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)…
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48 results for “log symplectic manifolds”
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.
A log symplectic manifold is a Poisson manifold which is generically nondegenerate. We develop two methods for constructing the symplectic groupoids of log symplectic manifolds. The first is a blow-up construction, corresponding to the notion of an elementary modification of a Lie algebroid along a subalgebroid. The se…
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentu…
We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry.
Study isotopy of rational cuspidal curves in 4-manifolds.
problem Isotopy of rational cuspidal curves in 4-manifolds.
method Tame symplectic curves, pseudo-holomorphic curves, log pairs, 4-dimensional topology.
result Every rational cuspidal curve is isotopic to a complex curve in degrees up to 5.
We compute the Poisson cohomology of a class of Poisson manifolds that are symplectic away from a collection of hypersurfaces. These Poisson structures induce a generalization of symplectic and cosymplectic structures, which we call a k-cosymplectic structure, on the intersection of hypersurfaces in .
Log-symplectic structures are Poisson structures on for which vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the -tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…
Let M be a compact, connected symplectic 2n-dimensional manifold on which an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic orbits, we show that the Duistermaat-Heckman measure of the T-action is log-concave.…
We construct a compact symplectic manifold with a Hamiltonian circle action for which the Duistermaat-Heckman function is not log-concave.
Lagrangian branes with boundary and symplectic methods for stable generalized complex manifoldsmath.DG
This paper introduces new Lagrangian branes in stable generalized complex manifolds.
problem Understanding stable generalized complex manifolds and their properties.
method Using log symplectic geometry and Floer theory techniques.
result Lagrangian branes with boundary are introduced and their properties are studied.
Defines log Floer cohomology for symplectic surfaces with a degenerate part.
problem Extending Floer cohomology to degenerate symplectic structures.
method Definition of log Floer cohomology for oriented log symplectic surfaces.
result Log Floer cohomology is invariant under isotopies and isomorphic to log de Rham cohomology for a single Lagrangian.
New Lie groups found for Poisson diffeomorphisms.
problem Finding Lie group structures on Poisson diffeomorphism groups.
method Using Poisson groupoids, develop Lie group structures.
result Poisson diffeomorphism groups of various Poisson manifolds are regular Lie groups.
Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed points set is the disjoint union of two copies of . In this article, for any closed symplectic four manifold with greater than 1, we show that there is a…
New Poisson structures defined from Lie algebroids, with conditions for existence.
problem Existence conditions for a new class of Poisson structures.
method Definition of algebroid desingularizable Poisson manifolds and infinitesimal obstruction.
result Characterization of desingularizable Poisson structures in terms of Lie algebra properties.
We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian…
We describe the space of Poisson bivectors near a log-symplectic structure up to small diffeomorphisms.
We consider Lagrangian-like submanifolds in certain even-dimensional 'symplectic-like' Poisson manifolds. We show, under suitable transversality hypotheses, that the pair consisting of the ambient Poisson manifold and the submanifold has unobstructed deformations and that the deformations automatically preserve the Lag…
The paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.
problem Growth rate of Reeb orbits on star-shaped hypersurfaces.
method Analyzing fiberwise star-shaped hypersurfaces in cotangent bundles with topological conditions.
result The number of Reeb orbits with period at most T grows at least like T/log(T).
The paper explores obstructions for symplectic Lie algebroids on surfaces.
problem Obstacles to the existence of symplectic Lie algebroids.
method Analysis through characteristic classes of symplectic Lie algebroids.
result Full obstructions for surfaces to carry symplectic Lie algebroids.
This paper studies Poisson structures defined by divisor ideals.
problem Understanding Poisson structures with degeneracy captured by divisor ideals.
method Developed a framework using divisor ideals and Lie algebroids.
result Effective methods for studying Poisson structures of divisor-type.
The paper tackles isotropy of symplectic forms using Hodge flows.
problem Whether symplectic forms in a given class are isotropic.
method Introduces nonlinear Hodge heat flows to study isotropy.
result The flow converges to the symplectic form smoothly for any initial symplectic form in the class.
Unique optimal symplectic connections found for submersions.
problem Finding unique optimal symplectic connections for submersions.
method Analytic results and geometric partial differential equations.
result Optimal symplectic connections are unique up to automorphism group.
Abstract: Surveying aspects of complex dimension two anti-canonical pairs.
problem Smooth topology, algebraic geometry, symplectic geometry, and contact geometry of anti-canonical pairs.
method Survey and review of existing work.
result Survey of various geometric properties of anti-canonical pairs.
