Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.     

Real loci of symplectic reductions

Let be a compact, connected symplectic manifold with a Hamiltonian action of a compact -dimensional torus . Suppose that is equipped with an anti-symplectic involution compatible with the -action. The real locus of is the fixed point set of . Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction has the same ordinary cohomology as its real locus , with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.

     

Enumerating Rational Curves: The Rational Fibration Method

A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich's formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F₃.     

Parameter spaces for curves on surfaces and enumeration of rational curves

Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevich's formula for plane curves.     

Frankel's theorem in the symplectic category

We prove that if an -dimensional torus acts symplectically on a -dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kähler circle action has a fixed point if and only if it is Hamiltonian. The case of is the well-known theorem by McDuff.

     

The degree of the variety of rational ruled surfaces and Gromov-Witten invariants

We compute the degree of the variety parametrizing rational ruled surfaces of degree in by relating the problem to Gromov-Witten invariants and Quantum cohomology.

     

Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds

Assume is a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case . We give a complete list of the possible manifolds, and determine their equivariant cohomology rings and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.

     

TWO NEW CONSTRUCTIONS OF CARTESIAN AUTHENTICATION CODES FROM SYMPLECTIC GEOMETRY

ＴＷＯＮＥＷＣＯＮＳＴＲＵＣＴＩＯＮＳＯＦＣＡＲＴＥＳＩＡＮＡＵＴＨＥＮＴＩＣＡＴＩＯＮＣＯＤＥＳＦＲＯＭＳＹＭＰＬＥＣＴＩＣＧＥＯＭＥＴＲＹＧＡＯＹＯＵＡＮＤＺＯＵＺＥＮＧＪＩＡ（ＤｅｐａｒｔｍｅｎｔｏｆＢａｓｉｃＳｃｉｅｎｃｅ，Ｎｏｒｔｈｅａｓｔ...     

Adjoint orbits of the odd real symplectic group

In this paper we classify the adjoint orbits of the odd symplectic group over the field of real numbers. We need a noneigenvalue modulus to classify certain orbits.      

Einstein Metrics,Symplectic Minimality,and Pseudo-Holomorphic Curves

An alternative proof of the uniformization theorem for compact two-dimensional riemannian surfaces of genus 0 is proved via the direct methods of the calculus of variation as a minimum problem of the metric-dependent Dirichlet energy in the class of those maps with degree 1. Because of the conformal invariance of the energy-functional and the non compactness of the conformal group on S², concentration of mass for sequences can occur and spheres can split off. Nevertheless this is not the case for appropriate conformaly retransformed minimizing sequences. The main argument is the verification of a condition of the Douglas-type.     

On symplectic mean curvature flows

In this paper we review all the main known results about mean curvature flows with initial surfaces symplectic in a Kähler-Einstein surface, including published results and new results obtained recently. We also propose some problems that we think are very interesting.     

On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions

Let be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian action such that the fixed point set consists of isolated points or surfaces. Assume dim . In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types'. In this paper, we construct such manifolds with these ``types'. As a consequence, we have a precise list of the values of these invariants.

     

Rational invariants of certain classical groups over finite fields

Let be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over and be the rational function field over . We seek to understand the structure of the rational invariant subfield . In this paper, we prove that is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.      

Fundamental groups of complements of plane curves and symplectic invariants

Introducing the notion of stabilized fundamental group for the complement of a branch curve in , we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples.     

Complex symplectic geometry with applications to ordinary differential operators

Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems.

     

On homology of real algebraic varieties

Let be a commutative ring with unity and an -oriented compact nonsingular real algebraic variety of dimension . If is any nonsingular complexification of , then the kernel, which we will denote by , of the induced homomorphism is independent of the complexification. In this work, we study and give some of its applications.

     

Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the structure in which the non-zero parallel spinor lives is equivalent to the canonical spin structure associated to the Kähler structure.

     

关于辛同痕问题的一个注记

结合Gromov的思想和McDuff的工作，本文证明了在具有正第一陈类的有理直纹面的给定同调类中的任意两个辛嵌入球面都辛同痕，这样扩充了Gromov的一个经典结果．