On The Geometric Monodromy Of The Milnor Fibre
Nikolai A. Krylov
本文证明了Milnor纤维的几何单值$ϕ$具有有穷级q的充要条件是算子$ϕ^q_*$对同调作用平凡.证据是基于经典的外科手术方法。
In this paper we show that the geometric monodromy $ϕ$ of the Milnor fibre, has finite order q if and only if the operator $ϕ^q_*$ acts trivially on the homology. The proof is based on the classical methods of surgery.
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We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.
Eigenvalues for the monodromy of the Milnor fibers of arrangements
本文用组合数学的方法给出了Milnor纤维单值排列的本征值阶数的上界。
On the monodromy of Milnor fibers of hyperplane arrangements
我们描述了一个一般的设置单值行动的第一个上同调群的Milnor纤维的超平面安排是身份。
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narv\' ez Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one p…
Using recent results by A. Macinic, S. Papadima and R. Popescu, and a refinement of an older construction of ours, we determine the monodromy action on $H^1(F(G),C)$, where $F(G)$ denotes the Milnor fiber of a hyperplane arrangement associated to an irreducible complex reflection group $G$.
A result about Picard-Lefschetz monodromy
令$f$和$g$在不同的变量集中约化为齐次多项式。我们建立了一个简单的公式，将单值算子在$fg$的Milnor纤维上同调的本征空间分解分别与$f$和$g$的本征空间分解联系起来。我们利用D. Cohen和A.苏丘
Motivic Milnor fibre for nondegenerate function germs on toric singularities
研究了复曲面簇上的函数芽，这些复曲面簇因其牛顿图而非退化。我们用他们的牛顿图来表示他们的动机米尔诺纤维。我们扩展的motivic附近的纤维的情况下，环形变性的公式。我们通过一些例子来说明这一点。
Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements
Milnor纤维monodromy算子的中央超平面安排的顺序示出组合确定。特别地，给出了该单值算子平凡的一个充要条件。已知复超平面排列的补是上同调的，并且如果该排列在$\Q$上定义，则具有多项式计数。我们表明，这些属性持有相应的Milnor纤维，如果单值是平凡的。构造了一个定义在$\Q$上的超平面排列，其Milnor纤维具有非平凡的monodromy算子，上同调Tate，且没有多项式计数.这样的例子不存在于低维。
Fibre de Milnor motivique a l'infini
给定一个从光滑复簇到仿射直线的正则映射f，我们定义了一个无穷远的运动Milnor纤维，并在无穷远的Newton多面体的非退化Laurent多项式的情况下计算了它.
On the monodromy action on Milnor fibers of graphic arrangements
我们分析了单值行动，在有理数，第一个同调群的Milnor纤维，任意子安排的Coxeter安排。我们提出了一个组合公式的monodromy行动，涉及Aomoto复合物的积极特征。在A、B、D三种情况下，验证了公式的正确性.
Let $ Φ: ({\mathbb C}^2, 0) \to ( {\mathbb C}^3, 0) $ be a finitely determined complex analytic germ and let $(\{f=0\},0)$ be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of $f$ from the double-point-geometry of $Φ$.
Intersection Cohomology, Monodromy, and the Milnor Fiber
我们说一个复解析空间$X$是一个交上同调流形当且仅当$X$上的移位常数层同构于交上同调;这很快就被看作等价于$X$是一个同调流形。给定一个交上同调流形上的解析函数$f$，我们描述了$V(f)$是一个交上同调流形与$f$的消失圈Milnor单值性之间的一个简单关系.然后，我们描述了如何让我们很容易地产生任意奇异集的相交上同调流形的Ratiani-Thom同构。最后，作为一个简单的应用，我们得到了具有特殊类型的一维临界轨迹的超曲面的Milnor纤维的上同调的限制。
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.
On homotopy types of complements of analytic sets and Milnor fibres
证明了对$\CC^n$中的复解析集的任一芽，存在一个超曲面奇点，其Milnor纤维化具有平凡的几何单值性，且纤维同伦于复解析集芽的补.作为一个应用程序，我们展示了一个例子的准齐次超曲面奇点，平凡的几何monodromy和简单的连接和非正式的Milnor纤维。
Invariant Subspaces of the Monodromy
我们表明，存在某些类型的不变子空间的Milnor monodromy的障碍，这地方的非孤立超曲面奇点的Milnor纤维的上同调的限制。
In case of one-dimensional singular locus, we use deformations in order to get refined information about the Betti numbers of the Milnor fibre.
