Searcharxiv⌕ Search

arXiv subjects

Vincent Grandjean

Publications and source records attributed to Vincent Grandjean.

At least 19 recordsLinked to original sources
arXiv2024

Remarks on Lipschitz geometry on globally conic singular manifolds

We study metric properties of manifolds with conic singularities and present a natural interplay between metrically conic and metrically asymptotically conic behaviour. As a consequence, we prove that a singular sub-manifold is Lipschitz normally embedded, i.e. its inner and outer metric structures are equivalent, in an ambient singular manifold, wheneverthe singularities are conic and the ends of the manifold are asymptotically conic, which answers positively a question raised in our last work.

math.MG↗
arXiv2023

Characterization of Lipschitz Normally Embedded complex curves

The main result of the paper states that a connected complex affine algebraic curve is Lipschitz normally embedded (shortened to LNE afterwards) in $\mathbb{C}^n$ if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in $\mathbb{CP}^n$ with the property of its general affine traces being LNE in $\mathbb{C}^n$. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

math.AG↗
arXiv2023

One point compactification and Lipschitz normally embedded definable subsets

A closed subset of $\mathbb{R}^q$, definable in some given o-minimal structure, is Lipschitz normally embedded in $\mathbb{R}^q$ if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere ${\bf S}^q$($ = \mathbb{R}^q \cup \{\infty \}$), i.e. the closure of its image by the inverse of the stereographic projection is Lipschitz normally embedded in ${\bf S}^q$. This implies that any closed connected unbounded definable subset of an Euclidean space is definably inner bi-Lipschitz homeomorphic to a Lipschitz normally embedded definable set.

math.AG↗
arXiv2023

Global Lipschitz geometry of conic singular sub-manifolds with applications to algebraic sets

The main result states that a connected conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: the outer and inner metric space structures are metrically equivalent. We also show that a closed subset of $\mathbb{R}^n$ is a conic singular sub-manifold if and only if its closure in the one point compactification ${\bf S}^n =\mathbb{R}^n\cup \infty$ is a conic singular sub-manifold. Consequently the connected components of generic affine real and complex algebraic sets are conic at infinity, thus are Lipschitz Normally Embedded.

math.DG↗
arXiv2023

Stereographic compactification and affine bi-Lipschitz homeomorphisms

Let $σ_q : \mathbb{R}^q \to {\bf S}^q \setminus N_q$ be the inverse of the stereographic projection with centre the north pole $N_q$. Let $W_i$ be a closed subset of $\mathbb{R}^{q_i}$, for $i=1,2$. Let $Φ:W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $σ_{q_2}\circ Φ\circ σ_{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded. As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: 1) Sampaio's tangent cone result; 2) Links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette.

math.MG↗
arXiv2023

Equi-singularity of real families and Lipschitz Killing curvature densities at infinity

Fix an o-minimal structure expanding the ordered field of real numbers. Let $(W_y)_{y\in\mathbb{R}^s}$ be a definable family of closed subsets of $\mathbb{R}^n$ whose total space $W = \cup_y W_y\times y$ is a closed connected $C^2$ definable sub-manifold of $\mathbb{R}^n\times\mathbb{R}^s$. Let $φ:W \to\mathbb{R}^s$ be the restriction of the projection to the second factor. After defining $K(φ)$, the set of generalized critical values of $φ$, showing that they are closed and definable of positive codimension in $\mathbb{R}^s$, contain the bifurcation values of $φ$ and are stable under generic plane sections, we prove that all the Lipschitz-Killing curvature densities at infinity $y \mapsto κ_i^\infty(W_y)$ are continuous functions over $\mathbb{R}^s\setminus K(φ)$. When $W$ is a $C^2$ definable hypersurface of $\mathbb{R}^n\times\mathbb{R}^s$, we further obtain that the symmetric principal curvature densities at infinity $y \mapsto σ_i^\infty(W_y)$ are continuous functions over $\mathbb{R}^s\setminus K(φ)$.

math.AG↗
arXiv2020

Double transgressions and Bott-Chern duality

We present a general framework for obtaining currential double transgression formulas on complex manifolds which can be seen as manifestations of Bott-Chern Duality. These results complement on one hand the simple transgression formulas obtained by Harvey -Lawson and on the other hand the double transgression formulas of Bismut-Gillet-Soulé. Among the applications we mention a Gysin isomorphism for Bott-Chern cohomology, an abstract Poincaré-Lelong formula for sections of holomorphic and Hermitian vector bundles implying Andersson's generalization of the standard Poincaré-Lelong, a Bott-Chern duality formula for the Chern-Fulton classes of singular varieties or a refinement of the first author's simple transgression formula for the Chern character of a Quillen superconnection associated to a self-adjoint, odd endomorphism. The existence of a Bismut-Gillet-Soulé double transgression without the hypothesis of degeneration along a submanifold stands out and is based on an extension to linear correspondences of the operation of morphism addition. Finally, as a by-product we also obtain a statement about the {pointwise localization} of the Samuel multiplicity of an analytic subvariety of a complex manifold along an irreducible component.

math.DG↗
arXiv2019

Multiplicity and degree relative to a set

The multiplicity (resp. degree) of a function $f$ relative to a semianalytic subset $S$ of $\mathbb{R}^n$ is the greatest (resp. smallest) exponent among numbers $j$ such that the inequality $|f(x)|\leq C\|x\|^j$ holds on $S$ near $0$ (resp. near infinity) for some constant $C$. We show that there exists a family of curves $\{Γ_d\}_{d\in \mathbb{N}}$, determined only by the set, such that the relative multiplicity of any polynomial of degree $d$ is equal to its relative multiplicity with respect to $Γ_d$. Moreover, a semianalytic family $(S_t)_{t\in\mathbb{R}^m}$ of sets given by inequalities $f_i+t_ig_i\geq 0$ for $i=1,\dots, m$ admits a stratification of the parameter space $\mathbb{R}^m$ such that on each component of the top-dimensional stratum the relative multiplicity function on $\mathcal{O}_n$ does not change. Analogous results, assuming the data are algebraic, hold in the relative degree case.

math.AG↗
arXiv2019

Re-parameterizing and reducing families of normal operators

We present a new proof of results of Kurdyka & Paunescu, and of Rainer, about real-analytic multi-parameters generalizations of classical results by Rellich and Kato about the reduction in families of univariate deformations of normal operators over real or complex vector spaces of finite dimensions. Given a real analytic family of normal operators over a finite dimensional real or complex vector space, there exists a locally finite composition of blowings-up with smooth centers re-parameterizing the given family such that at each point of the source space of the re-parameterizing mapping, there exists a neighbourhood of any given point over which exists a real analytic orthonormal frame in which the pull back of the operator is in reduced form at every point of the neighbourhood. A free by-product of our proof is the local real analyticity of the eigen-values, which in all prior works was a prerequisite step to get local regular reducing bases.

math.AG↗
arXiv2019

Monomialization of singular metrics on real surfaces

We present here a result of Monomialization of real analytic two-symmetric tensor fields over regular real analytic surfaces. We apply it to the (extension of the pull-back of the) inner metric of a resolved surface of a real analytic surface singularity. Doing so we recover Hsiang & Pati property at each point of the resolved surface.

math.AG↗
arXiv2019

Gauss-Kronecker Curvature and equisingularity at infinity of definable families

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be a globally definable one parameter family of $C^2$-hypersurfaces of $\mathbb{R}^n$. