arXiv research

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

Study focuses on classifying special geometric structures.

Resolves singular fibers of a 5-manifold with $S^1$ action.

Minimal singular fibers found in nonorientable Lefschetz fibrations.

Study on hyperelliptic Lefschetz fibrations, finding singular fiber counts.

Takamura established a theory on splitting families of degenerations of complex curves. He introduced a powerful method for constructing a splitting family, called a barking family, in which there appear not only a singular fiber over the origin but also singular fibers over other points, called subordinate fibers. In …

Study Stein and Milnor fillings of links from surface singularities.

Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.

Floer homology linked to Milnor fibers for certain singularities.

New Stein fillings found for rational surface singularities.

Divides help construct fibered links from singularities.

In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.

We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.

Study surgery obstructions for Seifert fibered homology spheres using knot and Heegaard Floer homology.

Researchers create metrics collapsing to a line, detailing near-fiber behavior.

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with bo…

The E_8-manifold has several natural framed link descriptions, and we give an efficient method (via `grapes') for showing that they are indeed the same 4-manifold. This leads to explicit handle pictures for the perturbation of singular fibers in an elliptic surface to a collection of fishtails. In the same vein, we sho…

Study plane curve singularities to determine vanishing cycles and monodromy groups.

The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…

The study explores conditions for realizing lens spaces as boundaries of Milnor fibers of hypersurface singularities.

Homotopy equivalence found between Milnor-Lê fibers of specific singularities.

We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the sing…

Compute symplectic cohomology of cAn singularities.

Classifies certain 3D knots with specific properties.

This paper extends previous work on genus two fibrations by studying and resolving singular fibers.

A (compact) manifold with fibered $P$-singularities is a (possibly) singular pseudomanifold $M_Σ$ with two strata: an open nonsingular stratum $\mathring M$ (a smooth open manifold) and a closed stratum $βM$ (a closed manifold of positive codimension), such that a tubular neighborhood of $βM$ is a fiber bundle with fib…

There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of tw…

Paper shows how to join Milnor fibers of real analytic functions.

Given some type of fibration on a 4-manifold $X$ with a torus regular fiber $T$, we may produce a new 4-manifold $X_T$ by performing torus surgery on $T$. There is a natural way to extend the fibration to $X_T$, but a multiple fiber (non-generic) singularity is introduced. We construct explicit generic fibrations (with…

Study shows Lefschetz fibrations on Milnor fibers of certain singularities.

Study symplectic fillings of sandwiched singularities.

Study of twisted Kähler-Einstein metrics on Calabi-Yau spaces with singularities.

Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.

We show that for a C^infty stable map of an oriented 4-manifold into a 3-manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4-manifold.

Study finds negatively curved spheres in elliptic surfaces and their modifications.

The Milnor fiber conjecture is proven for splice type singularities.

We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associat…

The note proves positive currents induced by VKE with mixed singularities.

Theorems prove upper bounds for foliations on closed Alexandrov spaces.

Given a Kähler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular Kähler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted Kähler-Eins…

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…

Researchers found a way to measure the complexity of Seifert fibered spaces with boundaries.

Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.

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.

Sharp decay estimate for Yang-Mills-Higgs fields near singular points.