平滑环面交叉空间

关于法向交叉簇的全局光滑化的霍奇结构

On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

本文研究半稳定退化中一般纤维的霍奇结构，给出满足∂∂-引理的判定准则和用中心纤维拓扑不变量计算霍奇指标的公式，并应用于ODP光滑化及非凯勒卡拉比-丘流形等例子。

具有简单正规交叉点的代数簇的投射光滑化

Projective smoothing of varieties with simple normal crossings

该研究提出基于态射形变理论与ribbons概形的新方法，证明投射空间中带两个Fano分支的简单正规交叉簇的存在性与光滑化，给出三维簇的显式构造及对应slc对的光滑化结果。

环面、环面型和环几何中实定向吹胀的介绍

An introduction to real oriented blowups in toric, toroidal and logarithmic geometries

本文介绍了复对数空间取整（Kato-Nakayama实现）在奇点理论中的应用，通过极坐标转换引入仿射环面簇、实定向吹胀、对数结构等概念，并涵盖Nakayama-Ogus关于特殊对数态射取整局部平凡性的定理。

Global smoothings of degenerate K3 surfaces with triple points

Let $X$ be a normal crossing compact complex surface with triple points. We prove that there exists a family of smoothings of $X$ when $X$ satisfies suitable conditions. Since our differential geometric proof also includes the case where $X$ is neither Kählerian nor $H^1(X, \mathcal O_X)=0$, this generalizes Friedman's result on degenerations of $K3$ surfaces in algebraic geometry.

如何构造log结构

How to make log structures

本文引入了广义Gorenstein锥形交叉空间的概念，并定义了与之相关的sheaf，建立了log结构与全局截面之间的双射关系，推广了相关理论。

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。

A local Torelli theorem for log symplectic manifolds

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing di…

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…

Non-Collapsible Dual Complexes and Fake del Pezzo Surfaces

We propose the new construction of complex surfaces with $h^{1,0} = h^{2,0} = 0$ from smoothings of normal crossing surfaces with non-collapsible dual complexes and carry it out for the simplest case of the duncehat complex, obtaining the surface with $h^{1,1} = 9$ (presumably Barlow surface).

A Surface Degeneration With Non-Collapsible Contractible Dual Complex

We construct a smoothing of a particular normal crossing surface, which dual complex is a duncehat, thus obtaining an example of a surface degeneration with non-collapsible, contractible dual complex. The general fiber of this family has $h^{1,0} = h^{2,0} = 0$ and $h^{1,1}=9$, we conjecture that it is in the deformation class of the Barlow surface. This text is a complete rewrite of author's previous work [14].

On the symplectic cohomology of log Calabi-Yau surfaces

This article studies the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex computing symplectic cohomology, and compute enough differentials to identify a basis for the degree-zero part of the symplectic cohomology. This basis is indexed by integral points in a certain integral affine manifold, providing a relationship to the theta functions of Gross--Hacking--Keel. Included is a discussion of wrapped Floer cohomology of Lagrangian su…

有理椭圆曲面模空间的K-模壁跨越与自守形式

K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces

该研究利用对数拟映射的K-模空间构造模插值，确定其壁跨越，通过Borcherds乘积构造新自守形式，证明特定区间的K-模空间为半环面紧化。

Log Smooth Deformation Theory

This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduce…

Weak Coupling, Degeneration and Log Calabi-Yau Spaces

We establish a new weak coupling limit in F-theory. The new limit may be thought of as the process in which a local model bubbles off from the rest of the Calabi-Yau. The construction comes with a small deformation parameter $t$ such that computations in the local model become exact as $t \to 0$. More generally, we advocate a modular approach where compact Calabi-Yau geometries are obtained by gluing together local pieces (log Calabi-Yau spaces) into a normal crossing variety and smoothing, in analogy with a similar cutting and gluing approach to topological field theories. We further argue fo…

对数共形射影流形

Log-Conformal Projective Manifolds

研究对数共形张量的射影流形，探讨其几何性质及在不同条件下的结构特征。