Lefschetz numbers of iterates of the monodromy and truncated arcs

勒夫谢茨束的特征簇上的单值群作用

Monodromy action on character varieties for Lefschetz pencils

该数学研究确定了来自勒夫谢茨束的单值群在黎曼曲面基本群的特征簇上的作用性质，解答了Katzarkov等人的相关猜想。

Integration over spaces of non-parametrized arcs and motivic versions of the monodromy zeta function

We elaborate notions of integration over the space of arcs factorized by the natural $C^*$-action and over the space of non-parametrized arcs (branches). There are offered two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on a smooth space. We indicate a direct formula which connects the naive motivic zeta function of J. Denef and F. Loeser with the classical monodromy zeta function.

Distinguished bases and monodromy of complex hypersurface singularities

We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of monodromy.

向量丛的Noether-Lefschetz定理

A Noether-Lefschetz theorem for vector bundles

本文利用单值论证明了向量丛的一个Noether-Lefschetz定理。

The Picard-Lefschetz formula and a conjecture of Kato, the case of Lefschetz fibrations

A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration.

Singular fibers in algebraic fibrations of genus 2 and their monodromy factorizations

Kodaira's classification of singular fibers in elliptic fibrations and its translation into the language of monodromies and Lefschetz fibrations has been a boon to the study of 4-manifolds. In this article, we begin the work of translating between singular fibers of genus 2 families of algebraic curves and the positive Dehn twist factorizations of Lefschetz fibrations for a certain subset of the singularities described by Namikawa and Ueno in the 70s. We look at four families of hypersurface singularities in $\mathbb{C}^3$. Each hypersurface comes equipped with a fibration by genus 2 algebraic…

Monodromy

Let $(X,x)$ be an isolated complete intersection singularity and let $f : (X,x) \to (\CC,0)$ be the germ of an analytic function with an isolated singularity at $x$. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to $f$. We give a survey on what is known about this operator. In particular, we review methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity.

扭曲Milnor超曲面的Motivic莱夫谢茨定理

Motivic Lefschetz theorem for twisted Milnor hypersurfaces

本研究针对Milnor超曲面内扭曲形式的光滑超平面截面，证明其Grothendieck-Chow motive可分解为相关循环代数的Severi-Brauer簇与极大交换子域的motive平移拷贝的直和，证明依赖于Galois作用在等变Chow群上的非平凡性。

On an equivariant version of the zeta function of a transformation

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W.Lück and J.Rosenberg. Here we offer another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta …

A degree doubling formula for braid monodromies and Lefschetz pencils

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz …

An equivariant version of the monodromy zeta function

We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the Grothendieck ring of finite G-sets tensored by the field of rational numbers. Main two ingredients of the definition are equivariant Lefschetz numbers and the lambda-structure on the Grothendieck ring of finite G-sets. We give an A'Campo type formula for the equivariant zeta function.

Moduli spaces of surfaces and monodromy invariants

This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P^1 x P^1, see (arXiv:0910.2142). The results being surveyed explore various fundamental questions about complex deformation vs. symplectomorphism vs. diffeomorphism from the perspective of one explicit family of examples; the paper also contains some remarks which were not in earlier papers.

分层Picard-Lefschetz理论

Stratified Picard--Lefschetz theory

研究了Morse函数在层奇异解析簇上的水平集同调中的单值作用。由这些函数的临界值周围的环所定义的标准同调群和交同调群中的局部变分算子被简化为相应地层的横截切片的同调群中的相似算子。

局部单值性、Behrend 函数与 $\mathbb{C}^3$ 上点的 Hilbert 概形

Local monodromy, Behrend function and Hilbert scheme of points on $\mathbb{C}^3$

本文通过局部单值性和喷射空间迹公式，给出了 $\mathbb{C}^3$ 上 Hilbert 概形 Behrend 函数值的判定准则，并基于 Milnor 纤维证明了其非常数性。

On the Topology of Foliations with a First Integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type $F^p/G^q$ in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle $δ$ generates the first homology group of a generic fiber of $F^p/G^q$. In particular, we will prove that for any two Lefschetz vanishing cycles $δ_0$ and $δ_1$ in a regular compact fiber of $F^p/G^q$, there exists a monodromy $h$ such that $h(δ_0)=\pm δ_1$.

Thom条件与单调性

Thom condition and Monodromy

本文定义了Thom条件，证明在特定条件下复解析函数芽存在无不动点的几何局部单调性，推广了已有定理并应用于奇点不合并的研究。

Families of ICIS with constant total Milnor number

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well known result of Gabrielov, Lazzeri and Lê for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

Sur les varietes de Hodge

Let $Y$ be a smooth complex projective variety of dimension $N+1$, $L$ an invertible sufficiently ample sheaf, $X\in |L|$ a smooth hypersurface and $λ\in F^kH^N(X,C)$ a vanishing cohomology class, where $F^{*}$ is the Hodge filtration and $k\in\{1,...,[N/2]\}$. Assume that $L$ is sufficiently ample and that the codimension in $|L|$ of the Hodge variety associated to $λ$ (locally defined as the locus where the image of $λ$ by flat transport over $|L|$ remains in $F^k$) is sufficiently small. I show that this forces $N$ to be even and $k=[N/2]$, and that the class $λ$ is a linear combination wit…

Lowest non-zero vanishing cohomology of holomorphic functions

We study the vanishing cycle complex $φ_fA_X$ for a holomorphic function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the monodromy eigenvalue decomposition may hold after a localization of $A$). Assuming the perversity of the shifted constant sheaf $A_X[d_X]$, we show that the lowest possibly-non-zero vanishing cohomology at $0\in X$ can be calculated by the restriction of $φ_fA_X$ to an appropriate nearby curve in the singular locus $Y$ of $f$, which is given by intersecting $Y$ with …

有效同调与复射影超曲面的周期

Effective homology and periods of complex projective hypersurfaces

本文提出了一种基于Picard-Lefschetz理论的新算法，用于高效计算复射影超曲面的周期，通过结合同调基计算与monodromy作用分析，实现了高精度的周期计算。