Related papers: Du Bois complex and extension of forms beyond rati…

We construct some version of the trace morphism between the Du Bois complexes, with applications towards the behavior of the local cohomological dimension and some Hodge theoretic aspects of singularities under finite morphisms.

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…

Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the…

We show that k-rational singularities of local complete intersections are k-Du Bois. For hypersurfaces, we characterize k-rationality in terms of the minimal exponent. We also establish some local vanishing results for k-rational and k-Du…

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…

We study the singularities of the secant variety $\Sigma(X,L)$ associated to a smooth variety $X$ embedded by a sufficiently positive adjoint bundle $L$. We show that $\Sigma(X,L)$ is always Du Bois singular. Examples of secant varieties…

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 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 show that some properties of log canonical centers of a log canonical pair (X,D) also hold for certain subvarieties that are close to being a log canonical center. As a consequence, we obtain that if one works with deformations of pairs…

Let $g$ be a finite dimensional complex reductive Lie algebra and <.,.> an invariant non degenerated bilinear form on $g\times g$ which extends the Killing form of $[g,g]$. We define a subcomplex $E\_{\bullet}(g)$ of the canonical complex…

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of…

Rational pairs generalize the notion of rational singularities to reduced pairs $(X,D)$. In this paper we deal with the problem of determining whether a normal variety $X$ has a rationalizing divisor, i.e. a reduced divisor $D$ such that…

The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and…

We show that it is possible to utilize the Hirzebruch-Milnor classes of projective hypersurfaces in the classical sense to detect higher du Bois or rational singularities only in some special cases. We also give several remarks clarifying…

The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well…

Let $X$ be a normal variety over a perfect field of positive characteristic and $B$ a reduced divisor on $X$. We prove that if the Cartier isomorphism on the log smooth locus of $(X,B)$ extends to the entire $X$, then $(X,B)$ satisfies 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…

We prove Hodge-theoretic formulas for the $\mathbb{Q}$-factoriality defect of a normal projective variety, and for the local analytic $\mathbb{Q}$-factoriality defect of an analytic germ of a normal variety. These formulas lead to…

Under certain assumptions, we show that unitary rational $\mathcal{N}=(2,2)$ conformal field theories together with a certain generating set of Cardy boundary states in the associated boundary conformal field theories give rise to rational…

We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration $\pi\colon X \to \Delta$ whose central fiber $X_0$ has $r$ ordinary double…