Related papers: Completeness and compactness for varieties over lo…

A topological group $G$ is {\em sequentially $h$-complete} if all the continuous homomorphic images of $G$ are sequentially complete. In this paper we give necessary and sufficient conditions on a complete group for being compact, using the…

We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid cohomology with compact supports H^i_c(X,E) are finite…

The tilting correspondence is a fundamental property of perfectoid fields. In this note, we show that the tilting construction can also be used to detect perfectoid fields among nonarchimedean fields. In particular, for $K$ a complete…

A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical…

We show that a compact Lorentzian locally symmetric space is geodesically complete if the Lorentzian factor in the local de Rham-Wu decomposition is of Cahen-Wallach type or if the maximal flat factor is one-dimensional and time-like. Our…

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the…

We prove that an isometric action of a compact Lie group on a compact symmetric space is variationally complete if and only if it is hyperpolar.

In this paper we have studied the idea of ideal completeness of function spaces Y to the power X with respect to pointwise uniformity and uniformity of uniform convergence. Further involving topological structure on X we have obtained…

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space $(X, \tau)$ is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the…

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity…

We study the k-forms of almost homogeneous varieties over perfect base fields k. First, we discuss criteria for the existence of k-forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to…

An extension $K/k$ of analytic (i.e. real valued complete) fields is called small if it is topologically-algebraically generated by finitely many elements. We prove that this property is inherited by subextensions and hence topological…

A variety X over a field K is of Hilbert type if the set of rational points X(K) is not thin. We prove that if f: X\to S is a dominant morphism of K-varieties and both S and all fibers f^{-1}(s), s in S(K), are of Hilbert type, then so is…

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological…

Let X be a Zariski open subset of a compact Kaehler manifold. In this paper, we study the set $\Sigma^k(X)$ of one dimensional local systems on X with nonvanishing kth cohomology. We show that under certain conditions (X compact, X has a…

For an arbitrary field $K$ and $K$-variety $V$, we introduce the \'etale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably…

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with…

A topology on a nonempty set $X$ specifies a natural subset of $\mathcal{P}(X)$. By identifying $\mathcal{P}(\mathcal{P}(X))$ with the totally disconnected compact Hausdorff space $2^{\mathcal{P}(X)}$, the lattice $Top(X)$ of all topologies…

The main result is that, for any projective compact analytic subset A of dimension q>0 in a reduced complex space X, there is a neighborhood U of A such that, for any covering space Z of X in which the lifting B of A has no noncompact…

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is…