Related papers: Arcwise Analytic Stratification, Whitney Fibering …

We use Kiehl-Verdier's and Houzel's finiteness theorems in the setting of local analytic geometry, and the Whitney-Thom theory of stratified spaces, to prove that fibrewise constructible complex of sheaves have coherent direct images. We…

Our aim is to construct fibrewise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized…

We investigate conditions for "simultaneous normalizability" of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions)…

Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn's work, we refine the stratification of iterated singular loci to a Whitney (a) stratification, which…

Understanding how singularities behave under small perturbations is a central theme in singularity theory. In this paper we establish sufficient conditions for families of analytic function-germs on a germ of a complex analytic space to…

We prove the analogue of Viehweg's hyperbolicity conjecture for Whitney equisingular families of projective varieties with Gorenstein rational singularities whose geometric generic fiber has a good minimal model. Namely, for such families…

For a smooth toric variety X over a field of positive characteristic, a T-equivariant \'{e}tale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to…

On Zariski Main Theorem in Algebraic Geometry and Analytic Geometry. We fill a surprising gap of Complex Analytic Geometry by proving the analogue of Zariski Main Theorem in this geometry, i.e. proving that an holomorphic map from an…

Given a Cayley-Hamilton smooth order A in a central simple algebra $\Sigma$, we determine the flat locus of the Brauer-Severi fibration of the smooth order. Moreover, we give a classification of all (reduced) central singularities where the…

We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a…

We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but…

This paper is devoted to study multiplicity and regularity as well as to present some classifications of complex analytic sets. We present an equivalence for complex analytical sets, namely blow-spherical equivalence and we receive several…

Guillet and Soul\'e have shown that, for a fibration $\pi: Y \to X$ with fibre $Z$, locally trivial in the Zariski topology, we have a decomposition \[ [Y] = [X] \cdot [Z], \] where $[\cdot]$ denotes a class in the Grothendieck group…

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the…

{\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\it bundle} $G_F \subset T^{*}K^n $. We…

We investigate a scheme-theoretic variant of Whitney condition a. If X is a projec-tive variety over the field of complex numbers and Y $\subset$ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y…

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine…

A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families…

For any (not necessarily perfect) field $k$ we obtain equivalences of $\infty$-categories \[\mathbf{H}^{\mathrm{fr},\mathrm{gp}}(k)\simeq \mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(k) \text{ and }…

This text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical…