Logarithmic deformations of normal crossing Enriques surfaces in characteristic two

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.

Toroidal crossings and logarithmic structures

We generalize Friedman's notion of d-semistability, which is a necessary condition for spaces with normal crossings to admit smoothings with regular total space. Our generalization deals with spaces that locally look like the boundary divisor in Gorenstein toroidal embeddings. In this situation, we replace d-semistability by the existence of global log structures for a given gerbe of local log structures. This leads to cohomological descriptions for the obstructions, existence, and automorphisms of log structures. We also apply toroidal crossings to mirror symmetry, by giving a duality constru…

Logarithmic Gromov-Witten theory with expansions

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there exist proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$, equipped with virtual classes. Each point in such a moduli space parameterizes a map from a nodal curve to an expanded degeneration of $X$ that is dimensionally transverse to the strata. In the context of maps to a simple normal crossings degeneration, the virtual fundamen…

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。

局部/对数对应关系与准映射的退化公式

The local/logarithmic correspondence and the degeneration formula for quasimaps

本文通过准映射理论研究局部几何与对数曲线计数的关系，提出新的退化公式并揭示不同相对Gromov-Witten理论表述间的联系。

对数 Donaldson-Thomas 理论

Logarithmic Donaldson-Thomas theory

本文构造了带有单正规交叉除子的光滑三维簇对的 Donaldson-Thomas 理论，通过研究目标几何膨胀中的子概形紧化模空间，并提出与标准理论平行的点态求值、有理性和穿墙猜想。

Smoothings of Fano varieties with normal crossing singularities

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.

对数小平维数为1的光滑仿射曲面的对数多重典范系

Logarithmic multicanonical systems of smooth affine surfaces of logarithmic Kodaira dimension one

本文研究对数小平维数为1的光滑仿射曲面的对数多重典范系，证明当m≥8时，该多重典范系可给出曲面到光滑射影曲线的ℙ¹-纤维化，推进了对数曲面纤维化理论的相关结果。

对数叶复形与叶状分解的d-半稳定性

The logarithmic leaf complex and foliated d-semistability

研究正常交叉品种上全纯叶状分解的半稳定性，通过log结构分析d-半稳定性，识别局部和全局障碍，并发展叶状变形理论以证明模态functor具有versal hull。

Characterizing normal crossing hypersurfaces

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a hypersurface has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito's theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found and and finally another one in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.

对退化情形的枚举几何的邀请

An invitation to the enumerative geometry of degenerations

本文介绍了通过简单正常交叉退化研究光滑项目ive变种的对数Gromov-Witten理论及其在曲线模空间上的应用。

Simple normal crossing Fano varieties and log Fano manifolds

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r\geq n/2 with ρ(X)\geq 2 or r\geq n-2. This result is a partial generalization of the classification of logarithmic Fano threefolds by Maeda.

On Saito's normal crossing condition

Kyoji Saito defined a residue map from the logarithmic differential 1-forms along a reduced complex analytic hypersurface to the meromorphic functions on the hypersurface. He studied the condition that the image of this map coincides with the weakly holomorphic functions, that is, with the functions on the normalization. With Michel Granger, the author proved that this condition is equivalent to the hypersurface being normal crossing in codimension one. In this article, the condition is given a natural interpretation in terms of regular differential forms beyond the hypersurface case. For redu…

对数上同调理论的构造 I

Construction of logarithmic cohomology theories I

本文借助 $\mathbb{F}_1$ 上的对数动机同伦论构造严格正规交叉边界对数概形的上同调理论，恢复开补集 K-理论与对数 THH，并证明非正则概形 K-理论可表且引入对数分圆迹。

A note on the cycle of curves in a product of pairs

We prove that the cycle-valued logarithmic Gromov--Witten theory of a product of simple normal crossings pairs $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, provided that the decomposition is considered over a blowup of the moduli space of curves. The result is established by applying the toroidal semistable reduction theorem to the stabilization morphism on the stack of maps to the Artin fans. Since products of smooth pairs are naturally simple normal crossings pairs, the result provides a direct avenue of access to questions in logarithmic Gromov--Witten theory via…

对数负切线与根堆栈

Logarithmic negative tangency and root stacks

本文研究稳定映射到正常交叠对的负切线阶问题，通过点状Gromov-Witten理论和根堆栈的Orbifold Gromov-Witten理论，定义了新的虚拟类，揭示了点状不变量与Orbifold不变量的联系，为零 genus 的对数-Orbifold 对比提供了完整图景。

通过多重加权爆破的对数分辨率

Logarithmic resolution via multi-weighted blow-ups

本文引入多重加权爆破概念，并系统构造了特征零域上Hironaka意义下的函子性对数分辨的显式高效算法。

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…

Logarithmic vanishing theorems for effective $q$-ample divisors

Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,Ω^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$

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。

简单正规交叉对的局部–轨形对应

The local-orbifold correspondence for simple normal crossings pairs

本文针对光滑射影簇上的简单正规交叉除子，通过秩约化证明局部丛与多根叠最大接触理论在亏格零闭链层面的对应，并厘清对数与轨形 Gromov–Witten 不变量的关系。

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。

On logarithmic extension of overconvergent isocrystals

In this paper, we establish a criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor. This is a generalization of a result of Kedlaya, who treated the case of unipotent monodromy. Our result is regarded as a $p$-adic analogue of the theory of canonical extension of regular singular integrable connections on smooth varieties of characteristic 0.

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 …

motivic、对数和拓扑 Milnor 纤维化

Motivic, logarithmic, and topological Milnor fibrations

本文通过引入完全 Milnor 纤维化，比较了正则复函数的拓扑 Milnor 纤维化和 motivic Milnor 纤维，给出两种等价构造并应用于恢复 A'Campo 模型，还证明 motivic Milnor 纤维在无需 Lefschetz motive 可逆时仍可良好定义。

叶状结构对数指标的全局残差公式

Global residue formula for logarithmic indices of foliations

本文研究一维全纯叶状结构的对数指标，建立全局残差公式及奇点总和公式，并应用于复射影空间与奇异正规射影簇。

Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan-Tian Perturbations

In a previous paper [FT1], for any logarithmic symplectic pair (X,D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural set up for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove…

曲线和三维流形中曲面的等奇异变形

Equisingular Deformations of Curves and Surfaces in Threefolds

本文研究曲线和曲面在三维流形中的等奇异变形问题，提出基于变形理论的标准，证明奇点数量由全局可实现性决定，并探讨其在退化情况下的应用。

On Kähler-Einstein surfaces with edge singularities

In this paper we characterize logarithmic surfaces which admit Kähler-Einstein metrics with negative scalar curvature and small edge singularities along a normal crossing divisor.

对数莱夫谢茨不动点公式与共振边界指标

Logarithmic Lefschetz fixed point formulae and resonant boundary indices

该研究针对带简单正常交叉边界且有孤立不动点的紧复流形严格自映射，证明了对数莱夫谢茨不动点公式，明确了边界点处特殊化系数的来源及共振与非共振边界项的不同表现。

Arrangements of Hyperplanes and Vector Bundles on $P^n$

Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector bundles on $P^n$. It turns out that all such bundles are stable. The study of jumping lines of these bundles gives a unified treatment of several classical constructions associating a curve to a collection of points in $P^n$. The main result of the paper is "Torelli theorem" which says that the collection of hyperplanes can be recovered from the isomorphism cl…

Pseudoholomorphic Curves Relative to a Normal Crossings Symplectic Divisor: Compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than…

henselian局部环的正常交叉簇的gersten型猜想

Gersten-type conjecture for henselian local rings of normal crossing varieties

本文证明了关于henselian局部环的正常交叉簇的gersten型猜想，并推广了artin关于brauer群的定理。

泊松连接的Poincaré–Dulac定理

A Poisson--Poincaré--Dulac for Poisson Connections

研究具有对数极点的泊松平坦连接，在非共振条件下证明其可规范等价于具有常数交换留数的纯Euler–泊松标准型，并建立泊松设置下的对数Riemann–Hilbert对应。

Čech-Stokes 推出群体：Stokes 蕴含物的 Log/Kummer Betti 呈现

A Čech--Stokes Pushout Groupoid: a Log/Kummer Betti Presenter for Stokes Torsors

本文提出了一种显式的Betti呈现，用于描述具有预设不规则类型沿简单正常交叉除子的 meromorphic 平展连接的 Stokes 蕴含物，通过严格的1-范畴和基于覆盖的构造，利用小Čech-Stokes 群体和显式推出构造，获得全局Stokes对象的群体呈现。