohomology ring of M , the total Chern class of M , and the sets of weights of the S -actionat all the fixed points. We will see that all these data are isomorphicto those of known examples, CP n or e G ( R n +2 ), equipped with standardcircle actions. Introduction
Let the circle act on a compact 2 n -dimensional symplectic manifold ( M, ω )with moment map φ . The moment map φ is a perfect Morse-Bott functionwith critical set being the fixed point set M S of the S -action. Since theeven Betti numbers of M , b i ( M ) ≥ ≤ i ≤ n , M S contains atleast n + 1 points. When M S consists of exactly n + 1 isolated points, wesay that the action has minimal isolated fixed points; when M S consistsof exactly n + 2 isolated points, we say that the action has almost minimal isolated fixed points.Known examples of compact Hamiltonian S -manifolds with minimal iso-lated fixed points are CP n , and e G ( R n +2 ) with n ≥ S -manifolds with al-most minimal isolated fixed points are e G ( R n +2 ) with n ≥ S -actions on compact symplectic manifoldswith minimal and almost minimal isolated fixed points, we study the inter-play between the geometry and topology of the symplectic manifolds andthe S -actions. Key words and phrases.
Symplectic manifold, Hamiltonian circle action, equivariantcohomology, Chern classes, K¨ahler manifold, biholomorphism, symplectomorphism.2010 classification. 53D05, 53D20, 55N25, 57R20, 32H02.The author is supported by the NSFC grant K110712116.
For a symplectic S -manifold ( M, ω ) of dimension 2 n with isolated fixedpoints, a neighborhood of each fixed point P is S -equivariantly diffeomor-phic to a neighborhood of the tangent space at P with an S -linear action.The tangent space at P splits into n copies of C , on each of which S actsby multiplication by λ w i , where λ ∈ S and w i ∈ Z for i = 1 , · · · , n . Theintegers w i ’s are well defined, and are called the weights of the S -action at P . In the set of all the weights at all the fixed points, the largest positiveweight is called the largest weight at the fixed points on M .For a compact Hamiltonian S -manifold ( M, ω ) with moment map φ , if[ ω ] ∈ H ( M ) is an integral class, then for any two fixed points P and Q , wehave φ ( P ) − φ ( Q ) ∈ Z ; if w is a weight of the S -action at a fixed point, thenthere exist fixed points P and Q such that w | (cid:0) φ ( P ) − φ ( Q ) (cid:1) (see Lemma 2.2).The topologic and geometric data we concern are the integral cohomologyring and total Chern class of the manifold, and the data on the circle actionwe concern are the sets of weights at the fixed points. The first Chern classof the manifold is an important data, especially when the manifold is K¨ahler,in certain cases, it can determine the biholomorphism type of the manifold.First, we consider the case when the action has minimal isolated fixedpoints. By Lemma 3.3, we may denote M S = { P , P , · · · , P n } , where thepoint P i has Morse index 2 i for each i , and we have φ ( P ) < φ ( P ) < · · · < φ ( P n ) . Our first main result gives equivalent conditions on the first Chern class c ( M ) and the largest weight of the S -action at the fixed points. Note thatin the following two cases, these data respectively coincide with those inExamples 3.1 and 3.2. Theorem 1.1.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . Then c ( M ) = ( n + 1)[ ω ] if and onlyif φ ( P n ) − φ ( P ) is the largest weight of the S -action at the fixed points; c ( M ) = n [ ω ] if and only if φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largestweight of the S -action at the fixed points. In the two cases of the largest weights, we obtain Theorems 1.2 and 1.3.Theorem 1.2 follows from Propositions 7.8, 8.1 and 8.7. Theorem 1.3 followsfrom Propositions 9.10, 10.1 and 10.10.
Theorem 1.2.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . If φ ( P n ) − φ ( P ) is a weight of the S -action at a fixed point, then all the following are true. (1) The integral cohomology ring of M is isomorphic to that of CP n . (2) The total Chern class of M is isomorphic to that of CP n . (3) The sets of weights of the S -action at all the fixed points on M are isomorphic to those of a standard circle action on CP n (as inExample 3.1). AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 3
Theorem 1.3.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . If φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) isthe largest weight of the S -action at the fixed points, then n ≥ is odd, andall the following are true. (1) The integral cohomology ring of M is isomorphic to that of e G ( R n +2 ) . (2) The total Chern class of M is isomorphic to that of e G ( R n +2 ) . (3) The sets of weights of the S -action at all the fixed points on M areisomorphic to those of a standard circle action on e G ( R n +2 ) (as inExample 3.2). Next, we consider the case when the action has almost minimal isolatedfixed points. By Lemma 4.2, n ≥ M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) , where the point P i has Morse index 2 i for each i , and the point P ( n ) ′ hasMorse index n . Moreover, φ ( P ) < · · · < φ ( P n − ) < φ ( P n ) ≤ φ ( P ( n ) ′ ) < φ ( P n +1 ) < · · · < φ ( P n ) . We obtain the following equivalent condition on c ( M ) and the largest weightof the S -action at all the fixed points. Note that these data are the sameas in Example 4.1. Theorem 1.4.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) . Then c ( M ) = n [ ω ] if and only if φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight ofthe S -action at the fixed points. In dimension , for the “if” part to hold, ω needs to be chosen suitably. In the above case of largest weight, we obtain Theorem 1.5, which followsfrom Propositions 11.4, 12.17 and 12.18.
Theorem 1.5.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) . Then n ≥ must be even. If φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight ofthe S -action at the fixed points, then all the following are true. (1) The integral cohomology ring of M is isomorphic to that of e G ( R n +2 ) . (2) The total Chern class of M is isomorphic to that of e G ( R n +2 ) . (3) The sets of weights of the S -action at all the fixed points on M areisomorphic to those of a standard circle action on e G ( R n +2 ) (as inExample 4.1). Clearly, if we know the total Chern class of M , we know the first Chernclass of M ; if we know the sets of weights of the S -action at all the fixed HUI LI points, we know the largest weight of the S -action at all the fixed points.Hence, the above theorems give a few equivalent conditions. It would beinteresting to show that if the integral cohomology ring of M is isomorphicto that of CP n or e G ( R n +2 ), then we can know one of the other conditionsappeared in the theorems. This is closely related to Petrie’s conjecture([16]), which states that if a homotopy complex projective space X admitsa nontrivial circle action, then the total Pontryagin class of X is isomorphicto that of CP n , i.e., p ( X ) ∼ = p ( CP n ), where n = dim X . Let ( M, ω ) bea compact Hamiltonian S -manifold of dimension 2 n with isolated fixedpoints, by [9], M is simply connected. Furthermore, if M has the integralcohomology ring of CP n , then M is homotopy equivalent to CP n . Moreover,by Morse theory, this cohomology ring of M implies that the S -action hasprecisely n + 1 fixed points. Hence Petrie’s conjecture, in one case, can bestated as follows: Conjecture 1.
Let ( M, ω ) be a compact symplectic manifold whose integralcohomology ring is isomorphic to that of CP n . If M admits a Hamilton-ian S -action with isolated fixed points, then the total Chern class of M isisomorphic to that of CP n , i.e., c ( M ) ∼ = c ( CP n ) . Hence, in Theorem 1.2, if starting from condition (1), i.e., H ∗ ( M ; Z ) ∼ = H ∗ ( CP n ; Z ) as rings, we can get c ( M ) = ( n + 1)[ ω ] or the largest weight is φ ( P n ) − φ ( P ), then we can prove Conjecture 1 holds. Based on the positiveevidence on the conjecture through this work, we similarly pose the followingconjecture: Conjecture 2.
Let ( M, ω ) be a compact symplectic manifold whose integralcohomology ring is isomorphic to that of e G ( R n +2 ) . If M admits a Hamil-tonian S -action with isolated fixed points, then the total Chern class of M is isomorphic to that of e G ( R n +2 ) , i.e., c ( M ) ∼ = c (cid:0) e G ( R n +2 ) (cid:1) . The method and techniques we develope, and the results we obtain inthis paper, besides standing on their own interests, we hope will also help tosolve these conjectures and help to deal with more general related problems.Now let us look at our work and related works in the literature. In [4], fora compact almost complex S -manifold M of dimension 2 n with n +1 isolatedfixed points, Hattori uses equivariant K-theory and equivariant cohomologyto show that if c ( M ) = ( n + 1) x or c ( M ) = nx for some x ∈ H ( M ; Z ),then the sets of weights at the fixed points are respectively isomorphic tothose of a standard circle action on CP n , or on e G ( R n +2 ) with n ≥ n + 2 isolated fixed points, with additional con-ditions , he shows that if c ( M ) = nx , then the sets of weights at the fixedpoints are isomorphic to those of a standard circle action on e G ( R n +2 ) with n ≥ S -manifold ( M, ω ) of di-mension 2 n with n + 1 isolated fixed points, Tolman shows that the sets ofweights at all the fixed points determine the integral cohomology ring andtotal Chern class of M . In this paper, for compact Hamiltonian S -manifolds AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 5 ( M, ω ) of dimension 2 n with n + 1 and n + 2 isolated fixed points, usingtechniques for symplectic (Hamiltonian) S -manifolds, we prove equivalenceof the conditions on c ( M ) and on the largest weight of the S -action at thefixed points; and starting from the single largest weight, we directly deter-mine the generators and relations of the integral cohomology ring structureof M , find the sets of weights of the S -action at all the fixed points, andgive constructive proof on the total Chern class of M . In particular, for thecase of n + 1 fixed points, we determine the integral cohomology ring of M before we know the sets of weights at all the fixed points. Our methods ofproofs emphasize on demonstrating the close and direct relations betweenthe largest weight and all the other data, c ( M ), c ( M ), the ring H ∗ ( M ; Z ),and the sets of weights at all the fixed points. I expect that our approachwill be useful for solving Conjectures 1 and 2.Recent works on compact Hamiltonian S -manifolds with fixed point setminimal or almost minimal in a certain sense include [17, 14, 13, 10, 11, 12,3]. These works deal with manifolds of dimensions no bigger than 8 or thenumber of fixed point set components no bigger than 3.Using a theorem by Kobayashi and Ochiai [6], in [10, Prop. 4.2 andSec. 5], we prove the following facts: Let ( M, ω, J ) be a compact K¨ahlermanifold of complex dimension n , which admits a holomorphic Hamiltoniancircle action. Assume that [ ω ] is an integral class. If c ( M ) = ( n + 1)[ ω ],then M is S -equivariantly biholomorphic and symplectomorphic to CP n ,and if c ( M ) = n [ ω ], then M is S -equivariantly biholomorphic and sym-plectomorphic to e G ( R n +2 ). Hence, for the case of K¨ahler manifolds, whenthe S -action has minimal and almost minimal isolated fixed points, ourtheorems imply that, we may use c ( M ) or the largest weight to determineuniquely the manifold in the complex and symplectic categories.The paper is organized as follows. In Section 2, we present preliminarymaterials for the whole paper. In Sections 3 and 4, we respectively giveexamples of Hamiltonian S -manifolds with minimal and almost minimalisolated fixed points, state and prove preliminary and key facts for the cor-responding case. In Section 5, we study the relation between the first Chernclass of the manifold and the largest weight of the S -action at the fixedpoints, and prove Theorems 1.1 and 1.4. In Section 6, we prove certainproperties of the weights of the S -action at the fixed points. We will usethese properties in the next sections. In Section 7, for the case of minimalisolated fixed points, when φ ( P n ) − φ ( P ) is the largest weight, we determinethe integral cohomology ring of the manifold. In Section 8, for the case justmentioned, we determine the sets of weights of the S -action at all the fixedpoints, and the total Chern class of the manifold. In Section 9, for the caseof minimal isolated fixed points, when φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) isthe largest weight, we determine the integral cohomology ring of the man-ifold. In Section 10, for the case just mentioned, we determine the sets ofweights of the S -action at all the fixed points, and the total Chern class of HUI LI the manifold. In Section 11, for the case of almost minimal isolated fixedpoints, when φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight, wedetermine the sets of weights of the S -action at all the fixed points. In Sec-tion 12, for the case just mentioned, we determine the integral cohomologyring and total Chern class of the manifold.2. Preliminaries
