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.     

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.     

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.     

Symplectic automorphisms on Kummer surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface X by a finite abelian group G of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of H²(X,\mathbbZ){H^2(X,\mathbb{Z})} which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

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     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

Symplectic Automorphisms and the Picard Group of a K3 Surface

We consider the symplectic action of a finite group G on a K3 surface X. The Picard group of X has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then investigate the classification of symplectic actions by a fixed finite group, using moduli spaces of K3 surfaces with symplectic G-action.     

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.     

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).     

Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R³, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R⁴, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.     

Nilpotent flows of S1-invariant Hamiltonian systems on 4-dimensional symplectic manifolds

We investigate S¹-invariant Hamiltonian systems on compact 4-dimensional symplectic manifolds with free symplectic action of a circle. We show that, in a rather general case, such systems generate ergodic flows of types (quasiperiodic and nilpotent) on their isoenergetic surfaces. We solve the problem of straightening of these flows. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 122–140, January, 1997.     

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.     

Contact structures on product five-manifolds and fibre sums along circles

Two constructions of contact manifolds are presented: (i) products of S ¹ with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the corresponding contact structure. In particular, we prove that \mathbb CP²×S¹{{\mathbb {CP}}^2\times S^1} admits a contact structure in every homotopy class of almost contact structures. The existence of contact structures is also established for a large class of 5-manifolds with fundamental group \mathbbZ₂{{\mathbb{Z}}_2} .     

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.     

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).     

The Cohomology Ring of Weight Varieties and Polygon Spaces

We use a theorem of S. Tolman and J. Weitsman (The cohomology rings of Abelian symplectic quotients, math. DG/9807173) to find explicit formulæ for the rational cohomology rings of the symplectic reduction of flag varieties in ⁿ, or generic coadjoint orbits of SU(n), by (maximal) torus actions. We also calculate the cohomology ring of the moduli space of n points in P^k, which is isomorphic to the Grassmannian of k planes in ⁿ, by realizing it as a degenerate coadjoint orbit.     

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     

Multiplicity-free Hamiltonian actions need not be Kähler

Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible K?hler structures, and are therefore toric varieties. In this note we show that Delzant's result does not generalize to the non-abelian case. Our examples are constructed by applying U(2)-equivariant symplectic surgery to the flag variety U(3)/T ³. We then show that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible K?hler structure. Oblatum IX-1995 & 21-IV-1997     

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.