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Remarks on Symplectic Geometry

We survey the progresses on the study of symplectic geometry past four decades. We briefly deal with the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of Gromov-Witten invariants.


Introduction
The relatively-new subject, symplectic geometry, has been studied past four decades. In the 1970s, Arnold, Marsden and Weinstein took this subject and began to study symplectic manifolds. In the early 1980s, the convexity properties of the moment map were investigated by Atiyah, Guillemin, Sternberg, Mumford and Kirwan [1,2,30,31,36].
In 1985 Gromov introduced the pseudo-holomorphic (or J-holomorphic) curve technique into symplectic geometry and the symplectic capacity to prove the famous Nonsqueezing Theorem in his seminal paper, "Pseudoholomorphic curves in symplectic manifolds" [27]. Thereafter pseudo-holomorphic curves motivated the occurrence of the new mathematical subject, symplectic topology which has been developed by the McDuff school [45,46]. Pseudo-holomorphic curves are used as a tool in the four-dimensional symplectic topology, the study of the symplectic embeddings, and in the theory of the Gromov-Witten invariants. Taubes [62,63] discovered that in four dimensional symplectic manifolds, the Gromov-Witten invariants coincide with the Seiberg-Witten invariants. The theory of Floer homology (cf. [73]) is based on pseudo-holomorphic curves with boundary lying on a Lagrangian submanifold. This led on to the notion of the Fukaya category which is closely related to the homological mirror symmetry formulated by Kontsevich. Later the symplectic capacity which is a notion of monotonic symplectic invariant was pioneered by Ekeland and Hofer [17], and then has been developed by Hofer and his collaborates (cf. [33,34]) from the angle of dynamical systems and Hamiltonian dynamcis. After the middle 1980s there have been many major developments on symplectic geometry, for example, symplectic embedding problems, classification of symplectic and Hamiltonian group actions, the theory of Gromov-Witten invariants, quantum cohomolgy, Floer theory, symplectic capacities, Fukaya category, homological mirror symmetry etc.
The purpose of this article is to survey briefly various results related on symplectic geometry past four decades. The author had an opportunity to learn recent progresses on symplectic geometry and symplectic topology while he was writing this survey paper. This article is organized as follows. In section 2, we give basic notions, definitions, and examples of symplectic manifolds. We mention Marsden-Weinstein-Meyer theorem and Duistermaat-Heckman theorem. In section 3, we review some fundamental results, which are Darboux theorem, Weinstein Lagrangian neighborhood theorem, Weinstein tubular neighborhood theorem, Gromov's non-squeezing theorem, Poincaré-Birkhoff theorem and Arnold's conjecture. In section 4, we review the convexity properties of a moment map obtained by Atiyah, Guillemin, Sternberg and Kirwan. Using the convexity properties of a moment map, Delzant [12] classified all symplectic-toric manifolds in terms of a set of very special polytopes, so-called the Delzant polytopes. We state the Delzant's work. In the final section we survey various results related to symplectic geometry obtained during past three decades. We briefly deal with the following subjects: the classification problem of symplectic and Hamiltonian actions, the symplectic embedding problems and the theory of Gromov-Witten invariants. Finally we mention the recent work of Shen and Zhou on the Landau-Ginzburg/Calabi-Yau correspondence for elliptic orbifold projective lines.
Notations: We denote by Q, R and C the field of rational numbers, the field of real numbers and the field of complex numbers respectively. We denote by Z and Z + the ring of integers and the set of all positive integers respectively. Q + (resp. R + ) denotes the set of all positive rational (resp. real) numbers. We denotes by Z + (resp. Q + , R + ) the set of all non-negative integers (resp. rational numbers, real numbers). Q × (resp. R × , C × ) denotes the group of nonzero rational (resp. real, complex) numbers. The symbol ":=" means that the expression on the right is the definition of that on the left. For two positive integers k and l, F (k,l) denotes the set of all k × l matrices with entries in a commutative ring F . For a square matrix A ∈ F (k,k) of degree k, σ(A) denotes the trace of A. For any B ∈ F (k,l) , t B denotes the transpose of B. For a positive integer n, I n denotes the identity matrix of degree n. For a complex matrix A, A denotes the complex conjugate of A. diag(a 1 , · · · , a n ) denotes the n × n diagonal matrix with diagonal entries a 1 , · · · , a n . For a smooth manifold M, we denote by C c (M) (resp. C ∞ c (M) the algebra of all continuous (resp. infinitely differentiable) functions on M with compact support, and by X(M) the Lie algebra of all smooth vector fields on M. H denotes the Poincaré upper half plane and D denotes the Poincaré disk.

