Vanishing Cycles and Thom's $a_f$ Condition

Étale tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$

We develop a theory of tame vanishing cycles for schemes over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$ in the context of étale sheaves. We show some desired properties of this formalism, among which: a compatibility with tame vanishing cycles over a (strctly) henselian trait, a compatibility with the theory of tame vanishing cycles over $\mathbb{A}^1_{S}$, a compatibility with tensor product and with duality. In the last section, we prove that monodromy-invariant vanishing cycles, introduced by the second named author, are the homotopy fixed points with respect to a canonical continuous action of…

Vanishing Vanishing Cycles

If $\Adot$ is a bounded, constructible complex of sheaves on a complex analytic space $X$, and $f:X\to\C$ and $g:X\to\C$ are complex analytic functions, then the iterated vanishing cycles $ϕ_g[-1](ϕ_f[-1]\Adot)$ are important for a number of reasons. We give a formula for the stalk cohomology $H^*(ϕ_g[-1]ϕ_f[-1]\Adot)_x$ in terms of relative polar curves, algebra, and the normal Morse data and micro-support of $\Adot$.

Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink

Using iterated vanishing cycles and convolution, we prove a motivic version of a conjecture of Steenbrink concerning the spectrum of hypersurface singularities

Vanishing cycles in complex symplectic geometry

We study the vanishing cycles on the Milnor fibre of a holomorphic map germ with special kind of non-isolated singularities which appear in symplectic geometry. We show, under assumptions given in the text, that the Lefschetz vanishing cycles freely generate the corresponding homology group of the Milnor fibre. We study two examples of applications: involutive systems in symplectic geometry (as characteristic varieties of $\hbar$-differential equations are of this type) and ajoint orbits in Lie algebras.

Some consequences of perversity of vanishing cycles

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate it explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the monodromy.

Nearby and vanishing cycles for perverse sheaves and D-modules

We survey nearby and vanishing cycles for both perverse sheaves and D-modules under analytic setting. Following ideas of A. Beilinson, M. Kashiwara and M. Saito, we explain in detail the proof of the comparison theorem between them in the sense of Riemann-Hilbert correspondence.

Motivic vanishing cycles as a motivic measure

We show that the motivic vanishing cycles introduced by J. Denef and F. Loeser give rise to a motivic measure on the Grothendieck ring of varieties over the affine line. We discuss the relation of this motivic measure to the motivic measure we constructed earlier using categories of matrix factorizations.

Control by the lowest degree vanishing cycles

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the stalk cohomology of the Milnor fiber in lowest degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant.

Vanishing cycles under base change and the integral Hodge conjecture

In this paper we discuss an obstruction to the integral Hodge conjecture, which arises from certain behavior of vanishing cycles. This allows us to construct new counter-examples to the integral Hodge conjecture. One typical such counter-example is the product of a very general hypersurface of odd dimension and an Enriques surface. Our approach generalizes the degeneration argument of Benoist-Ottem.

Vanishing cycles of smoothable isolated Cohen-Macaulay codimension 2 singularities of type 2

We extend the results from the previous paper by A. Frühbis-Krüger and the author [arXiv:1501.01915] to the vanishing topology of those singularities in the title. Studying the case of possibly non-isolated singularities in the Tjurina- transform, we reveal that in dimension 3 and 2 there always is exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.

On surfaces with zero vanishing cycles

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic.

A Thom-Sebastiani Theorem in Characteristic p

Let $k$ be a perfect field of characteristic $p$, let $f_i:X_i\to\mathbb A_k^1$ $(i=1,2)$ be two $k$-morphism of finite type, and let $f:X_1\times_k X_2\to \mathbb A_k^1$ be the morphism defined by $f(z_1,z_2)=f_1(z_1)+f_2(z_2)$. For each $i\in\{1,2\}$, let $x_i$ be a $k$-rational point in the fiber $f_i^{-1}(0)$ such that $f_i$ is smooth on $X_i-\{x_i\}$. Using the $\ell$-adic Fourier transformation and the stationary phase principle of Laumon, we prove that the vanishing cycle of $f$ at $x=(x_1,x_2)$ is the convolution product of the vanishing cycles of $f_i$ at $x_i$ $(i=1,2)$.

