Michel Raibaut
Given a regular map f from a smooth complex variety to the affine line, we define a motivic Milnor fiber at infinity and we compute it in the case of a non degenerate Laurent polynomial for its Newton polyhedra at infinity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We define motivic Milnor fiber of cyclic $L_\infty$-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by $L$ is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that Behrend function depends only on the formal neighborhood of the moduli space.
Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.
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.
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.
We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular fibers. Then for a family of polynomials we detect parameters where the topology of the polynomials can change. Implementation and examples are given with the computer algebra system Singular.
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.
Given a real polynomial function and a point in its zero locus, we defined a set consisting of algebraic real Puiseux series naturally attached to these data. We prove that this set determines the topology and the geometry of the real Milnor fibre of the function at this point. To achieve this goal, we balance between the tameness properties of this set of Puiseux series, considered as a real algebraic object over the field of algebraic Puiseux series, and its behaviour as an infinite dimensional object over the real numbers.
We give a global version of Le-Ramanujam mu-constant theorem for polynomials. Let f_t, (t in [0,1]), be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number, the Milnor number at infinity, the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the f_t is a constant, then the polynomials f_0 and f_1 are topologically equivalent. For n>3 we suppos…
Let $L$ be a cyclic $L_\infty$-algebra of dimension $3$ with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic $L_\infty$-algebra structure on the cohomology $H^*(L)$. The inner product plus the higher products of the cyclic $L_\infty$-algebra defines a superpotential function $f$ on $H^1(L)$. We associate with an analytic Milnor fiber for the formal function $f$ and define the Euler characteristic of $L$ is to be the Euler characteristic of the étale cohomology of the analytic Milnor fiber. In this paper we prove a Thom-Sebastiani type form…
Let $(X,0) \subset (\mathbb{R}^n,0)$ be the germ of a closed subanalytic set and let $f$ and $g : (X,0) \rightarrow (\mathbb{R},0)$ be two subanalytic functions. Under some conditions, we relate the critical points of $g$ on the real Milnor fibre $X \cap f^{-1}(δ) \cap B_ε$, $0 <| δ| \ll ε\ll 1$, to the topology of this fibre and other related subanalytic sets. As an application, when $g$ is a generic linear function, we obtain an "asymptotic" Gauss-Bonnet formula for the real Milnor fibre of $f$. From this Gauss-Bonnet formula, we deduce "infinitesimal" linear kinematic formulas.
Hyperplane arrangements with non-formal Milnor fibers
本文提出基于multinet结构的组合条件，用于判定超平面排列的Milnor纤维非形式，并构造了具有该性质的无限族单项排列。
By introducing motivic Milnor fibers at infinity of polynomial maps, we propose some methods for the study of nilpotent parts of monodromies at infinity. The numbers of Jordan blocks in the monodromy at infinity will be described by the Newton polyhedron at infinity of the polynomial.
We use the notion of Milnor fibres of the germ of a meromorphic function and the method of partial resolutions for a study of topology of a polynomial map at infinity (mainly for calculation of the zeta-function of a monodromy). It gives effective methods of computation of the zeta-function for a number of cases and a criterium for a value to be atypical at infinity.
We give a topological model for a polynomial map from $\C^n$ to $\C$ in the neighborhood of a fiber with isolated singularities. This is motivated out of the ``unfolding of links'' described earlier by the first author and Lee Rudolph. The topological model gives a useful encoding of the local and global monodromy for the map. The new ingredients --- the topology of the singularities at infinity of an irregular fiber, as encoded by the Milnor fibers at infinity and the monodromy maps on these Milnor fibers --- are recoverable for n=2 in full from the link at infinity of the irregular fiber. Th…
Based on Balmer's tensor triangular Chow group [2], we propose (Milnor)K-theoretic Chow groups of derived categories of schemes. These Milnor K-theoretic Chow groups recover the classical ones [6] for smooth projective varieties and can detect nilpotent, while the classical ones can't do. As an application, we extend Soulé's variant of Bloch-Quilled identification from smooth projective varieties to their trivial infinitesimal thickenings. This answers affirmatively a question by Green-Griffiths for trivial deformations, see Question 1.1 below.
We consider a continuous family $(f_s)$, $s\in[0,1]$ of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant (or equivalently the sum of the affine Milnor number and the Milnor number at infinity $μ(s)+λ(s)$ is constant). We firstly prove that the set of critical values at infinity depends continuously on $s$, and secondly that the degree of the $f_s$ is constant (up to an algebraic automorphism of $\Cc^2$).
A lower bound for the Milnor number of vector fields
本文研究了包含局部完全交光滑正维成分的全纯向量场，推导了其Milnor数的下界，并通过显式族展示最优性及奇点的重新分布机制。
Given a cycle module M with a ring structure we show that the cycle complex with coefficients in M of a smooth scheme of finite type over a field has a A-infinity algebra structure. In the case of Milnor K-theory this gives a homotopy model for the classical intersection theory of algebraic cycles.
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…
A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szabó, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification o…
Let $F$ be a polynomial mappping from $\mathbb{C}^n$ to $\mathbb{C}^q$ with $n>q$. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre $F^{-1}(\infty)$ "at infinity" and its cohomology. Let us fix a weighted homogeneous degree on $\mathbb{C}[x_1,...,x_n]$ with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of $F$. We introduce the cohomology groups $H^k(F^{-1}(\infty))$ of $F$ at infinity. These groups enable us to compute all the other cohomology groups of $F$. For i…
In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k." The outcome of this approach is the number of solutions for "generic" systems in terms of their "Newton polytopes," and an explicit characterization of what makes a system "generic." The pioneering work in this field was done in the 1970s by Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of counting solutions of generic sys…
Given a family $X$ of complex varieties degenerating over a punctured disc, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of $X$. Our first main result is that the motivic nearby fiber of $X$ can be computed by first stratifying $X$ into locally closed subvarieties that are non-degenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。