Related papers: New Open-book Decompositions in Singularity Theory

We say that the the germ of a singular holomorphic foliation on $(\mathbb{C}^2,0)$ is algebraizable whenever it is holomorphically conjugate to the singularity of a foliation defined globally on a projective algebraic surface. The object of…

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group…

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if…

In a previous paper \cite{SV}, the authors studied the isolated singular fibers that can occur in algebraic fibrations of certain genus two fibrations. There the goal was to determine their monodromy factorizations with the goal of…

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More…

By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over $\mathbb{S}^{2}$. We show that $K3\#2 \overline{\mathbb{CP}}{}^{2}$ can be rationally blown down along seven disjoint copies…

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for…

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes…

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded…

We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain…

This article is concerned with the geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of {\it Bishop quadrics}. There are two kinds of…

As we known, the {\it Seifert-Van Kampen theorem} handles fundamental groups of those topological spaces $X=U\cup V$ for open subsets $U, V\subset X$ such that $U\cap V$ is arcwise connected. In this paper, this theorem is generalized to…

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups…

From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field…

It is well known that the diffeomorphism-type of the Milnor fibration of a (Newton) non-degenerate polynomial function $f$ is uniquely determined by the Newton boundary of $f$. In the present paper, we generalize this result to certain…

In case of one-dimensional singular locus, we use deformations in order to get refined information about the Betti numbers of the Milnor fibre.

For the link $M$ of a normal complex surface singularity $(X,0)$ we ask when a knot $K\subset M$ exists for which the answer to whether $K$ is the link of the zero set of some analytic germ $(X,0)\to (\mathbb C,0)$ affects the analytic…

Using nonstandard analysis we define a topology on the ring of germs of functions: $(mathbb R^n,0)\rightarrow(mathbb R,0)$. We prove that this topology is absolutely convex, Hausdorff, that convergent nets of continuous germs have…

Let $f: X \to S$ be a unipotent degeneration of projective complex manifolds over a disc such that the reduction of the central fibre $Y=f^{-1}(0)$ is simple normal crossings, and let $X_\infty$ be the canonical nearby fibre. Building on…

We introduce a new type of singularity for smooth maps from $4$-manifolds to surfaces, called an $M$-singularity, whose critical locus is a circle contained in a single fiber. We show that the monodromy around an $M$-singularity is a…