关于具有二维消没循环空间的射影三维簇

On projective threefolds with two-dimensional space of vanishing cycles

本文完整分类了复数域上消没循环空间维数为2的光滑射影三维簇，以及光滑射影曲面上第二陈类为3的秩2非常充足向量束。

Vanishing cycles on Poisson varieties

We extend slightly the results of Evens-Mirković, and "compute" the characteristic cycles of Intersection Cohomology sheaves on the transversal slices in the double affine Grassmannian and on the hypertoric varieties. We propose a conjecture relating the hyperbolic stalks and the microlocalization at a torus-fixed point in a Poisson variety.

刚性解析簇上Zariski可构造层的消失环

Vanishing Cycles for Zariski-Constructible Sheaves on Rigid Analytic Varieties

本文在非阿基米德域上建立了Zariski可构造层的近环与消失环理论，证明其保持可构造性、具有Milnor纤维解释、满足Beilinson粘合、perverse t-精确且与Verdier对偶交换。

Tube classes over elementary vanishing cycles

Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube mapping. We generalize this result to the first integral homology group by relating the tube mapping with the topological Abel--Jacobi mapping. By making use of the mapping class group action, we prove that all tube classes constructed from the elementary vanishing cycles form a cofinite subgroup of the first integral homology group of $X$.

Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete intersections contained in $X$, we prove that the vanishing cycles in the middle homology group of $X$ are represented by topological cycles whose support is contained in a proper Zariski closed subset $T\subset X$ of certain codimension. In some cases, we can find such a Zariski closed subset $T$ with codimension equal to the upper bound obtained from the Hodge struc…

通过消没圈构造虚拟类

Virtual classes via vanishing cycles

本文提出了一种利用消没圈反常层为拟光滑导出概形构造虚拟基类的新方法，并证明其与现有类一致，同时探讨了$-2$-移位辛导出概形的DT4虚拟类构造方法。

Wild ramification and a vanishing cycles formula

In this paper we prove an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germ of curves over a complete discrete valuation ring of inequal characteristics (0,p). This formula can be easily generalised to the case of a Galois cover with group which is nilpotent or which has a normal p-Sylow subgroup. The results of this paper play a key role in [Sa-1] where is studied the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p…

关于正特征下平展层消失圈的稳定性

On the stability of vanishing cycles of étale sheaves in positive characteristic

研究正特征下平展层消失圈的稳定性，表明光滑曲面上其对测试函数的依赖一般仅取决于有限阶喷流，猜想高维也成立，还研究了消失圈稳定性最强的层的类别及相关性质。

A remark on vanishing cycles with two strata

Suppose that the critical locus $Σ$ of a complex analytic function $f$ on affine space is, itself, a space with an isolated singular point at the origin $\0$, and that the Milnor number of $f$ restricted to normal slices of $Σ-\{\0\}$ is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of $f$ at $\0$, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.

Deformation quantisation for (-1)-shifted symplectic structures and vanishing cycles

We formulate a notion of $E_0$ quantisation of $(-1)$-Poisson structures on derived Artin $N$-stacks, and construct a map from $E_0$ quantisations of $(-1)$-shifted symplectic structures to power series in de Rham cohomology. For a square root of the dualising line bundle, this gives an equivalence between even power series and self-dual quantisations. In particular, there is a canonical quantisation of any such square root, which localises to recover the perverse sheaf of vanishing cycles on derived DM stacks, thus giving a form of derived categorification of Donaldson--Thomas invariants.

$\ell$-adic消失环的非交换性质

Non-commutative nature of $\ell$-adic vanishing cycles

本文证明特殊纤维的奇点范畴可恢复$\ell$-adic消失上同调，并利用相对完美复形的dg范畴计算嵌入的紧支撑K-理论。

Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations

We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the…

修正对角闭链的消失性结果 II：志村曲线

Vanishing results for the modified diagonal cycles II: Shimura curves

本文证明了志村曲线三重积的Chow群中修正对角闭链的消失性结果，并构造了具有平凡自同构群且修正对角闭链消失的曲线例子。

$p$-adic vanishing cycles as Frobenius-fixed points

Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.