Basic notions, definitions and examples
Let (M, ω) be a symplectic manifold of dimension 2n, that is, a smooth manifold of dimension 2n equipped with a closed (dω = 0), nondegenerate (ω n = 0) 2−form ω. The notion of symplectic structures arose in the Hamiltonian formulation of the theory of classical mechanics. A classical mechanical system can be modelled by the phase space which is a symplectic space. On the other hand, a quantum mechanical system is modelled by a Hilbert space. Each state of the system corresponds to a line in a Hilbert space.
(CJ2) The set of all ω-compatible almost complex structures on a symplectic manifold (M, ω) is path-connected and contractible.
(CJ3) Let (M, ω) be a symplectic manifold equipped with a ω-compatible almost complex structure J. Then any almost complex submanifold N of (M, J) is a symplectic submanifold of (M, ω).
Let (M, h) be a Kähler manifold of dimension n with hermitian metric h = (h ij ). Then its Kähler form ω is given by Then ω is a symplectic form on M. Thus (M, ω) is a symplectic manifold. According to the positivity of h, we see that the symplectic form ω satisfies the positivity condition Moreover the complex structure J satisfies the property J * ω = ω, where J * ω(X, Y ) := ω(JX, JY ) for all X, Y ∈ X(M). The symplectic volume form is We may say that a Kähler manifold is a symplectic manifold (M, ω) equipped with an integral ω-compatible almost complex structure J. Then there exist a neighborhood U of p and a Kähler potential ϕ ∈ C ∞ (U; R) such that Definition 2.4. Let G be a connected Lie group and let (M, ω) be a symplectic manifold. A smooth G-action φ : G × M −→ M is said to be symplectic if G acts by symplectomorphisms, that is, for each g ∈ G the diffeomorphism φ g : M −→ M given by φ g (p) := φ(g, p), p ∈ M satisfies the condition φ * g ω = ω. The triple (M, ω, φ) is called a symplectic G-manifold.
Definition 2.5. Let G be a connected Lie group and let (M, ω) be a symplectic manifold. Let g be the Lie algebra of G. Let φ : G × M −→ M be an action of (M, ω). For any X ∈ g, we denote by X M the vector field on M generated by the one-parameter subgroup of global diffeomorphisms p → φ(exp(tX), p), p ∈ M, that is, Given a smooth funcftion H : (M, ω) −→ R, let X H be the vector field on M defined by the Hamilton's equation (HA1) For each X ∈ g, where µ X : M −→ R is a function defined by µ X (p) := µ(p), X and X M is the vector field on M generated by the one-parameter subgroup {exp tX | t ∈ R} of G.
(HA2) µ is equivariant with respect to the given action σ and the coadjoint action Ad * : G −→ GL(g * ), that is, In this case, the quadruple (M, ω, G, µ) is called a Hamiltonian G-space and µ is called the moment map. (2) Any symplectic G-action on a simply connected symplectic manifold (M, ω) is Marsden and Weinstein [40], and Meyer [48] proved the following.
(c) there exists a symplectic form ω red on M red such that i * ω = π * ω red .
Definition 2.8. Let (M, ω) be a symplectic manifold of dimension 2n. The symplectic measure (or Liouville measure) of a Borel subset U of M is defined to be where ω n /n! is the symplectic volume form of M.
Definition 2.9. Let G be a torus of dimension n. Let (M, ω, G, µ) be a a Hamiltonian G-space of dimension 2n such that the moment map µ is proper. The Duistermaat-Heckman measure (briefly D-H measure), m DH , on g * is defined to be the push-forward of m ω by µ : M −→ g * . More precisely, for any Borel subset W of g * , we have where ω n /n! is the symplectic volume form of M.
For a function f ∈ C ∞ c (g * ), we define its integral with respect to the D-H measure to be On g * ≃ R n , there is also the Lebesque measure m 0 . The relation between m DH and m 0 is governed by the Radon-Nikodym derivative, denote by dm DH dm 0 , which is a generalized function satisfying Duistermaat and Heckman [15] proved the following. Theorem 2.4. (Duistermaat-Heckman [1982]). Let G be a torus of dimension n. Let (M, ω, G, µ) be a a Hamiltonian G-space of dimension 2n such that the moment map µ is proper. The D-H measure m DH on g * is a piecewise polynomial multiple of the Lebesque measure m 0 on g * ≃ R n , that is, the Radon-Nikodym derivative ϑ = dm DH dm 0 is a piecewise polynomial. More precisely, for any Borel subset W of g * , where dx = dm 0 is the Lebesque volume form on W and ϑ : g * −→ R is a polynomial on any region consisting of regular values of µ. In particular, for the standard Hamiltonian action of S 1 on (S 2 , ω), we have m DH = 2π m 0 with a constant polynomial ϑ = 2π. Remark 2.4. As far as I know, the terminology, Symplectic Geometry, first appeared in Siegel's paper [61] in 1943. In that paper, Siegel studied the geometry of the Siegel upper half plane H g which is biholomorphic to the Hermitian symmetric space Sp(2g, R)/U(g). He discovered the explicit fundamental domain with respect to the Siegel modular group Sp(2g, Z). In fact, H g is a symplectic manifold of dimension g(g + 1) because it is a Kähler manifold. Now we roughly state the geometry of H g .
