Related papers: Invariant Subspaces of the Monodromy
Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism…
This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and…
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…
We derive a formula for the Milnor class of scheme-theoretic global complete intersections (with arbitrary singularities) in a smooth variety in terms of the Segre class of its singular scheme. In codimension one the formula recovers a…
It is known that the lens space $L(2n,1)$ supports a virtually overtwisted contact structure arising as the boundary of the Milnor fiber of a complex hypersurface singularity. In this article we study the problem of realizing other…
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of…
Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber $F$ can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit…
We are solving for the case of flat superspace some homological problems that were formulated by Berkovits and Howe. (Our considerations can be applied also to the case of supertorus.) These problems arise in the attempt to construct…
In arXiv:1911.08213 it was conjectured that the compactly supported cohomology of the $m$-th restricted contact locus of an isolated hypersurface singularity coincides, up to a shift, with the Floer cohomology of the $m$-th iterate of the…
Given a nonconstant polynomial map over the reals having an isolated critical point in the origin and with zero locus of positive dimension, we establish a formula for the singular homology groups of a Milnor fibre relative to its boundary.
The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2\pi i c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture…
We find and describe unexpected isomorphisms between two very different objects associated to hypersurface singularities. One object is the Milnor algebra of a function, while the other object associated to a singularity is the local ring…
We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3).…
Let f be a hypersurface surface local singularity whose zero set has 1-dimensional singular locus. We develop an explicit procedure that provides the boundary of the Milnor fibre of f as an oriented plumbed 3-manifold. The method provides…
We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception.
It is known that the structure of invariant subspaces of the Hardy space $H^2(\mathbb D^n)$ on the polydisc $\mathbb{D}^n$ is very complicated; hence, we need good examples help us to understand the structure of invariant subspaces of…
We discuss and prove a number of cohomological results for Milnor fibers, real links, and complex links of local complete intersections with singularities of arbitrary dimension.
This is a review article on the combinatorial aspects of the mixed Hodge structure of a Milnor fibre of the isolated hypersurface singularity. We give a purely combinatorial method to compute spectral pairs of the singularity under the…
In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally…
In this paper, we prove that the Milnor fibre of a singularity over an i.c.i.s. of dimension 3 has the homotopy type of a bouquet of spheres, provided that the function that defines the singularity has finite extended codimension with…