We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show th…
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48 results for “symplectic non-fillability”
The paper introduces new contact structure modifications via round surgery.
problem Studying higher-dimensional contact structures and their modifications.
method Contact round surgery to introduce three types of modifications: overtwisted, weakly symplectically non-fillable, and strongly symplectically non-fillable.
result The introduced modifications can be realized by sequences of contact round surgeries and contribute to the Euler classes of contact structures.
We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fil…
The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions and for under certain conditions.
Study confoliations' symplectic fillability, finding obstructions.
problem Understanding confoliations' symplectic fillability in higher dimensions.
method Generalized Massot-Niederkrüger-Wendl's bordered Legendrian open book for confoliations.
result Obstructions to weak symplectic fillability for confoliations.
Study non-fillable curves in a hyperbolic surface with a real line.
problem Non-fillable curves in
method Asymptotic Plateau problem in
result First examples of non-fillable finite curves with no thin tail.
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S^{2n-1}, n>1, an exotic contact structure ξ_- that also contains a plastikstufe. …
A geometric obstruction, the so called "plastikstufe", for a contact structure to not being fillable has been found by K. Niederkruger. This generalizes somehow the concept of overtwisted structure to dimensions higher than 3. This paper elaborates on the theory showing a big number of closed contact manifolds with a "…
In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.
On generalizing Lutz twistsmath.SG
We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that has at least three…
We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology …
New augmentations of twist knots found that can't be filled.
problem Finding augmentations of twist knots that cannot be filled by orientable Lagrangian fillings.
method Using a Floer-theoretic version of a result from microlocal sheaf theory, showing augmentations cannot be induced by algebraic tori.
result Established new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings.
Study on Legendrian and transverse realizations of negative torus knots.
problem Classification of transverse and Legendrian realizations of negative torus knots.
method Analysis of contact structures, knot Floer homology, Legendrian surgeries.
result Classification of strongly non-loose transverse and Legendrian realizations.
Defines symplectic sectional curvature and its properties.
problem Defines symplectic sectional curvature and its properties.
method Defines and characterizes symplectic sectional curvature in terms of curvature tensor and its covariant derivatives.
result Characterizes symplectic sectional curvature in terms of curvature tensor and its covariant derivatives.
Log-symplectic structures on fibrations are explored using symplectic techniques.
problem Existence of log-symplectic structures on fibrations.
method Using Lie algebroid and symplectic techniques, introduce -hyperfibrations.
result Log-symplectic structures on fibrations are linked to achiral Lefschetz fibrations and folded-symplectic structures.
Study blow-ups of locally conformal symplectic manifolds.
problem Blow-ups of locally conformal symplectic manifolds.
method Show existence of locally conformal symplectic structure on blow-ups.
result Existence of locally conformal symplectic structure on blow-ups.
A symplectic manifold 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 symple…
Constructs curves in log-symplectic manifolds, classifying and obstructing certain structures.
problem Classifying and understanding curves in log-symplectic manifolds.
method Constructs moduli spaces of curves, uses symplectic field theory.
result Classifies symplectically ruled log-symplectic 4-manifolds, obstructs contact boundary components.
We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #m CP^2 # n bar(CP^2)…
Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be -approximated by PL symplectic manifolds.
Symplectic 4-manifolds can be divided into three parts with a special structure.
problem Understanding the structure of symplectic 4-manifolds.
method Proved the existence of a trisection compatible with the symplectic structure.
result Symplectic 4-manifolds admit a trisection compatible with the symplectic structure.
The paper defines pre-symplectic algebroids and their applications.
problem Understanding the geometric structure of symplectic Lie algebroids.
method Introducing pre-symplectic algebroids and establishing their correspondence with symplectic Lie algebroids.
result Pre-symplectic algebroids are geometric structures underlying symplectic Lie algebroids.
We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…
Solves symplectic and conformal symplectic group actions equivalence problem.
problem Equivalence problem for symplectic and conformal symplectic group actions.
method Computing differential invariants via the Lie-Tresse theorem.
result Solves equivalence problem for symplectic and conformal symplectic group actions.
