相关论文: Higher Du Bois singularities of hypersurfaces

We study the Du Bois complex $\underline{\Omega}_Z^\bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $\widetilde{\alpha}(Z)$. The latter is an invariant of singularities, defined as the…

We establish a characterization of the Du Bois complex of a reduced pair $(X,Z)$ when $X\smallsetminus Z$ has rational singularities. As an application, when $X$ has normal Du Bois singularities and $Z$ is the locus of non-rational…

In this note, we give several equivalent characterizations of higher Du Bois and higher rational singularities in the context of globally defined hypersurfaces. As a key input, we characterize these singularities using the Hodge filtration…

We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tld Y \to Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If…

We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $\pi_*\omega_{X'}(G) \simeq \omega_X$ for a log resolution $\pi: X' \to X$, where $G$ is the reduced exceptional divisor of $\pi$. Many basic…

We prove that the higher direct images $R^qf_*\Omega^p_{\mathcal Y/S}$ of the sheaves of relative K\"ahler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have $k$-Du Bois local…

Let $X$ be a normal complex variety and $\pi:\tilde X \to X$ a resolution of singularities. We show that the inclusion morphism $\pi_*\Omega_{\tilde X}^p\hookrightarrow \Omega_X^{[p]}$ is an isomorphism for $p <…

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…

We compare a couple of notions of differential form on singular complex algebraic varieties, and relate them to the outermost associated graded spaces of the Hodge filtration of ordinary and intersection cohomology. In particular, we…

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…

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…

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…

We study the relationship between higher Du Bois singularities and $K$-regularity, a notion that measures the $\mathbb{A}^1$-invariance of the algebraic $K$-groups. Building on this relationship, we establish a strengthened form of Vorst's…

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…

Recent progress in the deformation theory of Calabi-Yau varieties $Y$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $k$-liminal…

Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb{Q}$-algebras. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our…

Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated…

We study the weighted spectrum and vanishing cohomology for several classes of isolated hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of…

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…

Let $X$ be a complex algebraic variety. With $\mathbb{Q}$-coefficients, the compactly supported cohomology groups $H^{i}_{c}(X, \mathbb{Q})$ and the compactly supported intersection cohomology groups $IH^{i}_{c}(X, \mathbb{Q})$ have mixed…