Frédéric Campana, Hubert Flenner
A contact singularity is a normal singularity $(V,0)$ together with a holomorphic contact form $η$ on $V\backslash$ Sing $V$ in a neighbourhood of 0, i.e. $η\wedge (dη)^r$ has no zero, where dim $V=2r+1$. The main result of this paper is that there are no isolated contact singularities.
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We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].
On fillings of contact links of quotient singularities
该研究利用Floer理论探讨商奇点接触链环的各类填充问题，证明多类孤立终端商奇点的接触链环不存在恰当填充并解决相关猜想，同时得到部分情形下恰当轨形填充的轨形微分同胚类型唯一性。
Rigidity properties of holomorphic Legendrian singularities
本文在复解析范畴中研究接触流形的勒让德奇点，证明了约化切锥奇点与切锥的接触同胚双全纯等价性，以及正规勒让德奇点的形变刚性，方法基于无穷小接触同胚与自然线丛截面的联系。
It is known that the lens space $L(2n,1)$ supports a virtually overtwisted contact structure arising as the boundary of the Milnor fiber of a complex hypersurface singularity. In this article we study the problem of realizing other $(L(p,q),ξ)$ in such a way, obtaining a series of necessary conditions for this to happen.
Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterate…
Contact loci of semihomogeneous singularities
本文研究半齐次奇点的接触轨迹几何，在特定维数和次数条件下肯定回答弧-弗洛尔猜想，并完整解决嵌入纳什问题。
We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.
The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global s…
We construct coarse moduli spaces of semiquasihomogeneous hypersurface singularities with respect to right equivalence and contact equivalence. We have to fix the principal part of the semiquasihomogeneous singularities. For the moduli spaces with respect to contact equivalence we also fix the Hilbert function of the Tjurina algebra induced by the weights.
Idealistic Flowers in the Reduction of Singularities
本文基于“理想花朵”思想，遵循Maximal Contact Theory，为经典奇点化简提供了证明，并解决了由最大接触局部性质引发的全球化问题。
Let X be an affine normal variety with a C^*-action having only positive weights. Assume that X_{reg} has a symplectic 2-form w of weight l. We prove that, when l is not zero, the w is a unique symplectic 2-form of weight l up to C^*-equivariant automorphism When $l = 0$, we have a counter-example to this statement. In the latter half of the article, we associate to X a projective variety P(X) and prove that P(X) has a contact orbifold structure. Moreover, when X has canonical singularities, the contact orbifold structure is rigid under a small deformation. By using the contact structure on P(…
We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of $S^2 \times S^3$.
Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and $\D_X$ the complex vector bundle associated to $\cal X$. We prove that the bundle $\D_X$ is trivial if and only if the Milnor number of $X$ satisfies $μ(X,0) \equiv (-1)^{n-1}$ modulo $(n-1)!$. This follows from a general theorem stating that the complex orthogonal complement of a vector field in $X$ with an isolated singularity at 0 is trivial iff the GSV-index of $v$ is a multiple of $(n-1)!$. We have also an application to foli…
We study loci of arcs on a smooth variety defined by order of contact with a fixed subscheme. Specifically, we establish a Nash-type correspondence showing that the irreducible components of these loci arise from (intersections of) exceptional divisors in a resolution of singularities. We show also that these loci account for all the valuations determined by irreducible cylinders in the arc space. Along the way, we recover in an elementary fashion-without using motivic integration-results of the third author relating singularities to arc spaces. Moreover, we extend these results to give a jet-…
The main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real an complex hypersurfaces in the 1970's with respect to right equivalence, several new phenomena occur in positive characteristic. Already the notion of isolated singularity is different for right resp. contact equivalence over fields of characteristic other than zero. The heart of this paper consists of the study of different notions of non-degeneracy and the associated piecewise f…
We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the exceptional line; we also find the degree of the locus defined by imposing a leaf of total contact with the exceptional line.
We study the sheaf of locally square integrable holomorphic section of vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.
