Maximal contact and normal crossings in resolution of singularities

Desingularization by blowings-up avoiding simple normal crossings

It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points where the transformed variety together with the exceptional divisor has only snc singularities. The proof follows the philosophy of arXiv:1107.5595 that the desingularization invariant can be used together with natural geometric information to compute local normal forms of singularities.

Analytic equivalence of normal crossing functions on a real analytic manifold

By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a suitable modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions $C^{\infty}$ right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of equivalence classes is zero or countable.

代数曲面中A-奇点最大数的一些下界. II

Some lower bounds for the maximal number of A-singularities in algebraic surfaces. II

本文扩展了先前构造，为某些额外情形下代数曲面中A型奇点的最大数提供了下界。

Conic singularities metrics with prescribed scalar curvature: a priori estimates for normal crossing divisors

The purpose of this paper is to prove the a priori estimates for constant scalar curvature Kaehler metrics with conic singularities along normal crossing divisors. The zero order estimates are proved by a reformulated version of Alexandrov's maximum principle. The higher order estimates follow from Chen-Cheng's frame work, equipped with new techniques to handle the singularities. Finally, we extend these estimates to the twisted equations.

Maximal rank for planar singularities of multiplicity two

We prove that general unions of singularity schemes of multiplicity two in the projective plane have maximal rank.

Normal Crossings Singularities for Symplectic Topology, II

In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological smoothability criterion for them. The present paper extends these notions to arbitrary normal crossings singularities, from both local and global perspectives, and shows that they are also equivalent to the corresponding geometric notions. In subsequent papers, we extend our smoothability criterion to arbitrary normal crossings symplectic varieties and construct a …

群循环奇点与保持正常交叉的局部除 Singularities

Group-circulant singularities and partial desingularization preserving normal crossings

本文提出了一种基于群循环矩阵组合学的方法，用于局部除 Singularities并保持正常交叉结构。

Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of logarithmic vector fields along an algebraic hypersurface singularity we indentify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one-to-one correspondence to maxmimal tori in the degree zero jet of the embedded automorphism group. The result is motivated by Kyoji Saito's characterization of quasihomogeneity for isolated hypersurface singularities and extends its formal version and a result of Hauser and Mueller.

Essentials of the method of maximal singularities

A consistent exposition of the arguments and constructions of the method of maximal singularities, the aim of which is to describe birational iso/automorphisms of Fano varieties and Fano fibrations. The principal elements of the method are considered: N{\" o}ther-Fano inequality, maximal cycles, infinitely near maximal singularities, exclusion and untwisting. In a detailed way the crucial technical points are discussed. We also give a new version of the proof of Sarkisov theorem which is ideologically more close to the original arguments of V.A.Iskovskikh and Yu.I.Manin.

Resolution except for minimal singularities I

The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the small…

Moduli of regular holonomic D-modules with normal crossing singularities

This paper solves the global moduli problem for regular holonomic D-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to ``pre-D-modules''), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-D-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces.

代数曲面中A-奇点最大数量的若干下界

Some lower bounds for the maximal number of A-singularities in algebraic surfaces

本文构造具有大量A型奇点的代数曲面，利用二元多项式和Belyi多项式族，部分情形给出Jacobi多项式的显式表达，从而得到奇点数量的下界。

除极小奇点外的消解 II：四元变量情形

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。

Semistable Reduction of a Normal Crossing $\mathbb{Q}$-Divisor

In a previous work we have introduced the notion of embedded $\mathbf{Q}$-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities, and A'Campo's formula was calculated in this setting. Here we study the semistable reduction associated with an embedded $\mathbf{Q}$-resolution so as to compute the mixed Hodge structure on the cohomology of the Milnor fiber in the isolated case using a generalization of Steenbrink's spectral sequence. Examples of Yomdin-Lê surface singularities are presented as an application.

Resolution of singularities of pairs preserving semi-simple normal crossings

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X,D) is said to be semi-simple normal crossings (semi-snc) at a point a of X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f: X' -> X such that the transformed pair (X',D') is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal…

Simple normal crossing varieties with prescribed dual complex

We prove that every simplicial complex is the dual complex of some simple normal crossing divisor in a smooth variety. As an application, we simplify and extend the results of Kapovich--Kollár (math.AG:1109.4047) on the existence of singularities with given dual complex.