In this section, we review equivariant cohomology, state and prove certainfacts for compact Hamiltonian S -manifolds. We will use them in the proofof our main theorems.Let us first introduce equivariant cohomology. Let M be a smooth S -manifold. The equivariant cohomology of M in a coefficient ring R is H ∗ S ( M ; R ) = H ∗ ( S ∞ × S M ; R ), where S acts on S ∞ freely. If p is apoint, then H ∗ S ( p ; R ) = H ∗ ( CP ∞ ; R ) = R [ t ], where t ∈ H ( CP ∞ ; R ) isa generator. If S acts on M trivially, i.e., it fixes M , then H ∗ S ( M ; R ) = H ∗ ( M ; R ) ⊗ R [ t ] = H ∗ ( M ; R )[ t ]. The projection map π : S ∞ × S M → CP ∞ induces a pull back map π ∗ : H ∗ ( CP ∞ ) → H ∗ S ( M ) , so H ∗ S ( M ) is an H ∗ ( CP ∞ )-module.Let ( M, ω ) be a compact symplectic manifold. There exists an almostcomplex structure J : T M → T M which is compatible with ω , i.e., ω ( J ( · ) , · )is a Riemannian metric. The space of compatible almost complex structureson ( M, ω ) is contractible, hence there is well defined total Chern class c ( M ) = 1 + c ( M ) + · · · + c n ( M ) ∈ H ∗ ( M ; Z ) , where c i ( M ) ∈ H i ( M ; Z ) is the i -th Chern class of M . Similarly, if ( M, ω )is a compact symplectic S -manifold (the action preserves the symplecticform ω ), on the normal bundle of each connected component F of the fixedpoint set, there is a well defined set of nonzero integers, called the (nonzero) weights of the action. Moreover, the normal bundle of M at F natu-rally splits into complex line bundles, one line bundle corresponding to oneweight. If ( M, ω ) is a compact Hamiltonian S -manifold with moment map φ : M → R , then the map φ is a perfect Morse-Bott function, and its crit-ical set coincides with the fixed point set of the action. At each connectedcomponent F of the fixed point set, the negative normal bundle of F is thesub-bundle with negative weights; if λ F is the number of negative weights at F , counted with multiplicities, then the Morse index of F is λ F , whichis the dimension of the negative normal bundle to F . Similarly, the Morsecoindex of F is 2 λ + F , where λ + F is the number of positive weights at F .Let ( M, ω ) be a compact 2 n -dimensional symplectic S -manifold. Assumethe fixed points are isolated. Let P be a fixed point, and let { w , · · · , w n } be the set of weights at P . We denote the equivariant total Chern class AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 7 of M as c S ( M ) = 1 + c S ( M ) + · · · + c S n ( M ) ∈ H ∗ S ( M ; Z ) , where c S i ( M ) ∈ H iS ( M ; Z ) is the i -th equivariant Chern class of M . Therestriction of c S ( M ) to P is c S ( M ) | P = 1 + n X i =1 c S i ( M ) | P = 1 + n X i =1 σ i ( w , · · · , w n ) t i , where σ i ( w , · · · , w n ) is the i -th symmetric polynomial in the weights at P .Next we state and prove some basic results. First, the symplectic class [ ω ]of a Hamiltonian S -manifold can be extended to an equivariant cohomologyclass e u : Lemma 2.1. [13, Lemma 2.7]
Let the circle act on a compact symplecticmanifold ( M, ω ) with moment map φ : M → R . Then there exists e u =[ ω − φt ] ∈ H S ( M ; R ) such that for any fixed point set component F , e u | F = [ ω | F ] − φ ( F ) t. If [ ω ] is an integral class, then e u is an integral class. For an S -manifold M , when there exists a finite stabilizer group Z k ⊂ S ,where k >
1, the set of points, M Z k , which is fixed by Z k but not fixed by S , is called a Z k -isotropy submanifold . If a Z k -isotropy submanifold isa sphere, it is called a Z k -isotropy sphere . For k = 1, we denote Z = 1,and we regard M Z = M . Lemma 2.2.
Let the circle act on a connected compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integral class. Thenfor any two fixed point set components F and F ′ , φ ( F ) − φ ( F ′ ) ∈ Z . If Z k is the stabilizer group of some point on M , then for any two fixed point setcomponents F and F ′ on the same connected component of M Z k , we have k | ( φ ( F ′ ) − φ ( F )) .Proof. Since M is compact, the action has at least two fixed components.Since [ ω ] is integral, by Lemma 2.1, we may assume φ ( F ) ∈ Z for any fixedcomponent F . Hence φ ( F ′ ) − φ ( F ) ∈ Z for any two fixed components F and F ′ .The isotropy submanifold M Z k is a compact Hamiltonian S -manifold,hence contains at least two fixed components. Consider the S / Z k ≈ S action on M Z k , whose moment map is φ ′ = φ/k . Since [ ω | M Z k ] is integral,by the first paragraph, for any two fixed components F and F ′ on the sameconnected component of M Z k , we have φ ′ ( F ′ ) − φ ′ ( F ) ∈ Z , i.e., φ ( F ′ ) k − φ ( F ) k ∈ Z . (cid:3) For a compact Hamiltonian S -manifold, Tolman gives an inequality onthe index of a fixed point set component [17, Lemma 3.1]. We only state itfor the case of isolated fixed points: HUI LI
Lemma 2.3.
Let the circle act on a compact symplectic manifold ( M, ω ) with isolated fixed points and with moment map φ : M → R . Then for anyfixed point P , λ P ≤ l , where l is the number of fixed points Q ’s such that φ ( Q ) < φ ( P ) . For a compact Hamiltonian S -manifold M with isolated fixed points,using standard method in equivariant cohomology (see [7] and [18]), we havethe following general result on the basis of H ∗ S ( M ; Z ) as an H ∗ ( CP ∞ ; Z )-module. Proposition 2.4.
Let the circle act on a compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume M S consists of isolated points.Then for each P ∈ M S with index λ P , there exists a class α P ∈ H λ P S ( M ; Z ) such that α P | P = Λ − P t λ P and α P | P ′ = 0 for any P ′ ∈ M S with φ ( P ′ ) ≤ φ ( P ) , where Λ − P is the product of the negative weights at P . Such classes α P ’s with P ∈ M S form a basis for H ∗ S ( M ; Z ) as an H ∗ ( CP ∞ ; Z ) -module. The projection π : S ∞ × S M → CP ∞ induces a natural push forward map π ∗ : H ∗ S ( M ; Q ) → H ∗ ( CP ∞ ; Q ), which is given by “integration over the fiber M ”, denoted R M . We have the following theorem due to Atiyah-Bott, andBerline-Vergne [1, 2]. Theorem 2.5.
Let the circle act on a compact oriented manifold M . As-sume the fixed points are isolated. Fix a class α ∈ H ∗ S ( M ; Q ) . Then aselements of Q ( t ) , Z M α = X P ⊂ M S α | P e S ( N P ) , where the sum is over all the fixed points, and e S ( N P ) is the equivariantEuler class of the normal bundle to P . When we use equivariant cohomology, we often use the following nota-tions. For convenience, we list them here, not repeating their meaningseach time we use them. Let P i be an isolated fixed point for a symplectic S -action, where i belongs to some index set. Then • Γ i denotes the sum of the weights of the S -action on the normalbundle to P i , • Λ i denotes the product of the weights of the S -action on the normalbundle to P i , • Λ − i denotes the product of the negative weights of the S -action onthe normal bundle to P i , • Λ + i denotes the product of the positive weights of the S -action onthe normal bundle to P i .Finally, we state a lemma which we will use in our proofs. AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 9
Lemma 2.6. [17, Lemma 2.6]
Let the circle act on a compact symplecticmanifold ( M, ω ) . Let p and q be fixed points which lie on the same componentof M Z k for some k > . Then the S weights at p and q are equal modulo k . when the action has minimal isolated fixed points In this section, we consider the case when a compact Hamiltonian S -manifold has minimal isolated fixed points. We give examples, prove andset up key basic things. Example 3.1.
Consider CP n . It naturally arises as a coadjoint orbit of SU ( n + 1), hence it has a K¨ahler structure and a Hamiltonian SU ( n + 1)action. Consider the S ⊂ SU ( n + 1) action on CP n given by λ · [ z , z , · · · , z n ] = [ λ b z , λ b z , · · · , λ b n z n ] , where the b i ’s are mutually distinct integers. This action has n + 1 isolatedfixed points, P i = [0 , · · · , , z i , , · · · , i = 0 , , · · · , n . The momentmap values of the fixed points of the S -action are φ ( P i ) = b i , i = 0 , , · · · , n .The set of weights of the S -action at any P i is (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i . As a ring, H ∗ ( CP n ; Z ) = Z [ x ] /x n +1 , where deg( x ) = 2. The total Chernclass c ( CP n ) = (1 + x ) n +1 , in particular, c ( CP n ) = ( n + 1) x . Example 3.2.
Let e G ( R n +2 ) be the Grassmanian of oriented 2-planes in R n +2 , with n ≥ SO ( n +2), hence it has a K¨ahler structure and a Hamiltonian SO ( n + 2) action.Consider the S ⊂ SO ( n + 2) action on e G ( R n +2 ) induced by the S -actionon R n +2 = R × C n +12 given by λ · (cid:0) t, z , · · · , z n − (cid:1) = (cid:0) t, λ b z , · · · , λ b n − z n − (cid:1) , where the b i ’s, with i ∈ { , · · · , n − } , are mutually distinct non-zero inte-gers. This action has n + 1 isolated fixed points, denoted P , P , · · · , and P n , where for each i ∈ { , · · · , n − } , P i and P n − i are given by the plane(0 , · · · , , z i , , · · · ,
0) respectively with two different orientations. The mo-ment map values of the fixed points are respectively φ ( P ) = − b , · · · , φ (cid:0) P n − (cid:1) = − b n − , φ (cid:0) P n +12 (cid:1) = b n +12 , · · · , φ ( P n ) = b , assuming in the order of increasing. The set of weights of the S -action atany P i , where i ∈ { , , · · · , n } , is (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i,n − i ∪ (cid:8) w i,n − i = 12 (cid:0) φ ( P n − i ) − φ ( P i ) (cid:1)(cid:9) . As a ring, H ∗ (cid:0) e G ( R n +2 ); Z (cid:1) = Z [ x, y ] / ( x n +12 − y, y ), where deg( x ) = 2and deg( y ) = n + 1. The total Chern class c (cid:0) e G ( R n +2 ) (cid:1) = (1+ x ) n +2 x , inparticular, c (cid:0) e G ( R n +2 ) (cid:1) = nx .Next, we prove a key basic lemma for the case we are considering. Lemma 3.3.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume M S consists of n + 1 isolated points, denoted M S = { P , P , · · · , P n } . Then the points in M S can be labelled so that P i has Morse index i , and we have H i ( M ; Z ) = Z and H i − ( M ; Z ) = 0 for all ≤ i ≤ n . Moreover, (3.4) φ ( P ) < φ ( P ) < · · · < φ ( P n ) . Proof.
