Related papers: Srinivas' Problem for Rational Double Points

Let A be the local ring at a point of a normal complex variety with completion R. Srinivas has asked about the possible images of the induced map from Cl A to Cl R over all geometric normal domains A with fixed completion R. We use…

We use our extension of the Noether-Lefschetz theorem to describe generators of the class groups at the local rings of singularities of very general hypersurfaces containing a fixed base locus. We give several applications, including (1)…

We prove that the Nash problem holds for two-dimensional rational double points in all characteristics. The proof is based on a direct computation of the families of arcs through these singularities.

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions…

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a singular germ of a surface $S$ coincides with the number of irreducible components of the exceptional divisor in the minimal…

We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed…

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL($n$). As an…

The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the…

This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $\mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR…

Rational double points are the simplest surface singularities. In this essay we will be mainly concerned with the geometry of the exceptional set corresponding to the resolution of a rational double point. We will derive the classification…

We study the holographic dual of two-point correlation functions for nonconformal field theories. We first take into account a Lifshitz geometry as the dual of a Lifshitz field theory which may appear at a critical or IR fixed point. We…

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a…

We show that every point in a uniformly $2$-nondegenerate CR hypersurface is canonically associated with a model $2$-nondegenerate structure. The $2$-nondegenerate models are basic CR invariants playing the same fundamental role as quadrics…

We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic…

This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic…

By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…

We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains…

We describe solutions of 10-dimensional supergravity comprising null deformations of $AdS_5\times S^5$ with a scalar field, which have $z=2$ Lifshitz symmetries. The bulk Lifshitz geometry in 3+1-dimensions arises by dimensional reduction…

In this survey, we explain a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various $L^2$-vanishing theorems for the $\overline\partial$-equation on…

In the recent articles by Alper, Eastwood and Isaev, it was conjectured that all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of…