Resolution except for minimal singularities II. The case of four variables
本文是奇点消解研究续篇，刻画了四元变量下无法通过避免吹胀正规交叉奇点的双有理态射消除的、作为三重正规交叉奇点极限的最小奇点类。
Singular contact varieties
本研究将光滑流形的全纯切触结构推广到有理奇点簇，证明其基本性质，给出幂零轨道闭包射影化等实例，建立辛簇与切触簇的对应及Kaledin型分层，还证明射影情形下 crepant 与切触解消等价、簇的单直纹性并完成三维分类。
Planar multilinks and rational singularities
本文研究了平面多链与有理奇点的关系，证明了若表面奇点的canonical接触结构由平面多链开书支撑，则该奇点必为有理奇点，并通过平面多链的组件多重性1来特征化嵌套奇点。同时，文章还展示了平面开书的一些拓扑性质可扩展到平面多链：辛填充为负定且不包含正亏格的辛表面，以及Heegaard Floer接触不变量在HF_{red}中消失。
Currently there are several approaches to resolution of singularities in positive characteristic all of which have hit some obstruction. One natural idea is to try to construct new meaningful examples at this point to gain a wider range of experience. To produce such examples we mimic the characteristic zero approach and focus on cases where it fails. In particular, this short note deals with an example-driven study of failure of maximal contact and the search for an appropriate replacement.
Thom series in negative relative codimension
本文建立接触函数奇点的Thom级数理论，证明其两项核心性质，阐明三类Thom多项式的关系并完成多类奇点的相关计算，成果将应用于计数几何问题研究。
The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory
本文证明加权齐次型二维孤立超曲面奇点（非有理二重点）的Milnor纤维单值微分同胚在光滑映射类群中具有无限阶，并借助等变Seiberg--Witten--Floer同调中的切触不变量类比，将该结果归结为负定向Seifert纤维化有理同调3-球面辛填充边界Dehn扭转的无限阶性质。
The paper is motivated on the open problem of resolution of singularities in positive characteristic. The aim is to present a form of induction which is different from that used by Hironaka. In characteristic zero induction is formulated by restriction to smooth hypersurfaces (hypersurfaces of maximal contact). Our alternative approach, introduced here, replaces restrictions to smooth sub-schemes by generic projections on smooth schemes of smaller dimension. We also introduce a generalization of the discriminant, and our result makes use of the elimination theory. In the case of fields of char…
The work is a detailed study of rational singularities of multiplicity 3 (RTP-singularities, for short). We give a list of nonisolated hypersurface singularities of which normalisations are the RTP-singularities, and construct their minimal resolution graphs by means of a subdivision of Newton polygons of those -- a method introduced by M. Oka for isolated complete intersection singularities. We show that nonisolated forms of RTP-singularities and their normalisations are both Newton non-degenerate.
We study holomorphic function germs under equivalence relations that preserve an analytic variety. We show that two quasihomogeneous polynomials, not necessarily with isolated singularities, having isomorphic relative Milnor algebras are relative right equivalent. Under the condition that the module of vector fields tangent to the variety is finitely generated, we also show that the relative Tjurina algebra is a complete invariant for the classification of arbitrary function germs with respect to the relative contact equivalence. This is the relative version of a well known result by Mather an…
Normal reduction numbers of normal surface singularities
本文研究了正常表面奇点的正常还原数，综述了相关结果并证明了Brieskorn完全交集的新结论。
Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as isolated, non Cohen-Macaulay threefold singularities. They arise by a small contraction of a smooth rational curve, whose normal bundle has a sufficiently positive subbundle. We study such singularities from their nonnormal general hyperplane section.
In the study of normal surface singularities the relation between analytical and topological properties and invariants of the singularity is a very rich problem. This relation is particularly close for surface singularities constructed from families of curves. We use these Kulikov singularities to reexamine results of Némethi-Okuma and Tomaru.
In this paper we give a new point of view for optimizing the definitions related to the study of singularities of normal varieties, introduced in [dFH09] and further studied in [Urb12a] and [Urb12b], in relation to the Minimal Model Program. We introduce a notion of discrepancy for normal varieties, and we define log terminal+ singularities. We use finite generation to relate these new singularities with log terminal singularities (in the sense of [dFH09]).
We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form $F_{k,n}=\sum_{i=1}^n x_i^k$ invariant under the natural $S_n$-action.
In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence classes of the unimodal real singularities of corank 2. In other words, we explicitly answer the question which normal forms of different type are equivalent, and how a normal form can be transformed within the same equivalence class by changing the value of the parameter. This provides new theoretical insights into these singularities and has important cons…
In this article we study the triangulated category of singularities associated with a non-commutative resolution of singularities. In particular, we give a complete description of this category in the case of a curve with nodal singularities, classifying its indecomposable objects and computing its Auslander-Reiten quiver and K-group.
This is a survey article on $F$-singularities and their applications.
This is a continuation of our paper math.AG/0111298. We prove an explicit formula for the geometric genus p_g of a quasihomogeneous isolated surface singularity in terms of the Seiberg-Witten invariant of the link and other topological data which can be read from a resolution graph of the singularity. Moreover, we also determine all the Reidemeister-Turaev sign-refined torsion of the link (associated with any spin^c structure) in terms of the Seifert invariants.
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