Sándor J Kovács, Karl Schwede
This is a survey of some recent developments in the study of singularities related to the classification theory of algebraic varieties. In particular, the definition and basic properties of Du Bois singularities and their connections to the more commonly known singularities of the minimal model program are reviewed and discussed.
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We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing, and reflexive differentials associated to Z.
In this note we show that the sheaf $R^1 f_* \mathcal{O}_X$ is an anti-nef vector bundle (i.e., its dual is nef), where $f : X \to Y$ is a family of Du Bois schemes of pure dimension.
A Hodge module construction of relative Du Bois complexes
本文通过混合 Hodge 模和 Grothendieck 对偶构造相对 Du Bois 复形，并建立有限滤过及纤维限制与符号 m-Du Bois 奇点的联系。
Q-factoriality and Hodge-Du Bois theory
该研究证明了正规射影簇及正规簇解析芽的$\text{Q}$-因子性缺陷的霍奇理论公式，其结果涵盖Samuel猜想局部解析版本、有理同调三维簇刻画及霍奇-杜布瓦数在翻转下的不变性等。
We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is quite nontrivial, because the theory of mixed Hodge module does not work on a simplicial scheme. Some application to the Du Bois singularity is given.
It is proved that for projective varieties having Du Bois singularities is equivalent to the condition that the coherent cohomology groups of the structure sheaf coincide with the appropriate Hodge components of the singular cohomology groups.
Higher Du Bois and higher rational singularities of hypersurfaces
本研究针对全局定义的超曲面，利用消失循环混合霍奇模的Hodge滤过给出高阶Du Bois与高阶有理奇点的等价刻画，并提出基于谱Hirzebruch-Milnor特征类的同调检测准则。
Hodge adjacency conditions for singularities
本文研究孤立 $k$-Du Bois 奇点及其平坦射影族，证明纤维混合 Hodge 数与泛形变间的相容关系，并将极化关系推广到正规交叉边界之外，分析弱极化关系图的组合性质。
Trace for the Du Bois complex
本文构造 Du Bois 复形的迹态射，研究有限态射下局部上同调维数及奇点的 Hodge 理论性质。
Differential Forms and Hodge Structures on Singular Varieties
研究奇异复代数变种上的微分形式概念及其与Hodge滤波器的关联，引入并研究称为准有理的奇异性，探讨孤立奇异性条件。
We study the Du Bois complex $\underlineΩ_Z^\bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $\widetildeα(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $\widetildeα(Z)\geq p+1$, then the canonical morphism $Ω_Z^p\to \underlineΩ_Z^p$ is an isomorphism, where $\underlineΩ_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, …
Singularities of secant varieties from a Hodge theoretic perspective
本文从Hodge理论视角，利用双有理几何方法研究光滑射影簇割线簇的奇点，给出了p-Du Bois奇点的充要条件，并证明其奇点非更高有理，进而推导出Kodaira-Akizuki-Nakano型消失定理。
For a complex algebraic variety $X$, we introduce higher $p$-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms $Ω_X^q$ and the shifted graded pieces of the Du Bois complex $\underlineΩ_X^q$ for $q\le p$. If $X$ is a reduced hypersurface, we show that higher $p$-Du~Bois singularity coincides with higher $p$-log canonical singularity, generalizing a well-known theorem for $p=0$. The assertion that $p$-log canonicity implies $p$-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems assertin…
Hodge symmetry and Lefschetz theorems for singular varieties
该研究证明奇点代数簇拓扑与霍奇理论新结果，揭示奇点复杂度条件与霍奇-杜波依斯菱形对称性的关联，关联有理同调流形理论并在低维有理奇点代数簇中刻画了它。
Partial Cohomologically Complete Intersections via Hodge Theory
本文通过Hodge理论引入不变量c(X)，度量孤立奇点簇上对偶Kodaira-Akizuki-Nakano消失的失效，并由Hodge-Lyubeznik数等刻画，且在锥和行列式簇上可计算。
Du Bois complex and extension of forms beyond rational singularities
本文刻画了约化对的 Du Bois 复形，推广了全纯（对数）形式延拓定理，并给出 Kollár-Kovács 定理的新证明，证明基于混合 Hodge 模理论。
Symmetries and vanishing theorems for symplectic varieties
针对带孤立奇点的奇异辛簇的局部与Steenbrink零化问题，构造特定条件下的态射并证明其拟同构性与对称性下延性质，给出辛芽相关性质及本原辛4-fold上同调的应用结果。
Higher Du Bois and higher rational singularities
本文证明平坦族中相对 Kähler 微分高次直接像的局部自由性，提出 $k$-有理奇点定义并证明其蕴含 $k$-Du Bois 性质，应用于 Hodge 数与形变理论。
Global smoothing of singular Fano and Calabi-Yau varieties
本文研究了Fano和Calabi-Yau品种的全局平滑问题，证明了在特定奇点条件下，这些品种可以通过变形获得更光滑的结构，并推广了已有结论。
Restrictions of mixed Hodge modules using generalized V-filtrations
本文利用Sabbah定义的广义V-滤子，通过循环覆盖与通常V-滤子的比较，计算了零截面包含映射的限制函子，并应用于研究完全交子簇的奇点，分类了加权齐次孤立完全交奇点的k-Du Bois和k-理性性质。
Differential forms on varieties with pre-$k$-rational singularities
本文针对具有预k-有理奇点的复代数簇，比较两类紧支上同调群的混合霍奇结构，研究其微分形式相关概念，重点考察Du Bois复形分次部分的深度。
A Hodge Theoretic generalization of $\mathbb{Q}$-Homology Manifolds I: General Case
研究通过不变量$\HRH(Z)$对$\mathbb{Q}$-同调流形进行霍奇理论推广，利用局部上同调混合霍奇模刻画$\HRH(Z)$，给出对部分庞加莱对偶性的应用，还引入一般局部上同调缺陷并关联到$\HRH(Z)$。
The GIT stability and Hodge structures of hypersurfaces via minimal exponent
本文通过最小指数刻画超曲面的GIT稳定性，证明其与半稳定性的临界条件，并引入m-极限源以推广周期映射行为，从而由奇点类型决定极限混合Hodge结构。
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