Arnaud Bodin
For a polynomial $f$ in two complex variables, we prove that the multi-link at infinity of the 0-fiber $f^{-1}(0)$ is a fibred multi-link if and only if all the values different from 0 are regular at infinity.
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In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and indices (topological degree) of appropriated vector fields defined on spheres of radii small or big enough.
We give a necessary condition for a meromorphic function in several variables to give rise to a Milnor fibration of the local link (respectively of the link at infinity). In the case of two variables we give some necessary and sufficient conditions for the local link (respectively the link at infinity) to be fibred.
In the present paper, we study deformations of polar weighted homogeneous polynomials which are also polar weighted homogeneous polynomials. We describe a round handle decomposition of the Milnor fibration of a deformation of a polar weighted homogeneous polynomial concretely and give the number of round handles by the number of positive and negative components of the links of singularities appearing before and after the deformation. We also give a formula of characteristic polynomials of these singularities by using the decomposition of the monodromy of the Milnor fibration induced by a round…
We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which we call the LF-property. We will show that the shadowed polyhedron constructed from a divide satisfies this property and the Lefschetz fibration of this polyhedron is isomorphic to the Lefschetz fibration of the divide. Furthermore, applying the same technique to certain free divides we will show that the links of those free divides are fibered with positive monodromy.
Slopes of trigonal fibred surfaces and of higher dimensional fibrations
本文推广Cornalba-Harris方法，在GIT半稳定条件下给出高维纤维化斜率下界，并证明偶数亏格三角纤维化斜率的锐利下界，从而证实Harris-Stankova猜想。
We consider a real analytic map $F=(f_1,...,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)$, $2 \le k \le n-1$, that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic $f_I=(f_{i_1},...,f_{i_l}) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^l,0)$, induced from $F$ by projections where $1 \le l \le n-2$ and $I=\{i_1,...,i_l\}$, also satisfies Milnor's conditions (a) and (b). We give several relations between the Euler characteristics of the Milnor fibre of $F$, the Milnor fibres of the maps $f_I$, the link of $F^{-1}(0)$ and the links of …
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.
We consider a mixed function of type $H(\mathbf z,\bar {\mathbf z})=f(\mathbf z)\bar g(\mathbf z)$ where $f$ and $g$ are convenient holomorphic functions which have isolated critical points at the origin and we assume that the intersection $f=g=0$ is a complete intersection variety with an isolated singlarity at theorigin. We assume also that $H$ satisfies the multiplicity condition.We will show that $H$ has a tubular Milnor fibration and also a spherical Milnor fibration. We give examples which does not satisfy the Newton multiplicity condition where one does not have Milnor fibration and 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 $H$ has a tubular Milnor fibration and a spherical Milnor fibration. We show also two fibrations are equivalent.
In this article we investigate mixed polynomials and present conditions that can be applied on a specific class of polynomials in order to prove the existence of the Milnor Fibration, Milnor-Lê Fibration and the equivalence between them. We prove for this class the of functions that the Milnor-Lê fiber on a regular value is homeomorphic to the Milnor-Lê fiber on a critical value. We develop a criterion to verify the transversality property and apply it to a special case of the class of mixed polynomial.
In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.
We give a strong version of a classic inequality of Łojasiewicz; one which collapses to the usual inequality in the complex analytic case. We show that this inequality for a pair, quadruple, or octuple of real analytic functions allows us to construct a real Milnor fibration inside a ball.
Given a relatively minimal non locally trivial fibred surface f: S->B, the slope of the fibration is a numerical invariant associated to the fibration. In this paper we explore how properties of the general fibre of $f$ and global properties of S influence on the lower bound of the slope. First of all we obtain lower bounds of the slope when the general fibre is a double cover. We also obtain a lower bound depending as an increasing function on the relative irregularity of the fibration, extending previous results of Xiao. We construct several families of examples to check the assimptotical sh…
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
We prove a Milnor-Lê type fibration theorem for a subanalytic map $f: X \to Y$ between subanalytic sets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$. Moreover, if $f$ extends to an analytic map $\mathbb{R}^m \to \mathbb{R}^n$, we define the singular set and the discriminant set of $f$, in a stratified sense. Then we give Milnor-Lê type fibration theorems for $f$ outside its discriminant, as well as over it. We give examples of each situation approached in this paper.
Consider a fibration \[p:E\to S^{p-1}\] with fiber $F$. We have the following natural question: Under what conditions does this fibration admit a cross-section? Our purpose is to discuss this problem when the fibration $p$ is the Milnor fibration $f_{\mid}:f^{-1}(S^{p-1}_δ)\cap D^{n}_ε\to S^{p-1}_δ$ with Milnor fiber $F_f$ and the Milnor fibration of arrangements $Q:\mathcal{M}(\mathcal{A})\to \mathbb{C}^\ast$ with fiber $F=F(\mathcal{A})$. Furthermore, we use our results to study the tasking planning problem for the Milnor fibration as a work map and we give the tasking algorithms.