一维中增强的近邻与消失循环及傅里叶变换

Enhanced nearby and vanishing cycles in dimension one and Fourier transform

本文在一维增强反常对象上精确化近邻与消失循环，并利用拓扑方法证明仿射直线上全纯D-模在某点的消失循环同构于其傅里叶-拉普拉斯变换在无穷远处Stokes滤过的相应分次分量。

Around the Thom-Sebastiani theorem

For germs of holomorphic functions $f : (\mathbf{C}^{m+1},0) \to (\mathbf{C},0)$, $g : (\mathbf{C}^{n+1},0) \to (\mathbf{C},0)$ having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $Φ^{m+n+1}(f \oplus g)$ (and its monodromy) as a tensor product $Φ^m(f) \otimes Φ^n(g)$, where $(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. We prove algebraic variants and generalizations of this result in étale cohomology over fields of any characteristic, where the tensor product is replaced by a certain local co…

Monodromy of the perverse sheaf of vanishing cycles of some simple Shimura varieties and applications

In the geometric situation of the simple Shimura varieties of Kottwitz studied in Harris and Taylor's book, we describe the monodromy filtration of the vanishing cycles complex and the spectral sequence associated to it. We prove in particular that this filtration coincides with the weight one up to shift. Thanks to the Berkovich-Fargues' theorem, we deduce the description of the local monodromy filtration of the Deligne-Carayol model. In application, we obtain the description of the local components of cohomological automorphic representations of certains unitary groups, a global Jacquet-Lang…

Perverse Cohomology and the Vanishing Index Theorem

The characteristic cycle of a complex of sheaves on a complex analytic space provides weak information about the complex; essentially, it yields the Euler characteristics of the hypercohomology of normal data to strata. We show how perverse cohomology actually allows one to extract the individual Betti numbers of the hypercohomology of normal data to strata, not merely the Euler characteristics. We apply this to the ``calculation'' of the vanishing cycles of a complex, and relate this to the work of Parusiński and Briançon, Maisonobe, and Merle on Thom's $a_f$ condition.

On the vanishing cycles of a meromorphic function on the complement of its poles

We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor…

A Lagrangian sphere which is not a vanishing cycle

We give examples of Calabi-Yau threefolds containing Lagrangian spheres which are not vanishing cycles of nodal degenerations, answering a question of Donaldson in the negative.

Vanishing cycles of pencils of hypersurfaces

We prove an extended Lefschetz principle for a large class of pencils of hypersurfaces having isolated singularities, possibly in the axis, and show that the module of vanishing cycles is generated by the images of certain variation maps.

Polar degree and vanishing cycles

We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which represent local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.

Quadratic vanishing cycles, reduction curves and reduction of the monodromy group of plane curve singularities

The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.

消失环的混合Hodge结构与orbifold上同调

Mixed Hodge structures for vanishing cycles and orbifold cohomology

研究消失环的混合Hodge结构与orbifold上同调的极化关系，探讨Hodge-Tate情况下的上同调结构。

La monodromie Hamiltonienne des cycles évanescents

We study the monodromy of vanishing cycles for map-germs $f:(C^{2n},0) \to (\CM^k,0)$ whose components are in involution. Although the singular fibres of such maps have non-isolated singularities, it is shown that the regular fibres are $2(n-k)$-connected and that the vanishing homology group of rank $2(n-k)+1$ is freely generated by the vanishing cycles. As corollaries, we get that the multiplicity of the discriminant is equal to the dimension of the vanishing homology group of rank $2(n-k)+1$ and that the Variation operator is an isomorphism. These results are proved under two assumptions: 1…

Vanishing cycles and Hermitian duality

We show the compatibility between the moderate or nearby cycle functor for regular holonomic $\mathcal{D}$-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara.

