arXiv subjects

Mutsuo Oka

Publications and source records attributed to Mutsuo Oka.

At least 37 records · Page 2Linked to original sources

arXiv2015

Non-compact Newton boundary and Whitney equisingularity for non-isolated singularities

In an unpublished lecture note, J. Briançon observed that if $\{f_t\}$ is a family of isolated complex hypersurface singularities such that the Newton boundary of $f_t$ is independent of $t$ and $f_t$ is non-degenerate, then the corresponding family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular (and hence topologically equisingular). A first generalization of this assertion to families with non-isolated singularities was given by the second author under a rather technical condition. In the present paper, we give a new generalization under a simpler condition.

math.AG↗

arXiv2015

On the roots of an extended lens equation and an application

We consider a certain mixed polynomial which is an extended Lens equation $L_{n,m}=\bar z^m-p(z)/q(z)$ with $\text{degree}\, q=n$, $\text{degree}\, p<n$ whose numerator is a mixed polynomial of degree $(n+m;n,m)$. Then we consider its deformation of type $L_{n,m}+ε/z^m$ to construct a special mixed polynomial of degree $(n+2m;n+m,m)$ with $5n$ zeros. This generalizes an example of Rhie. We give an application to the number of connected components of the moduli space of strongly mixed weighted homogeneous polynomials of two variables with isolated singularity at the origin.

math.AG↗

arXiv2015

Topology of strongly polar weighted homogeneous links

We consider a canonical $S^1$ action on $S^3$ which is defined by $(ρ,(z_1,z_2))\mapsto (z_1ρ^p,z_2ρ^q)$ for $ρ\in S^1$ and $(z_1,z_2)\in S^3\subset {\mathbb C}^2$. We consider a link consisting of finite orbits of this action, which some of the orbits are reversely oriented. Such a link appears as a link of a certain type of mixed polynomials. We study the space of such links and show smooth degeneration relations.

math.AG↗

arXiv2014

On Milnor fibrations of mixed functions, $a_f$-condition and boundary stability

Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed functions. We also study geometric properties such as Thom's $a_f$ condition, the transversality of the nearby fibers and stable boundary property of the Milnor fibration and their relations.

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arXiv2013

On the fundamental groups of non-generic $\mathbb{R}$-join-type curves

An \emph{$\mathbb{R}$-join-type curve} is a curve in $\mathbb{C}^2$ defined by an equation of the form \begin{equation*} a\cdot\prod_{j=1}^\ell (y-β_j)^{ν_j} = b\cdot\prod_{i=1}^m (x-α_i)^{λ_i}, \end{equation*} where the coefficients $a$, $b$, $α_i$ and $β_j$ are \emph{real} numbers. For generic values of $a$ and $b$, the singular locus of the curve consists of the points $(α_i,β_j)$ with $λ_i,ν_j\geq 2$ (so-called \emph{inner} singularities). In the non-generic case, the inner singularities are not the only ones: the curve may also have \emph{`outer'} singularities. The fundamental groups of (the complements of) curves having only inner singularities are considered in \cite{O}. In the present paper, we investigate the fundamental groups of a special class of curves possessing outer singularities.

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arXiv2012

Contact structure on mixed links

A strongly non-degenerate mixed function has a Milnor open book structures on a sufficiently small sphere. We introduce the notion of {\em a holomorphic-like} mixed function and we will show that a link defined by such a mixed function has a canonical contact structure. Then we will show that this contact structure for a certain holomorphic-like mixed function is carried by the Milnor open book.

math.AG↗

arXiv2012

Mixed functions of strongly polar weighted homogeneous face type

Let $f(\bf z,\bar{\bf z})$ be a mixed polynomial with strongly non-degenerate face functions. We consider a canonical toric modification $π:\,X\to \Bbb C^n$ and a polar modification $π_{\Bbb R}:Y\to X$. We will show that the toric modification resolves topologically the singularity of $V$ and the zeta function of the Milnor fibration of $f$ is described by a formula of a Varchenko type.

math.AG↗

arXiv2011

Intersection theory on mixed curves

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect transversely at $P$, the intersection number $I_{top}(C,C';P)$ is topologically defined. We will generalize this definition to the case when the intersection is not necessarily transversal or either $C$ or $C'$ may be singular at $P$ using the defining mixed polynomials.