A finiteness theorem on symplectic singularities

For positive integers N and d, there are only finite number of conical symplectic varieties of dimension 2d with maximal weights N, up to isomorphism. The maximal weight of a conical symplectic variety X is, by definition, the maximal weight of the minimal homogeneous generators of the coordinate ring R of X.

利用加权爆破在特征0中实现奇点的对数消解

Logarithmic resolution of singularities in characteristic 0 using weighted blow-ups

本文研究特征零域上的奇点消解问题，核心方法是使用加权爆破，构造出例外因子呈简单正常交叉的对数消解。

除正常交叉定位外的解析性及伽罗瓦作用

Resolution Except for the Normal-Crossing Locus and Galois actions

本文在零特征下构造了一个基于加权吹脱的通用解析算法，严格保持正常交叉定位，解决Kollar问题。通过加权吹脱框架和对数细化，实现鲁棒解析算法，应用于正常交叉Deligne-Mumford堆的canonical compactification和子簇与堆的functorial解析。

Normal Crossings Singularities for Symplectic Topology

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in …

Normal Crossings Singularities for Symplectic Topology: Structures

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures hav applications in constructions and analysis of various moduli spaces. As a coroll…

Combinatorial Aspects of Classical Resolution of Singularities

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.

New Ideas for Resolution of Singularities in Arbitrary Characteristics

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…

Logarithmic deformations of normal crossing Enriques surfaces in characteristic two

Working in characteristic two, I classify nonsmooth Enriques surfaces with normal crossing singularities. Using Kato's theory of logarithmic structures, I show that such surfaces are smoothable and lift to characteristic zero, provided they are d-semistable.

Singularities and Semistable Degenerations for Symplectic Topology

We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.

在辛奇异性上构造极小Cohen-Macaulay层

Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities

本文研究了辛奇异性上极小Cohen-Macaulay层的构造与性质，通过反射层和上同调消失理论，推导了不可分解层的生成方法。

形式分裂与堆叠理论中的正常交叉除 singularities

Formal splitting and stack-theoretic normal crossings desingularization

本文通过形式分裂定理和Abramovich等人算法，展示如何用加权吹胀获得保持正常交叉的堆叠理论除 singularities。

Maximal Cohen-Macaulay modules over surface singularities

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss various results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we describe matrix factorizations corresponding to the indecomposable Cohen-Macaulay modules over the non-isolated singularities $A_\infty$ and $D_\infty$.

Polyhedral Groups and Pencils of K3-Surfaces with Maximal Picard Number

A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each pencil has Picard number 19 and each pencil contains four surfaces with Picard number 20. These surfaces are obtained as the minimal resolution of quotients X/G, where $G\subset SO(4,\R)$ is some finite subgroup and $X\subset ¶_3(\C)$ denotes a G-invariant surface. The singularities of X/G come from fix points of G on X or from singularities of X. In any case the…

On the maximal number of du Val singularities for a K3 surface

A complex K3 surface or an algebraic K3 surface in characteristics distinct from $2$ cannot have more than $16$ disjoint nodal curves.

奇点化简中的理想花朵

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。

$1$-generic 矩阵的极大子式具有有理奇点

Maximal minors of $1$-generic matrices have rational singularities

本文证明 $1$-generic 矩阵极大子式理想的商环具有有理奇点，证实 Eisenbud 的正规性猜想并推广 generic Hankel 行列式环的相关结论。

Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of $k\llbracket x,y,z\rrbracket/(xyz)$ as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new …

Normal crossing properties of complex hypersurfaces via logarithmic residues

We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, describe all free divisors with Gorenstein singular locus.

