A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of Whitney says that any complex analytic set has a stratification. This result was extended by Lojasiewicz to real (semi)analytic sets. In this paper we present a short geometric proof of existence of stratifications based on Thom's transversality theorem and Milnor's curve selection lemma and not relying on difficult results of Lojasiewicz.
We prove that local Lipschitz Killing curvatures of definable sets in a polynomially bounded o-minimal structure are continuous along strata of Whitney stratifications and locally Lipschitz if the stratifications are (w)- regular.
We specify the canonical stratifications satisfying respectively Whitney (a)-regularity, Whitney (b)-regularity, Kuo-Verdier (w)-regularity, and Mostowski (L)-regularity for the family of surfaces y^a = t^b x^c + x^d, where a, b, c, d are positive integers, in both the real and complex cases.
We define "t-stratifications", a strong notion of stratifications for Henselian valued fields $K$ of equi-characteristic 0, and prove that they exist. In contrast to classical stratifications in Archimedean fields, t-stratifications also contain non-local information about the stratified sets. For example, they do not only see the singularities in the valued field, but also those in the residue field. Like Whitney stratifications, t-stratifications exist for different classes of subsets of $K^n$, e.g. algebraic subvarieties or certain classes of analytic subsets. The general framework are defi…
In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of Zariski equisingularity, w…
Swansea University（斯旺西大学）；National and Kapodistrian University of Athens（雅典国立卡波迪斯特里亚斯大学）；City, University of London（伦敦城市大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步辐射中心）；University of Oxford（牛津大学）；Trinity College, Dublin（都柏林三一学院）
We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratify…
{\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 restrict $ G_F $ over the critical points $Sing(F)$ of $ F$ in $ F^{-1}(0)$ and partition $Sing(F)$ into {\it 'quasistrata'} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications 'near' $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the qu…
We give a simple geometric proof of the decomposition theorem in terms of Thom-Whitney stratifications by reduction to fibrations by normal crossings divisors over the strata and explain the relation with the local purity theorem an unpublished result of Deligne and Gabber.
We show that the way in which Betti cohomology varies in a proper family of complex algebraic varieties is controlled by certain "higher discriminants" in the base. These discriminants are defined in terms of transversality conditions, which in the case of a morphism between smooth varieties can be checked by a tangent space calculation. They control the variation of cohomology in the following two senses: (1) the support of any summand of the pushforward of the IC sheaf along a projective map is a component of a higher discriminant, and (2) any component of the characteristic cycle of the pro…
In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of isolated hypersurface singularities is Whitney equisingular if and only if the mu^*-sequence for a hypersurface is constant in the family. In this paper we generalize to non-isolated hypersurface singularities. By assuming non-contractibility of strata of a Whitney stratification of the non-isolated singularities outside the origin we show that Whitney equisingularity of a family is equivalent to constancy of a certain selection of invariants fr…
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 review the notions of stratification theory which are needed.
This paper concerns an analytical stratification question of real algebraic and semi-algebraic sets. For Whitney's stratification in 1957, it partitions a real algebraic set into partial algebraic manifolds\cite{W}. In 1975 Hironaka reproved that a real algebraic set is triangulable and also generalized it to semi-algebraic sets, following the idea of Lojasiewicz's triangulation of semi-analytic sets in 1964. Following their examples and wondering how geometry looks like locally. this paper tries to come up with a stratification, in particular a cell decomposition, such that it satisfies the f…
This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the derived category of a derived local quotient stack which admits a stratification of the kind arising in geometric invariant theory. The use of derived algebraic geometry leads to results with pleasingly few hypotheses, even when the stack is not smooth. Using the same methods, we establish a "virtual non-abelian localization theorem" which is a K-theoretic analog of the virtual localization theorem in…
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 locally finite partition of a reduced complex analytic space $X$ into nonsingular strata with the property that the local geometry of $X$ is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given poin…
Let $X\subset\Bbb C^n$ be an affine variety and $f:X\to\Bbb C^m$ be the restriction to $X$ of a polynomial map $\Bbb C^n\to\Bbb C^m$. In this paper, we construct an affine Whitney stratification of $X$. The set $K(f)$ of stratified generalized critical values of $f$ can be also computed. We show that $K(f)$ is a nowhere dense subset of $\Bbb C^m$, which contains the set $B(f)$ of bifurcation values of $f$ by proving a version of the isotopy lemma for non-proper polynomial maps on singular varieties.
In this paper we show that the equivalences between certain properties of closed subanalytic sets proved by E. Bierstone and P. Milman in \cite{[BM-1]} hold for closed sets definable in quasianalytic o-minimal structures. In particular we prove that uniform Chevalley estimate implies a stratification by the diagram of initial exponents and further, Zariski semicontinuity of the diagram of initial exponents. We also show that the stratification by the diagram implies Zariski semicontinuity of Hilbert-Samuel function.
An integrable system is a dynamic system characterized by the existence of constants of motion and the existence of algebraic invariants, having an origin in algebraic geometry. In the 1970s, Mumford introduced a new completely integrable system defined on a smooth hyperelliptic curve. In the 2000s, Vanhaecke completed the description of the Munford integrable system by defining a Poisson structure on the phase space of the Mumford system. In this article we will study the singular Mumford system. The starting point is to determine when and why the Mumford system is singular. For this we will …
In this paper, we establish the following version at infinity of Whitney's theorem [6, 7]: Geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of geometric tangent cones at infinity and using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. We also show that tangent cones at infinity of complex algebraic varieties can be computed using Gröbner bases.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the "beyond geometric invariant theory" program for analyzing moduli problems in algebraic geometry. We discuss methods for analyzing stability in general moduli problems, focusing on the moduli of coherent sheaves on a smooth projective scheme as a toy example. We describe several applications: a general structure theorem for the derived category of coherent sheaves on an algebraic stack; some results on the topology of moduli stacks; and a "v…
The local Euler obstructions and the Euler characteristics of linear sections with all hyperplanes on a stratified projective variety are key geometric invariants in the study of singularity theory. Despite their importance, in general it is very hard to compute them. In this paper we consider a special type of singularity: the recursive group orbits. They are the group orbits of a sequence of $G_n$ representations $V_n$ satisfying certain assumptions. We introduce a new intrinsic invariant called the $c_{sm}$ invariant, and use it to give explicit formulas to the local Euler obstructions and …
A stratified variety has a Kazhdan-Lusztig atlas if it can be locally modelled with Kazhdan-Lusztig varieties stratified by Schubert varieties in some Kac-Moody flag manifold via stratified isomorphisms. In this paper, we show that the partial flag manifold $G/P$ with the projected Richardson stratification has a Kazhdan-Lusztig atlas, with each chart stratified-isomorphic to a Kazhdan-Lusztig variety in the affine flag manifold of the formal loop group $\widehat{G}$ of $G$. This result generalizes that of Snider's on $Gr(k,n)$ with the positroid stratification, and is a geometric counterpart …
We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $Γ_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Flo…
For linear actions of real reductive Lie groups we prove the Kempf-Ness Theorem about closed orbits and the Kirwan-Ness Stratification Theorem of the null cone. Since our completely self-contained proof focuses strongly on geometric and analytic methods, essentially avoiding any deep algebraic result, it applies also to non-rational linear actions.
