A cocycle on the group of symplectic diffeomorphisms

We define a cocycle on the group of symplectic diffeomorphisms of a symplectic manifold and investigate its properties. The main applications are concerned with symplectic actions of discrete groups. For example, we give an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.Comment: 19 pages, no figures, corrected versio

Types de familles, conditions de vie, fonctionnement du système familial et inadaptation sociale au cours de la latence et de l’adolescence dans les milieux défavorisés

Les données manquent concernant l'impact de certains types de familles, et les résultats sont souvent discordants concernant l'inadaptation. Après avoir décrit les variations de l'activité délictueuse et des troubles de comportement selon les types de familles, nous analysons les difficultés de fonctionnement du système familial. Six types de familles sont comparés pour 763 garçons de 10 ans, 319 adolescentes et 426 adolescents de 14 et 15 ans: les familles intactes, les familles monoparentales patricentriques et matricentriques, les familles recomposées patricentriques et matricentriques et les familles substituts. Les données présentées montrent qu'en cette fin des années 1980, près de 40 % des enfants et des adolescents des quartiers à faible statut socio-économique de Montréal vivent dans des familles désunies. Les données confirment en outre une observation classique: les familles désunies, en comparaison aux familles intactes, sont défavorisées sur le plan des conditions de vie, déficientes sur le plan du fonctionnement psychosocial et propices aux troubles de comportement et à l'activité délictueuse. Par ailleurs, il est établi que certains types de familles désunies constituent un facteur de risque considérable. L'effet dommageable de la structure de la famille s'accroît dans l'ordre suivant: familles intactes, familles monoparentales matricentriques, familles recomposées matricentriques, familles substituts, familles recomposées patricentriques et familles monoparentales patricentriques. Pour terminer, quelques pistes d'intervention pour la prévention des difficultés comportementales et familiales sont proposées.Available data on the impact of certain types of families is lacking, and the results are often misleading with respect to maladjustment. Following a description of variations in delinquent activity and behaviour problems according to family type, the authors analyse the difficulties in the operation of family systems. Comparisons of six family types apply to data from 763 boys aged 10, 319 female and 426 male adolescents, aged 14 and 15: intact families, father-based and mother-based single- parent families, father-based and mother-based reconstituted families and substitute families. The article's data show that in the late eighties, nearly 40 per cent of children and adolescents living in low-income districts in Montreal belonged to disunited families. In addition, the data confirm a classic observation: in comparison with intact families, disunited families are underprivileged in relation to living conditions, deficient in relation to psychosocial functioning, and propitious to behaviour problems and delinquent activity. In addition, it has been established that certain disunited family types represent a considerable risk factor. The damaging effect of family structure increases in the following order: intact families, mother-based single-parent families, mother-based reconstituted families, substitute families, father-based reconstituted families and father-based single-parent families. Finally, certain intervention methods are suggested to help prevent behaviour and family problems

Symplectic geometry on moduli spaces of J-holomorphic curves

Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J.Comment: 16 page

Global symplectic coordinates on gradient Kaehler-Ricci solitons

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,\omega)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional complete Kaehler manifolds with non-negative sectional curvature globally symplectomorphic to $R^{2n}$ through a symplectomorphism satisfying Ciriza's property.Comment: 8 page

Inequivalent contact structures on Boothby-Wang 5-manifolds

We consider contact structures on simply-connected 5-manifolds which arise as circle bundles over simply-connected symplectic 4-manifolds and show that invariants from contact homology are related to the divisibility of the canonical class of the symplectic structure. As an application we find new examples of inequivalent contact structures in the same equivalence class of almost contact structures with non-zero first Chern class.Comment: 27 pages; to appear in Math. Zeitschrif

Seidel elements and mirror transformations

The goal of this article is to give a precise relation between the mirror symmetry transformation of Givental and the Seidel elements for a smooth projective toric variety $X$ with $-K_X$ nef. We show that the Seidel elements entirely determine the mirror transformation and mirror coordinates.Comment: 36 pages. We corrected several issues as pointed out by the refere

Discontinuous symplectic capacities

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.Comment: We include generalizations to higher dimensions due to the unknown referee and Janko Latschev. We add examples of open sets with connected boundary on which the shell capacity is not continuous. 3rd and 4th version: minor changes, to appear in J. Fixed Point Theory App

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat

Massey products in symplectic manifolds

The paper is devoted to study of Massey products in symplectic manifolds. Theory of generalized and classical Massey products and a general construction of symplectic manifolds with nontrivial Massey products of arbitrary large order are exposed. The construction uses the symplectic blow-up and is based on the author results, which describe conditions under which the blow-up of a symplectic manifold X along its submanifold Y inherits nontrivial Massey products from X ot Y. This gives a general construction of nonformal symplectic manifolds.Comment: LaTeX, 48 pages, 2 figure

Gauged vortices in a background

We discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism. At critical self-coupling, and for low temperatures, it has been argued that the configuration space for vortex dynamics in each topological class of the abelian Higgs model approximately truncates to a finite-dimensional moduli space with a Kaehler structure. For the case where the vortices live on a 2-sphere, we explain how localisation formulas on the moduli spaces can be used to compute explicitly the partition function of the vortex gas interacting with a background potential. The coefficients of this analytic function provide geometrical data about the Kaehler structures, the simplest of which being their symplectic volume (computed previously by Manton using an alternative argument). We use the partition function to deduce simple results on the thermodynamics of the vortex system; in particular, the average height on the sphere is computed and provides an interesting effective picture of the ground state.Comment: Final version: 22 pages, LaTeX, 1 eps figur