Smooth resolutions found for quotient of R^2 by infinite discrete groups.
problem Symplectic resolutions of quotient spaces by infinite discrete subgroups.
method Constructing smooth symplectic resolutions for R^2 under infinite discrete subgroups of GL_2(R).
result Minimal resolutions of Du Val singular varieties are symplectic resolutions of R^2/G.
We investigate the relationship between stability and the existence of extremal Kähler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. We introduce a space of symplectic potentials for toric manifolds, which …
Local index theorem for cofinite hyperbolic Riemann surfaces derived from computational perspective.
problem Deriving the local index theorem for cofinite Riemann surfaces.
method Using Ahlfors' variational formulas and projection formulas, deriving integral formulas for variations of determinants.
result Explicit integral formulas for variations of and .
Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.
problem Understanding the moduli spaces of quasimaps and Calabi-Yau fibrations.
method Constructing a projective K-moduli space of quasimaps and investigating relationships with Calabi-Yau fibrations.
result Entire quasi-projectivity and ampleness of the CM line bundle on the normalization of the K-moduli space of Calabi-Yau fibrations.
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
Let X be a complex manifold with strongly pseudoconvex boundary M. If u is a defining function for M, then -log u is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form s = i \del \delbar(-log u) is a symplectic structure on the complement of M in a neighborhood in X of M; it blows up along M. The Poiss…
Given a compact oriented surface, we classify log Poisson bi-vectors whose degeneracy loci are locally modeled by a finite set of lines in the plane intersecting at a point. Further, we compute the Poisson cohomology of such structures and discuss the relationship between our classification and the second Poisson cohom…
A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic technique…
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.
Study symplectic embeddings of 4-manifolds using Lefschetz fibrations.
problem Proper symplectic and iso-symplectic embeddings of 4-manifolds in 6-manifolds.
method Use Lefschetz fibrations to study symplectic embeddings.
result Closed orientable smooth 4-manifolds admitting Lefschetz fibrations over CP^1 can be embedded symplectically in (CP^1 × CP^1 × CP^1, ω_pr).
Geodesics in Kähler metrics connect metrics with constant scalar curvature.
problem Deriving geodesics for relatively Kähler metrics on fibrations.
method Deriving geodesic equation, proving uniqueness, convexity of log-norm functional.
result Fibrations with optimal symplectic connections are polystable.
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 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.
This paper provides a new method to construct -symplectic toric manifolds from toric manifolds.
problem Classifying and constructing -symplectic toric manifolds.
method A new method to construct -symplectic toric manifolds from toric manifolds.
result This new method allows for the decomposition of -symplectic toric manifolds into toric manifolds.
Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.
problem Classifying and understanding different types of contact metric manifolds.
method Investigating metric structures on symplectizations and proving the existence of a unique metric symplectization.
result Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.
Proves h-principle for symplectic foliations on closed manifolds.
problem Symplectic foliations on closed manifolds.
method Proves h-principle for regular symplectic foliations.
result Establishes h-principle for symplectic foliations on closed manifolds.
The study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.
problem Existence of conformal symplectic foliations on closed manifolds.
method Analysis of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5.
result Construction of conformal symplectic foliations on closed, simply-connected, almost contact manifolds in dimension 5.
On the realization of symplectic algebras and rational homotopy types by closed symplectic manifoldsmath.AT
Study realizes symplectic algebras and homotopy types on manifolds.
problem Realizing symplectic algebras and homotopy types on manifolds.
method Addressing questions on realizability of symplectic algebras and rational homotopy types by closed symplectic manifolds.
result Realization of symplectic algebras and homotopy types in various dimensions.
Proves Arnold conjecture for singular symplectic manifolds using novel techniques.
problem Hamiltonian dynamics on singular symplectic manifolds.
method Introducing smooth symplectic forms to singular symplectic structures under mild conditions, using Floer homology.
result Proves a lower bound on the number of 1-periodic Hamiltonian orbits for -symplectic manifolds.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
New method constructs symplectic structures on 4-manifolds from trisections.
problem Characterize 4-manifolds admitting symplectic structures.
method Explicit criteria on trisections allow construction of symplectic structures.
result New characterization of 4-manifolds with symplectic structures.
Develops a diagrammatic method for symplectic filling classifications.
problem Classifying exact/weak symplectic fillings of 3D contact manifolds.
method Symplectic JSJ decomposition applied to contact surgery diagrams.
result Recover symplectic fillings for certain lens spaces and torus bundles, and classify fillings for a large class of plumbed 3-manifolds.
We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of th…