We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.
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 Némethi and Zaharia's fibration theorem. In particular, we obtain a global version of Oka's fibration theorem for strongly non-degenerate and convenient mixed polynomials.
In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations.
We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.
In this article we extend Milnor's fibration theorem for complex singularities to the case of singularities $f \bar g:(X,P) \to (C,0))$ defined on a complex analytic singularity germ $(X,P)$, with $f, g$ holomorphic and $f \bar g$ having an isolated critical value at $0 \in C$. This can also be regarded as a result for meromorphic germs. Then we strenghten this fibration theorem when $X$ has complex dimension 2, obtaining a fibration theorem for multilinks that extends previous work by Pichon. We prove that the multilink $L_{f \bar g}$ in $L_X$ (the link of $X$), is fibred iff the map $f \bar …
Milnor fibrations and oriented matroids
该研究利用定向拟阵为复化实排列构造组合模型，得到与米尔诺纤维同伦等价的有限正则CW复形，证明其同伦类型仅依赖定向拟阵结构，且适用于任意定向拟阵。
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 degenerate functions, namely we show that the diffeomorphism-type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots, f^{k_0}$ if $\{f^{k_1}=\cdots=f^{k_m}=0\}$ is a non-degenerate complete intersection variety for any $k_1,\ldots,k_m\in \{1,\ldot…
We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor-Hamm tube fibration. Under natural conditions, we prove the existence of open book structures with singularities and solve the equivalence problem.
We prove fibration theorems à la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the existence of a Milnor fibration on the sphere. We also explain how one can modify the target space by homeomorphisms to linearize a general discriminant. Whenever the composed map is d-regular one has fibration on the sphere. Plenty of examples are discussed along the text.
We prove that every map-germ ${f \bar g}: (\C^n,\0) {\to}(\C,0)$ with an isolated critical value at 0 has the Thom $a_{f \bar g}$-property. This extends Hironaka's theorem for holomorphic mappings to the case of map-germs $f \bar g$ and it implies that every such map-germ has a Milnor-Lê fibration defined on a Milnor tube. One thus has a locally trivial fibration $ϕ: \mathbb S_\e \setminus K \to \mathbb S^1$ for every sufficiently small sphere around $\0$, where $K$ is the link of $f \bar g$ and in a neighbourhood of $K$ the projection map $ϕ$ is given by $f \bar g / | f \bar g|$.
Motivic, logarithmic, and topological Milnor fibrations
本文通过引入完全 Milnor 纤维化，比较了正则复函数的拓扑 Milnor 纤维化和 motivic Milnor 纤维，给出两种等价构造并应用于恢复 A'Campo 模型，还证明 motivic Milnor 纤维在无需 Lefschetz motive 可逆时仍可良好定义。
We find natural and convenient conditions which allow us to produce classes of genuine real map germs with Milnor tube fibration, either with Thom regularity or without it.
We study from a geographical point of view fibrations of threefolds over smooth curves, such that the general fibre is of general type. We prove the non-negativity of certain relative invariants under general hypotheses and give lower bounds for the self-interssection of the relative canonical divisor of the fibration, depending on other relative invariants. We also study the influence of the relative irregularity on these bounds. A more detailed study of the lowest cases of the bounds is given.
The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid. Another tool is a polarization construction, ba…
Topological complexity of Milnor fibration
本文将 Milnor 纤维化理论与拓扑机器人学结合，研究解析映射芽的 Milnor 纤维化的任务规划问题，发现规划算法依赖纤维化定理，并推测其拓扑复杂度与基一致。
In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in \mathbb{Z}^+$ and $r \geq 2$. We describe the link $L_F$ as a graph manifold using its natural open book decomposition, related to the Milnor fibration of the map-germ $f \bar{g}$ and the description of its monodromy as a quasi-periodic diffeomorphism through its Nielsen invariants. Furthermore, such a germ $F$ gives rise to a Milnor fibration $\frac{F}{|F|…
Milnor fibrations of arrangements with trivial algebraic monodromy
本文研究具有平凡代数单值性的超平面配置 Milnor 纤维化，重点分析上同调跳跃轨迹与基本群下中心序列商，并构造出 Betti 数相同但基本群不同构的一对配置。
On four-dimensional Dehn twists and Milnor fibrations
该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。
We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surface…
The total spine of the Milnor fibration of a plane curve singularity
本文提出了一种方法，通过构造脊柱来实现平面曲线奇点的Milnor纤维的消失拓扑，并在商空间上建立了等价的光滑纤维结构。
Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=χ_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the analysis of the relative canonical algebra $\cal{R}(f)$.
Tameness conditions and the Milnor fibrations for composite singularities
本文通过引入新的正则性条件精确刻画复合奇点的驯顺性，建立各分量 Milnor 纤维拓扑之间的联系，研究驯顺性在各类等价下的不变性，并给出 Euler 示性数之间的关系。
Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand,…
We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.
