Related papers: A Simple Characterization of Du Bois Singularities

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 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 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…

Let $Y$ be a compact Gorenstein analytic space with only isolated singularities and trivial dualizing sheaf. A recent paper of Imagi studies the deformation theory of $Y$ in case the singularities of $Y$ are weighted homogeneous and…

We continue to study and present concrete examples in characteristic 2 of compound Du Val singularities defined over an algebraically closed field which have one dimensional singular loci but cannot be written as products (a rational double…

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface $(X,D)$ we define and…

Assume that $X$ and $Y$ are arithmetic schemes, i.e., integral schemes of finite types over $Spec(\mathbb{Z})$. Then $X$ is said to be quasi-galois closed over $Y$ if $X$ has a unique conjugate over $Y$ in some certain algebraically closed…

We prove that if X is a locally complete intersection variety, then X has all the jet schemes irreducible if and only if X has canonical singularities. After embedding X in a smooth variety Y, we use motivic integration to express the…

We introduce new notions of $k$-Du Bois and $k$-rational singularities, extending the previous definitions in the case of local complete intersections (lci), to include natural examples outside of this setting. We study the stability of…

This paper investigates the geometry of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic $p>0$ in the case when the automorphism scheme of $X$ is not smooth. This is a situation that…

Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part…

Let $Y$ be a generic link of a subvariety $X$ of a nonsingular variety $A$. We give a description of the Grauert-Riemenschneider canonical sheaf of $Y$ in terms of the multiplier ideal sheaves associated to $X$ and use it to study the…

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $\pi:X\to Y$ a resolution of singularities, $G:=\pi^{-1}\left(\rm{Sing}(Y)\right)$ the exceptional locus. From the Decomposition Theorem one knows that…

We prove an injectivity theorem for the cohomology of the Du Bois complexes of varieties with isolated singularities. We use this to deduce vanishing statements for the cohomologies of higher Du Bois complexes of such varieties. Besides…

We prove semi-rationalification and semi-log-canonicalization for Gorenstein demi-normal surfaces. That is, given a Gorenstein demi-normal surface X with semi-rational (respectively, semi-log canonical) singularities in an open set U with…

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that…

Let k be a field, and let {\pi}:\tilde{X} -> X be a proper birational morphism of irreducible k-varieties, where \tilde{X} is smooth and X has at worst quotient singularities. When the characteristic of k is zero, a theorem of Koll\'ar in…

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…

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…

This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called…