Singularities and Enriched Cycles

Cycles of Singularities appearing in the Resolution Problem in positive Characteristic

We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It i…

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。

Explicit Resolutions of Double Point Singularities of Surfaces

Locally analytically, any isolated double point occurs as a double covering of a smooth surface. It can be desingularized via the canonical resolution, as it is well-known. In this paper we explicitly compute the fundamental cycle of both the canonical and minimal resolution of a double point singularity and we classify those for which the fundamental cycle differs from the fiber cycle. Finally we compute the conditions that a double point imposes to pluricanonical systems.

小平次数的正规曲面奇点

Normal surface singularities of small degrees

本文通过Yau序列推广Yau椭圆序列，利用Yau环计算二次曲面奇点的典范环，并给出算术亏格公式、几何亏格上界及加权对偶图分类性质。

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.

The integral monodromy of the cycle type singularities

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.

近Gorenstein有理表面奇点

Nearly Gorenstein rational surface singularities

本文研究了近Gorenstein有理表面奇点的性质，通过分析canonical trace理想和基本循环，给出了其判据并进行了分类。

Pure sheaves and Kleinian singularities

Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity $A_n$. We first study the analogue for the other Kleinian singularities except for $A_n$. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau $2$-dimensional category.

带权的光滑表面奇点与布里森完全交点奇点的同胚

Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections

本文研究了与布里森完全交点奇点同胚的带权光滑表面奇点，探讨了几何亏格最大值及极小良好解析中最大理想环与基本环重合的条件。

Cleanliness and log-characteristic cycles for vector bundles with flat connections

Let $X$ be a proper smooth algebraic variety over a field $k$ of characteristic zero and let $D$ be a divisor with simple normal crossings. Let $M$ be a vector bundle over $X-D$ equipped with a flat connection with possible irregular singularities along $D$. We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of $D$. When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-varia…

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.

Vanishing cycles, plane curve singularities, and framed mapping class groups

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 …

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.

Hypersurface Singularities and the Swing

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

Limits and Singularities of Normal Functions

We construct a collection of higher Chow cycles on certain surfaces which degenerate to an arrangement of planes in general position. When its degree is 4, this construction gives a new explicit proof of the Hodge-D-Conjecture for a certain type of K3 surfaces. As an application, we also construct a certain type of threefolds with non trivial Griffiths group.

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 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.

The vanishing cohomology of non-isolated hypersurface singularities

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it …

Invariants of plane curve singularities and Plücker formulas in positive characteristic

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $μ(f)=2δ(f)-r(f)+1$ and that $κ(f)=2δ(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $μ(f)\geq 2δ(f)-r(f)+1$ by showing that $μ(f)-\left( 2δ(f)-r(f)+1\right)$ measures the wild vanishing cycles. By introducing new invariants $γ,\tildeγ$, we prove in this note that $κ(f)\geq γ(f)+\mathrm{mt}(f)…

邻近周期的动机欧拉示性数与广义二次导子公式

Motivic Euler Characteristic of Nearby Cycles and a Generalized Quadratic Conductor Formula

本文通过半稳定约化的层计算 Ayoub 邻近周期谱的动机欧拉示性数，证明 Levine–Pepin Lehalleur–Srinivas 超曲面导子公式的局部性，并将其推广到多奇点情形。

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line…

对称双线性形式与正特征中孤立奇点的局部epsilon因子

Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic

研究孤立奇点的对称双线性形式与vanishing cycles复形的局部epsilon因子关系，通过双线性形式的判别式确定符号，改进Milnor公式。

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.

Zero cycles on certain surfaces in arbitrary characteristic

Let $k$ be a field of arbitrary characteristic. Let $S$ be a singular surface defined over $k$ with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation $\tilde{S}$ is finite dimensional. We give numerical conditions under which the Chow group of zero cycles of $S$ is finite dimensional.

