Related papers: On a resolution of singularities with two strata

We prove that certain asymptotically flat initial data sets with nontrivial topology and/or differentiable structure collapse to form singularities. The class of such initial data sets is characterized by a new smooth invariant, the maximal…

Let $S$ be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations $\rho:\pi_1(S) \to \text{PSL}_2(\mathbb{C})$ that arise as the monodromy of the Schwarzian equation on $S$ with…

Let $X$ be a surface with an ADE-singularity and let $\widetilde{X}$ be its crepant resolution. In this paper, we show that there exists a Bridgeland stability condition $\sigma_X$ on ${\rm D}^b(X)$ and a weak stability condition…

Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}\pi{_*}\mathbb{Q}$, where $\pi$ denotes the projection from the universal hyperplane family of…

Let $X$ be a connected, orientable, 5-dimensional Poincar\'{e} duality complex with torsion-free $H_1(X;\mathbb{Z})$. We show that $\Sigma X$ is homotopy equivalent to a wedge of recognisable spaces and study to what extent its homotopy…

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting…

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an…

In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to…

Let $X$ be a perfectoid space with tilt $X^\flat$. We construct a canonical map $\theta:\operatorname{Pic} X^\flat\to\lim\operatorname{Pic} X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an…

Let $X$ be an 1-convex surface and $p:\tilde X\to X$ an (unbranched) covering map. We prove that if $\tilde X$ does not contain an infinite Nori string of rational curves then $\tilde X$ satisfies the discrete disk property.

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This…

For the toric variety X associated to the Bruhat poset of Schubert varieties in the Grassmannian, we describe the singular locus in terms of the faces of the associated polyhedral cone. We also determine the tangent cones at the maximal…

Let $(X,o)$ be a complex normal surface singularity. We fix one of its good resolutions $\widetilde{X}\to X$, an effective cycle $Z$ supported on the reduced exceptional curve, and any possible (first Chern) class $l'\in…

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects…

We give a description of the cohomology groups of the structure sheaf on smooth compactifications $\overline{X}(w)$ of Deligne--Lusztig varieties $X(w)$ for ${\rm GL}_n$, for all elements $w$ in the Weyl group. As a consequence, we obtain…

Any two homologous surfaces of the same genus embedded in a smooth 4-manifold X with simply-connected complements are shown to be smoothly isotopic in the connected sum of X and the product of a 2-sphere with itself, if the surfaces are…

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered fourfolds defined by Weierstrass models admitting a split A_4 singularity over a divisor of the discriminant locus. Such varieties are…

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…

A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces…

Given a normal $\mathbb{Q}$-Gorenstein complex variety $X$, we prove that if one spreads it out to a normal $\mathbb{Q}$-Gorenstein scheme $\mathcal{X}$ of mixed characteristic whose reduction $\mathcal{X}_p$ modulo $p$ has normal $F$-pure…