Real Milnor Fibres and Puiseux Series
给定一个实多项式函数和它的零轨迹上的一个点，我们定义了一个由自然地附着在这些数据上的代数实多项式级数组成的集合。我们证明，这一套确定的拓扑结构和几何的真正的米尔诺纤维的功能在这一点上。为了实现这一目标，我们之间的驯服属性的这一套的Puietrix系列，被认为是一个真正的代数对象在该领域的代数Puietrix系列，其行为作为一个无限维的对象在实数。
The Milnor fibre signature is not semi-continuous
考虑$(\mC^3,0)$中孤立曲面奇点的芽。相应的Milnor纤维具有同调格（具有自然对称相交形式的积分中同调）。A.Durfee（1978）的一个老问题是：这种形式的符号在退化下是非增的吗？本文否定地回答：我们给出了牛顿非退化族的例子，其中签名在退化下增加。
Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a multiplicative structure on the Whitney homology of $\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperpl…
Combining recent results by A. Macinic, S. Papadima and R. Popescu with a spectral sequence and computer aided computations, we determine the monodromy action on $H^1(F,\mathbb{C})$, where $F$ denotes the Milnor fiber of the hyperplane arrangement associated to the exceptional irreducible complex reflection group $G_{31}$. This completes the description given by the first author of such monodromy operators for all the other irreducible complex reflection groups.
The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up f…
Stratified critical points one the real Milnor fibre and integral-geometric formulas
设$(X,0) \subset (\mathbb{R}^n,0)$是闭次解析集的芽，$f$和$g : (X,0) \rightarrow (\mathbb{R},0)$是两个次解析函数.在一定条件下，我们将实Milnor纤维$X \cap f^{-1}(δ) \cap B_ε$上的$g$的临界点，$0 <| δ| \ll ε\ll 1$，与该纤维的拓扑及其它相关的子解析集联系起来.作为应用，当$g$是一般线性函数时，我们得到了$f$的实Milnor纤维的“渐近”Gauss-Bonnet公式.从这个高斯-博内公式，我们推导出“无穷小”的线性运动公式。
In this note we consider the Milnor fiber $F$ associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.
Monodromy Eigenvalues of Milnor Fibers for Line Arrangements
本文通过组合消失准则与 Hirzebruch 不等式，证明线排列 Milnor 纤维单值特征值阶数至多五（实排列至多四），推进并证实了相关猜想。
Integral Restrictions on the Monodromy
给定一个复解析函数，在原点具有一维临界轨迹，我们研究了单值作用对Milnor纤维的积分上同调的影响。我们将这种单值性与通过原点的通用超平面切片和原点附近的通用超平面切片相关联。因此，我们得到数论限制的monodromy和上同调的原始米尔诺纤维。
The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on C^3
研究了仿射三维空间上四点Hilbert格式的一个自然Hodge模，它是Hilbert格式的Donaldson-Thomas不变量的一个分类.我们明确地确定了Hodge模上的权滤子，并计算了其上同调的E-多项式。的计算使必不可少的使用的描述的奇异性的希尔伯特计划的退化轨迹的Pfiran函数。
Equisingularity and The Euler Characteristic of a Milnor Fibre
研究了超曲面奇点的Milnor纤维的欧拉示性数.这个不变量是根据原始奇点和其Milnor纤维之间的纤维的欧拉示性数以及根据与中间纤维的层相关联的欧拉示性数给出的。由此我们可以推出Massey和Siersma关于具有一维临界轨迹的奇点的一个结果。这一结果也应用于等奇异性的研究。著名的Briançon-Speder-Teissier结果指出，一族孤立的超曲面奇点是等奇异的当且仅当它的$μ^*$ -序列是常数。本文证明了如果从n-空间到（n+1）-空间的余秩1复解析映射族的相似序列是常数，则映射族的象是等奇异的.对于从3-空间到4-空间的corank 1映射族，我们证明了反之亦然。
The boundary of the Milnor fibre of complex and real analytic non-isolated singularities
设f和g是全纯函数-芽在一个三维复解析芽的原点处为零。设它们没有共同的不可约分支，且由f与g的共轭相乘得到的实解析映射芽在0处有一个孤立的临界值。本文给出了实解析映射芽有Milnor纤维化的充要条件，并证明了在这种情况下其Milnor纤维的边界是Waldhausen流形.作为一个中间的里程碑，我们描述几何的Milnor纤维的映射芽的乘法给出的全纯和反全纯函数定义在一个复杂的表面芽，我们证明了一个A 'Campo型公式的zeta函数的monodromy。
The degeneration of the boundary of the Milnor fibre to the link of complex and real non-isolated singularities
本文研究了实解析奇点$f: (\bR^m,0) \to (\bR^k,0)$，$m\geq k$的Milnor纤维的边界问题，该纤维具有孤立的临界值和Thom $a_f$ -性质.我们定义了$f$的消失区，并给出了它是$f^{-1}(0)$的奇异集的环上的纤维丛的充分必要条件。在$\fgbar: (\bC^n,0) \to (\bC,0)$型奇点的情况下，与一个孤立的临界值，$f, g$全纯的，我们进一步描述的退化的边界的Milnor纤维的链接的$\fgbar$。作为一个里程碑，我们还构造了一个Lê多面体的类型$\fgbar: (\bC^2,0) \to (\bC,0)$，使真正的解析奇点的$f$或$g$只依赖于一个变量。
On the Milnor fibres of initial forms of topologically equivalent holomorphic functions