Let f: X->B be a fibred surface of genus g whose general fibre is a double cover of a smooth curve of genus gamma. We show that, for g > 4gamma+1, the number 4(g-1)/(g-gamma) is a sharp lower bound for the slope of f, proving a conjecture of Barja. Moreover, we give a characterisation of the fibred surfaces that reach the bound. In the case g = 4gamma+1 we obtain the same sharp bound under the assumption that the involutions on the general fibres glue to a global involution on X.
We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.
In [22] Milnor proved that a real analytic map $f\colon (R^n,0) \to (R^p,0)$, where $n \geq p$, with an isolated critical point at the origin has a fibration on the tube $f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1}) \to S_δ^{p-1}$. Constructing a vector field such that, (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he "inflates" the tube to the sphere, to get a fibration $φ\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$, but the projection is not necessarily given by $f/ \|f\|$ as in the complex case. In the case $f$ has isolated critical value, in [9] it was proved th…
Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_ε$ be a ball in $U$ of sufficiently small radius $ε>0$, centred at $\underline{0}\in\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_θ$, with axis $V$, which leads to a fibration $Φ$ of the whole space $(X \cap \mathbb{B}_ε) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_ε) \setminus V$ is the usual Milnor fibrati…
In this article, we study the topology of the family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ given by $F(x,y,z)=\bar{xy}(x^p+y^q)+z^r$ with $p,q,r \in \mathbb{N}$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ has isolated singularity at 0 and gives rise to a Milnor fibration $\frac{F}{|F|} \colon \mathbb{S}^{5} \setminus L_F \to \mathbb{S}^{1}$. We describe the link $L_F$ as a Seifert manifold and we show that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of $\mathbb{S}^{5}$ …
We study and obtain Slope inequalities for fibred irregular varieties of non-maximal Albanese dimension. We give a comparison theorem between Clifford-Severi and Slope inequalities for this type of fibrations. We also obtain a set of Slope inequalities considering the geometry of the Albanese map and the associated eventual maps.
Milnor Fiber Consistency via Flatness
本文提出研究 Milnor 纤维化的新代数几何视角，证明了全纯函数芽形变满足 Thom 条件并保持原 Milnor 纤维化的准则，推论出固定次数齐次多项式空间的划分方法及完全交临界轨迹函数形变的良态判据。
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
In this article, we study the topology of real analytic germs $F \colon (\C^3,0) \to (\C,0)$ given by $F(x,y,z)=\overline{xy}(x^p+y^q)+z^r$ with $p,q,r \in \N$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ gives rise to a Milnor fibration $\frac{F}{\mid F \mid} \colon \Sp^5\setminus L_F \to \Sp^1$. We describe the link $L_F$ as a Seifert manifold and we show that in many cases the open-book decomposition of $\Sp^5$ given by the Milnor fibration of $F$ cannot come from the Milnor fibration of a complex singularity in $\C^3$.
Sandwiched singularities and nearly Lefschetz fibrations
通过引入脊柱开书和近Lefschetz纤维化，研究夹层奇点的Milnor纤维与辛填充，对比代数几何形变理论与辛拓扑，发现一类夹层奇点链环存在非预期的Stein填充。
We give a concrete formula for the total Milnor number of the weighted-Le-Yomdin-at-infinity polynomial in most of the interesting cases. As an application, we give a description of the monodromy fibration at infinity for such kind of polynomial.
Fibred surfaces and their unitary rank
本文研究亏格至少为2的复纤维化曲面的酉秩，证明多个新斜率不等式，并由此得到非等平凡纤维化酉秩的Xiao型上界及Coleman-Oort猜想方向的新约束。
Isotrivially fibred surfaces and their numerical invariants
本文综述了光滑射影曲面到曲线的相对极小等平凡纤维化相关研究，重点分析了涉及两个数值不变量的不等式，并辅以多组例子和反例进行阐释。
We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko and Varchenko etc. concerning polynomials on $\CC^n$ will be generalized to polynomial functions on any toric variety.
The Picard number of fibred Mori dream surfaces
研究光滑复射影曲面的皮卡数计算，提出利用纤维化可约纤维不可约分支计数的方法，证明特定几类曲面不是 Mori 梦空间。
In the present article we determine and characterize completely the support genus, the binding number and the norm of a page of an open book under the following restrictions: M is a rational homology sphere which can be realized as the link of a surface singularity. Moreover, we restrict ourselves to the collection of those open book decompositions which can be realized as Milnor fibrations determined by some analytic germ (the so-called Milnor open books).
We prove that for any germ of complex analytic set in $\CC^n$ there exists a hypersurface singularity whose Milnor fibration has trivial geometric monodromy and fibre homotopic to the complement of the germ of complex analytic set. As an application we show an example of a quasi-homogeneous hypersurface singularity, with trivial geometric monodromy and simply connected and non-formal Milnor fibre.
Certaines fibrations en surfaces quadriques réelles
该研究探讨了实二次曲面纤维化是否稳定有理的问题，通过反例和两种新方法揭示了连通空间下的分解对角线问题。
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