Related papers: Arcwise Analytic Stratification, Whitney Fibering …

Let X ->Y be a Zariski locally trivial fibration of smooth complex projective varieties, with fiber F. We give a structure theorem for the derived category of X provided both F and Z have a full strongly exceptional collection of line…

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…

The weak Hilbert property (WHP) for varieties over fields of characteristic zero was introduced by Corvaja and Zannier in 2017. There exist integral variants of WHP for arithmetic schemes. We present new fibration theorems for both the WHP…

When we have a proper action of a Lie group on a manifold, it is well known that we get a stratification by orbit types and it is known that this stratification satisfies the Whitney (b) condition. In a previous article we have seen that…

Let $G$ be a simply connected semi-simple complex algebraic group. Fix a maximal torus $T$ and a Borel subgroup $B$ such that $T\subset B\subset G$. Let $W$ the Weyl group of $G$ relative to $T$. For any $w$ in $W$, let $X_w=\bar {BwB/B}$…

Given a finite cocommutative Hopf algebra $A$ over a commutative regular ring $R$, the lattice of localising tensor ideals of the stable category of Gorenstein projective $A$-modules is described in terms of the corresponding lattices for…

We show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology onto the weight complex of McCrory and Parusi\'nski. The group…

Let $f : X \to S$ be a smooth projective family defined over $\mathcal{O}_{K}[\mathcal{S}^{-1}]$, where $K \subset \mathbb{C}$ is a number field and $\mathcal{S}$ is a finite set of primes. For each prime $\mathfrak{p} \in…

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein's theory of analytic continuation of holonomic…

Let $A$ be a finite-dimensional algebra with two simple modules. It is shown that if the derived category of $A$ admits a stratification with simple factors being the base field $k$, then $A$ is derived equivalent to a quasi-hereditary…

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_\epsilon$ is a stabilization of…

Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and…

We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable…

We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable,…

Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of…

We introduce a new moduli stack $\mathscr{E}_{g,n}$ of ``equinormalized curves," closely related to the moduli space of all reduced, connected algebraic curves. We construct a stratification $\bigsqcup_\Gamma \mathscr{E}_\Gamma$ of…

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a…

Let X be a normal connected complex algebraic variety equipped with a semisimple complex representation of its fundamental group. Then, under a maximality assumption, we prove that the covering space of X associated to the kernel of the…

In this work, we consider a pair $(\textbf{X},0)$ and $(\textbf{Y},0)$ of hypersurfaces in $(\mathbb{C}^{n+1},0)$ parametrized by finitely determined, quasihomogeneous map germs $f$ and $g,$ respectively. Zariski asked whether the…

A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…