Related papers: Fibered Multilinks and singularities $f \bar g$

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset…

We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in ${\bf C}^3$. We study irreducible (i.e. $gcd (m,k,l) = 1$) non-isolated (i.e. $1 \leq k < l$)…

Let $f:X@>>>\Bbb P^1$ be a fibered surface with fibers of genus g>1. If f is semistable and non isotrivial we prove that X of non negative Kodaira dimension implies that the number s of singular fibers is at least 5. Information about the…

We consider a mixed function of type $H(z,\bar z)=f(z)\bar g(z)$ where $f,g$ are non-degenerate but they are not assumed to be convenient. We assume that $f=0$ and $g=0$ and $f=g=0$ are non-degenerate and locally tame. We will show that $H$…

The Arakelov--Parshin rigidity theorem implies that a holomorphic Lefschetz fibration $\pi: M \to S^2$ of genus $g \geq 2$ admits only finitely many holomorphic sections $\sigma:S^2 \to M$. We show that an analogous finiteness theorem does…

We study the topology of the boundaries of the Milnor fibers of real analytics map-germs $f: (\mathbb{R}^M,0) \to (\mathbb{R}^K,0)$ and $f_{I}:=\Pi_{I}\circ f : (\mathbb{R}^M,0) \to (\mathbb{R}^I,0)$ that admit Milnor's tube fibrations,…

We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary…

Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the…

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L)…

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra $\mathfrak{g}$. Their theory associates a maximal Poisson-commutative subalgebra of…

Let $f : X\to \Delta$ be a $1$-parameter family of $2$-dimensional isolated hypersurface singularities. In this paper, we show that if the Milnor number is constant, then any semistable model, obtained from $f$ after a sufficiently large…

For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of type $\mathcal{V}$, is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense…

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 study the existence of Milnor fibration on a big enough sphere at infinity for a mixed polynomial $f: \bR^{2n} \to \bR^2$. By using strong non-degeneracy condition, we prove a counterpart of N\'emethi and Zaharia's fibration theorem. In…

We give a new proof of Zariski's multiplicity conjecture in the case of isolated hypersurface singularities; this was first proved by de Bobadilla-Pe\l ka \cite{BobadillaPelka}. Our proof uses the TQFT structure of fixed-point Floer…

In this paper, we shall prove Beauville's conjecture: if $f:S \to P^1$ is a non-trivial semistable fibration of genus g>1, then $f$ admits at least 5 singular fibers. We have also constructed an example of genus 2 with 5 singular fibers.…

This survey is the continuation of a series of works aimed at applying tools from Singularity Theory to Differential Equations. More precisely, we utilize the powerfull Milnor's Fibration Theory to give geometric-topological classifications…

Let $X$ be a complex affine variety in $\mathbb{C}^N$, and let $f:\mathbb{C}^N\to \mathbb{C}$ be a polynomial function whose restriction to $X$ is nonconstant. For $g:\mathbb{C}^N \to \mathbb{C}$ a general linear function, we study the…

We initiate a systematic construction of real analytic Lagrangian fibrations from integer matrices. We prove that when the matrix is of full column rank, the perverse filtration associated with the Lagrangian fibration matches the mixed…

We discuss a new perspective on the dualities among seven-dimensional M-theory on elliptically fibered K3 surfaces, eight-dimensional (8D) heterotic strings on $T^2$, and 8D F-theory on elliptic K3 surfaces. There are several distinct…