本研究针对全纯函数拓扑等价相关猜想，在平面曲线情形下证明了拓扑等价全纯函数初始形式的Milnor纤维同伦等价，并指出该结论可推出Zariski重数猜想在两类情形下成立。
Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres
我们定义了一个Grothendieck环的基本实半代数公式，即系统的实代数方程和不等式。在这个环类的一个公式考虑到代数性质的一组点满足这个公式，并包含作为一个环通常Grothendieck环的真正的代数公式。我们给出了我们的环的一个实现，它允许将一个类表示为实代数公式类的Z[1/2]-线性组合，因此这种实现产生了基本半代数公式的虚Poincaré多项式的概念。然后，我们定义zeta函数的系数在我们的环，建立在弧空间的半代数公式。我们表明，他们是合理的，并将它们与真正的Milnor纤维的拓扑结构。
We give a description of the Milnor fiber and the monodromy of a singularity of the form f+zg = 0 where f and g define plane curves and have no common components. The description depends only on the topological type of the two plane curve germs defined by f and g. In particular, this gives a description of the boundary of the Milnor fiber. As a corollary, we give a simple formula for the monodromy zeta function and the Euler characteristic of the fiber.
A central question in arrangement theory is to determine whether the characteristic polynomial $Δ_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $Δ_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $β_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $…
Monodromy of plane curves and quasi-ordinary surfaces
我们开发的递归公式的水平和垂直monodromies的准普通表面。这些都是单值性与米尔诺纤维的切片横向的一个组成部分的奇异轨迹。在计算这些递推关系的过程中，我们发现了一种新的方法来表示奇异平面曲线的Milnor纤维化的单值性。
Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as ``the non-crossing algebra'', which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $\widetilde{A}$, which seems to be ``dual'' to the Fomin-Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection bet…
Milnor fibre boundary of a non-isolated surface singularity
设f是超曲面局部奇点，其零点集具有一维奇异轨迹。我们开发了一个明确的程序，提供了边界的Milnor纤维的f作为一个定向铅垂3-流形。该方法也给出了代数单值性的特征多项式。此外，对于任意的解析芽g，使得（f，g）是孤立的完全相交奇点，还计算了由g确定约束边界和由g的辐角确定页数的开卷分解的重数系.为了做到这一点，我们必须建立关键结果的横向型奇点与奇异轨迹的f和ICIS（f，g）的水平和垂直的monodromies。这个理论有许多例子支持。例如，齐次奇点的情况（包括安排的情况）是详细的。一个列表，特别是奇特的例子，并列出了相关的开放问题。
In this article we prove two results concerning the motivic Milnor fibres $S^ε(f)$ associated to a map germ $f: (\mathbb{R}^n,0)\to(\mathbb{R},0)$, defined by G. Comte and G. Fichou. Firstly, we prove that if $f,g:(\mathbb{R}^n,0)\to(\mathbb{R},0)$ are arc-analytically equivalent germs of Nash functions then the virtual Poincaré polynomial of the corresponding motivic Milnor fibres are equal. This extends (and provides a new proof) of a result of G. Fichou. Secondly, let $T\subset\mathbb{R}^m$ be a real algebraic set and $f: T\times(\mathbb{R}^n,0)\to\mathbb{R}$ a polynomial function of polyno…
We study the topology of a line singularity, which is a complex hypersurface with non-isolated singularity given by a complex line. We describe the degeneration of its Milnor fibre to the singular hypersurface by means of a pair of polyhedra, one in the Milnor fibre and other in the singular fibre, which are deformation retracts of the corresponding fibres; and a continuous map taking the Milnor fibre to the singular fibre and the first polyhedron to the second one, which restrict to a homeomorphism outside the polyhedra. In the same sense, we also study the topology of a complex isolated sing…
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。
Milnor fibers of real line arrangements
我们研究复杂的实线排列的米尔诺纤维。给出了一种计算第一上同调单值特征空间的新算法。该算法基于对补空间同伦的最小CW-复形的描述，并利用了腔室的实图形即邻接关系。推广了Libgober的一个零化结果，给出了新的上界，并利用Milnor单值的非平凡性刻画了$A_3$ -排列.
This note is the observation that a simple combination of known results shows that the usual analytic invariants of a finitely determined multi-germ $f\colon (\mathbb{C}^2,S)\to(\mathbb{C}^3,0)$ ---namely the image Milnor number $μ_I$, the number of crosscaps and triple points, $C$ and $T$, and the Milnor number $μ(Σ)$ of the curve of double points in the target--- depend only on the embedded topological type of the image of $f$. As a consequence one obtains the topological invariance of the sign-refined Smale invariant for immersions $j\colon S^3\looparrowright S^5$ associated to finitely det…
This article and its successor concern the topology of real isolated hypersurface singularities. We prove that after attaching a certain number of handles the real Milnor fibres become contractible, with each handle corresponding to a critical point of a morsification; in particular one recovers the classical formula of Khimshiashvili for the Euler characteristic of the Milnor fibres. We then give sufficient conditions for having that the integer homology groups of the real Milnor fibres are isomorphic to the homology groups of a bouquet of spheres. This is followed by a discussion on real van…
On the cohomology of the Milnor fibre of a hyperplane arrangement
我们调查的Milnor纤维的反射安排作为一个模块的组$Γ$产生的反射，连同循环monodromy的上同调。虽然我们成功地完全只为酉反射群的秩二，我们建立了一些一般性的结果，其中有关的同型成分的monodromy上同调，霍奇结构和上同调度。利用反射群的本征空间理论，证明了等变权多项式和与欧拉示性数有关的Hodge-Deligne多项式等可加函数的求和公式.我们还使用monodromy特征空间，以确定在某些情况下，这反过来又抛出光的霍奇结构的上同调的频谱。这些方法使我们能够计算完整的故事，包括$Γ$的Hodge组件在每个上同调度的表示，对于一些组的低秩。
The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${\mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.