We consider aspherical manifolds with torsion-free virtually polycyclic fundamental groups, constructed by Baues. We prove that if those manifolds are cohomologically symplectic then they are symplectic. As a corollary we show that cohomologically symplectic solvmanifolds are symplectic.
We present some methods to construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or non-symplectic cyclic isotropy point sets. All such actions are not compatible with any symplectic form.
New insights into symplectic loops and their flux groups.
problem Understanding symplectic loops and their properties.
method Analyzing symplectic diffeomorphisms and their orbits.
result Flux of symplectic loops vanishes for contractible orbits.
Anti-symplectic involutions connect a sphere in a symplectic surface.
problem Understanding involutions on Lagrangian spheres in symplectic quadrics.
method Using Hamiltonian isotopy to show connections between involutions.
result Anti-symplectic involutions are Hamiltonian isotopic.
We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite (-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds n…
Method constructs complex symplectic Lie algebras from simpler ones.
problem Classifying complex symplectic Lie algebras of various dimensions.
method Complex symplectic oxidation method
result Classification of eight-dimensional nilpotent complex symplectic Lie algebras.
Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
Characterizes flat affine symplectic Lie groups and their properties.
problem Characterizing flat affine symplectic Lie groups.
method Using symplectic étale affine representations and central translations.
result Obtains nontrivial examples of flat affine symplectic Lie groups in every even dimension.
A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…
Study on symplectic semi-characteristic using cohomology and vector fields.
problem Defining and calculating the symplectic semi-characteristic of symplectic manifolds.
method Defined using even-degree primitive cohomology and proved a counting formula using vector fields.
result Established a counting formula for symplectic semi-characteristic and derived vanishing properties.
Extends symplectic flow results to foliations.
problem Applying hard Lefschetz theorem to symplectic foliations.
method Generalizes results from symplectic flows to foliations.
result Extends transversal hard Lefschetz theorem to transversely symplectic foliations.
The study creates symplectic examples with non-trivial homotopy groups.
problem Constructing symplectic manifolds with specific homotopy groups.
method Examples of symplectic manifolds with non-trivial second homotopy groups in various dimensions.
result Examples of symplectic manifolds with non-trivial homotopy groups in dimensions 4 and greater than 6.
Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.
problem Characterizing solvable symplectic Lie algebras.
method Symplectic double extension process.
result Classifies Lie algebras of dimensions up to 6 and proves structural theorems.
We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symple…
New models for symplectic structures on classifying stacks.
problem Building models for symplectic structures on classifying stacks.
method Introducing -shifted symplectic Lie -groupoids and constructing explicit symplectic Morita equivalences.
result Explicit symplectic Morita equivalences between models of the 2-shifted symplectic structure on classifying stacks.
Symplectic Rational Blow-upmath.SG
Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear plumbing configuration of spheres is replaced with a rational homology ball , . Subsequently, Symington [Sy] defined this procedure in the symplectic category, where a symplectic (given…
Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.
Symplectic singularitiesmath.SG
We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic and singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
Symplectic structures simplified for compact manifolds.
problem Locally conformally symplectic structures on compact manifolds.
method Symplectic analogue of Vaisman's theorem.
result Locally conformally symplectic structures become globally symplectic.
Flat symplectic Lie algebrasmath.DG
The study of flat symplectic Lie algebras and groups.
problem Characterizing and understanding flat symplectic Lie algebras and groups.
method Analyzing the derived ideal, curvature, and double extension process.
result Every flat symplectic Lie algebra is obtained by a sequence of double extensions starting from the trivial algebra.
Study of symplectic and Poisson reduction, proposing Poisson implosion.
problem Understanding and generalizing symplectic reduction to Poisson manifolds.
method Recalled and reviewed symplectic and Poisson reduction, proved cross-section theorem for Poisson manifolds.
result Generalized Guillemin-Sternberg theorem for Poisson manifolds, identified Poisson transversals.
New symplectic barriers found in ball embeddings.
problem Existence of symplectic embeddings with intersections.
method Proving obligatory intersections with symplectic planes.
result Existence of symplectic barriers in ball embeddings.