The study describes Lefschetz-Bott fibrations on symplectic manifolds and their applications.
problem Understanding Lefschetz-Bott fibrations on symplectic manifolds.
method Explicit construction and analysis of Lefschetz-Bott fibrations over line bundles.
result Construction of strong symplectic fillings of symplectic manifolds.
New Poisson and near-symplectic structures found on 6D wrinkled fibrations.
problem Finding compatible geometric structures on 6D wrinkled fibrations.
method Extending wrinkled fibrations to 6D and proving the existence of rank-2 Poisson structures and near-symplectic structures.
result 6D wrinkled fibrations can be equipped with Poisson and near-symplectic structures.
The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.
problem Creating symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.
method Producing simply connected, minimal, symplectic Lefschetz fibrations and rationally blowing down Lefschetz fibrations with clustered nodal fibers.
result New constructions of small symplectic exotic 4-manifolds.
We show that Poisson fibrations integrate to a special kind of symplectic fibrations, called fibered symplectic groupoids.
The paper discusses symplectic lifts for actions on Lagrangian fibrations.
problem Understanding symplectic lifts for actions on Lagrangian fibrations.
method Describes symplectic cotangent lifts in terms of 1-cocycles and generalizes to complete Lagrangian fibrations.
result Symplectic lifts of actions for complete Lagrangian fibrations are understood in terms of cohomology.
Construct symplectic Lefschetz fibrations with any signature and spin type.
problem Existence of symplectic Lefschetz fibrations with arbitrary signature.
method Develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature and spin type.
result Solve a long-standing conjecture on the existence of such fibrations with positive signature.
We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces…
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).
We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact struc…
Log-symplectic structures are Poisson structures π on X2n for which ⋀nπ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the b-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…
The study creates Lefschetz fibrations for symplectic fillings of specific surface singularities.
problem Creating Lefschetz fibrations for symplectic fillings of quotient surface singularities.
method Constructing genus-0 or genus-1 positive allowable Lefschetz fibrations. result Minimal symplectic fillings can be derived from rational blowdowns of their resolutions.
Study Lagrangian fibrations using Jacobians and Prym varieties.
problem Understanding Lagrangian fibrations in symplectic geometry.
method Survey of holomorphic symplectic varieties with Jacobians and Prym varieties as fibres.
result Characterization of Lagrangian fibrations using Jacobians and Prym varieties.
New symplectic 4-manifolds created from genus 2 fibrations.
problem Creating new symplectic 4-manifolds with small topology.
method 2-nodal spherical deformations of singular fibers of genus 2 fibrations.
result Construction of exotic symplectic 4-manifolds homeomorphic but not diffeomorphic to known manifolds.
The study shows that symplectic Lefschetz fibrations can have infinitely many sections.
problem The finiteness of sections in Lefschetz fibrations.
method General criterion and examples for symplectic Lefschetz fibrations with infinitely many sections.
result Symplectic Lefschetz fibrations can have infinitely many homologically distinct sections.
The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefsch…
Develops a correspondence between symplectic orbits and Grassmannians.
problem Understanding the homotopy types of Grassmannians of linear subspaces in symplectic vector spaces.
method Uses orbit fibrations and linear symplectic reduction to compute homotopy types.
result Recover observations from Arnold, Oh-Park, and Lee-Leung in different cases.
The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.
problem Analyzing the isotopy of C-symplectic structures and their applications.
method Proves an analogue of Moser's isotopy theorem for families of C-symplectic structures.
result Locally trivial degenerate twistorial deformation over the base of holomorphic Lagrangian fibrations.
In this paper we give an explicit construction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study the numerical properties of the sections in symplectic Lefschetz fibrations and their relation to the structure of the mon…
Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow …
Study intersection cohomology and Lagrangian fibrations in symplectic varieties.
problem Understanding the intersection cohomology and perverse filtration of Lagrangian fibrations in symplectic varieties.
method Analyzes the deformation equivalence class, computes the border of the perverse diamond, and identifies perverse and Hodge numbers.
result Complete description of intersection cohomology and invariant cohomology classes of fibers.
Proves unique symplectic Lefschetz fibration from Morse functions.
problem Mapping Morse functions to symplectic Lefschetz fibrations.
method Homotopically unique complex-valued symplectic Lefschetz fibration on cotangent bundles.
result Existence and uniqueness of symplectic Lefschetz fibrations.
New method constructs stable generalized complex structures.
problem Creating stable generalized complex structures.
method Developed Gompf-Thurston symplectic techniques adapted to Lie algebroids, used to construct stable structures from log-symplectic structures.
result Introduced boundary Lefschetz fibrations to obtain stable structures from genus one Lefschetz fibrations.
The well-known fact that any genus g symplectic Lefschetz fibration X4→S2 is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manif…
In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurf…
Quantizes symplectic fibrations to analyze vector bundles and metrics.
problem Quantizing higher rank vector bundles and understanding their metrics.
method Relates Berezin-Toeplitz quantization to hybrid systems and symplectic fibrations.
result Established refined estimates for computing balanced metrics on Kähler manifolds.
Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
problem Understanding the structure of holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds.
method Analyzing the properties of Lagrangian fibrations and special Kähler structures.
result Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
Extends Kähler metrics theory to symplectic manifolds with toric actions.
problem Extending invariant Kähler metrics theory to symplectic manifolds with toric actions.
method Using Delzant subspaces and Lagrangian fibrations, establishing a correspondence between metrics and connections.
result Characterizes extremal invariant Kähler metrics as those with scalar curvature on base integral affine manifold.
In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-man…
In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.
6D symplectic manifold with many homologous but inequivalent submanifolds.
problem Finding many symplectic submanifolds in a 6D manifold.
method Hyperelliptic Lefschetz fibrations on 4-manifolds.
result Infinitely many mutually homologous but homotopy inequivalent submanifolds.
Spin Lefschetz fibrations can represent any group and lattice point.
problem Representing groups and lattice points using Lefschetz fibrations.
method Proving any finitely presented group and admissible lattice point can be realized as a spin Lefschetz fibration.
result Spin Lefschetz fibrations can represent any group and lattice point.
Constructs families of Toeplitz operators for symplectic fibrations.
problem Quantization of symplectic fibrations.
method Smooth families of Szegö projections and Toeplitz operators.
result Deformation quantization of prequantizable symplectic fibrations.
The paper connects moment maps to the stability of holomorphic fibrations.
problem Stability of holomorphic fibrations.
method Use of moment maps and K-stability criteria.
result Existence of optimal symplectic connections implies stability of fibrations.
We show that any 4-manifold, after surgery on a curve, admits an achiral Lefschetz fibration. In particular, we show that the connected sum of any simply connected 4-manifold with a 2-sphere bundle over the 2-sphere will admit an achiral Lefschetz fibration. We also show these surgered manifolds admit near-symplectic s…
New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.
problem Deriving new symplectic forms from existing ones.
method Proving existence of degenerate twistorial deformations.
result Existence of degenerate twistorial deformations preserving complex structures.
We present a generalized Weinstein Tubular Neighbourhood theorem for Lagrangian subbundles of Symplectic fibrations. This is then used to study the space of Lagrangian fibrations of symplectic manifolds.
The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.
problem Understanding and constructing new symplectic 4-manifolds.
method Explicitly constructing Lefschetz pencils on hyperelliptic Lefschetz fibrations.
result Infinite families of symplectic 4-manifolds are diffeomorphic to fiber sums of standard hyperelliptic Lefschetz fibrations.
Constructs inequivalent Lefschetz fibrations on various surfaces.
problem Finding inequivalent Lefschetz fibrations on surfaces.
method Explicit construction of fibrations on rational and ruled surfaces.
result Explicit construction of inequivalent fibrations on all rational and ruled surfaces.
The paper studies deformations of Lagrangian fibrations on symplectic manifolds.
problem Understanding deformations of Lagrangian fibrations on holomorphic symplectic manifolds.
method Analyzes degenerate twistor deformations and meromorphic sections.
result Compact hyperkahler manifolds with primitive fibers admit meromorphic sections.
New method proves achiral Lefschetz fibrations exist using Riemannian geometry.
problem Existence of achiral Lefschetz fibrations.
method Riemannian geometry and Smith signature formula extension.
result Existence of achiral Lefschetz fibrations proven.
Study symplectic structures on open 4-manifolds with distinct primitive cohomologies.
problem Identifying symplectic structures with non-isomorphic primitive cohomology.
method Analyze fibrations and compute monodromy explicitly for specific link complements.
result Symplectic structures on open 4-manifolds can have different primitive cohomology dimensions.
The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.
problem Formulating Hamiltonian actions of Lie 2-groups on orbifolds.
method Constructing automorphisms of Lie groupoids, using 2-group actions and Kan fibrations.
result Symplectic reductions of Lie 2-group actions on orbifolds are Lie 2-groupoids under certain conditions.
Introduces stability for families of K-polystable varieties and connects it to optimal symplectic connections.
problem Forming correct moduli for fibrations and understanding their geometry.
method Introduces a new stability condition for fibrations and relates it to the existence of optimal symplectic connections.
result Proves that the existence of an optimal symplectic connection implies semistability of the fibration.
New symplectic caps and embeddings found in complex projective plane.
problem Embeddings of homology balls in complex projective plane.
method Handlebody construction of symplectic caps and embeddings.
result First examples of symplectic handlebody decompositions of a closed symplectic 4-manifold.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with b2≥7 admits only finitely…
Study symplectic fillings of sandwiched singularities.
problem Contrast deformation theory and symplectic topology of Milnor fibers.
method Develop an analog of de Jong--van Straten's theory in the symplectic setting using spinal open books and nearly Lefschetz fibrations.
result Minimal symplectic fillings of links are generated by certain immersed disk arrangements.
A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove…