Nikolaos Tziolas
We describe the sheaf T^1(X) of first order deformations of a reduced scheme with normal crossing singularities. In particular, we obtain a formula for T^1(X) in a suitable log resolution of X.
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We study the deformation theory of a Fano variety X with normal crossing singularities of dimension at most three. We obtain a formula for the sheaf T^1(X) of first order deformations of X in a suitable log resolution of X and its singular locus Z and we obtain explicit criteria for the existence of smoothings of X.
This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in [EGA,III.2.6.2] and raised again by Viehweg.
This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of logarithmic structures and deformations is used in order to prove that X is smoothable by a smooth variety, if and only if T^1(X)=O_D, where D is the singular locus of X.
The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X,D), where D is a divisor on X), we construct a functorial desingularization of all but stable simple normal crossings (stable-snc) singularities, by smooth blowings-up that preserve such singularities. A variety has stable simple normal crossings at a point if, locally, its irreducible components are smooth and tranverse in some smooth embedding variety. We also show that our main assertion is false for more general simple norm…
We develop a Hodge theory for relative simple normal crossing varieties over an Artinian base scheme. We introduce the notion of a mixed Hodge structure over an Artin ring, which axiomatizes the structure that is found on the cohomology of such a variety. As an application we prove that the maps between the graded pieces of the Hodge bundles have constant rank.
Equivariant deformation of minimally elliptic singularities
本文研究了有限群作用下极小椭圆曲面奇点的等变变形分量，探讨了简单椭圆与尖点奇点的商的例子，并在局部完全交情形下识别出具有完美障碍理论的一般型曲面的等变变形分量。
In this article, we prove that any $\Bbb Q$-factorial weak Fano 3-fold with only terminal singularities has a smoothing.
We prove the constancy of the Łojasiewicz exponent in non-degenerate $μ$-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.
Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.
Deformation Classification of Quartic Surfaces with Simple Singularities
本文对作为 $K3$ 曲面的简单空间四次曲面进行了完整的等奇点形变分类。
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.
In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.
Deformations of isolated cyclic quotient singularities in arbitrary characteristic
该研究证明了孤立循环商奇点在任意特征下均可变形，并验证了相关猜想。
The purpose of this paper is to prove dimension formulas for $T^1$ and $T^2$ for rational surface singularities. These modules play an important role in the deformation theory of isolated singularities in analytic and algebraic geometry. The first may be identified as the Zariski tangent space of the versal deformation of the singularity; i.e. it is the space of infinitesimal deformations. The second contains the obstruction space -- in all known cases it is the whole obstruction space for rational surface singularities.
We show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. In Theorem 1.1 we list integers can be attained as Milnor numbers of a given semi-quasi-homogeneous singularity.
In this paper we study $(ε,δ)$-lc singularites, i.e. $ε$-lc singularities admitting a $δ$-plt blow-up. We prove that $n$-dimensional $(ε,δ)$-lc singularities are bounded up to a deformation, and $2$-dimensional $(ε,δ)$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(ε,δ)$-lc singularities are not bounded in higher dimensions, even in the analytic sense.
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.
In this paper, we consider the generalized isomonodromic deformations of rank two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank two irregular connections by apparent singularities. By the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local …
We describe the versal deformation of two-dimensional cyclic quotient singularities in terms of equations, following Arndt, Brohme and Hamm. For the reduced components the equations are determined by certain systems of dots in a triangle. The equations of the versal deformation itself are governed by a different combinatorial structure, involving rooted trees.
We investigate the correspondence between three theories of deformations of rational surface singularities: de Jong and van Straten's picture deformations, Kollár's P-resolutions, and Pinkham's smoothings of negative weights. We provide an explicit method for obtaining, from a given deformation in one theory, deformations in other theories that parameterize the same irreducible components of the deformation space of the singularity. We employ the semi-stable minimal model program significantly for this purpose. We prove Kollár conjecture for various sandwiched surface singularities as an appli…
Blow-$ADE$ singularities and $μ^*$-constant deformations
本研究引入复曲面的blow-$ADE$奇点，证明其在若干特殊情形下对$μ^*$-常形变具有稳定性，且研究覆盖连通$μ^*$-常层任意两元素的形变族，突破小参数限制。
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.
The jump of the Milnor number of quasihomogeneous singularities for linear deformations
本文研究拟齐次奇点在线性形变类中 Milnor 数的最小非零变化，并给出了该跳跃值的确定方法。
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter space. The methods used in paper are very elementary.
We give a $δ$-constant criterion for equinormalizability of deformations of isolated (not necessarily reduced) curve singularities over smooth base spaces of dimension $\geq 1$. For one-parametric families of isolated curve singularities, we show that their topologically triviality is equivalent to the admission of weak simultaneous resolutions.
We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $δ$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details defor…
We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different invariants, such as the Milnor number, the Tjurina number, and the dimension of a smoothing component.
Deformations of weighted homogeneous surface singularities with big central node
本研究证明了带大中心结点的加权齐次曲面奇点的Kollár猜想，指出此类奇点形变空间的各不可约分支均可由$P$-消解这种部分消解来参数化。
The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. Main result states that if $f$ is irreducible and nondegenerate, by deforming $f$ one can attain all Milnor numbers ranging from $μ(f)$ to $μ(f)-r(p-r)$, where $r$ and $p$ are easily computed from the Newton diagram of $f$.
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.
We prove that the cohomology sheaves of the relative dualizing complex of a flat family of varieties with semi-log-canonical or Du Bois singularities are flat and commute with base change. This is a local version of our earlier similar result where the family was assumed to be projective. We also derive several consequences for deformations of semi-log-canonical and of Du Bois singularities.
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 …
We explicitly describe infintesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollár-Shepherd-Barron and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.
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。
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…
In this paper we study the deformation and Q-Gorenstein deformation theory of schemes with non-isolated singularities. We obtain obstruction spaces for the existence of deformations and also for local deformations to exist globally. Finally we obtain explicit criteria in order for a pure and reduced scheme of finite type over a field $k$ to have smoothings and Q-Gorenstein smoothings.
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.
Given a polyhedral cone sigma with smooth two-dimensional faces and, moreover, a lattice point R in the dual cone of sigma, we describe the part of the versal deformation of the associated toric variety TV(sigma) that is built from the deformation parameters of multidegree R. Let Q the polyhedron obtained by intersecting sigma with the hyperplane R=1. Then the base space is (the germ of) an affine scheme that reflects certain possibilities of splitting Q into Minkowski summands.
Non-Commutative Deformations of Derived McKay Correspondence for A(n) singularities
本文证明导出McKay对应可扩展到A(n)环面曲面奇点的交换与非交换crepant解消的半泛非交换形变之间，从而推广了经典的导出等价结果。
We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.
Let $C$ be an isolated plane curve singularity, and $(C,l)$ be a decorated curve. In this article we compare the equisingular deformations of $C$ and the sandwiched singularity $X(C,l)$. We will prove that for $l \gg 0$ the functor of equisingular deformations of $C$ and $(C,l)$ are equivalent. From this we deduce a proof of a formula for the dimension of the equisingular stratum. Furthermore we will show how compute the equisingularity ideal of the curve singularity $C$, given the minimal (good) resolution of $C$.
Resolutions and deformations of cyclic quotient surface singularities
本文研究形如 C^2/G 的循环商奇点的最小分辨、G-Hilbert 方案、广义 McKay 对应、奇点形变和箭图簇之间的关系，提供统一阐述和丰富例子。
Inspired by an article of Cotti, Dubrovin and Guzzetti arXiv:1706.04808, we extend to a degenerate case a result of Malgrange on integrable deformations of irregular singularities. We give an application to integrable deformations of the solution of some Birkhoff problem and apply it to the construction of Frobenius manifolds.
We prove that small deformations of canonical singularities are canonical.
Deformations of I-surfaces with elliptic singularities
本文研究具有简单椭圆或尖点奇点的I曲面（满足$K_S^2=1$，$p_g=2$的一般型代数曲面），在一般位置假设下证明其形变对奇点形变是versal的，并详细分析两个例外情况。
Deformations, local freeness, and base change for higher Du Bois singularities
该研究证明严格高阶Du Bois奇点在小形变下不变，针对严格高阶Du Bois纤维的相对Du Bois复形建立基变换定理，展示1-Du Bois纤维的相关失效并推广局部自由性定理、证明Hodge数不变性。
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。
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.
Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops.
We study the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability conditions. We use wall-crossing techniques to describe its geometric structure and singularities, which reproves the classical result of Piene and Schlessinger.
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…
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.
We study the deformations of the minimally elliptic surface singularity $N_{16}$. A standard argument reduces the study of the deformations of $N_{16}$ to the study of the moduli space of pairs $(C,L)$ consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of $K3$ surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via $K3$ surfaces and for another we obtain the full deformation space of $N_{16}…
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。
The obstruction space T^2 and the cup product T^1 x T^1 -> T^2 are computed for toric singularities.
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。
We investigate some coboundary map associated to a $3$-dimensional terminal singularity which is important in the study of deformations of singular $3$-folds. We prove that this map vanishes only for quotient singularities and a $A_{1,2}/4$-singularity, that is, a terminal singularity analytically isomorphic to a $\mathbb{Z}_4$-quotient of the singularity $ (x^2+y^2 +z^3+u^2=0)$. As an application, we prove that a $\mathbb{Q}$-Fano $3$-fold with terminal singularities can be deformed to one with only quotient singularities and $A_{1,2}/4$-singularities. We also treat the $\mathbb{Q}$-smoothabi…
The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)$. We give a formula for the jump in some class of surface singularities in the case deformations are non-degenerate.
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