Vanishing cycles and mutation

This is the writeup of a talk given at the European Congress of Mathematics, Barcelona. It considers Picard-Lefschetz theory from the Floer cohomology viewpoint.

Vanishing Cycles and Wild Monodromy

Let K be a complete discrete valuation field of mixed characteristic (0,p) with algebraically closed residue field, and let f: Y --> P^1 be a three-point G-cover defined over K, where G has a cyclic p-Sylow subgroup P. We examine the stable model of f, in particular, the minimal extension K^{st}/K such that the stable model is defined over K^{st}. Our main result is that, if g(Y) \geq 2, the ramification indices of f are prime to p, and |P| = p^n, then the p-Sylow subgroup of Gal(K^{st}/K) has exponent dividing p^{n-1}. This extends work of Raynaud in the case that |P| = p.

Monodromy and vanishing cycles in toric surfaces

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear system, and giving a precise determination of this monodromy group.

More about vanishing cycles and mutation

The paper continues the discussion of symplectic aspects of Picard-Lefschetz theory begun in "Vanishing cycles and mutation" (this archive). There we explained how to associate to a suitable fibration over a two-dimensional disc a triangulated category, the "derived directed Fukaya category" which describes the structure of the vanishing cycles. The present second part serves two purposes. Firstly, it contains various kinds of algebro-geometric examples, including the "mirror manifold" of the projective plane. Secondly there is a (largely conjectural) discussion of more advanced topics, such a…

修正对角闭链的周群消没结果

Vanishing results in Chow groups for the modified diagonal cycles

本文给出了曲线上三重积周群中修正对角闭链消没的一个充分条件，并构造了无穷多个满足该条件的非超椭圆曲线实例。

退化的 Hodge 理论（III）：非孤立奇点的消没圈演算

Hodge theory of degenerations, (III): a vanishing-cycle calculus for non-isolated singularities

本文研究退化 Hodge 理论第三部分，针对奇点轨迹为一维的非孤立奇点情形进行具体计算，涵盖 $k$-log-canonical 奇点、slc 与非 slc 曲面奇点及 Feynman integrals 中的五维奇点。

Galois covers of type (p,...,p), vanishing cycles formula, and the existence of torsor structures

In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type (p,p,...,p) between formal germs of p-adic curves and which generalises the formula proven by the first author in the case of Galois covers of degree p. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type (p,p,...,p) between p-adic schemes.

The vanishing cycles of curves in toric surfaces II

We resume the study initiated in \cite{CL}. For a generic curve $C$ in an ample linear system $\vert \mathcal{L} \vert$ on a toric surface $X$, a vanishing cycle of $C$ is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of $C$ to a nodal curve in $\vert \mathcal{L} \vert$. The obstructions that prevent a simple closed curve in $C$ from being a vanishing cycle are encoded by the adjoint line bundle $K_X \otimes \mathcal{L}$. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructio…

Singularities and vanishing cycles in number theory over function fields

This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example, and then give three examples which are relevant to current research. The focus will be a general explanation of which sorts of problems this method can be applied to.

Gopakumar-Vafa invariants via vanishing cycles

In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where …

单例和消失环对于足够 ample 线性系统在单连通表面上的 Monodromy

Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces

研究了单连通表面中足够 ample 线性系统的 Monodromy 与 r-spin 映射类群的关系，并探讨了消失环在节点退化中的应用。

Motivic Realizations of Singularity Categories and Vanishing Cycles

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over a complete discrete valuation ring. By using noncommutative motives, we first construct a motivic $\ell$-adic realization functor for dg-categories. Our main result, then asserts that, given a Landau-Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the $\ell$-adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and $\infty$-categorical…

四维复射影空间中超曲面的本原消没环的单调性

Monodromy of Primitive Vanishing Cycles for Hypersurfaces in $\mathbb P^4$

本文证明了在四维复射影空间中，超曲面的中维本原上同调可以由节点退化中单个消没球面类上的管状类生成，从而肯定回答了Clemens提出的问题。

Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles

We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of…

The A_f condition and relative conormal spaces for functions with non-vanishing derivative

We introduce a join construction as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom's A_f condition.