A homology theory for tropical cycles on integral affine manifolds and a perfect pairing

We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\mathbb{Q}$ in degree one when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi-Yau degenerations. We also give an intersection theoretic application for Strominger-Yau-Zaslow fibrations. The treatment of the cap product and Poincaré-Lefschetz by simplicial methods for constructible sheaves might be of i…

二维正常奇点中的良好理想与 $p_g$-理想

Good ideals and $p_g$-ideals in two-dimensional normal singularities

本文研究了二维正常奇点中的良好理想与 $p_g$-理想，证明了二维Gorenstein正常局部环存在良好理想，并对简单椭圆奇点的Ulrich理想进行了分类。

Singularities of metrics on Hodge bundles and their topological invariants

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.

Invariants of Hypersurface Singularities in Positive Characteristic

We study singularities f in K[[x_1,...,x_n]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positive characteristic. Moreover, we consider different non-degeneracy conditions of Kouchnirenko, Wall and Beelen-Pellikaan in positive characteristic, and we show that planar Newton non-degenerate singularities satisfy Milnor's formula mu=2 delta-r+1. Th…

关于稳定牛顿退化奇点的猜想

Conjectures on stably Newton degenerate singularities

本文讨论Arnold问题，提出存在函数不能通过稳定化变为牛顿图非退化，并给出反例；同时猜想特征有限时非退化奇点没有wild消失圈，并论证不可约平面曲线稳定非退化。

Catégories des singularités, factorisations matricielles et cycles évanescents

The aim of this thesis is to study the dg categories of singularities Sing(X,s) of pairs (X,s), where X is a scheme and s is a global section of some vector bundle over X. Sing(X,s) is defined as the kernel of the dg functor from Sing(X_0) to Sing(X) induced by the pushforward along the inclusion of the (derived) zero locus X_0 of s in X. In the first part, we restrict ourselves to the case where the vector bundle is trivial. We prove a structure theorem for Sing(X,s) when X=Spec(B) is affine. Roughly, it tells us that every object in Sing(X,s) is represented by a complex of B-modules concentr…

Complex Intersections of Real Cycles in Real Algebraic Varieties and Generalized Arnold-Viro Inequalities

Consider a real algebraic variety, $\R X$, of dimension $d$. If its complexification, $\C X$, is a rational homology manifold (at least in a neighborhood of $\R X$), then the intersection form in $\C X$ defines a bilinear form in $d$-homologies of $\R X$. Analizing it, one can obtain an information about $\R X$, as it was done by V.I.Arnold in the case of non-singular double planes and then generalized by O.Ya.Viro and V.M.Kharlamov to the nodal surfaces. I present an integration (based on the Euler characteristic) formula, which expresses this form in terms of a certain local inveriant of the…

Choking horns in Lipschitz Geometry of Complex Algebraic Varieties

We study the Lipschitz Geometry of Complex Algebraic Singularities. For this purpose we introduce the notion of choking horns. A Choking horn is a family of cycles on the family of the sections of an algebraic variety by very small spheres centered at a singular point, such that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness as we can see with some classical isolated hypersurfaces singularities which we prove are not metrically conic. We also show that there exist infinitely countably many singular varieties, which are lo…

Nearby cycles and characteristic classes of singular spaces

In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and fun…

On the nearly smooth complex spaces

We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection …

Topological equivalence of complex polynomials

The following numerical control over the topological equivalence is proved: two complex polynomials in $n\not= 3$ variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions $f_s \colon \mathbb{C}^n \to \mathbb{C}$ with isolated singularities such that the degree, the number of vanishing cycles and the number of atypical values are constant in the family.

A generalization of Springer theory using nearby cycles

Let g be a complex semisimple Lie algebra, and f : g --> g/G the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f. We give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of complex reductive groups, for which the invariant theory works by analogy with the adjoint representations. Let G|V be such a representation, f : V --> V/G the quotient map, and P the sheaf of nearby cycles of f. We show that the Fourier transform of P is an intersection homology sheaf on V…

奇点谱、解析挠率与Calabi--Yau退化

Spectrum of singularities, analytic torsion and Calabi--Yau degenerations

本文通过解析挠率将奇点谱与Calabi--Yau退化几何相联系，提出高阶Durfee--Saito猜想，并利用挠率公式给出双有理光滑填充的障碍，推广了Voisin的结果。