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。
Lê Modules and Traces
我们展示了我们最近的一些结果是如何阐明非孤立超曲面奇点的Milnor纤维的Lê数和上同调之间的关系的。Lê数实际上是自由阿尔贝群--Lê模--在复形中的秩，复形的上同调是Milnor纤维。此外，Milnor单值作用于Lê模复形，我们用临界轨迹的拓扑描述了这些单值作用的迹。
Lê cycles and Milnor classes
这项工作的目的是建立一个联系的理论陈类奇异品种和几何的品种问题。也就是说，我们证明了，如果$Z$是紧致复流形中的一个超曲面，由一个非常充分的线丛的非零全纯截面的零格式定义，那么它的Milnor类，作为$Z$的Chow群中的元素，确定$Z$的整体Lê圈;反之亦然：Lê圈确定Milnor类。从道义上讲，这意味着米尔诺尔类决定了$Z$中每一点的局部米尔诺尔纤维的拓扑，而局部米尔诺尔纤维的几何形状决定了相应的米尔诺尔类。
Topology of real Milnor fibration for non-isolated singularities
本文考虑一个实解析映射$F=(f_1,...,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)$，$2 \le k \le n-1$，满足D.梅西这意味着每个由$F$通过投影导出的实解析$f_I=(f_{i_1},...,f_{i_l}) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^l,0)$，其中$1 \le l \le n-2$和$I=\{i_1,...,i_l\}$也满足Milnor条件（a）和（b）。本文给出了$F$的Milnor纤维的Euler特征线、$f_I$的Milnor纤维、$F^{-1}(0)$的环和$f_I^{-1}(0)$的环之间的关系。
Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements
研究了齐次多项式的Milnor纤维上的单值性和Deligne混合Hodge结构之间的相互作用。在超平面Milnor纤维的情况下，我们得到了关于可能权的一个新结果。对于线的安排，我们证明了在一个新的方式的事实，由于Budur和斋藤的频谱是由弱组合数据，并表明，这样的结果失败的Hodge-Deligne多项式。
The monodromy conjecture for plane meromorphic germs
在[GZLM 1]中引入了亚纯函数的Milnor纤维化的概念以及相应的单值性和单值zeta函数的概念。本文定义了亚纯芽的拓扑zeta函数，并在平面情形下研究了它的极点。我们表明，极点不表现为在全纯的情况下，但仍然满足推广的单值猜想。
We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn lattice. For this, we use adiabatic techniques and well-known results on spectral flow and Maslov index. Furthermore we interpret the inequality of a certain member of this family of spectral flows with a spectral flow induced by a Reeb flow on the boundary of the Milnor fibre as g…
On bifurcation braid monodromy of elliptic fibrations
我们定义了一个单值同态不可约家庭的定期椭圆纤维化的映射类群的穿孔球。我们只计算椭圆纤维化的单值性，它不包含除I_1和I*_0以外的奇异纤维。存在极大群，它们可以是代数的单值群，也可以是代数的单值群。可微家庭的椭圆曲面，我们给出了一个代数准则的平等，我们可以检查适用的情况下，I_1纤维的数量是最多6。
Milnor fibers over singular toric varieties and nearby cycle sheaves
我们提出了一个新的层理论方法计算的单值zeta函数的Milnor纤维化。作为应用，Kushnirenko和Varchenko等人关于$\CC^n$上的多项式的经典公式将推广到任何复曲面簇上的多项式函数。
Finite Galois covers, cohomology jump loci, formality properties, and multinets
我们探讨有限伽罗瓦覆盖上同调跳轨迹，形式性质和代数单值作用之间的关系。证明了在底空间是1-形式的且1次单值作用是平凡的条件下，底空间和全空间的跳跃轨迹基本相同.我们使用减少多网结构线安排，以构建组件的第一个特征品种的Milnor纤维的度1，并证明，monodromy行动是非平凡的度1。对于一个任意的线的安排，我们证明了平凡的monodromy度1可以检测到一个精确的方式，共振品种。
Vanishing cycles in complex symplectic geometry
研究了辛几何中一类具有特殊非孤立奇点的全纯映射芽的Milnor纤维上的消失圈。我们表明，在文中给出的假设下，莱夫谢茨消失周期自由生成相应的同调群的米尔诺纤维。我们研究两个应用的例子：对合系统在辛几何（作为特征品种的$\hbar$ -微分方程是这种类型）和联合轨道在李代数。
Monodromy at infinity of polynomial maps and Newton polyhedra (with Appendix by C. Sabbah)
通过引入多项式映射无穷远点的运动Milnor纤维，给出了研究无穷远点单值映射幂零部分的一些方法.无穷远处单值的若当块的个数将由多项式无穷远处的牛顿多面体来描述。
Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements
本文给出了复线型Milnor纤维的单值本征空间的一个消灭定理。利用Papadima-Suciu给出的局部系上同调群的模界，得到了有限域上Aomoto复形的上同调为零的结果.为了证明这一点，我们引入Orlik-Solomon代数的退化同态。
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。
The local monodromy as a generalized algebraic correspondence
本文证明了对于以$X$为泛纤维的局部域的整数环上的正规交叉退化$Z$，局部单值算子及其幂决定了积$X \times X$的上同调分解群下的不变上循环类.更确切地说，他们还定义了代数循环的特殊纤维的决议$Z \times Z$。本文给出了最坏三重点为奇点的退化环的一个明确的描述。这些周期在几何上解释了与$X \times X$相关的L函数的特定局部因子上极点的存在。
Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_n$ and $D_n$, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of …
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