Since M is compact and symplectic, 0 = [ ω i ] ∈ H i ( M ; R ), so(3.5) dim H i ( M ) ≥ , ∀ ≤ i ≤ dim( M ) . The moment map φ is a perfect Morse function. Since φ has n + 1 criticalpoints P , P , · · · , P n , each of which has even index, they must respectivelyhave index 0, 2, · · · , and 2 n to make (3.5) to hold. Moreover, by Morsetheory, the negative disk bundle of P i is a 2 i -cell for the CW-structureof M given by the Morse flow for any invariant metric, and there are noodd dimensional cells. By cellular cohomology theory, H i ( M ; Z ) ∼ = Z and H i +1 ( M ; Z ) = 0 for all 0 ≤ i ≤ n . The inequality (3.4) follows fromLemma 2.3. (cid:3) Remark 3.6.
From now on, when we use the notation M S = { P , P , · · · , P n } ,we mean that P i is the fixed point of index 2 i , for any 0 ≤ i ≤ n . We willuse this fact and (3.4) so frequently as not to refer to Lemma 3.3.By Proposition 2.4, for the case we consider, we get a basis of H ∗ S ( M ; Z )as follows. Moreover, since H ∗ ( M S ; Z ) has no torsion, we have H ∗ ( M ; Z ) = H ∗ S ( M ; Z ) / ( t ) (Sec. 2 of [13]), hence the restriction of a basis of H ∗ S ( M ; Z )to ordinary cohomology is a basis of H ∗ ( M ; Z ). Proposition 3.7.
Let the circle act on a compact n -dimensional sym-plectic manifold ( M, ω ) with moment map φ : M → R . Assume M S = { P , P , · · · , P n } . Then as an H ∗ ( CP ∞ ; Z ) -module, H ∗ S ( M ; Z ) has a basis (cid:8)e α i ∈ H iS ( M ; Z ) | ≤ i ≤ n (cid:9) such that e α i | P i = Λ − i t i , and e α i | P j = 0 , ∀ j < i. Moreover, (cid:8) α i = r ( e α i ) ∈ H i ( M ; Z ) | ≤ i ≤ n (cid:9) is a basis for H ∗ ( M ; Z ) ,where r : H ∗ S ( M ; Z ) → H ∗ ( M ; Z ) is the natural restriction. A direct corollary of Proposition 3.7 is:
AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 11
Corollary 3.8.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume M S = { P , P , · · · , P n } .Let e α ∈ H iS ( M ; Z ) be a class such that e α | P j = 0 for all j < i . Then e α = a i e α i for some a i ∈ Z . when the action has almost minimal isolated fixed points In this section, we consider the case when a compact Hamiltonian S -manifold has almost minimal isolated fixed points. We give examples, proveand set up key basic things. Example 4.1.
Let e G ( R n +2 ) be the Grassmanian of oriented 2-planes in R n +2 , with n ≥ n -dimensional manifold naturally arises as acoadjoint orbit of SO ( n + 2), hence it has a natural K¨ahler structure and aHamiltonian SO ( n + 2) action.Consider the S ⊂ SO ( n + 2) action on e G ( R n +2 ) induced by the S action on R n +2 = C n +22 given by λ · (cid:0) z , z , · · · , z n (cid:1) = (cid:0) λ b z , λ b z , · · · , λ b n z n (cid:1) , where the b i ’s, with i = 0 , , · · · , n , are mutually distinct integers. Thisaction has n + 2 isolated fixed points, denoted P , P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n , where for each 0 ≤ i ≤ n , P i and P n − i are given by the plane (0 , · · · , , z i , , · · · , n − n = ( n ) ′ . Let φ be the moment map of this S -action. Then the φ ( P i )’sare respectively − b , · · · , − b n , b n , · · · , b , assuming in the order of nondecreasing. Note that if b n = 0, then φ ( P n ) <φ ( P ( n ) ′ ), otherwise, φ ( P n ) = φ ( P ( n ) ′ ). The set of weights of the S -actionat any P i , where i ∈ (cid:8) , · · · , n , ( n ) ′ , · · · , n (cid:9) , is (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i,n − i . The ring H ∗ (cid:0) e G ( R n +2 ); Z (cid:1) is generated by x , y and z , where deg( x ) = 2and deg( y ) = deg( z ) = n . The relations are x n = y + z , xy = xz = x n +1 , and y = z = − n x n y = − n x n z . The total Chern class c (cid:0) e G ( R n +2 ) (cid:1) = (1+ x ) n +2 x , in particular, c (cid:0) e G ( R n +2 ) (cid:1) = nx .Next, we prove a key basic lemma. Lemma 4.2.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume M S consists of n + 2 isolated points. Then n must be even, and we can denote M S = { P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n } , where the points can be labelled so that P i has Morse index i for all i , and P ( n ) ′ has Morse index n . Moreover,the cohomology groups of M are H k ( M ; Z ) = Z , if k is even, ≤ k ≤ n , and k = n, Z ⊕ Z , if k = n, , if k is odd , and (4.3) φ ( P ) < · · · < φ ( P n ) ≤ φ ( P ( n ) ′ ) < φ ( P n +1 ) < · · · < φ ( P n ) . Proof.
Similar to the proof of Lemma 3.3, there is at least one fixed point,namely P i , of index 2 i for each 0 ≤ i ≤ n , which contributes a copy of R to H i ( M ; R ). By assumption, there is a remaining fixed point, let 2 k be itsMorse index. Since dim H i ( M ) = dim H n − i ( M ) for all 0 ≤ i ≤ n byPoincar´e duality, we must have 2 k = 2 n − k , which gives 2 k = n , i.e., n iseven. We denote the remaining fixed point as P ( n ) ′ , which has Morse index n . By Morse theory, M has a natural CW-structure, with cells given by thenegative disk bundles of the fixed points. By cellular cohomology theory,the groups H k ( M ; Z ) are as claimed.By Lemma 2.3, φ ( P ) < φ ( P ) < · · · < φ ( P n ) ≤ φ ( P ( n ) ′ ). Similarly,using − φ , − φ ( P n ) < − φ ( P n − ) < · · · < − φ ( P ( n ) ′ ) ≤ − φ ( P n ). The twoinequalities give (4.3). (cid:3) Remark 4.4.
From now on, when we use the notation M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) , we mean each point has the index as in Lemma 4.2, and these points satisfy(4.3). We will use these facts tacitly.Similar to the last section, we get a basis of the equivariant and ordinarycohomology of the Hamiltonian S -manifold we consider. Proposition 4.5.
Let ( M, ω ) be a compact n -dimensional Hamiltonian S -manifold with moment map φ : M → R . Assume M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) . Let I = (cid:8) , , · · · , n , ( n ) ′ , · · · , n (cid:9) . Then as an H ∗ ( CP ∞ ; Z ) -module, H ∗ S ( M ; Z ) has a basis (cid:8)e α i | i ∈ I (cid:9) such that for any i ∈ I , e α i | P i = Λ − i t i , e α i | P j = 0 , ∀ P j with index( P j ) ≤ index( P i ) . Moreover, (cid:8) α i = r ( e α i ) | i ∈ I (cid:9) is a basis for H ∗ ( M ; Z ) , where r : H ∗ S ( M ; Z ) → H ∗ ( M ; Z ) is the natural restriction.Proof. By Proposition 2.4, we get the basis satisfying the other conditionsexcept the condition that e α n | P ( n ′ = 0. If e α n | P ( n ′ = a e α ( n ) ′ | P ( n ′ = 0,where a ∈ Z , then replace e α n by e α n − a e α ( n ) ′ , still denoting it e α n . (cid:3) AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 13
Corollary 4.6.
Let ( M, ω ) be a compact n -dimensional Hamiltonian S -manifold with moment map φ : M → R . Assume M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) . If e α ∈ H kS ( M ; Z ) is a class such that e α | P i = 0 for all P i ’s with φ ( P i ) < a ,then e α = X i a i e α i , where the sum is over the indices i ’s such that φ ( P i ) ≥ a and deg( e α i ) ≤ k ,and a i ∈ H ∗ ( CP ∞ ; Z ) . Here, i ∈ I = (cid:8) , , · · · , n , ( n ) ′ , · · · , n (cid:9) . c ( M ) and the largest weight — the proofs of Theorems 1.1and 1.4 In this section, first, for compact Hamiltonian S -manifold ( M, ω ) withisolated fixed points, we study the relation between c ( M ) and the largestweight of the S -action at the fixed points. Then we use the results to proveTheorems 1.1 and 1.4.In a symplectic S -manifold ( M, ω ) with isolated fixed points, if w > S -action at a fixed point P , − w is a weight of the S -actionat a fixed point Q , and P and Q are on the same connected component of M Z w , we say that there is a weight w from P to Q . When the signs of w at P and at Q are clear, we will use these terminologies interchangeably: there is a weight ± w between P and Q , or between Q and P , or ± w is a weight between P and Q or between Q and P , or P has a weight ± w with Q .It is known ([4], [8], [15]) that for a compact symplectic (or almost com-plex) S -manifold with isolated fixed points, if W + and W − are respectivelythe set of positive weights and negative weights at all the fixed points, then W − = − W + . So if w is a weight at some fixed point, then − w is also a weight at somefixed point. We will use this fact tacitly. Lemma 5.1.
Let the circle act on a compact symplectic manifold ( M, ω ) with moment map φ : M → R . If c ( M ) = k [ ω ] , then for any two fixed pointset components F and F ′ , we have Γ F − Γ F ′ = k (cid:0) φ ( F ′ ) − φ ( F ) (cid:1) , where Γ F and Γ F ′ are respectively the sums of the weights at F and F ′ .Proof. We have an equivariant extension c S ( M ) = k e u + a t of c ( M ) = k [ ω ],where a ∈ R . Let f ∈ F and f ′ ∈ F ′ be points. Then c S ( M ) | f = Γ F t = − kφ ( F ) t + a t, and c S ( M ) | f ′ = Γ F ′ t = − kφ ( F ′ ) t + a t. Subtracting the two equalities, we obtain the claim. (cid:3)
Lemma 5.2.
Let the circle act on a compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume M S consists of isolated points. Let P, Q ∈ M S , and P = Q , where index( P ) = 2 i and index( Q ) = 2 j with i ≤ j . Assume there is a weight w from P to Q , − w is not a weight at P , − w has multiplicity at Q , and w is the largest among the absolute valuesof all the weights at P and Q . If c ( M ) = k [ ω ] , then j − i + 1 = k φ ( Q ) − φ ( P ) w .Proof. Let W − P = (cid:8) a , · · · , a i (cid:9) and W + P = (cid:8) b , · · · , b n − i (cid:9) be respectively the set of negative weights and positive weights at P , andlet W − Q = (cid:8) a ′ , · · · , a ′ j (cid:9) and W + Q = (cid:8) b ′ , · · · , b ′ n − j (cid:9) be respectively the set of negative weights and positive weights at Q . Assume b n − i = w ∈ W + P , and a ′ j = − w ∈ W − Q . By Lemma 2.6,(5.3) W − P ∪ W + P = W − Q ∪ W + Q mod w. Since w is the largest among the absolute values of all the weights at P and Q , − w does not occur at P and − w occurs once at Q , up to a reordering ofindices, (5.3) can only yield the following relations: a = a ′ , · · · , a m = a ′ m , where m ≤ i when i < j and m < i when i = j,b = b ′ , · · · , b l = b ′ l , where l ≤ n − j when i < j and l < n − j when i = j,a m +1 = b ′ l +1 − w, · · · , a i = b ′ n − j − w,b l +1 = a ′ m +1 + w, · · · , b n − i − = a ′ j − + w, and b n − i = a ′ j + 2 w, where ( i − m ) + l = n − j . HenceΓ P − Γ Q = X a i + X b i − X a ′ i − X b ′ i = 2 w − ( i − m ) w + ( n − i − l − w = ( j − i + 1) w. Combining Lemma 5.1, we obtain the claim of the lemma. (cid:3)
Lemma 5.4.
