Formule de Cauchy-Crofton pour la densité des ensembles sous-analytiques

For the density of subanalytic sets, we give a formula analogous to the classical Cauchy-Crofton formula for the volume. To do this we define the local polar profiles of a subanalytic set and the corresponding finite sequence of local multiplicities which is the real counterpart of the multiplicity of complex analytic sets.     

Flattening and Subanalytic Sets in Rigid Analytic Geometry

Let K be an algebraically closed field endowed with a completenon-archimedean norm with valuation ring R. Let f: Y X be amap of K-affinoid varieties. In this paper we study the analyticstructure of the image f(Y) X; such an image is a typical exampleof a subanalytic set. We show that the subanalytic sets areprecisely the D-semianalytic sets, where D is the truncateddivision function first introduced by Denef and van den Dries.This result is most conveniently stated as a Quantifier Eliminationresult for the valuation ring R in an analytic expansion ofthe language of valued rings. To prove this we establish a Flattening Theorem for affinoidvarieties in the style of Hironaka, which allows a reductionto the study of subanalytic sets arising from flat maps, thatis, we show that a map of affinoid varieties can be renderedflat by using only finitely many local blowing ups. The caseof a flat map is then dealt with by a small extension of a resultof Raynaud and Gruson showing that the image of a flat map ofaffinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in thezero characteristic case, a Uniformization Theorem for subanalyticsets: a subanalytic set can be rendered semianalytic using onlyfinitely many local blowing ups with smooth centres. As a corollarywe obtain the fact that any subanalytic set in the plane R2is semianalytic. 2000 Mathematical Subject Classification: 32P05,32B20, 13C11, 12J25, 03C10.     

Complements of subanalytic sets and existential formulas for analytic functions

We show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra. This result has an equivalent formulation in logic: Consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers. Such an expression is equivalent to an existential expression involving functions from the same class, provided that the variables approach neither infinity nor the boundary of the domain. Oblatum 17-VIII-1995     

Flattening and analytic continuation of affinoid morphisms: remarks on a paper of Gardener and Schoutens

We give an example of an affinoid curve without analytic continuation.We use this to produce an example of an affinoid morphism thatcannot be flattened by a finite sequence of local blow-ups.Thus the global rigid analogue of Hironaka's complex analyticflattening theorem given by T. Gardener and H. Schoutens, inTheorem 2.3 of ‘Flattening and subanalytic sets in rigidanalytic geometry’, Proc. London Math. Soc (3) 83 (2001)681–707, is not true. Since this is a key step in theproof of the affinoid elimination theorem (loc. cit. Theorem3.12), that proof contains a serious gap. We also give an exampleof an affinoid subset of the plane that is not the image undera proper rigid analytic map of a set that is globally semianalyticin the domain of that map. This clarifies the relationship amongseveral natural categories of rigid subanalytic sets.2000 MathematicsSubject Classification 32P05 (primary), 03C10, 32B20 (secondary).     

Multiplicity mod 2 as a Metric Invariant

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of ℝ ⁿ . In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S ⁿ is preserved by the antipodal map. The study of such subsets of S ⁿ enables us to deduce the subanalytic metric invariance of the multiplicity modulo 2 under some extra assumptions on the tangent cone. We also prove a real version of a theorem of Comte, and yield that the multiplicity modulo 2 is preserved by arc-analytic bi-Lipschitz homeomorphisms.     

Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

     

Polynomial recurrence with large intersection over countable fields

Applying a local Gauss–Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz–Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstruction. As a corollary, we give a positive answer to a question of Fu on the Euler obstruction and the Gauss–Bonnet measure.     

Global complexification of real analytic globally subanalytic functions

We show that a globally subanalytic function that is real analytic can be extended in a globally subanalytic way to a holomorphic function. We also establish a parametric version of this result. For other important ominimal structures, we show that unary definable real analytic functions can be extended definably to a holomorphic function.     

Subanalytic bundles and tubular neighbourhoods of zero-loci

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector spaceP of sections) on a subanalytic subsetX of a real analytic manifoldM, and prove that whenM is compact, there is a Baire subsetU of sections inP whose zero-loci inX have tubular neighbourhoods, homeomorphic to the restriction of the given bundle to these zero-loci.     