For a given fixed positive integer g, we let be the Siegel upper half plane of degree g and let be the symplectic group of degree 2g, where F (k,l) denotes the set of all k × l matrices with entries in a commutative ring F for two positive integers k and l, t α denotes the transpose matrix of a matrix α and Then Sp(2g, R) acts on H g transitively by and Ω ∈ H g . Let be the Siegel modular group of degree g. This group acts on H g properly discontinuously. C. L. Siegel investigated the geometry of H g and automorphic forms on H g systematically. Siegel [61] found a fundamental domain F g for H g with respect to Γ g and described it explicitly. Moreover he calculated the volume of F g . Let A g = Γ g \H g be the Siegel modular variety of degree g. In fact, A g is one of the important arithmetic varieties in the sense that it is regarded as the moduli of principally polarized abelian varieties of dimension g.
For Ω = (ω ij ) ∈ H g , we write Ω = X + iY with X = (x ij ), Y = (y ij ) real. We put dΩ = (dω ij ) and dΩ = (dω ij ). The Bergman metric ds 2 g on H g which is a Sp(2g, R)-invariant Kähler metric is given by [kl] dω ij dω kl and its Kähler form is The volume form is given by The invariant metric ds 2 g is Kähler-Einstein, that is, is the Kähler potential of ω g . A natural question arises as follows: Question. Is there a Hamiltonian action of a group G on (H g , ω g ) which is important arithmetically and geometrically ?
Examples: (1) Let x 1 , · · · , x n , x n+1 , · · · , x 2n be linear coordinates on R 2n . Then is a symplectic form. Thus (R 2n , ω 0 ) is a symplectic manifold. Now we consider the case n = 1. The one-dimensional torus acts on R 2 by rotations. This is a Hamiltonian action. The Lie algebra of S 1 is given by the vector field J ♯ on R 2 generated by J is given by The moment map µ : R 2 −→ so(2) * satisfies ı(J ♯ )ω 0 = x dx + y dy = d µ( · ), J .
Thus we obtain (2) The standard symplectic form ω on the two dimensional sphere S 2 is induced by Thus (S 2 , ω) is a symplectic manifold. Let ω st = dθ ∧ dh be the standard area form in cylindrical polar coordinates θ, h (0 ≤ θ ≤ 2π and −1 ≤ h ≤ 1) on S 2 . Endow (S 2 , ω st ) with the rotational S 1 -action about the z-axis. This is a Hamiltonian action with the moment map µ : S 2 −→ R given by µ(θ, h) = h, and the momentum polytope is µ( 1]. But S n (n > 2) has no symplectic structures.
(3) Let M be a smooth manifold of dimension n. Let x 1 , · · · , x n be local coordinates on an open neighborhood U of x ∈ M. If α ∈ T * x X, then Then (T * U, x 1 , · · · , x n , α 1 , · · · , α n ) is a local coordinate chart for T * M. We define Then it is easily seen that α is intrinsically defined and Ω can = −dα. Thus (T * M, Ω can ) is a symplectic manifold of dimension 2n. The zero section of T * M given by } is a Lagrangian submanifold of (T * M, Ω can ). The 1-form α is called the tautological form and Ω can is called the canonical symplectic form on T * M.
Suppose the Lie group G acts on a smooth manifold M. Then G acts naturally on T M and T * M. We can show that the action of G on (T * M, Ω can ) is Hamiltonian. The moment map µ : T * M −→ g * is given by where Y ♯ is the vector field on T * M generated by Y .
(4) For any submanifold Y of a smooth manifold M, the conormal bundle N * Y is a Lagrangian submanifold of (T * M, Ω can ).
(5) Let (M 1 , ω 1 ) and (M 2 , ω 2 ) be two symplectic manifolds of dimension 2n. Let 2) be the natural projections. For any nonzero real numbers a and b, the 2-form a π * 1 ω 1 + b π * 2 ω 2 is a symplectic form on M 1 × M 2 . We consider the twisted product form Then we can show that φ is a symplectomorphism if and only if the graph of φ is a Lagrangian submanifold of (M 1 × M 2 , ω).
This Heisenberg group is a 2-step nilpotent Lie group and is important in the study of smooth compactifications of the Siegel modular variety. In fact, H as before. Let g be the Lie algebra of G and let g * be the dual space of g. We observe that g can be regarded as the subalgebra consisting of all 2(m + n) × 2(m + n) real matrices of the form of the Lie algebra sp(2(m + n), R) of the symplectic group Sp(2(m + n), R). An easy computation yields The dual space g * of g can be identified with the vector space consisting of all 2(m + n) × 2(m + n) real matrices of the form The adjoint representation Ad of G is given by Ad G (g)X = gXg −1 for g ∈ G and X ∈ g. For g ∈ G and F ∈ g * , gF g −1 is not of the form F (a, b, c). We denote by Then it is easy to see that the coadjoint representation Ad * G : . Therefore the coadjoint orbits of G in g * fall into two classes: (I) The single point Ω a,b a, b ∈ R (m,n) located in the plane c = 0. (II) The affine planes Ω c c = t c ∈ R (m,m) , c = 0 parallel to the homogeneous plane c = 0.
In other words, the orbit space O(G) of coadjoint orbits is parametrized by The single point coadjoint orbits of the type Ω a,b are said to be the degenerate orbits of G in g * . On the other hand, the flat coadjoint orbits of the type Ω c are said to be the non-degenerate orbits of G in g * .
It is well known that each coadjoint orbit is a symplectic manifold. We will state this fact in detail. For the present time being, we fix an element F of g * once and for all. We consider the alternating R-bilinear form B F on g defined by F, X , X, Y ∈ g, where ad * g : g −→ End(g * ) denotes the differential of the coadjoint representation Ad * G : G −→ GL(g * ). More precisely, if F = F (a, b, c), X = X(α, β, γ), and Y = X(δ, ǫ, ξ), then We denote by g F the Lie subalgebra of g corresponding to G F . Then it is easy to show that Here rad B F denotes the radical of B F in g. We letḂ F be the non-degenerate alternating R-bilinear form on the quotient vector space g/rad B F induced from B F . Since we may identify the tangent space of the coadjoint orbit Ω F ∼ = G/G F with g/g F = g/rad B F , we see that the tangent space of Ω F at F is a symplectic vector space with respect to the symplectic formḂ F . Now we are ready to prove that the coadjoint orbit Ω F = Ad * G (G)F is a symplectic manifold. We denote by X the smooth vector field on g * associated to X ∈ g. That means that for each ℓ ∈ g * , we have

Proof.
Let X be the smooth vector field on g * associated to X ∈ g such that Proof. If X 1 , X 2 , and X 3 are three smooth vector fields on g * associated to X 1 , X 2 , X 3 ∈ g, then (by the Jacobi identity).
Therefore B Ω F is closed.
In summary, (Ω F , B Ω F ) is a symplectic manifold of dimension 2mn or 0. We remark that there is a one-to-one correspondence between the collection of the coadjoint orbits . We refer to [35,71,72] for more details.
Then the natural product action of G on M ×O(F ) − is Hamiltonian with the following moment map µ F : If the above action of G is free, we obtain a reduced space with respect to the coadjoint orbit O(F ).
(8) For a positive integer n, let T 1 := (R/Z) 2n−1 and T 2 := (R/Z) n be two tori. It is easily seen that the T 1 -action on (M, ω) = ((R/Z) 2n , n i=1 dx i ∧ dy i ) with coordinates (x 1 , y 1 , · · · , x n , y n ) in (R/Z) 2n by translation on the first 2n − 1 components (x 1 , y 1 , · · · , x n−1 , y n−1 , x n ) is free and symplectic. According to Remark 2.3, this action is not Hamiltonian. The T 1 -orbits are coisotropic submanifolds of M diffeomorphic to T 1 . The T 2 -action on (M, ω) by translations on (x 1 , · · · , x n ) is free, symplectic and hence not Hamiltonian. Its (9) Let N := ((R/Z) 2 ×S 2 , dx∧dy +dθ ∧dh) be a 4-dimensional symplectic manifold. The 2-torus T := (R/Z) 2 acts freely by translations on the left factor of N. Let Z 2 := Z/2Z act on S 2 by rotating each point horizontally by π radians, and let Z 2 act on (R/Z) 2 by the antipodal action on the first circle R/Z. Then the diagonal action of Z 2 on N id free. So the quotient space M := (R/Z) 2 × Z 2 S 2 is a smooth manifold equipped with the symplectic form ω and the T -action inherited from the ones on N. We see easily that the action of T on M is symplectic but not free, and the T -orbits are symplectic tori of dimension 2. The orbit space M/T = S 2 /Z 2 is an orbifold with two singular points of order 2, the south and north poles of S 2 .
The quotient space of (M, ω) by the diagonal action of Z 2 is a compact connected symplectic manifold, called the Kodaira variety, Consider the action of the 2-torus T := R/Z × R/Z on (KT, ω) such that the first circle acts on the x 1 -component and the second circle acts on the y 2 -component. It is easily checked that this action is symplectic and free. The T -orbits in (KT, ω) are Lagrangian orbits.
is the usual coordinate on C. Its Kähler form ω FS is given by The function K(z) := log(|z| 2 + 1) is the Kähler potential, that is, Since M is diffeomorphic to S 2 by stereographic projection, we obtain (2)).

Some fundamental theorems
According to the Darboux's theorem, a symplectic form (or symplectic structure) ω can always be written in the following form (3.1) ω = n k=1 dp k ∧ dq k in suitable canonical coordinates p 1 , · · · , p n , q 1 , · · · , q k . However these canonical coordinates are not uniquely determined. This theorem says that locally all symplectic forms are the same, i.e., all symplectic structures are indistinguishable. We recall the Moser's stability theorem which says that if ω t (t ∈ [0, 1]) is a smooth path of symplectic forms such that the cohomology class [ω 0 ] = [ω t ] ∈ H 2 (M, R) for all t ∈ [0, 1], then all these symplectic forms are the same in the sense that one can make them coincide by moving the points of M in a suitable way. In other words, one cannot change the symplectic form in any way by deforming it, provided that the cohomology class [ω] ∈ H 2 (M, R) is unchanged.
In 2-dimensions, a symplectomorphism can be characterized as an area preserving diffeomorphism. More precisely, if S is a region in R 2 that is diffeomorphic to a disc D and has the same area as D, then there exists a symplectomorphism φ : S −→ D.
Weinstein Lagrangian Neighborhood Theorem: Let M be a smooth manifold of dimension 2n, X a compact n-dimensional submanifold, i : X ֒→ M the inclusion map, and ω 1 and ω 2 symplectic forms on M such that i * ω 1 = i * ω 2 = 0, i.e., X is a Lagrangian submanifold of both (M, ω 1 ) and (M, ω 2 ). Then there exist neighborhoods U 1 and U 2 of X, and a diffeomorphism φ : Proof. The proof can be found in [66].
Weinstein Tubular Neighborhood Theorem: Let M be a smooth manifold of dimension n, X is a submanifold of M, NX the normal bundle of X in M, i 0 : X ֒→ NX the zero section, and i : X ֒→ M inclusion. Then there exist neighborhoods U 0 of X in NX, U 2 of X in M and a diffeomorphism φ : Proof. The proof can be found in [67].
In 1985 using the pseudo-holomorphic curve technique and constructing the Gromov radius, Gromov proved the famous Gromov ′ s nonsqueezing theorem which is also called the principle of the symplectic camel.
Proof. The proof can be found in [27].
Eliashberg's Principle: An obstruction to symplectic embeddings (beyond the volume condition) can be described by a J-holomorphic curve.
We refer to [18] for more details on Eliashberg's Principle.
An influential precursor in the study of global aspects of symplectic geometry is Arnold's conjecture which is a high-dimensional analogue of the classical fixed point theorem of H. Poincaré and G. Birkhoff. Proof. The proof can be found in [5].
Arnold's conjecture was proved by Conley-Zehnder, Floer, Hofer-Salamon, Ono, Fukaya-Ono, Liu-Tian using Floer homology which is an infinite dimensional analogue of Morse theory. Now Arnold's conjecture is described in the form of the following theorem: Proof. The proof can be found in [1,30]. We briefly sketch the proof of Atiyah [1] (cf. We leave the proof of Claim 1 to the reader.  Any p 0 , p 1 ∈ M can be approximated arbitrarily by points q 0 and q 1 in M with µ(q 0 ) − µ(q 1 ) ∈ ker ( t A) for some matrix A ∈ Z (m,m−1) of rank m − 1. Taking the limits q 0 −→ p 0 and q 1 −→ p 1 , we see that µ(M) is convex. This completes the proof of the statement (M2).
Here , is a T m -invariant positive definite inner product on R m . By the irrationality of ζ, the set {exp tζ | t ∈ R } is dense in T m . Hence the zeros of the vector field X ζ on M generated by ζ are the fixed points of the T m -action ρ. Since the function µ( · ), ζ attains its maximum value on one of the sets N j , we have the relation ξ, ζ > sup p∈M µ(p), ζ .
Theorem 4.2. Let M be a connected nonsingular complex projective variety with its Kähler form ω and G a compact connected subgroup of the group of complex symplectic transformations of (M, ω). Suppose that for some point p ∈ M, the stabilizer G p of p in G is finite. Let t be a Cartan subalgebra of the Lie algebra g of G, and let t * be the subspace of the dual space g * of g corresponding to t. Let t * + be a Weyl chamber in t * , and let µ : M −→ g * be the moment map. Then the intersection µ(M) ∩ t * is a convex polytope of dimension equal to the rank of G.
Kirwan [36] generalized the above convexity theorem to the case of a Hamiltonian G-space where G is a compact Lie group. Proof. We give a sketch of Kirwan's proof. We first fix a G-invariant inner product , on g and use it to identify g * with g, and t * with t. Here t is the Lie algebra of a maximal torus T of G. Let t + be a positive Weyl chamber in t. Let , be the associated norm on g. Claim 2. For any sufficiently small ε > 0, there exists an element α ∈ t + such that the ball B(α, ε) of radius ε and center α meets µ(M) ∩ t + in precisely two points α 1 and α 2 neither of which lies in the interior of B(α, ε).
Claim 3. The function µ (α) 2 on M × G/H takes its minimum value precisely at those points (x, gH) such that Here α 1 and α 2 are two points in Claim 2.
Using Claim 1, Claim 2 and Claim 3, Kirwan proved the above theorem as follows: Suppose µ(M) ∩ t + is not convex. Let µ (α) 2 be the function M × G/H where α satisfies the conditions of Claim 2. According to Claim 3, the set is the disjoint union of the following two non-empty closed subsets This contradicts Claim 1. Thus µ(M) ∩ t + is convex. Hence we complete the proof of the above theorem.
Though we do not have a classification of symplectic manifolds so far, fortunately we have a classification of symplectic-toric manifolds which are very special Hamiltonian torus-spaces in terms of combinatorial data. We recall that a symplectic-toric manifold is a compact connected symplectic manifold (M, ω) of dimension 2n equipped with an effective Hamiltonian action of an n-dimensional torus T n and with a corresponding moment map µ : M −→ R n . It can be seen that every symplectic-toric manifold is simply connected. For instance, according to Example (2) in section 2, (S 2 , dθ ∧ dh, S 1 ) is a symplectic-toric manifold. In 1988 T. Delzant classified all symplectic-toric manifolds in terms of a set of very special polytopes. We describe his classification roughly. (DP2) it is rational, i.e., the edges meeting at the vertex p are rational in the sense that each edge is of the form (DP3) it is smooth, i.e., for each vertex, the corresponding α 1 , · · · , α n can be chosen to be a Z-basis of Z n .
First we give several definitions for the reader's convenience. A facet of a polytope ∆ with dim ∆ = n in R n is a (n − 1)-dimensional face. Let ∆ be a Delzant polytope with dim ∆ = n and d = the number of facets of ∆. A lattice vector v ∈ Z n is said to be primitive if it cannot be written as v = ku with u ∈ Z n , k ∈ Z with |k| > 1.
We consider the map where j : Z ֒→ C d is the inclusion. Therefore (M ∆ , ω ∆ ) is a compact symplectic manifold of dimension 2n. Furthermore we can show that the torus T n acts on M ∆ in a Hamiltonian fashion and its moment map µ n : M ∆ −→ Hom(R n , R) such that µ n (M ∆ ) = ∆ (we refer to [6, pp. 185-186] for the detailed proof). Finally the quadruple (M ∆ , ω ∆ , T n , µ n ) is the required symplectic toric manifold of dimension 2n corresponding to ∆ ⊂ Hom(R n , R) = (R n ) * . Then (M, ω, S 1 ) is S 1 -equivariantly diffeomorphic to a complex surface with a holomorphic S 1 -action which is obtained from P 2 (C), a Hirzebruch surface, or a P 1 (C)bundle over a Riemann surface with appropriate circle actions by a sequence of blowips at the fixed points.
We refer to [3,4] for some details. Definition 5.2. Suppose that a torus T acts effectively and symplectically on a compact connected symplectic manifold. The T -action is said to be coisotropic if there is a coisotropic T -orbit. Definition 5.3. Suppose that a torus T acts effectively and symplectically on a compact connected symplectic manifold. If there is a dim T -dimensional symplectic Torbit, we say that the T -action is a maximal symplectic action.
Over past twenty years the theory of the symplectic actions on symplectic manifolds has been developed by some experts in symplectic geometry. In the good survey article [54],Álvaro Pelayo described classifications on compact connected symplectic manifolds (M, ω):   (d) "Maximal symplectic case": there is a dim T -orbit symplectic orbit.
Here T denotes a torus. He outlined connections of these works with algebraic geometry, toric varieties, log-symplectic toric geometry, torus bundles over tori, nilpotent Lie groups, integral systems and the classification of semi-toric systems.
Let S 1 = U(1) be a torus of dimension one. A Hamiltonian S 1 -action on a compact connected symplectic manifold (M, ω) of dimension 2n has at least n + 1 fixed points. In fact, the number of fixed points is 2n k=0 rankH k (M, R) and 0 = [ω k ] ∈ H 2k (M, R) for 1 ≤ k ≤ n. The following natural questions arise: Under which conditions is a symplectic G-action Hamiltonian ? Describe the obstruction to being Hamiltonian. Here G is an m-dimensional torus, a compact connected Lie group, a non-compact abelian group or a semisimple Lie group.
Question B. Are there non-Hamiltonian symplectic S 1 -actions on compact connected symplectic manifolds with nonempty discrete fixed point sets ?