Let the circle act on a compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume M S consists of isolated points. Let w > be the largest weight of the S -action at the fixed points. Then thereare P, Q ∈ M S with φ ( P ) < φ ( Q ) , such that there is a weight w from P to Q , − w is not a weight at P and − w has multiplicity at Q .Proof. Relative to the value of φ , choose the lowest fixed point P which hasa weight w . Then − w cannot be a weight at P . Suppose − w is a weight at P , consider the connected component C of M Z w containing P . Since w isthe largest weight on M , each weight at the fixed points in C is ± w . Since P is of index at least 2 in C , C is compact symplectic, and S -Hamiltonianwith moment map φ | C , C must contain a minimum for φ | C , a fixed point P ′ which has a lower moment map value and a positive weight w , contradictingto the choice of P . AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 15
Next, relative to the value of φ , choose a closest fixed point Q to P suchthat − w is a weight at Q , and P and Q are on the same connected component C of M Z w . The moment map φ | C on C has a unique minimum (by theconnectivity theorems). Suppose φ ( Q ) ≤ φ ( P ), then P is not the minimumfor φ | C on C , hence has a weight − w , contradicting to the first paragraph.Hence φ ( P ) < φ ( Q ). Suppose − w has multiplicity bigger than 1 at Q , thenin C , P is the minimum, Q has index at least 4; since C is symplectic, C contains at least an index 2 fixed point Q ′ with φ | C ( P ) < φ | C ( Q ′ ) < φ | C ( Q )(see Lemma 2.3), so − w is a weight at Q ′ , contradicting to the choice of Q . (cid:3) Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
First, since [ ω ] is integral, by Lemma 2.2, φ ( P i ) − φ ( P j ) ∈ Z for any i and j , and if w is a weight between P i and P j , then w | (cid:0) φ ( P i ) − φ ( P j ) (cid:1) . Let w > M . By Lemma 5.4,there is a weight w from some P i to some P j with i < j .By Lemma 3.3, H ( M ; Z ) = Z , hence there exists k ∈ R such that c ( M ) = k [ ω ]. By Lemma 5.2, c ( M ) = ( n + 1)[ ω ] if and only if i = 0and j = n , and w = φ ( P n ) − φ ( P ). Now assume c ( M ) = n [ ω ]. We mayassume that n > n = 1, then there are 2 fixed points, and itcorresponds to the last case. Since n ∤ ( n + 1), we have i = 0, j = n − w = φ ( P n − ) − φ ( P ), or i = 1, j = n and w = φ ( P n ) − φ ( P ), and both musthold by symmetry. Conversely, if w = φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) isthe largest weight, then we can only have i = 0 and j = n − i = 1 and j = n ; and c ( M ) = n [ ω ]. (cid:3) To prove Theorem 1.4, note that when dim( M ) = 4, dim H ( M ) is 2. Inthis case, we need to choose [ ω ] suitably so that c ( M ) is a multiple of [ ω ].We consider this case separately as follows, at the same time, we obtain theset of weights at the fixed points. Lemma 5.5.
Let the circle act on a compact -dimensional symplectic man-ifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integral classand M S = (cid:8) P , P , P ′ , P (cid:9) . Assume φ ( P ′ ) − φ ( P ) = φ ( P ) − φ ( P ) is thelargest weight of the S -action at the fixed points. Then (5.6) φ ( P ) − φ ( P ) = φ ( P ) − φ ( P ′ ) , and [ ω ] is primitive integral. Assume furthermore that [ ω | S ] is primitiveintegral, where S is an invariant gradient sphere (for any invariant metric)from P to P . Then the set of weights at any P i is (5.7) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i, − i , where i, j ∈ { , , ′ , } , and we use the convention ′ = 2 ; moreover, c ( M ) = 2[ ω ] .Proof. The equality φ ( P ′ ) − φ ( P ) = φ ( P ) − φ ( P ) implies (5.6). If φ ( P ) < φ ( P ′ ), then w ′ = φ ( P ′ ) − φ ( P ) can only divide φ ( P ′ ) − φ ( P ), hence is a weight between P and P ′ . If φ ( P ) = φ ( P ′ ), by Lemma 6.6,each of P and P ′ has a weight with P , hence w ′ is also a weight between P and P ′ (not between P and P ). For any invariant metric, if S ′ is thegradient sphere from P ′ to P , then Z S ′ [ ω ] = φ ( P ′ ) − φ ( P ) w ′ = 1 . This means that [ ω | S ′ ] is primitive integral, hence [ ω ] is primitive integral.By Lemma 6.6, P has a weight, say w , with P . By assumption, [ ω | S ] isprimitive integral, similar as above, we obtain that | w | = φ ( P ) − φ ( P ). Wedenote w = φ ( P ) − φ ( P ). By Lemma 6.6, P ′ has a weight, say w ′ , with P . By Lemma 2.6, w = w ′ mod w ′ , which, combining (5.6), impliesthat w ′ = w = φ ( P ) − φ ( P ′ ). We have proved (5.7) for P and P ′ .Similar to the last paragraph, w = φ ( P ) − φ ( P ) is a weight between P and P . Then we obtain (5.7) for P and P .Now consider the basis element e α and e α ′ of H S ( M ; Z ) in Proposi-tion 4.5. It is easy to check, by restricting the following equality to P , P and P ′ , and using (5.7) for P and P ′ , that e u + φ ( P ) t = e α + e α ′ . Restricting it to ordinary cohomology, we obtain [ ω ] = α + α ′ . Similarly,we can check that c S ( M ) = 2 e α + 2 e α ′ + Γ t. Restricting this to ordinary cohomology, we obtain c ( M ) = 2( α + α ′ ). Sothe claim c ( M ) = 2[ ω ] follows. (cid:3) Proof of Theorem 1.4.
If dim( M ) >
4, by Lemma 4.2, H ( M ; Z ) = Z . So c ( M ) = k [ ω ] for some k ∈ R . The proof of this case is the same as that ofTheorem 1.1.Now consider dim( M ) = 4. If c ( M ) = 2[ ω ], then the same proof as thatof Theorem 1.1 gives that φ ( P ′ ) − φ ( P ) = φ ( P ) − φ ( P ) is the largestweight. The converse is by Lemma 5.5. (cid:3) certain properties of the weights of the S -action In this section, we prove certain properties of the weights of the S -actionat the fixed points. We will use these properties in the next sections. Wesplit the section into two parts. In the first part, we prove certain propertiesfor the general case when the S -action has isolated fixed points; in thesecond part, we prove certain properties for the case when the S -action hasminimal or almost minimal isolated fixed points. AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 17
When the action has isolated fixed points.Lemma 6.1.
Let the circle act on a compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integral class and M S consists of isolated points. Assume for some P, Q ∈ M S with φ ( P ) < φ ( Q ) , a > is a weight from P to some (unknown) R , b > is a weight fromsome (unknown) R to Q , where φ ( R ) , φ ( R ) ∈ (cid:0) φ ( P ) , φ ( Q ) (cid:1) , and (6.2) a = − b + (cid:0) φ ( Q ) − φ ( P ) (cid:1) . Then (as numbers) (6.3) a = φ ( R ) − φ ( P ) and − b = φ ( R ) − φ ( Q ) , where R = R or R . Moreover, in any of the following cases, the multiplicity m of such a and − b satisfies ≤ m ≤ r , where r is the number of fixed points in φ − ( φ ( R )) . (1) If a ≥ (cid:0) φ ( Q ) − φ ( P ) (cid:1) , then the submanifold M Z a has P as theminimum. (2) If b ≥ (cid:0) φ ( Q ) − φ ( P ) (cid:1) , then the submanifold M Z b has Q as themaximum.Proof. Since a is a weight from P to R , and b is a weight from R to Q , byLemma 2.2, there exist k, l ∈ N such that ka = φ ( R ) − φ ( P ) , and lb = φ ( Q ) − φ ( R ) . Together with (6.2), we see that it is not possible that k ≥ l ≥ k = 1 or l = 1 or k = l = 1, in either case, (6.3) follows.Assume a ≥ (cid:0) φ ( Q ) − φ ( P ) (cid:1) . Since φ ( R ) ∈ (cid:0) φ ( P ) , φ ( Q ) (cid:1) , a only divides φ ( P ) − φ ( R ), hence a can only be a weight from P to a fixed point in φ − (cid:0) φ ( R ) (cid:1) . If a has multiplicity bigger than r , then in M Z a , P is theminimum, and there exists R ′ ∈ φ − (cid:0) φ ( R ) (cid:1) next to P such that there are twoweights a from P to R ′ , contradicting to the fact that M Z a is a symplecticmanifold (may see Lemma 2.3). Hence the multiplicity m of a satisfies1 ≤ m ≤ r . So does the multiplicity of b by (6.2). Similarly, the claimfollows in the case when b ≥ (cid:0) φ ( Q ) − φ ( P ) (cid:1) by looking at − φ . (cid:3) Remark 6.4.
In Lemma 6.1, in (6.3), we do not claim that a is a weightbetween P and R , and b is a weight between R and Q . By assumption, weknow at least one of them is true.We will use the following result by Jang and Tolman. Lemma 6.5. [5]
Let the circle act on a closed n -dimensional almost com-plex manifold with isolated fixed points. Let w be the smallest positive weightthat occurs at the fixed points on M . Then given any k ∈ { , , ..., n − } ,the number of times the weight − w occurs at fixed points of index k + 2 isequal to the number of times the weight + w occurs at fixed points of index k . Next, we show that if P is an index 2 fixed point, and − w is the negativeweight at P , then − w is a weight between P and a fixed point with a lowermoment map value. Lemma 6.6.
Let the circle act on a compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume M S consists of isolated points. If P ∈ M S is of index and − w is the negative weight at P , then there exists Q ∈ M S with φ ( Q ) < φ ( P ) such that w is a weight between Q and P .Similarly, if P ′ is of coindex and w ′ is the positive weight at P ′ , then thereexists Q ′ ∈ M S with φ ( Q ′ ) > φ ( P ′ ) such that w ′ is a weight between P ′ and Q ′ .Proof. Let P ∈ M S be of index 2 and − w be the negative weight at P .The connected component C of M Z w containing P has P as an index 2 fixedpoint. Since C is compact and symplectic, it contains a unique fixed point Q as the minimum of φ | C , so φ ( Q ) < φ ( P ). Since w is the smallest positiveweight on C , using Lemma 6.5 on C , we get that w is a positive weight at Q . If w = 1, we regard C = M . The other claim follows similarly by using − φ . (cid:3) A case with a stronger claim than that in Lemma 6.6 is as follows.
Lemma 6.7.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is a primitiveintegral class and M S consists of isolated points. If P ∈ M S is the uniqueindex point, then there is a weight w = φ ( P ) − φ ( Q ) between the index point Q ∈ M S and P . Similarly, if P ′ ∈ M S is the unique coindex point, then there is a weight w ′ = φ ( Q ′ ) − φ ( P ′ ) between P ′ and the index n point Q ′ ∈ M S .Proof. Let − w be the negative weight at P . By Lemma 6.6, there is aweight w from Q to P . We need to prove w = φ ( P ) − φ ( Q ). Let M − = { m ∈ M | φ ( m ) ≤ φ ( P ) } . By Lemma 2.3, M − only contains two fixed points P and Q . Since the other fixed points all have Morse index bigger than 2, therestriction map H ( M ; Z ) → H ( M − ; Z ) is an isomorphism. So [ ω | M − ] isprimitive integral. Let S be the invariant gradient sphere (for any invariantmetric) from P to Q . Then M − is homotopy equivalent to S , so φ ( P ) − φ ( Q ) w = Z S ω = 1 . Here, the first equality is due to that S acts on S − { P, Q } with order w .Hence w = φ ( P ) − φ ( Q ). We can similarly prove the other claim. (cid:3) Lemma 6.8.
