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.
Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.
problem Classify and understand nondegenerate fibrations of Euclidean spaces.
method Topological and geometric analysis of nondegenerate fibrations, including continuity at infinity.
result Prove that every germ of a nondegenerate fibration extends to a global fibration.
Introduces new stability concept for Fano fibrations.
problem Stability of Fano fibrations with singularities.
method Introduces f-stability and shows its implications. result Fibered semi log canonical singularities are restricted.
Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.
problem Understanding and constructing singular fibrations over surfaces.
method Explains how to construct examples of singular fibrations with a single singularity and outlines previous results.
result Closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.
Study of line fibrations in R^3, focusing on non-skew cases.
problem Classify line fibrations in R^3, especially non-skew ones.
method Develop parallel plane pushoff technique to study non-skew fibrations.
result Developed a technique to generate nonskew fibrations.
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum G2 manifolds. Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.
problem Understanding achiral Lefschetz fibrations from non-orientable ones.
method Composition of standard orientation double covering map and non-orientable Lefschetz fibration.
result Specification of a monodromy factorization for the composition.
Contact structures are induced by nondegenerate skew fibrations of R^3.
problem Understanding contact structures induced by skew fibrations of R^3.
method Characterizing nondegenerate fibrations and using a recent result on tightness in contact metric 3-manifolds.
result The plane field induced by any nondegenerate fibration is a tight contact structure.
Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…
Same genus-2 fibration structures for specific types found by different researchers.
problem Identifying equivalent genus-2 fibrations of type (4, 3).
method Comparing Lefschetz fibration structures of genus-2 fibrations of type (4, 3).
result Lefschetz fibration structures are the same for genus-2 fibrations of type (4, 3).
Constructs new coassociative fibrations for G2 manifolds.
problem Tackles the construction of new coassociative fibrations for G2 manifolds.
method Constructs fibrations by coassociative 4-folds, relates to hypersymplectic geometry and Donaldson's work.
result Shows natural generalizations of known coassociative fibrations.
We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinit…
Study fibrations over S2 with same singularities, showing monodromies are equivalent up to direct sums.
problem Classifying torus fibrations over S2 up to fibre sum stabilisation. method Analyzing monodromies and using direct sums with certain torus Lefschetz fibrations.
result Global monodromies of fibrations with same singularities are Hurwitz equivalent after direct sums.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.
problem Understanding the topology of Lefschetz fibrations.
method Analyzes upper bounds for the first Betti number, examines monodromy transitivity, and discusses potential fibrations.
result Upper bounds for the first Betti number and insights into Lefschetz fibration properties.
We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resu…
We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …
A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …
Examines properties of holomorphic fibrations in complex geometry.
problem Holomorphic fibrations in complex geometry.
method Analyzes properties of holomorphic fibrations.
result Provides insights into the structure of holomorphic fibrations.
The paper embeds 4-manifolds into CP^2 x CP^1 using Lefschetz fibrations.
problem Embedding 4-manifolds into CP^2 x CP^1.
method Proving Lefschetz fibration embeddings of 4-manifolds into CP^2 x CP^1.
result Every closed, connected, orientable 4-manifold admits a smooth Lefschetz fibration embedding into CP^2 x CP^1.
It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. Th…
Study how singular fibrations affect Poisson cohomology in 4D.
problem Understanding Poisson cohomology in singular fibrations.
method Analyzing the monodromy effect on Poisson cohomology groups.
result Describe the effect of monodromy on Poisson cohomology groups.
The paper extends trisection construction for Lefschetz fibrations with (−n)-sections.
problem Constructing trisections for 4-manifolds with specific Lefschetz fibrations.
method Explicit construction of trisections using monodromies of Lefschetz fibrations.
result Trisections for various 4-manifolds constructed from Lefschetz fibrations.
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.
The paper classifies fibrations of 3-dimensional flat orbifolds.
problem Classifying fibrations of compact flat 3-orbifolds.
method Developed a theory for classifying fibrations of compact flat n-orbifolds, applying it to 3-orbifolds. result All geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
A fibration of Rn by oriented copies of Rp is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of R3 by skew oriente…
In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify…
Study on conditions for singular local tube fibrations.
problem Conditions for singular local tube fibrations.
method Analyzes the most general condition for singular local tube fibrations.
result Provides conditions for the existence of singular local tube fibrations.
Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.
Introduces Hurewicz fibrations for embedding maps of orbifold charts.
problem No specific problem stated; focuses on new concept definition.
method Defines E-fibration embedding and studies its properties.
result Introduces and studies properties of E-fibration embedding.
We construct universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.
The paper studies curvature properties of Monge-Ampère fibrations and their existence.
problem Curvature properties and existence of Monge-Ampère fibrations.
method Analyzes a special relative Kähler fibration with a homogenous Monge-Ampère equation.
result Explicit curvature formulas and bounds for holomorphic curvatures of Monge-Ampère fibrations.
New connection found between Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
problem Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
method Established a compact analog of the P = W conjecture for holomorphic symplectic varieties with Lagrangian fibrations.
result Perverse numbers match Hodge numbers of the total space.
A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…
Local formulas for Poisson structures on wrinkled fibrations are derived.
problem Understanding Poisson structures on specific geometric fibrations.
method Local formulæ for Poisson bivectors and symplectic forms on wrinkled fibrations.
result Local formulas for Poisson structures on wrinkled fibrations are derived.
Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
Paper proves existence of compatible Lefschetz fibrations on 6-ball and Stein domains.
problem Existence of compatible Lefschetz fibrations on Stein domains.
method Topological proof for 6-ball fibrations, construction of relative Stein pairs.
result Existence of compatible Lefschetz fibrations on Stein domains of dimension six.
We show that Poisson fibrations integrate to a special kind of symplectic fibrations, called fibered symplectic groupoids.
The paper connects fibrations to generalized complex structures in semi-toric geometry.
problem Understanding the relationship between fibrations and generalized complex structures.
method Using moment maps in semi-toric geometry and Gompf--Thurston methods for Lie algebroids.
result Constructs self-crossing stable generalized complex four-manifolds and proves compatibility with connected sums.
Fibered knots and disks extend fibrations in 4D space.
problem Extending fibrations from knot complements to disk complements.
method Constructing movies of singular fibrations and identifying sufficient conditions.
result Fibrations extend when disks have exactly two local minima.
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.
In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.
Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.
problem Classify singular fibers in genus 2 algebraic fibrations and relate them to Lefschetz fibrations.
method Analyze four families of hypersurface singularities in C^3, determine resolutions, and find flat deformations into simpler pieces.
result Establish a dictionary between configurations of curves and monodromy factorizations for some genus 2 fibrations.
Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.
problem K-stability of Calabi-Yau fibrations over curves.
method Adiabatic uniform K-stability and log-twisted K-stability of base curves.
result Uniform K-stability of Calabi-Yau fibrations if and only if base curves are K-stable.
Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.
We bound the index of a subgroup in iterated Kodaira fibrations.
problem Bounding the index of a subgroup in iterated Kodaira fibrations.
method Passing to a finite index subgroup of π_1(X) to achieve the desired structure.
result We provide a bound on the index of such a group.