Recently some affirmative answers were given as follows. Theorem 5.1. (Tolman and Weitsman [65]) Let (M, ω) be a compact connected symplectic manifold equipped with a semi-free symplectic S 1 -action with isolated fixed points. If there is at least one fixed point, the S 1 -action is Hamiltonian. Theorem 5.2. (Frankel [23]) Let (M, ω) be a compact connected Kähler manifold admitting an S 1 -action preserving the Kähler structure ω. If the the S 1 -action has some fixed points, then it is Hamiltonian. Alvaro Pelayo [54] proposed the following natural classification problem: Problem A: Let G be an m-dimensional compact connected Lie group. Construct symplectic invariants and classify, up to equivariant symplectomorphisms, effective symplectic G-actions on compact connected symplectic 2n-dimensional manifolds (M, ω) in terms of these invariants.
We propose the following problem: Pelayo [53,54] proved the following result. If the symplectic T -action is not Hamiltonian, then one and only one of the following cases occurs: (3) (M, ω) is equivariantly symplectomorphic to (T × t * )/Q with the induced form and the T -action, where Q ≤ T × t * is a discrete cocompact subgroup for the group structure on T × t * .
(4) (M, ω) is equivariantly symplectomorphic to a symplectic orbifold bundle P := Σ × π orb 1 ( ,p 0 ) T over a good orbisurface Σ, with symplectic form and T -action induced by the product ones. Here, in order to form the quotient P , the orbifold fundamental group π orb 1 (Σ) acts on Σ × T diagonally, and on T by means of a homomorphism µ : π orb 1 (Σ) −→ T.

The symplectic embedding problems
Given two open subsets U and V in R n , we often write U s ֒→ V instead of "there exists a symplectic embedding of U into V ". We denote by D(a) the open disk in R 2 of area a, centered at the origin, and P (a 1 , · · · , a n ) = D(a 1 ) × · · · × D(a n ) the open polydisk in R 2n . We let C 2n (a) := P (a, · · · , a) be the cube in R 2n and let Z 2n (a) := D(a) × C n−1 be the symplectic cylinder. Let E(a 1 , · · · , a n ) := (z 1 , · · · , z n ) ∈ C n n i=1 denote the open ellipsoid whose projection to the j-th complex coordinate plane is D(a j ) and let B 2n (a) = E(a, · · · , a) be the ball of radius a/π. We put T 4 (A) := T 2 (A) × T 2 (A), where T 2 (A) is the torus R 2 /(AZ ⊕ Z) endowed with the symplectic form dx ∧ dy inherited from R 2 .
We define the symplectic capacities intervals where c EC is a step made from two segments. The first of these steps has edge at (6, 4 7 ) and the last at (7, 15 8 ). Eliashberg's Principle [18]: An obstruction to symplectic embeddings (beyond the volume condition) can be described by a J-holomorphic curve.
Surprisingly J-holomorphic curves can be used to construct symplectic embeddings.
For two positive real numbers a and b, let (2) There exists a symplectic embedding P (1, a, a) s ֒→ P (2, 2, ∞) for all a ≥ 1.
For a positive integer k, we consider the function defined by R. K. Hind [32] proved that  Let K be a compact connected Lie group and let k * be the dual of the Lie algebra k of K. Each coadjoint orbit O in k is equipped with the Kostant-Kirillov-Souriau symplectic form ω canonically defined by where X ♯ , Y ♯ are the vector fields on k * corresponding to X, Y ∈ k induced by the coadjoint action. Each coadjoint orbit intersects a positive Weyl chamber in a single point. So there is a bijection between the coadjoint orbits and points in the positive Weyl chamber. Points in the interior of the positive Weyl chamber are called regular points. The orbits corresponding to regular points are called generic orbits that are diffeomorphic to K/T for T a maximal torus of K. Coadjoint orbits intersecting the positive Weyl chamber at its boundary are called degenerate orbits.
Caviedes Castro [7] proved the following: Theorem 5.16. Let K be a compact connected Lie group. The Gromov width of a coadjoint orbit O λ through a point λ lying on some rational line in t * , equipped with the Kostant-Kirillov-Souriau symplectic form ω λ , can not be greater than the following quantity (5.5) min { | λ, α ∨ | | α ∨ is a coroot and λ, α ∨ = 0 } .
Here t * is the dual of the Lie algebra of a maximal torus T of K.
Fang, Littelmann and Pabiniak [21] gave a uniform proof for the conjectured Gromov width of rational coadjoint orbits of all compact connected simple Lie groups by analyzing simplices in Newton-Okounkov bodies.
Here t * is the dual of the Lie algebra of a maximal torus T of K.
Furthermore they proved the following fact in [21].
Proposition 5.1. Let K be a compact connected Lie group, not of type G 2 , F 4 of E 8 and let (O λ , ω λ ) be its generic coadjoint orbit through a point λ lying on some rational line in t * , equipped with the Kostant-Kirillov-Souriau symplectic form ω λ . Then there exists a symplectic embedding of a ball of capacity (2.29) into (O λ , ω λ ).

The theory of the Gromov-Witten invariants
Let (M, ω, J) be a symplectic manifold of dimension 2n with a ω-compatible almost complex structure J. For two nonnegative integers g, k ≥ 0, let M g,k be the Deligne-Mumford moduli space of stable curves of genus g with k marked points, and M g,k,A be the moduli space of stable maps into M of homology class A ∈ H 2 (M, Z)/torsion. The elements of M g,k,A are of the form where Σ is a (not necessarily stable) curve of genus g with k marked points and f : Σ −→ M is a pseudoholomorphic curve. Let Then we have the evaluation map ev : M g,k,A −→ Y defined by ev(Σ, p 1 , · · · , p k , f ) = (st(C, p 1 , · · · , p k ), f (p 1 ), · · · , f (p k )), where st(C, p 1 , · · · , p k ) is the stabilization of C. By the Atiyah-Singer index theorem, we obtain d : The evaluation map send the fundamental class of M g,k,A to a d-dimensional rational homology class Φ A g,k ∈ H d (Y, Q). The homology class Φ A g,k ∈ H d (Y, Q) is called the Gromov-Witten invariant of M for the data (g, k, A). It is an invariant of the symplectic isotopy class of (M, ω).

E. Witten
where δ i ∈ H i (M, C) (0 ≤ i ≤ 2n), A runs over H 2 (M, Z) and unordered i 1 , · · · , i k ∈ {0, 1, · · · , 2n} with k ν=1 deg(δ iν ) = d. The convergence problem arises. The fact that the Gromov-Witten potential Φ M ω satisfies the WDVV equations was proved by Y. Ruan and G. Tian [55]. Maxim Kontsevich and Yuri Manin [37] described how the WDVV equations yields a potential Dubrovin structure on H * (M, C), understood as a supermanifold. Therefore each tangent space of H * (M, C) is endowed with a metric given by Poincaré duality, and a multiplication (5.9) x * y = i,j,k where the set {φ i } is a homogeneous basis of H * (X, C) with φ i · φ j = g ij , the (g ij ) denotes the inverse matrix (g ij ), x = i x i φ i and y = i y i φ i . This is the quantum deformation of the cup product and the WDVV equations are equivalent to the associativity of the multiplication. The ordinary cup product is the limit of Φ M tω with t → ∞.
Y. Ruan and G. Tang [55] nicely developed a theory of quantum cohomology which is related to symplectic topology, algebraic geometry, quantum field theory, mirror symmetry, and differential topology in 4-dimensional manifolds. They first constructed the Gromov-Witten theory for semi-Fano symplectic manifolds. Thereafter the Gromov-Witten theory was generalized by other people. The quantum cohomology ring endows the affine space H * (X, C) with the structure of a Frobenius manifold, namely, a Riemannian manifold with an associative product on the tangent spaces and various compatibilities. C. Taubes [62,63] proved that in 4-dimensional symplectic manifolds, certain Gromov-Witten invariants coincide with the gauge-theoretic Seiberg-Witten invariants. This relates symplectic topology and differential topology via the Gromov-Witten theory. W. Chen and Y. Ruan [8,9] defined the Gromov-Witten invariants for compact symplectic orbifolds extending the Gromov-Witten invariants for compact symplectic manifolds. Based on a proposal by E. Witten, H. Fan, T. Jarvis and Y. Ruan [22] introduced and developed the new Gromov-Witten type theory of geometric invariants, known as the FJRW theory that is the mathematically rigorous development of topological gravity coupled with A-type topological Landau-Ginzburg matter, as an intersection theory on the moduli space of solutions of the Witten equation. The FJRW theory is believed to be the counterpart of the Gromov-Witten theory in the Landau-Ginzburg model [10,11,38,58]. The relationship between these two theories is referred to as Landau-Ginzburg/Calabi-Yau (briefly LG/CY) correspondence.
The hypersurface X W defined by {W = 0} is a one-dimensional Calabi-Yau variety in a weighted projective space. Then G W acts on X W , and J W acts trivially. Thus we obtain the CY orbifold curve which is a global quotient (5.12) X W := X W /(G/ J W ).
Here both µ, ν run over a basis of H GW and η (·,·) GW is the inverse of the pairing η GW (·, ·). At the large volume limit t = −∞, the quantum multiplication ⋆ q becomes the Chen-Ruan product.
We have a Frobenius algebra (H FJRW , •), where the multiplication • is defined from the pairing η FJRW on H FJRW and the genus zero 3-point invariants through the following formula Here both γ, ζ run over a basis of H FJRW and η (·,·) FJRW is the inverse of the pairing η FJRW (·, ·). The quantum multiplication • u is a deformation of the multiplication • in (5.25) as • u=0 = •.
We refer to [58,59,60] for more precise details on H GW , η GW and H FJRW , η FJRW . The LG/CY correspondence [69,22] says that the two enumerative theories should be equivalent under an appropriate transformation.
Here ψ j ∈ H 2 (M g,k , Q) (1 ≤ j ≤ k) is the j-th ψ-class and q = e t where t was parametrized by a Kähler class P ∈ H GW .