Let the circle act on a connected compact symplectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integral class and M S consists of isolated points. Let P be the minimum of φ , and P = Q ∈ M S . Assume there is a weight a = φ ( Q ) − φ ( P ) from P to Q . Then for any AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 19 other weight a ′ at P , which could be equal to a , there exists R ∈ M S −{ P, Q } such that a ′ is a weight from P to R . A similar claim holds if P is themaximum.Proof. Any connected component C of M Z a ′ containing P (as the mini-mum) contains at least an index 2 fixed point (since it is symplectic). ByLemma 6.5, there exists an index 2 fixed point in C , say R , where R = P ,such that a ′ is a weight from P to R . We claim that R = Q . Suppose R = Q ,then by Lemma 2.2, a ′ | a , so index | C ( R ) ≥ index | C ( P ) + 4, a contradiction.We can similarly argue using − φ if P is the maximum. (cid:3) When the action has minimal or almost minimal isolated fixedpoints.
First, we have a consequence of Lemma 6.8.
Lemma 6.9.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . If (cid:8) w j = φ ( P j ) − φ ( P ) | < j ≤ n (cid:9) isthe set of weights at P , then w j is a weight from P to P j for each j = 0 .Similar claim holds for the set of weights at P n .Proof. Since [ ω ] is integral, by Lemma 2.2, φ ( P j ) − φ ( P i ) ∈ Z for any i and j ; and for any j , if w j is a weight from P to P l , then w j | (cid:0) φ ( P l ) − φ ( P ) (cid:1) .Hence if some w j > w n , then w j can only be a weight from P to P j .Now assume for a fixed i , each w k with i < k ≤ n is a weight from P to P k . Then by Lemma 6.8, w i is a weight from P to some P j with j ≤ i .Since among such j , w i only divides φ ( P i ) − φ ( P ), w i is a weight from P to P i . The claim follows by induction. (cid:3) We want to have a generalization of Lemma 6.9, Lemma 6.11 below. Forthis, we first prove a lemma similar to Lemma 6.8.
Lemma 6.10.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . For fixed i and j with i < j , assumethere is a weight w ij = φ ( P j ) − φ ( P i ) from P i to P j , and for any ≤ k < i ,there is precisely one weight w kl = φ ( P l ) − φ ( P k ) between P k and P l for any ≤ l ≤ j with k = l . Then for any other positive weight w at P i , whichcould be equal to w ij , there exists P m with m > i and m = j such that w isa weight from P i to P m . A similar claim holds if we consider − φ .Proof. Since by assumption the negative weights at any fixed point P k with0 ≤ k ≤ i are already matched with some positive weights at such fixedpoints, w is a weight from P i to some P m with m > i . So a connectedcomponent C of M Z w containing P i contains at least one P m with m > i .Applying Lemma 6.5 on C , we get that w is a weight from P i to some P m withindex | C ( P m ) = index | C ( P i ) + 2. Suppose m = j , then by Lemma 2.2, w | w ij ,so P j has index at least 4 in C . The assumption implies that the number of fixed points in C below P i determines the index of P i in C . Note that P k ∈ C with k < i if and only if w | (cid:0) φ ( P i ) − φ ( P k ) (cid:1) if and only if w | (cid:0) φ ( P j ) − φ ( P k ) (cid:1) .So if any P k with k < i is contained in C , then it contributes to the indexof P i and P j by the same number, hence index | C ( P j ) ≥ index | C ( P i ) + 4, acontradiction. Hence m = j . (cid:3) Using Lemma 6.10, similar to the proof of Lemma 6.9, we can prove thefollowing lemma.
Lemma 6.11.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . For fixed i and j , assume (cid:8) w ik = φ ( P k ) − φ ( P i ) | i < k ≤ j (cid:9) is a subset of positive weights at P i , where each w ik is aweight from P i to some (unknown) P l with i < l ≤ j ; and for any ≤ m < i ,there is precisely one weight w mr = φ ( P r ) − φ ( P m ) between P m and P r forany ≤ r ≤ j with r = m . Then w ik is a weight from P i to P k for each k with i < k ≤ j . When the S -action has minimal isolated fixed points, for the case whenthe largest weight is φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ), we will need thefollowing property. Lemma 6.12.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R , where n ≥ is odd. Assume [ ω ] is an integral class and M S = { P , P , · · · , P n } . Assume for a fixed i with ≤ i ≤ n − , the set of weights at each P k with k ≤ i and k ≥ n − i is (cid:8) w kj = φ ( P j ) − φ ( P k ) (cid:9) j = k,n − k ∪ (cid:8) w k,n − k = 12 (cid:0) φ ( P n − k ) − φ ( P k ) (cid:1)(cid:9) , and for each k < i and k > n − i , w kj is a weight between P k and P j foreach j = k . Then w ij is a weight between P i and P j for each j = i , and w n − i,j is a weight between P n − i and P j for each j = n − i .Proof. First assume i = 0. Clearly w n is a weight between P and P n .We claim that each w j in the subset of weights { w j } ≤ j ≤ n − is a weightbetween P and some P l with 1 ≤ l ≤ n −
1. If the claim holds, thenLemma 6.11 implies that each w j in { w j } ≤ j ≤ n − is a weight between P and P j . First, by Lemma 6.7, w is a weight between P and P . Clearly, if w j > (cid:0) φ ( P n ) − φ ( P ) (cid:1) = w n , then w j can only be a weight between P and P j . Now assume for some j with 1 < j ≤ n − w j < (cid:0) φ ( P n ) − φ ( P ) (cid:1) isthe first one (from bigger ones) such that it can only be a weight between P and P n , then w j > j > w j | (cid:0) φ ( P n ) − φ ( P ) (cid:1) . By symmetry, w n,n − j is a weight between P n and P . Since n ≥ j = n − j . Wemay assume j < n − j (since w j < w n ). By Lemma 2.6, { w k } k =0 = { w nk } k = n mod w j . Among the equalities, one correspondence is w ,n − j = w nj mod w j , whichimplies that w j | (cid:0) φ ( P n − j ) − φ ( P j ) (cid:1) which in turn implies that w j | w ,n − j AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 21 and w j | w nj . So the connected component C of M Z w j containing P and P n also contains P n − j and P j , so index | C ( P n ) ≥ index | C ( P ) + 4. Since w j is the smallest positive weight on C , by Lemma 6.5, there is an index 2 fixedpoint, say P m , in C such that there is a weight w j between P and P m ,contradicting to that w j has multiplicity 1 at P and that it can only be aweight between P and P n . Hence the claim of the lemma for i = 0 follows.Similarly, the claim of the lemma for n − n follows.For a fixed i with 0 < i ≤ n − , the assumption implies that for eachnegative weight at P i and for each positive weight w ij at P i with n − i
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R , where n > is even. Assume [ ω ] is an integral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) .Assume for a fixed i with ≤ i ≤ n , the set of weights at each P k with k ≤ i and k ≥ n − i is (cid:8) w kj = φ ( P j ) − φ ( P k ) (cid:9) j = k,n − k , and for each k < i and k > n − i , w kj is a weight between P k and P j for each j = k, n − k . Then w ij is a weight between P i and P j for each j = i, n − i ,and w n − i,j is a weight between P n − i and P j for each j = n − i, i . The case of minimal isolated fixed points — from thelargest weight to the integral cohomology ring of M I In this section, for the case when the action has minimal isolated fixedpoints, assuming φ ( P n ) − φ ( P ) is a weight of the S -action at a fixed point,we determine the integral cohomology ring of M , and show that it is iso-morphic to that of CP n .First, we obtain the sets of weights at P and P n . Lemma 7.1.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . Assume w n = φ ( P n ) − φ ( P ) is a weightof the S -action at a fixed point. Then the set of weights at P is (7.2) (cid:8) w i = φ ( P i ) − φ ( P ) (cid:9)
Lemma 7.5.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an inte-gral class and M S = (cid:8) P , P , · · · , P n (cid:9) . Then the following conditions areequivalent: (1) H ∗ ( M ; Z ) = Z [ x ] /x n +1 , where x = [ ω ] . (2) Λ − n = Q n − j =0 (cid:0) φ ( P j ) − φ ( P n ) (cid:1) .Proof. By Lemma 2.1, we have Q n − j =0 (cid:0)e u + φ ( P j ) t (cid:1) | P j = 0 for all j < n . ThenCorollary 3.8 gives(7.6) n − Y j =0 (cid:0)e u + φ ( P j ) t (cid:1) = a n e α n , with a n ∈ Z . Restricting (7.6) to P n and using Proposition 3.7, we get a n Λ − n = n − Y j =0 (cid:0) φ ( P j ) − φ ( P n ) (cid:1) . Restricting (7.6) to ordinary cohomology, we get the generator of H n ( M ; Z ): α n = 1 a n [ ω ] n . Hence (2) holds if and only if a n = 1, and if and only if [ ω ] n is a generator of H n ( M ; Z ). Together with the facts H i ( M ; Z ) = Z and H i − ( M ; Z ) = 0for all 0 ≤ i ≤ n in Lemma 3.3, the claim follows. (cid:3) Remark 7.7.
The conditions in Lemma 7.5 are also equivalent to Λ − i = Q i − j =0 (cid:0) φ ( P j ) − φ ( P i ) (cid:1) for ∀ i . The proof is by considering the class Q i − j =0 (cid:0)e u + φ ( P j ) t (cid:1) similarly as above.Using Lemmas 7.1 and 7.5, we obtain the ring H ∗ ( M ; Z ): Proposition 7.8.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = (cid:8) P , P , · · · , P n (cid:9) . Assume w n = φ ( P n ) − φ ( P ) is a weightof the S -action at a fixed point. Then the integral cohomology ring of M is H ∗ ( M ; Z ) = Z [ x ] /x n +1 , where x = [ ω ] . The case of minimal isolated fixed points — from thelargest weight to all the weights and to the total Chernclass of M I In this section, for the case when the action has minimal isolated fixedpoints, assuming φ ( P n ) − φ ( P ) is a weight of the S -action at a fixed point,we determine the sets of weights of the action at all the fixed points, showingthat they are exactly as those in Example 3.1; moreover, we determine thetotal Chern class of M , showing that it is isomorphic to that of CP n . Forthe proof of these, we use Lemma 7.1, we do not use Proposition 7.8. First, we find the sets of weights at all the fixed points.
Proposition 8.1.
Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = (cid:8) P , P , · · · , P n (cid:9) . Assume w n = φ ( P n ) − φ ( P ) is a weightof the S -action at a fixed point. Then the set of weights at any P i is (8.2) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i . That is, the sets of weights at the fixed points are isomorphic to those of thestandard circle action on CP n as in Example 3.1.Proof. First of all, by Lemma 7.1, the claim holds for P and P n ; moreover, w i is a weight between P and P i for each i = 0.We use induction. Assume for a fixed i , the claim (8.2) holds for all P k ’swith k < i , and for each such k , there is a weight w kj = φ ( P j ) − φ ( P k )between P k and P j for any j = k . We will prove all these hold for k = i .First, the inductive hypotheses imply that the set of negative weights at P i is(8.3) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) ≤ j P i and P n . By Lemma 2.6,(8.4) { weights at P i } = { weights at P n } mod w in . First, in (8.4), we have the correspondence(8.5) w ij = w nj + w in , ∀ j with 0 ≤ j < i. Next, the inductive hypothesis and Lemma 6.8 imply that each positiveweight at P i other than w in is a weight from P i to some P k with i < k < n and hence is less than w in . For each weight w nj at P n with i < j < n , sincethe negative weights at P i already occured in (8.5), there can only exist apositive weight at P i , denoted w ij , such that there is the correspondence in(8.4): w ij = w nj + w in , ∀ j with i < j < n, which yields the subset of positive weights at P i :(8.6) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) i Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = (cid:8) P , P , · · · , P n (cid:9) . Assume w n = φ ( P n ) − φ ( P ) is a AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 25 weight of the S -action at a fixed point. Then the total Chern class of M is c ( M ) = (1 + [ ω ]) n +1 , hence is isomorphic to c ( CP n ) .Proof. By the injectivity theorem ([7], [18]), the restriction map H ∗ S ( M ; Z ) → H ∗ S ( M S ; Z )is injective for any Hamiltonian S -manifold M with isolated fixed points.Consider the class α = Q ≤ j ≤ n (cid:0) e u + φ ( P j ) t (cid:1) . By Proposition 8.1, we cancheck that α | P i = c S ( M ) | P i , ∀ ≤ i ≤ n. Hence c S ( M ) = Q ≤ j ≤ n (cid:0) e u + φ ( P j ) t (cid:1) . Restricting it to ordinary coho-mology, we obtain our claim. (cid:3) The case of minimal isolated fixed points — from thelargest weight to the integral cohomology ring of M ΠIn this section, for the case when the action has minimal isolated fixedpoints, assuming φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight ofthe S -action at the fixed points, we determine the integral cohomology ringof M , and show that it is isomorphic to that of e G ( R n +2 ) with n ≥ P and P n − , similarly, at P n and P . Lemma 9.1. Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action at the fixed points. Then (9.2) φ ( P n ) − φ ( P n − ) = φ ( P ) − φ ( P ) , [ ω ] is primitive integral, and for i = 0 , , n − , n , the set of weights at P i is (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i,n − i ∪ (cid:8) w i,n − i = 12 (cid:0) φ ( P n − i ) − φ ( P i ) (cid:1)(cid:9) . Proof. Since φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ), (9.2) holds.Let w ,n − = φ ( P n − ) − φ ( P ). Since w ,n − only divides φ ( P n − ) − φ ( P ), w ,n − is a weight from P to P n − . It is easy to see that w ,n − is the strictlylargest weight between P and P n − . So there is a Z w ,n − -isotropy sphere S with poles P and P n − . Since Z S [ ω ] = φ ( P n − ) − φ ( P ) w ,n − = 1 , [ ω | S ] is primitive integral. Hence [ ω ] is primitive integral on M .Let { ¯ w i | i = 0 } be the set of weights at P and { ¯ w n − ,i | i = n − } theset of weights at P n − . Let¯ w ,n − = w ,n − and ¯ w n − , = w n − , = − w ,n − . By Lemma 6.7, we may let¯ w = w = φ ( P ) − φ ( P ) and ¯ w n − ,n = w n − ,n = φ ( P n ) − φ ( P n − ) , where w is a weight between P and P , and w n − ,n is a weight between P n − and P n .By Lemma 2.6,(9.3) { ¯ w i | i = 0 } = { ¯ w n − ,i | i = n − } mod w ,n − . For the positive weight ¯ w n − ,n at P n − , in (9.3), the correspondence canonly be(9.4) ¯ w = ¯ w n − ,n + 0 w ,n − . (which is the same as (9.2).) For each positive weight ¯ w i at P , with i = 0 , , n − 1, by (9.3) and (9.4), there exists a negative weight at P n − ,namely ¯ w n − ,i , with i = 0 , n − , n , such that(9.5) ¯ w i = ¯ w n − ,i + w ,n − . By Lemma 6.8, each ¯ w i in (9.5) is a weight between P and some P m with m = 0 , , n − 1. Each − ¯ w n − ,i in (9.5) can only be a weight between some P l and P n − with 1 ≤ l ≤ n − 2. We will discuss (9.5) in the following twopossibilities. Case (1). Assume 2 ≤ m ≤ n − 2. Then by Lemma 6.1, there is a k with1 ≤ k ≤ n − w i = φ ( P k ) − φ ( P ) and ¯ w n − ,i = φ ( P k ) − φ ( P n − ) . We rename ¯ w i as w k , and ¯ w n − ,i as w n − ,k , i.e.,(9.6) w k = φ ( P k ) − φ ( P ) and w n − ,k = φ ( P k ) − φ ( P n − ) , where 1 ≤ k ≤ n − . Case (2). Assume m = n , i.e., ¯ w i in (9.5) is a weight between P and P n .By assumption, ¯ w i < φ ( P n ) − φ ( P ). By Lemma 2.2, ¯ w i | (cid:0) φ ( P n ) − φ ( P ) (cid:1) ,so a ¯ w i = φ ( P n ) − φ ( P ) , for some a with 2 ≤ a ∈ N . For the corresponding ¯ w n − ,i in (9.5), we have b ¯ w n − ,i = φ ( P l ) − φ ( P n − ) , with 1 ≤ l ≤ n − b ∈ N . Case (2 a ). Assume b = 1. We rename ¯ w n − ,i as w n − ,l , and the correspond-ing ¯ w i as w l , i.e., we denote w l = φ ( P l ) − φ ( P ) and w n − ,l = φ ( P l ) − φ ( P n − ) with 1 ≤ l ≤ n − , where w l is a weight between P and P n , and − w n − ,l is a weight between P l and P n − . Case (2 b ). Assume b ≥ 2. Since we also have a ≥ 2, for (9.5) to hold, theonly possibility is that a = 2, b = 2, and l = 1, i.e.,¯ w i = 12 (cid:0) φ ( P n ) − φ ( P ) (cid:1) and ¯ w n − ,i = 12 (cid:0) φ ( P ) − φ ( P n − ) (cid:1) . AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 27 We rename ¯ w i as ¯ w n and ¯ w n − ,i as ¯ w n − , , i.e., we denote(9.7) ¯ w n = 12 (cid:0) φ ( P n ) − φ ( P ) (cid:1) and ¯ w n − , = 12 (cid:0) φ ( P ) − φ ( P n − ) (cid:1) . Moreover, if it appears, each of ¯ w n and ¯ w n − , has multiplicity 1. In fact,since ¯ w n can only divide φ ( P n ) − φ ( P ), if ¯ w n has multiplicity bigger than1, then there exists an isotropy submanifold of dimension at least 4 withonly 2 fixed points P and P n , a contradiction.Claim 1: For 2 ≤ j ≤ n − 2, if w n − ,j = φ ( P j ) − φ ( P n − ) = ¯ w n − , is aweight at P n − , then it has multiplicity 1.Claim 2: w n − , = φ ( P ) − φ ( P n − ) cannot have multiplicity bigger than1, and w n − , and ¯ w n − , cannot appear at the same time.Claim 3: Case (2 b ) must occur.Proof of Claim 1: If | w n − ,j | ≥ w ,n − , then w n − ,j can only be a weightbetween P n − and P j . If taking into account (9.2), we see that M Z | wn − ,j | has P n − as the maximum and P j the next fixed point to P n − , hence w n − ,j cannot have multiplicity bigger than 1 by Lemma 2.3 (applied to − φ on M Z | wn − ,j | ). If | w n − ,j | < w ,n − , then by Case (1) and Case (2 a ), w n = w j > w ,n − , so w j can only be a weight between P and P j , byLemma 6.1, w n − ,j (and w j ) has multiplicity 1.Proof of Claim 2: Suppose w n − , appears with multiplicity bigger than 1.By Case (1) and Case (2 a ), w n − , is a weight between P n − and P . Then M Z | wn − , | of dimension at least 4 either contains only two fixed points P and P n − , or it contains 4 fixed points P , P , P n − and P n . The first case is notpossible. In the latter case, we have w n − , | w , hence w n − , | w ,n − , so in M Z | wn − , | , w ,n − is a weight between P and P n − , then P n − has index atleast 6 ( P has index 2), again not possible. Now suppose w n − , and ¯ w n − , both appear. By Case (2 b ), ¯ w n − , is (also) a weight between P n − and P .The same argument, by considering M Z | ¯ wn − , | , gives a contradiction.Proof of Claim 3: Suppose Case (2 b ) does not occur. Since there are n − P n − , by Claims 1 and 2, the set ofweights at P n − is (cid:8) w n − ,j = φ ( P j ) − φ ( P n − ) (cid:9) j = n − . Similarly, by symmetry (or by using − φ ), the set of weights at P is (cid:8) w j = φ ( P j ) − φ ( P ) (cid:9) j =1 . Then Γ − Γ n − = ( n +1) (cid:0) φ ( P n − ) − φ ( P ) (cid:1) . This contradicts to Theorem 1.1and Lemma 5.1.By Claims 2 and 3, w n − , = φ ( P ) − φ ( P n − ) does not appear. We mayrename ¯ w n as w n and ¯ w n − , as w n − , . Then the claims mean that theset of weights at P n − is as claimed in the lemma, and correspondingly, theset of weights at P is as claimed. Similarly, the sets of weights at P n andat P are as claimed. (cid:3) Lemma 9.8. Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is a primitiveintegral class and M S = { P , P , · · · , P n } . Then the following two condi-tions are equivalent: (1) H ∗ ( M ; Z ) = Z ( x, y ) / (cid:0) x ( n +1) − y, y (cid:1) , where n ≥ is odd, x = [ ω ] ,and deg( y ) = n + 1 . (2) Λ − n = Q n − j =0 (cid:0) φ ( P j ) − φ ( P n ) (cid:1) .Proof. We proceed similarly as in the proof of Proposition 7.8. We get that(2) holds if and only if [ ω ] n is a generator of H n ( M ; Z ). So (1) implies (2).Now we prove (2) implies (1). Since [ ω ] ∈ H ( M ; Z ) is a generator, by thefacts H i ( M ; Z ) = Z and H i − ( M ; Z ) = 0 for all 0 ≤ i ≤ n (Lemma 3.3),and Poincar´e duality, the generators of the even degree integral cohomologygroups of M must be as follows:1 , [ ω ] , [ ω ] , · · · , [ ω ] n − , 12 [ ω ] n +12 , · · · , 12 [ ω ] n − , 12 [ ω ] n . So n ≥ x = [ ω ] and y = [ ω ] n +12 . We get the ring H ∗ ( M ; Z ) as claimed. (cid:3) Remark 9.9. The conditions in Lemma 9.8 are also equivalent to: Λ − i = Q i − j =0 (cid:0) φ ( P j ) − φ ( P i ) (cid:1) if i ≤ n − , and Λ − i = Q i − j =0 (cid:0) φ ( P j ) − φ ( P i ) (cid:1) if i ≥ n +12 . The proof is by considering the classes Q i − j =0 (cid:0)e u + φ ( P j ) t (cid:1) for ∀ i .Corollary 3.8 gives i − Y j =0 (cid:0)e u + φ ( P j ) t (cid:1) = a i e α i , where a i ∈ Z . Restricting this to ordinary cohomology and to P i , we can obtain what weneed.Now with Lemmas 9.1 and 9.8, we obtain the ring H ∗ ( M ; Z ): Proposition 9.10. Let the circle act on a compact n -dimensional symplec-tic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an integralclass and M S = { P , P , · · · , P n } . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action at the fixed points. Then the integralcohomology ring of M is H ∗ ( M ; Z ) = Z ( x, y ) / (cid:0) x ( n +1) − y, y (cid:1) , where n ≥ is odd, x = [ ω ] , and deg( y ) = n + 1 . The case of minimal isolated fixed points — from thelargest weight to all the weights and to the total Chernclass of M ΠIn this section, for the case when the action has minimal isolated fixedpoints, assuming φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 29 action at the fixed points, we determine the sets of weights at all the fixedpoints, showing that they are exactly as those in Example 3.2; moreover,we find the total Chern class of M , showing that it is isomorphic to that of e G ( R n +2 ) with n ≥ n ≥ Proposition 10.1. Let the circle act on a compact n -dimensional sym-plectic manifold ( M, ω ) with moment map φ : M → R . Assume M S = (cid:8) P , P , · · · , P n (cid:9) and [ ω ] is an integral class. Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action at the fixed points. Thenthe set of weights at any P i is (10.2) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i,n − i ∪ (cid:8) w i,n − i = 12 (cid:0) φ ( P n − i ) − φ ( P i ) (cid:1)(cid:9) . Moreover, (10.3) φ ( P k ) − φ ( P ) = φ ( P n ) − φ ( P n − k ) , ∀ ≤ k ≤ n − . Hence the sets of weights are isomorphic to those of the standard circleaction on e G ( R n +2 ) as in Example 3.2.Proof. First, by Lemma 9.1, the claim (10.2) holds for P , P n − , P n and P ,and (10.3) holds for k = 1.Note that by Proposition 9.10, n ≥ n + 1 ≥ i = 0 , , n − , n , each w ij is a weightbetween P i and P j with j = i . We use induction. Fix i with 2 ≤ i ≤ n − .Assume that for each k with 0 ≤ k < i , (10.2) holds for P k and P n − k , w kj isa weight between P k and P j for each j = k , and w n − k,j is a weight between P n − k and P j for each j = n − k , moreover, (10.3) holds for all 1 ≤ k ≤ i − k by i . The inductive hypothesesimplies that the set of negative weights at P i is(10.4) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) ≤ j ≤ i − . Since w ni is a weight between P n and P i , by Lemma 2.6,(10.5) { weights at P i } = { weights at P n } mod w in = − w ni . Note that in (10.5), we first have the equalities(10.6) w ij = w nj + w in , ∀ j with 1 ≤ j ≤ i − . The remaining negative weight w i at P i , by symmetry, we may assume ithas absolute value less than w in . All the remaining weights at P n (otherthan w ni ) also have absolute value less than w in . Note that any positiveweight at P i other than w in can only be a weight from P i to some P j with i < j < n (since any negative weight at any fixed point below P i is already paired with a positive weight at a fixed point below P i ), so is less than w in .Considering all these facts, the remaining correspondences in (10.5) are(10.7) w i = w n,n − i + 0 w in , and(10.8) w ij = w nj + w in if i + 1 ≤ j ≤ n − j = n − i , and w i,n − i = w n + w in , where for the known w nj with i + 1 ≤ j ≤ n − j = n − i , we denotethe corresponding weight at P i as w ij , and for the known w n , we denotethe corresponding weight at P i as w i,n − i . Equation (10.7) gives (10.3) for k = i . From (10.8), we get the subset of positive weights at P i :(10.9) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) i +1 ≤ j ≤ n − ,j = n − i ∪ (cid:8) w i,n − i = 12 (cid:0) φ ( P n − i ) − φ ( P i ) (cid:1)(cid:9) . Here, we used (10.3) for k = i to get w i,n − i . Equations (10.4) and (10.9),and w in = − w ni give the set of weights, claim (10.2) for P i . Similarly, claim(10.2) for P n − i follows. Finally, by Lemma 6.12, each w ij is a weight between P i and P j , and each w n − i,j is a weight between P n − i and P j . (cid:3) Using Proposition 10.1, we obtain c ( M ): Proposition 10.10. Let the circle act on a compact n -dimensional sym-plectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is anintegral class and M S = (cid:8) P , P , · · · , P n (cid:9) . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action at the fixed points. Thenthe total Chern class of M is c ( M ) = (1+[ ω ]) n +2 ω ] , hence is isomorphic to c (cid:0) e G ( R n +2 ) (cid:1) .Proof. Similar to the proof of Proposition 8.7, the restriction map H ∗ S ( M ; Z ) → H ∗ S ( M S ; Z )is injective. Consider the class α = (cid:0) e u + (cid:0) φ ( P n ) + φ ( P ) (cid:1) t (cid:1) Q ≤ j ≤ n (cid:0) e u + φ ( P j ) t (cid:1) (cid:0)e u + φ ( P ) t (cid:1) + (cid:0)e u + φ ( P n ) t (cid:1) . By Proposition 10.1, (cid:0) φ ( P ) − φ ( P i ) (cid:1) + (cid:0) φ ( P n ) − φ ( P i ) (cid:1) = φ ( P n − i ) − φ ( P i )for all 0 ≤ i ≤ n . Using this, and Proposition 10.1, we can check that α | P i = c S ( M ) | P i , ∀ ≤ i ≤ n. Hence c S ( M ) = α . Restricting this to ordinary cohomology, we obtain ourclaim. (cid:3) AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 31 The case of almost minimal isolated fixed points—from thelargest weight to all the weights In this section, we consider the case when the action has almost minimalisolated fixed points. Assuming φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is thelargest weight of the S -action at the fixed points, we find the sets of weightsat all the fixed points, showing that they are exactly as those in Example 4.1.First, we find the sets of weights at P , P , P n − and P n . Lemma 11.1. Let the circle act on a compact n -dimensional symplec-tic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an in-tegral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action atthe fixed points. Then (11.2) φ ( P ) − φ ( P ) = φ ( P n ) − φ ( P n − ) , [ ω ] is primitive integral, and for i = 0 , , n − , n , the set of weights at P i is (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i,n − i . Proof. Lemma 5.5 gives the claims for n = 2. So we assume now n > φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) implies (11.2). Let w ,n − = φ ( P n − ) − φ ( P ). Since w ,n − only divides φ ( P n − ) − φ ( P ), w ,n − is aweight between P and P n − .For the following steps, we proceed similarly as in the proof of Lemma 9.1.First we get that [ ω ] is primitive integral. Since n > 2, there is a unique index2 fixed point P and a unique coindex 2 fixed point P n − ; by Lemma 6.7, w = φ ( P ) − φ ( P ) is a weight between P and P , and w n − ,n = φ ( P n ) − φ ( P n − ) is a weight between P n − and P n . Then we use { weights at P } = { weights at P n − } mod w ,n − to get the possible cases: Case (1), Case (2 a ) and Case (2 b ) as in theproof of Lemma 9.1, and we have 3 claims:Claim 1: For j ∈ (cid:8) , · · · , n , ( n ) ′ , · · · , n − (cid:9) , if w n − ,j = φ ( P j ) − φ ( P n − ) =¯ w n − , appears as a weight at P n − , then it has multiplicity 1.Claim 2: w n − , = φ ( P ) − φ ( P n − ) cannot appear with multiplicity biggerthan 1, and w n − , and ¯ w n − , = (cid:0) φ ( P ) − φ ( P n − ) (cid:1) cannot appear at thesame time.Claim 3: Neither w n − , nor ¯ w n − , can appear.The proofs of Claims 1 and 2 are the same as in the proof of Lemma 9.1.Now we prove Claim 3. First assume w n − , appears. Then by Claim 2, w n − , has multiplicity 1 and ¯ w n − , does not appear. Since there are n − P n − , and there are n number of fixed pointsbelow P n − , by Claim 1, there exists one k ∈ (cid:8) , · · · , n , ( n ) ′ , · · · , n − (cid:9) such that w n − ,k = φ ( P k ) − φ ( P n − ) is not a weight at P n − . By symmetry, w ,n − k = φ ( P n − k ) − φ ( P ) is not a (positive) weight at P . Then we canwrite down the sets of weights at P n − and P , and compute(11.3) Γ − Γ n − = n (cid:0) φ ( P n − ) − φ ( P ) (cid:1) , which contradicts to Lemma 5.1 and Theorem 1.4. Hence w n − , does notappear. Next, assume ¯ w n − , appears, i.e., Case (2 b ) occurs. Then w n − , does not appear. Similar to the above, we can write down the sets of weightsat P n − and P as follows. The set of weights at P n − is (cid:8) w n − ,j = φ ( P j ) − φ ( P n − ) (cid:9) j =1 , n − , k ∪ ¯ w n − , for some k ∈ (cid:8) , · · · , n , ( n ) ′ , · · · , n − (cid:9) , and by symmetry the set of weightsat P is (cid:8) w j = φ ( P j ) − φ ( P ) (cid:9) j =1 , n − , n − k ∪ ¯ w ,n − , where ¯ w ,n − = − ¯ w n − , . Here we use the convention n − n = ( n ) ′ . First assume φ ( P n ) < φ ( P ( n ) ′ ).Then we always have (11.3) (no matter what k is), which contradicts toLemma 5.1 and Theorem 1.4. Hence for the cases φ ( P n ) < φ ( P ( n ) ′ ), ¯ w n − , never occurs, i.e., Case (2 b ) never occurs, and by the 3 claims, the set ofweights at P n − is as claimed in the lemma. Since the cases φ ( P n ) < φ ( P ( n ) ′ )are generic, the set of weights at P n − is as claimed. Correspondingly, theset of weights at P is as claimed. Similarly, the sets of weights at P and P n are as claimed. (cid:3) Next, we find the sets of weights at all the fixed points. Proposition 11.4. Let the circle act on a compact n -dimensional sym-plectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is anintegral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action atthe fixed points. Then the set of weights at any P i is (11.5) (cid:8) w ij = φ ( P j ) − φ ( P i ) (cid:9) j = i,n − i . Moreover, (11.6) φ ( P k ) − φ ( P ) = φ ( P n ) − φ ( P n − k ) , for any k with ≤ k ≤ n . Hence the sets of weights at the fixed points are isomorphic to those of thestandard circle action on e G ( R n +2 ) , with n ≥ even, as in Example 4.1.Proof. Lemma 5.5 gives the results for n = 2. For n > 2, Lemma 11.1 gives(11.5) for P , P n − , P n and P , and (11.6) for k = 1. By Lemma 6.13, forany i = 0 , , n − , n , w ij is a weight between P i and P j for each j = i, n − i .The proof is similar to that of Proposition 10.1. We use induction. Fix i with 2 ≤ i ≤ n . Assume that for each k with 0 ≤ k < i , (11.5) holds for P k and P n − k , w kj is a weight between P k and P j for each j = k, n − k , and w n − k,j is a weight between P n − k and P j for each j = n − k, k , moreover,(11.6) holds for all 1 ≤ k ≤ i − 1. We need to prove all these hold if we AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 33 replace k by i . Proceed similarly as in the proof of Proposition 10.1, anduse Lemma 6.13. (cid:3) The case of almost minimal isolated fixed points — fromweights to the integral cohomology ring and total Chernclass of M In this section, for the case when the S -action has almost minimal iso-lated fixed points, using the weights of the S -action at the fixed points,we determine the integral cohomology ring and the total Chern class of themanifold. Lemma 12.1. Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is a primitiveintegral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Let x =[ ω ] . Then the following conditions are equivalent: (1) There exist classes y, z ∈ H n ( M ; Z ) such that the generators of H i (cid:0) M ; Z (cid:1) with ≤ i ≤ n are , x , · · · , x n − , y and z , where y + z = x n . (2) For any i ∈ (cid:8) , , , · · · , n , ( n ) ′ (cid:9) , (12.2) Λ − i = Y index( P j ) < index( P i ) (cid:0) φ ( P j ) − φ ( P i ) (cid:1) . (3) For any i ∈ (cid:8) n , ( n ) ′ , n + 1 , · · · , n − , n (cid:9) , (12.3) Λ + i = Y index( P j ) > index( P i ) (cid:0) φ ( P j ) − φ ( P i ) (cid:1) . Proof. Let us prove (1) and (2) are equivalent. Let { e α i } i ∈{ , , ··· , n , ( n ) ′ , ··· ,n } be the basis of H ∗ S ( M ; Z ) as an H ∗ ( CP ∞ ; Z )-module as in Proposition 4.5.Let { α i } i ∈{ , , ··· , n , ( n ) ′ , ··· ,n } be the restriction of the above basis to ordinarycohomology.First, consider 0 ≤ i < n . Since Q j