Intersections of smooth polar unitals with secants are homeomorphic to spheres

In [5] and [6] we proved that (non-empty) sets of absolute points of smooth polarities, i.e. smooth polar unitals, in smooth projective planes of dimension 2l are smooth submanifolds of the point spaces homeomorphic to spheres of dimension . In this paper we show that the intersections of smooth polar unitals with secants are homeomorphic to spheres of dimension , respectively. Furthermore we prove that the condition of connectedness in [6, Theorem 1.2] may be omitted. This means that a closed (not necessarily connected) submanifold U of the point space of a smooth projective plane is homeomorphic to a sphere provided that there exists precisely one tangent at each point of U, and each secant intersects U transversally. If U has codimension 1 in the point space then the second condition follows from the first one, and also the intersections of U with secants are homeomorphic to spheres. This result may be generalized to compact hypersurfaces in the point spaces of smooth affine planes. Received: 1 July 2008     

Capacity Density of Subanalytic Sets in Higher Dimension

We show that capacity density exists for subanalytic sets in any dimension. Moreover, we state connections between capacity density and volume density for subanalytic sets.      

Examples of uniformly homeomorphic Banach spaces

We give several examples of separable Banach spaces which are nonisomorphic but uniformly homeomorphic. For example, we show that for every 1 < p ≠ 2 < ∞ there are two uniformly homeomorphic subspaces (respectively, quotients) of ? _p which are not linearly isomorphic; similarly c ₀ has two uniformly homeomorphic subspaces which are not isomorphic. We also give an example of two non-isomorphic separable L _∞-spaces which are coarsely homeomorphic (i.e. have Lipschitz equivalent nets).     

Choking Horns in Lipschitz Geometry of Complex Algebraic Varieties. Appendix by Walter D. Neumann

The paper studies the Lipschitz geometry of germs of complex algebraic varieties and introduces the notion of a choking horn. A choking horn is a family of cycles on an algebraic variety with the property that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness and the authors use this to prove that some classical isolated hypersurface singularities are not metrically conical. They also show that there exist countably infinitely many singular varieties, which are locally homeomorphic, but not locally subanalytically bi-Lipschitz equivalent with respect to the inner metric. The Appendix by W. Neumann uses separating sets to provide another example of the same phenomenon.     

Bi-Lipschitz Sufficiency of Jets

We give some theorems of bi-Lipschitz or C ¹ sufficiency of jets which are expressed by means of transversality with respect to some strata of a stratification satisfying the (L) condition of T. Mostowski. This enables us to prove that the number of metric types of intersection of smooth transversals to a stratum of an (a) regular stratification of a subanalytic set is finite.     

Nonprojecting isotopies and knots with homeomorphic coverings

In this paper, new examples of nonhomeomorphic knots and links which for certain r have homeomorphic r-sheeted cyclic branched coverings are constructed. In particular, it is proved that the two nonhomeomorphic knots with eleven crossings and with Alexander polynomial equal to one, have homeomorphic two-sheeted branched coverings, and that knots obtained from any knot by the Zeeman construction with p-fold and with q-fold twist have homeomorphic r-sheeted cyclic branched coverings ifp=±q(mod 2r). The construction of examples is based on regluing a link along a submanifold of codimension 1 by means of a homeomorphism which is covered by a homeomorphism which is isotopic to the identity only through nonprojecting isotopies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 133–147, 1976.     

Küchle fivefolds of type c5

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of “rough” Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.     

Rigid Subanalytic Subsets of Curves and Surfaces

Working over an algebraically closed, complete, non-Archimedean,nontrivially valued field of characteristic zero, it is shownthat (i) any subanalytic subset of a one-dimensional semialgebraicset is semialgebraic, (ii) any one-dimensional subanalytic setis semianalytic, and (iii) any subanalytic subset of a two-dimensionalsemianalytic set is semianalytic.     

Hölder-topology of the Heisenberg group

Aequationes mathematicae - The Heisenberg group is an example of a sub-Riemannian manifold homeomorphic, but not bi-Lipschitz equivalent to the Euclidean space. Its metric is derived from curves...     

The link of the germ of a semi-algebraic metric space

In this paper we investigate the metric properties of semi-algebraic germs. More precisely we introduce a counterpart to the notion of link for semi-algebraic metric spaces, which is often used to study the topology. We prove that it totally determines the metric type of the germ. We give a nice consequence for semi-algebraically bi-Lipschitz homeomorphic semi-algebraic germs.