Potentially Du Bois spaces

@article{Graf2014PotentiallyDB,
  title={Potentially Du Bois spaces},
  author={Patrick Graf and S'andor J. Kov'acs},
  journal={arXiv: Algebraic Geometry},
  year={2014}
}
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… Expand
Inversion of adjunction for rational and Du Bois pairs
We prove several results about the behavior of Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting whileExpand
Local cohomology of Du Bois singularities and applications to families
In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, thenExpand
On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs
We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and anExpand
Deformations of log canonical and $F$-pure singularities
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 aExpand
Deformations of log canonical singularities
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 aExpand
Potential log canonical centers
Given an ambient variety $X$ and a fixed subvariety $Z$ we give sufficient conditions for the existence of a boundary $\Delta$ such that $Z$ is a log canonical center for the pair $(X, \Delta)$. WeExpand
The generalized Lipman–Zariski problem
We propose and study a generalized version of the Lipman–Zariski conjecture: let $$(x \in X)$$(x∈X) be an $$n$$n-dimensional singularity such that for some integer $$1 \le p \le n - 1$$1≤p≤n-1, theExpand
On rationalizing divisors
  • L. Prelli
  • Mathematics, Computer Science
  • Period. Math. Hung.
  • 2017
TLDR
This paper gives a criterion for cones to have a rationalizing divisor, and relates the existence of such adivisor to the locus of rational singularities of a variety. Expand
EXTENSION THEOREMS FOR DIFFERENTIAL FORMS ON LOW-DIMENSIONAL GIT QUOTIENTS
In this paper we will show that the pull-back of any regular differential form defined on the smooth locus of a GIT quotient of dimension at most four to any resolution yields a regular differentialExpand
Aspects of the Geometry of Varieties with Canonical Singularities
This survey reports on recent developments regarding the global structure of complex varieties which occur in the minimal model program.

References

SHOWING 1-10 OF 40 REFERENCES
Du Bois singularities deform
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 familyExpand
The canonical sheaf of Du Bois singularities
We prove that a Cohen–Macaulay normal variety X has Du Bois singularities if and only if π∗ωX′(G)≃ωX for a log resolution π:X′→X, where G is the reduced exceptional divisor of π. Many basic theoremsExpand
Inversion of adjunction for rational and Du Bois pairs
We prove several results about the behavior of Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting whileExpand
Log canonical singularities are Du Bois
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 moreExpand
Rational, Log Canonical, Du Bois Singularities: On the Conjectures of Kollár and Steenbrink ?
Let X be a proper complex variety with Du Bois singularities. Then H(X,iℂ)→ Hi(X,\({\mathcal{O}}\)X) is surjective for all i. This property makes this class of singularities behave well with regardExpand
Categorical resolutions, poset schemes and Du Bois singularities
We introduce the notion of a poset scheme and study the categories of quasicoherent sheaves on such spaces. We then show that smooth poset schemes may be used to obtain categorical resolutions ofExpand
An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture
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 logarithmicExpand
Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties
Abstract Given a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities $\pi : \widetilde {Z} \to Z$, we study the problem of extending the pull-back π*(σ)Expand
F-injectivity and Buchsbaum singularities
Let (R,m) be a local ring that contains a field. We show that, when R has equal characteristic p>0 and when H_m^i(R) has finite length for all i<dimR, then R is F-injective if and only if every idealExpand
Differential forms on log canonical spaces
The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extendsExpand
...
1
2
3
4
...