椭圆表面奇点中理想上同调

Cohomology of ideals in elliptic surface singularities

本文研究了二维正常奇点的正常还原数，证明了椭圆奇点的正常还原数为二，并探讨了非有理奇点的Gorenstein性质与$p_g$-理想的关系。

部分去奇异化

Partial desingularization

本文研究保持法向交叉的部分去奇异化问题，通过循环矩阵定义极小奇点列表，并结合首一多项式分裂技术与奇点消解，在维数不超过4或仅保持至多三阶法向交叉时给出肯定回答。

Combinatorial structure of exceptional sets in resolutions of singularities

The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity. The homotopy type of the dual complex does not depend on the choice of a resolution and thus can be considered as an invariant of singularity. In this preprint we show that the dual complex is homotopy trivial for resolutions of 3-dimensional terminal singularities and for resolutions of Brieskorn singularities. We also review our earlier results on resoluti…

对数等纲平滑与最大节点退化

Logarithmic Equigeneric Smoothing and Maximal Nodal Degenerations

本文研究了曲线和曲面在代数族中的等纲变形问题，提出通过对数半正则性方法在退化情况下保持最大奇点数量的理论，并探讨了其在细化塞弗里计数中的应用。

Fundamental groups of links of isolated singularities

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to G. Lastly, we prove that a finitely-presented group G is Q-superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if G is iso…

Isolated non-normal crossings

We describe a multivariable polynomial invariant for certain class of non isolated hypersurface singularities generalizing the characteristic polynomial on monodromy. The starting point is an extension of a theorem due to Le Dung Trang and K.Saito on commutativity of the local fundamental groups of certain hypersurfaces. The description of multivariable polynomial invariants is given in terms of the ideals and polytopes of quasiadjunction generalizing corresponding data used in the study of the homotopy groups of the complements to projective hypersurfaces and Alexander invariants of plane red…

Local vanishing and Hodge filtration for rational singularities

Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. We prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization of the structure sheaf along D is at most n-3.

The dual complex of singularities

The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well defined up-to piecewise linear homeomorphism. This is derived from a more global result concerning dual complexes of dlt pairs. As an application, we also show that the dual complex of a log terminal singularity as well as the one of a simple normal crossing degeneration of a family of rationally connected manifolds are contractible.

On the classification of rational surface singularities

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely certain types of graphs. This allows to extend the classification of rational singularities to multiplicity 8. We also discuss the complexity of rational resolution graphs.

Maximal Modifications and Auslander-Reiten Duality for Non-isolated Singularities

We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modify…

Resolution of Singularities of Germs in Characteristic Positive Associated with Valuation Rings of Iterated Divisor Type

In this paper we show that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations in centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transformations with a normal crossing, Weierstrass representations, reduction sequences, and so forth.

具有正常交叉极限的退化凯勒-爱因斯坦流形族的射影嵌入

Projective embedding of degenerating family of Kähler-Einstein manifolds with normal crossing limit

本文研究负曲率凯勒-爱因斯坦流形退化族的伯格曼嵌入，针对中央纤维仅含简单正常交叉奇点的情况，构造标准正交基实现伯格曼嵌入的收敛性。

曲线奇点与δ不变量的等价性

Equivalence of Curve Singularities and delta-Invariants

本文证明了完备既约诺特曲线参数化的等价条件，以及既约不可约曲线奇点的同构性与完备化同构的关系，给出了优于Hironaka的新下界。

A型和D型商奇点的矩阵分解

Matrix factorization of quotient singularities of type A and D

本文扩展ADE奇点与极大库恩-麦考利模的关系至A型和D型商奇点，通过投影超曲面构造显式矩阵分解和模族，并恢复所有特殊库恩-麦考利模。

Rank two Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$

We describe, by matrix factorizations, all the rank two maximal Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$.

Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary r…

Generic Hypersurface Singularities

We give a new differential proof of our result on the maximal rank of generic unions of points of multiplicity two in projective space in degrees greater than five. This simplifies somewhat our proof of the Waring conjecture.

Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities

In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi-Yau type property. If C is such a category, we say that C is Calabi-Yau with dimension at most one. We extend the notion of Calabi-Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categor…

A characterization of nilpotent orbit closures among symplectic singularities

We prove that a conical symplectic variety with maximal weight 1 is isomorphic to one of the following: (i) an affine space with the standard symplectic form (ii) a nilpotent orbit closure of a complex semisimple Lie algebra with the Kirillov-Kostant form.