Lemma 12.6. Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Then for any i ∈ (cid:8) ( n ) ′ , n + 1 , · · · , n − (cid:9) , (12.7) e α i | P i +1 = − Λ i +1 Λ + i Y j>i +1 φ ( P j ) − φ ( P i ) φ ( P j ) − φ ( P i +1 ) t i . Here, the e α i ’s are the classes in Proposition 4.5, and we use the convention ( n ) ′ + 1 = n + 1 and t ( n ) ′ = t n .Proof. For any i ∈ (cid:8) ( n ) ′ , n + 1 , · · · , n − (cid:9) , note thatdeg (cid:0)e α i · Y j>i +1 (cid:0)e u + φ ( P j ) t (cid:1)(cid:1) < n. Using Theorem 2.5 to integrate this class on M , we get0 = e α i | P i · Q j>i +1 ( φ ( P j ) − φ ( P i ))Λ i + e α i | P i +1 · Q j>i +1 ( φ ( P j ) − φ ( P i +1 ))Λ i +1 . Solving this, we get (12.7). (cid:3) Lemma 12.8. Let the circle act on a compact n -dimensional symplecticmanifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is a primitive AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 35 integral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Thenfor any i ∈ (cid:8) ( n ) ′ , n + 1 , · · · , n − (cid:9) , [ ω ] α i = α i +1 if and only if (12.9) e α i | P i +1 = Λ − i +1 φ ( P i ) − φ ( P i +1 ) t i . Here, the e α i ’s and α i ’s are the classes in Proposition 4.5, and we use theconvention ( n ) ′ + 1 = n + 1 and t ( n ) ′ = t n .Proof. For any i ∈ (cid:8) ( n ) ′ , n + 1 , · · · , n − (cid:9) , note thatdeg (cid:0)(cid:0)e u + φ ( P i ) t (cid:1)e α i (cid:1) = deg( e α i +1 ) , and (cid:0)(cid:0)e u + φ ( P i ) t (cid:1)e α i (cid:1) | P j = 0 , ∀ P j with φ ( P j ) < φ ( P i +1 ) . By Corollary 4.6, (cid:0)e u + φ ( P i ) t (cid:1)e α i = a i +1 e α i +1 , with a i +1 ∈ Z . Restricting it respectively to ordinary cohomology and to P i +1 , we obtainthe claim. (cid:3) Proposition 12.10. Let the circle act on a compact n -dimensional sym-plectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is aprimitive integral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) .Let x = [ ω ] . Then the following conditions are equivalent: (1) There exist classes y, z ∈ H n ( M ; Z ) such that the generators of H ∗ ( M ; Z ) are , x , · · · , x n − , y , z , xy = xz = x n +1 , x y = x z = x n +2 , · · · , x n y = x n z = x n , where x n = y + z . (2) Lemma 12.1 (2) holds, and for any i with n + 1 ≤ i ≤ n , (12.11) Λ − i = Q index( P j ) < index( P i ) (cid:0) φ ( P j ) − φ ( P i ) (cid:1)(cid:0) φ ( P n ) − φ ( P i ) (cid:1) + (cid:0) φ ( P ( n ) ′ ) − φ ( P i ) (cid:1) . (3) Lemma 12.1 (3) holds, and for any i with ≤ i ≤ n − , (12.12) Λ + i = Q index( P j ) > index( P i ) (cid:0) φ ( P j ) − φ ( P i ) (cid:1)(cid:0) φ ( P n ) − φ ( P i ) (cid:1) + (cid:0) φ ( P ( n ) ′ ) − φ ( P i ) (cid:1) . Proof. Let { e α i } i ∈{ , , ··· , n , ( n ) ′ , ··· ,n } be the basis of H ∗ S ( M ; Z ) as an H ∗ ( CP ∞ ; Z )-module as in Proposition 4.5. Let { α i } i ∈{ , , ··· , n , ( n ) ′ , ··· ,n } be the restrictionof the above basis to ordinary cohomology.First, consider i = n + 1. Since deg (cid:16)Q j ≤ n (cid:0)e u + φ ( P j ) t (cid:1)(cid:17) = n + 2, and Y j ≤ n (cid:0)e u + φ ( P j ) t (cid:1) | P j = 0 , ∀ j ≤ n , by Corollary 4.6,(12.13) Y j ≤ n (cid:0)e u + φ ( P j ) t (cid:1) = a ( n ) ′ t e α ( n ) ′ + b n +1 e α n +1 , where a ( n ) ′ , b n +1 ∈ Z . Restricting (12.13) to ordinary cohomology, we get[ ω ] n +1 = b n +1 α n +1 . Restricting (12.13) to P ( n ) ′ , we get Y j ≤ n (cid:0) φ ( P j ) − φ ( P ( n ) ′ ) (cid:1) = a ( n ) ′ Λ − ( n ) ′ . Restricting (12.13) to P n +1 , we get Y j ≤ n (cid:0) φ ( P j ) − φ ( P n +1 ) (cid:1) = a ( n ) ′ e α ( n ) ′ | P n + b n +1 Λ − n +1 . Similarly, for any i ≥ n + 2, we can write(12.14) Y index( P j ) < i − ( e u + φ ( P j ) t ) = a i − t e α i − + b i e α i , where a i − , b i ∈ Z . Restricting (12.14) to ordinary cohomology, we get[ ω ] i = b i α i . Restricting (12.14) to P i − , we get Y index( P j ) < i − ( φ ( P j ) − φ ( P i − )) = a i − Λ − i − . Restricting (12.14) to P i , we get Y index( P j ) < i − ( φ ( P j ) − φ ( P i )) = a i − e α i − | P i + b i Λ − i . Let y = α n and z = α ( n ) ′ . We argue below that (1) and (2) are equivalent.Assume (1) holds. Then Lemma 12.1 (2) holds, b i = 2 for all n +1 ≤ i ≤ n ,and (12.9) holds. Using (12.2), we can get a ( n ) ′ , then get Λ − n +1 . Inductively,we can get a i − , and then get Λ − i for all i ≥ n + 2.Conversely, assume (2) holds. Then by Lemma 12.1, we have the genera-tors of H i ( M ; Z ) for 0 ≤ i ≤ n as in (1) and x n = y + z , and Lemma 12.1(3) holds. By (12.7), we get that (12.9) holds. Then Lemma 12.8 impliesthat x k z = α n + k for all 1 ≤ k ≤ n . Moreover, from the above equalities, wecan solve a ( n ) ′ and all a i − , and then get b i = 2 for all n + 1 ≤ i ≤ n . Notethat x k y + x k z = x n + k for all 1 ≤ k ≤ n . Hence x k y = x k z = x n + k = α n + k are generators of H n +2 k ( M ; Z ) for all 1 ≤ k ≤ n .Using − φ as a Morse function, we get the Poincar´e dual of the classes in(1), they form a new basis (satisfying the same relations). The existence ofthe new basis is equivalent to (1), and by the same arguments as above, isequivalent to (3). (cid:3) AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 37 Lemma 12.15. Let the circle act on a n -dimensional compact symplec-tic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is an in-tegral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action atthe fixed points. Then Proposition 12.10 (1) holds. Moreover, x n y = x n z = yz, y = z = 0 , when dim( M ) = 4 m with m odd , and x n y = x n z = y = z , yz = 0 , when dim( M ) = 4 m with m even . Proof. By Proposition 11.4, Proposition 12.10 (2) (and (3)) holds. HenceProposition 12.10 (1) holds. It remains to determine the relations of y , z and yz with the top degree generators. First, since x n = y + z , we have x n y = y + yz and x n z = z + yz . Since x n y = x n z , we have y = z .Recall that y = α n and z = α ( n ) ′ , where α n and α ( n ) ′ are respectivelythe restrictions of e α n and e α ( n ) ′ to ordinary cohomology. For any k ≥ n + 1,note that (cid:0)e α n · Y j ∈{ n , n +1 , ··· , k − } (cid:0)e u + φ ( P j ) t (cid:1)(cid:1) | P j = 0 , ∀ P j with φ ( P j ) < φ ( P k ) . By Corollary 4.6, e α n · Y j ∈{ n , n +1 , ··· , k − } (cid:0)e u + φ ( P j ) t (cid:1) = a k e α k , with a k ∈ Z . By restricting it to ordinary cohomology, we see that a k = 1. Then restrict-ing it to P k , we get e α n | P k = Λ − k Q j ∈{ n , n +1 , ··· , k − } (cid:0) φ ( P j ) − φ ( P k ) (cid:1) . Similarly, we have e α ( n ) ′ | P k = Λ − k Q j ∈{ ( n ) ′ , n +1 , ··· , k − } (cid:0) φ ( P j ) − φ ( P k ) (cid:1) , ∀ k ≥ n . By Theorem 2.5, Z M y = Z M (cid:0)e α n (cid:1) = Λ − n Λ − n Λ n + X k ≥ n +1 ( e α n ) | P k Λ k = Λ − n Λ + n + X k ≥ n +1 ( e α n ) | P k Λ k , and Z M yz = Z M e α n e α ( n ) ′ = X k ≥ n +1 e α n | P k · e α ( n ) ′ | P k Λ k . Since these integrals are constants, we may assume(12.16) φ ( P n ) = φ ( P ( n ) ′ ) . Then by the expressions above, e α n | P k = e α ( n ) ′ | P k , ∀ k ≥ n + 1, and byProposition 11.4, Λ − n = ( − n Λ + n . These together give Z M y − Z M yz = ( − n . While x n y = y + yz gives Z M y + Z M yz = Z M x n y = 1 . Hence Z M y = 1 + ( − n . (cid:3) Lemma 12.15 immediately gives the ring H ∗ ( M ; Z ): Proposition 12.17. Let the circle act on a n -dimensional compact sym-plectic manifold ( M, ω ) with moment map φ : M → R . Assume [ ω ] is anintegral class and M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 · · · , P n (cid:9) . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action atthe fixed points. Then as rings, H ∗ ( M ; Z ) ∼ = H ∗ (cid:0) e G ( R n +2 ); Z (cid:1) with n ≥ even. Finally, we use the weights in Proposition 11.4 to determine c ( M ). Proposition 12.18. Let ( M, ω ) be a compact n -dimensional Hamiltonian S -manifold with moment map φ : M → R . Assume [ ω ] is an integral classand M S = (cid:8) P , · · · , P n − , P n , P ( n ) ′ , P n +1 , · · · , P n (cid:9) . Assume φ ( P n − ) − φ ( P ) = φ ( P n ) − φ ( P ) is the largest weight of the S -action at the fixedpoints. Then the total Chern class of M is c ( M ) = (1+[ ω ]) n +2 ω ] , hence isisomorphic to c (cid:0) e G ( R n +2 ) (cid:1) .Proof. Similar to the proof of Proposition 8.7, the restriction map H ∗ S ( M ; Z ) → H ∗ S ( M S ; Z )is injective. Let I = { , , · · · , n , ( n ) ′ , · · · , n } . Consider the class α = Q j ∈ I (cid:0) e u + φ ( P j ) t (cid:1) (cid:0)e u + φ ( P ) t (cid:1) + (cid:0)e u + φ ( P n ) t (cid:1) . By Proposition 11.4, (cid:0) φ ( P ) − φ ( P i ) (cid:1) + (cid:0) φ ( P n ) − φ ( P i ) (cid:1) = φ ( P n − i ) − φ ( P i )for all i ∈ I . Using this, and Proposition 11.4, we can check that α | P i = c S ( M ) | P i , ∀ i ∈ I. Hence c S ( M ) = α . Restricting this to ordinary cohomology, we obtain ourclaim. (cid:3) AMILTONIAN CIRCLE ACTIONS WITH ISOLATED FIXED POINTS 39 References [1] M. Atiyah and R. Bott, The moment map and equivariant cohomology , Topology, (1984), 1-28.[2] N. Berline and M. Vergne, Classes caract´eristiques ´equivariantes, formule de locali-sation en cohomologie ´equivariante , C. R. Acad. Sci. Paris, (1982), 539-541.[3] L. Godinho and S. 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