math.AG↗

arXiv2010

On mixed plane curves of degree 1

Let $f(\bfz,\bar\bfz)$ be a mixed strongly polar homogeneous polynomial of $3$ variables $\bfz=(z_1,z_2, z_3)$. It defines a Riemann surface $V:=\{[\bfz]\in \BP^{2}\,|\,f(\bfz,\bar\bfz)=0 \}$ in the complex projective space $\BP^{2}$. We will show that for an arbitrary given $g\ge 0$, there exists a mixed polar homogeneous polynomial with polar degree 1 which defines a projective surface of genus $g$. For the construction, we introduce a new type of weighted homogeneous polynomials which we call {\em polar weighted homogeneous polynomials of twisted join type}.

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arXiv2009

On mixed projective curves

Let $f(\bfz,\bar\bfz)$ be a mixed polar homogeneous polynomial of $n$ variables $\bfz=(z_1,..., z_n)$. It defines a projective real algebraic variety $V:=\{[\bfz]\in \BC\BP^{n-1} | f(\bfz,\bar\bfz)=0 \}$ in the projective space $\BC\BP^{n-1}$. The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if $V$ is non-singular. We study a basic property of such a variety.

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arXiv2009

On Mixed Brieskorn Variety

Let $f_{{\bf a},\{bf b}}({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}+...+z_n^{a_n+b_n}\bar z_n^{b_n}$ be a polar weighted homogeneous mixed polynomial with $a_j>0,b_j\ge 0$, $j=1,..., n$ and let $f_{\bf a}({\bf z})=z_1^{a_1}+...+z_n^{a_n}$ be the associated weighted homogeneous polynomial. Consider the corresponding link variety $K_{{\bf a},{\bf b}}=f_{{\bf a},{\bf b}}\inv(0)\cap S^{2n-1}$ and $K_{\bf a}=f_{\bf a}\inv(0)\cap S^{2n-1}$. Ruas-Seade-Verjovsky \cite{R-S-V} proved that the Milnor fibrations of $f_{{\bf a},{\bf b}}$ and $f_{\bf a}$ are topologically equivalent and the mixed link $K_{{\bf a},{\bf b}}$ is homeomorphic to the complex link $K_{\bf a}$. We will prove that they are $C^\infty$ equivalent and two links are diffeomorphic. We show the same assertion for $ f({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}z_2+...+z_{n-1}^{a_{n-1}+b_{n-1}}\bar z_{n-1}^{b_{n-1}}z_n+z_n^{a_n+b_n}\bar z_n^{b_n}$ and its associated polynomial $ g({\bf z})=z_1^{a_1}z_2+...+ z_{n-1}^{a_{n-1}}z_n+z_n^{a_n}$.

math.AG↗

arXiv2009

Non-degenerate mixed functions

Mixed functions are analytic functions in variables $z_1,..., z_n$ and their conjugates $\bar z_1,..., \bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

math.AG↗

arXiv2008

Alexander-equivalent Zariski pairs of irreducible sextics

The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example.

math.AG↗

arXiv2008

Topology of polar weighted homogeneous hypersurfaces

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.

math.AG↗

arXiv2007

A plane sextic with finite fundamental group

We analyze irreducible plane sextics whose fundamental group factors to $D_{14}$. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities $3A_6$, the group is $D_{14}\times Z_3$.

math.AG↗

arXiv2006

Tangential Alexander polynomials and non-reduced degeneration

We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with $θ$^Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration $C_t \to D_0+jL$. We show that there exists a certain surjectivity of the fundamental groups and divisibility among their Alexander polynomials.

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arXiv2005

Zariski pairs on sextics I

We show the existence of sextics of non-torus type which is a Zariski partner of the tame sextics of torus type with simple singularities.

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arXiv2005

Zariski Pairs on sextics II

The complete list of reducible sextics of torus type with simple singularities is known in our previous paper. In this paper, we give a complete list of existence and non-existence of Zariski partner sextics of non-torus type corresponding to the list of sextics of torus type.

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