Spectral flexibility of symplectic manifolds <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">M</Emphasis>

We consider Riemannian metrics compatible with the natural symplectic structure on T ² × M, where T ² is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ₁. We show that λ₁ can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic question.     

The lagrange intersections for (ℂP n , ℝP n )

Summary If (M, ω) is a compact symplectic manifold andL ⊂M a compact Lagrangian submanifold and if φ is a Hamiltonian diffeomorphism ofM then the V. Arnold conjecture states (possibly under additional conditions) that the number of intersection section points ofL and φ (L) can be estimated by #{Lϒφ (L)}≥ cuplength +1. We shall prove this conjecture for the special case (L, M)=(ℝP ⁿ , ℂP ⁿ ) with the standard symplectic structure.     

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).     

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

Factorization and symplectic uniton numbers for harmonic maps into symplectic groups

It is proved that any harmonic map ϕ : Ω →Sp(N) from a simply connected domain Ω ⊆R ²⋃ | ∞ | into the symplectic groupSp(N) ⊂U(2N) with finite uniton number can be factorized into a product of a finite number of symplectic unitons. Based on this factorization, it is proved that the minimal symplectic uniton number of ϕ is not larger thanN, and the minimal uniton number of ϕ is not larger than 2N - 1. The latter has been shown in literature in a quite different way.     

Symplectic 4-manifolds as branched coverings of ℂℙ2

We show that every compact symplectic 4-manifold X can be topologically realized as a covering of ℂℙ² branched along a smooth symplectic curve in X which projects as an immersed curve with cusps in ℂℙ². Furthermore, the covering map can be chosen to be approximately pseudo-holomorphic with respect to any given almost-complex structure on X. Oblatum 9-III-1999 & 2-IX-1999 / Published online: 29 November 1999     

New symplectic 4–manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>+</Subscript>=1

In 1995 Dusa McDuff and Dietmar Salamon conjectured the existence of symplectic 4–manifolds (X,ω) which satisfy b₊=1, K²=0, K·ω>0, and which fail to be of Lefschetz type. This is equivalent to finding a symplectic, homology T²×S² manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of T²×S². They needed such examples to complete a list of possible symplectic 4–manifolds with b₊=1. In that same year Tian-Jun Li and Ai-ko Liu, working from a different point of view, questioned whether there existed symplectic 4–manifolds with b₊=1 with Seiberg- Witten invariants that did not depend on the chamber structure of the moduli space. The purpose of this paper is to construct an infinite number of examples which satisfy both requirements. The author was partially supported by NSF grant DMS-0406021.     

A stability property of symplectic packing

We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H ²(M, Q) there exists a number N ₀ such that for every N≥N ₀, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N ₀. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants. Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999     

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S² V S² V ... →N. The main result is that the images of these renormalized maps converge in L^1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.     

Symplectic resolutions for nilpotent orbits

In this paper, firstly we calculate Picard groups of a nilpotent orbit 𝒪 in a classical complex simple Lie algebra and discuss the properties of being ℚ-factorial and factorial for the normalization 𝒪tilde; of the closure of 𝒪. Then we consider the problem of symplectic resolutions for 𝒪tilde;. Our main theorem says that for any nilpotent orbit 𝒪 in a semi-simple complex Lie algebra, equipped with the Kostant-Kirillov symplectic form ω, if for a resolution π:Z𝒪tilde;, the 2-form π^*(ω) defined on π⁻¹(𝒪) extends to a symplectic 2-form on Z, then Z is isomorphic to the cotangent bundle T ^*(G/P) of a projective homogeneous space, and π is the collapsing of the zero section. It proves a conjecture of Cho-Miyaoka-Shepherd-Barron in this special case. Using this theorem, we determine all varieties 𝒪tilde; which admit such a resolution. Oblatum 6-V-2002 & 7-VIII-2002?Published online: 10 October 2002     

Normal Forms of Symplectic Matrices

Abstract In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix P such that P ⁻¹ MP is a symplectic matrix of the normal forms defined in this paper. Partially supported by the NSF, MCSEC of China, and the Qiu Shi Sci. Tech. Foundation * Associate Member of the ICTP     

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C ¹ (Theorem 3). Then we consider the case of the C ⁰ integrable symplectic twist maps and we prove that for such a map, there exists a dense G _δ subset of the set of its invariant curves such that every curve of this G _δ subset is C ¹ (Theorem 4).     

Semitoric integrable systems on symplectic 4-manifolds

Let (M,ω) be a symplectic 4-manifold. A semitoric integrable system on (M,ω) is a pair of smooth functions J,H∈C ^∞(M,ℝ) for which J generates a Hamiltonian S ¹-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce. A. Pelayo was partially supported by an NSF Postdoctoral Fellowship.     

Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions

We give a proof of the theorem of removing isolated singularities of pseudo-holomorphic curves with Lagrangian boundary conditions and bounded symplectic area. The proof is a combination of some L^p-type estimates, standard techniques of geometric P.D.E., and some ideas from symplectic geometry and calibration theory.     

A note on the geometry of M 3 and symplectic structures on S 1 × M 3

We investigate the relationship between the geometry of a closed, oriented 3-manifold M and the symplectic structures on S ¹ × M. In most cases the existence of a symplectic structure on S ¹ × M and Thurstonșs geometrization conjecture imply the existence of a geometric structure on M. This observation together with the existence of geometric structures on most 3-manifolds which fiber over the circle suggests a different approach to the problem of finding a fibration of a 3-manifold over the circle in case its product with the circle admits a symplectic structure. This work was supported in part by a GEBIP grant from the Turkish Academy of Sciences and a CAREER grant from the Scientific and Technological Research Council of Turkey.     

Quillen bundle and geometric prequantization of non-abelian vortices on a Riemann surface

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from L ² metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized by Ψ₀, a section of a certain bundle. The equivalence of these prequantum bundles are discussed.     

Pseudo-symplectic Runge-Kutta methods

Apart from specific methods amenable to specific problems, symplectic Runge-Kutta methods are necessarily implicit. The aim of this paper is to construct explicit Runge-Kutta methods which mimic symplectic ones as far as the linear growth of the global error is concerned. Such method of orderp have to bepseudo-symplectic of pseudosymplecticness order2p, i.e. to preserve the symplectic form to within ⊗(h ^2p )-terms. Pseudo-symplecticness conditions are then derived and the effective construction of methods discussed. Finally, the performances of the new methods are illustrated on several test problems.     

Symplectic semifield spreads of <Emphasis Type="Italic">PG</Emphasis>(5, <Emphasis Type="Italic">q</Emphasis>) and the veronese surface

In this paper we show that starting from a symplectic semifield spread S{\mathcal{S}} of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of S{\mathcal{S}}, and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s ², s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).