Karl Schwede
In this paper, we prove that singularities of $F$-injective type are Du Bois. This extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of Frobenius in positive characteristic.
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Let $(R,m,k)$ be an excellent local ring of equal characteristic. Let $j$ be a positive integer such that $H_m^i(R)$ has finite length for every $0\leq i <j$. We prove that if $R$ is $F$-injective in characteristic $p>0$ or Du Bois in characteristic $0$, then the truncated dualizing complex $τ_{>-j}ω_R^\bullet$ is quasi-isomorphic to a complex of $k$-vector spaces. As a consequence, $F$-injective or Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. Moreover, when $R$ has $F$-rational or rational singularities on the punctured spectrum, we obtain stronger results.
A short note on Du Bois singularities
研究Du Bois奇点在基变换和纤维积下的行为，证明了在特征零的可嵌入簇中Du Bois奇点从任何域扩张下降，以及具有有理奇点的簇与具有Du Bois奇点的簇的乘积仍具有Du Bois奇点。
Let $X$ be a variety and $H$ a Cartier divisor on $X$. We prove that if $H$ has Du Bois (or DB) singularities, then $X$ has Du Bois singularities near $H$. As a consequence, if $X \to S$ is a family over a smooth curve $S$ whose special fiber has Du Bois singularities, then the nearby fibers also have Du Bois singularities. We prove this by obtaining an injectivity theorem for certain maps of canonical modules. As a consequence, we also obtain a restriction theorem for certain non-lc ideals.
Pure subrings of Du Bois singularities are Du Bois singularities
本文研究了环的纯映射下杜布瓦奇点的保持性，证明了若S具有杜布瓦奇点，则R也具有杜布瓦奇点，且在忠实平坦情况下仍成立，同时探讨了素特征和混合特征下的应用。
We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $π_*ω_{X'}(G) \simeq ω_X$ for a log resolution $π: X' \to X$, where $G$ is the reduced exceptional divisor of $π$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for …
We prove several results about the behavior Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the non-pair setting. We also prove a new inversion of adjunction result for Du Bois and rational pairs. In the non-pair setting this asserts that if a family over a smooth base has a special fiber $X_0$ with Du Bois singularities and the general fiber has rational singularities, then the total space has rational singularities near $X_0$.
In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)\to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kovács and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing…
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.
We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $π: \tld Y \to Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If $E$ is the reduced pre-image of $X$ in $\tld Y$, then $X$ has Du Bois singularities if and only if the natural map $Ø_X \to R π_* Ø_E$ is a quasi-isomorphism. We also deduce Kollár's conjecture that log canonical singularities are Du Bois in the special case of a local complete intersection and prove other results related to adjunction.
Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and Kóvacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.
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.
Upper bound on the multiplicity of rational and Du Bois singularities
本文证明了Du Bois奇点重数的尖锐上界，并恢复了有理奇点的已知上界，解决了Huneke和Watanabe的问题。
We investigate properties of potentially Du Bois singularities, that is, those that occur on the underlying space of a Du Bois pair. We show that a normal variety $X$ with potentially Du Bois singularities and Cartier canonical divisor $K_X$ is necessarily log canonical, and hence Du Bois. As an immediate corollary, we obtain the Lipman-Zariski conjecture for varieties with potentially Du Bois singularities. We also show that for a normal surface singularity, the notions of Du Bois and potentially Du Bois singularities coincide. In contrast, we give an example showing that in dimension at leas…
We introduce the notion of a poset scheme and study the categories of quasi-coherent sheaves on such spaces. We then show that smooth poset schemes may be used to obtain categorical resolutions of singularities for usual singular schemes. We prove that a singular variety $X$ possesses such a resolution if and only if $X$ has Du Bois singularities. Finally we show that the de Rham-Du Bois complex for an algebraic variety $Y$ may be defined using any smooth poset scheme which satisfies the descent over $Y$ in the classical topology.
The higher Du Bois and higher rational properties for isolated singularities
本文研究孤立奇点的更高Du Bois与更高有理性质，在局部完全交情形下给出完整刻画，证明k-Du Bois蕴含(k-1)-有理，并建立不变量间新关系及伴随定理反转的快速证明。
Examples of Hirzebruch-Milnor classes of projective hypersurfaces detecting higher du Bois or rational singularities
本文表明经典 Hirzebruch-Milnor 类仅在特殊情形下能检测射影超曲面的更高 du Bois 或有理奇点，并澄清了作者早期论文中的若干要点。
A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the modu…
Injectivity and Vanishing for the Du Bois Complexes of Isolated Singularities
本文证明了具有孤立奇点的簇的 Du Bois 复形上同调的单射定理，并推导出更高阶 Du Bois 复形上同调的消没性，同时讨论了非孤立情形下的推广、猜想以及交复形的类似结果。
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。
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。
The Holomorphic Extension Property for Higher Du Bois Singularities
针对正规复簇的Du Bois奇点，证明了特定条件下解消诱导的微分形式层包含态射为同构，改进Flenner判别准则，并从全纯延拓视角研究$k$-Du Bois定义及相关层限制对奇点的影响。
On $k$-Du Bois and $k$-rational singularities
本文引入 $k$-Du Bois 与 $k$-有理奇点的新定义，推广至非 lci 情形，证明 $k$-有理奇点蕴含 $k$-Du Bois 性质，并识别仅依赖高阶上同调消失的理论方面。
Recently M. Mustata and V. Srinivas related a natural conjecture about the Frobenius action on the cohomology of the structure sheaf after reduction to characteristic $p > 0$ with another conjecture connecting multiplier ideals and test ideals. We generalize this relation to the case of singular ambient varieties. Additionally, we connect these results to a conjecture relating $F$-injective and Du Bois singularities. Finally, using an unpublished result of Gabber, we also show that $F$-injective and Du Bois singularities have a common definition in terms of smooth hypercovers.
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。
Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -- the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.
On higher Du Bois singularities and $K$-regularity
研究高阶 Du Bois 奇点与 $K$-正则性（衡量代数 $K$-群的 $\mathbb{A}^1$-不变性）的关系，并基于此建立特征零局部完全交的 Vorst 猜想的加强形式，同时提供构造新例子的工具。
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。
On k-rational and k-Du Bois local complete intersections
本文证明了局部完全交的k-有理奇点是k-Du Bois的，并用最小指数刻画超曲面的k-有理性，同时建立了相关局部消失定理。
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…
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。
We prove that every quasi-log canonical pair has only Du Bois singularities. Note that our arguments are free from the minimal model program.
Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities
本文将 Hirzebruch-Milnor 类推广到法丛非平凡且无整体定义函数的超曲面，并证明在射影奇点轨迹假设下，谱类可刻画最小指数，未规范化类可检测高阶 du Bois 与有理奇点。
Higher Du Bois and Higher Rational Pairs
本文提出将更高杜布瓦和更高理性奇点的概念扩展到对，并通过广义注入定理证明相关性质。
The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
本文证明了一个广义Briançon-Skoda型包含关系，并由此推出伪有理奇点（如正则环）和Du Bois奇点下的完整Briançon-Skoda包含，同时应用于证明有限维拟优秀环的一致Artin-Rees定理和一致Briançon-Skoda定理。
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.
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。
Higher F-injective singularities
该研究引入高阶$F$-内射性概念，证明特征零域孤立奇点的$k$-$F$-内射性与$k$-Du Bois性质的素约化关联及普通性猜想下的逆命题，并应用于Frobenius可提升超曲面研究。
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.
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。
Complexes of differential forms and singularities: The injectivity theorem
本文证明了对于具有(m-1)-Du Bois奇点的代数簇，从m阶Du Bois复形的Grothendieck对偶到其零阶上同调sheaf的Grothendieck对偶的自然映射在上同调上是单射的，验证了Popa, Shen和Vo的猜想G。
Regularity, singularities and $h$-vector of graded algebras
本文研究了标准分次代数的奇点对其 $h$-向量及重数的影响，利用对特定 Ext 模正则性的精确界限，证明了满足 Serre 条件且具有合理奇点的代数具有非负 $h$-向量及重数下界，并在等号成立时迫使代数成为 Cohen-Macaulay。
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。
We prove that strongly F-regular and F-pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F-injective singularities fail to satisfy even the most basic Bertini-type results.
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。
Singularities and syzygies of secant varieties of smooth projective varieties
本文研究光滑射影簇的高阶割线簇，证明曲面情形在正线丛下具有Du Bois奇点及线性合冲，并揭示其奇点类型由曲面结构层上同调决定，同时给出高维低阶情形的类似结果及高阶情形的否定性结论。
On the injective self-maps of algebraic varieties
本文证明了宫西猜想在非奇异簇、曲面及局部完全交集且余维数2处正则的簇上成立，并讨论了诱导非奇异轨迹自同构的实例及孤立奇点情形。
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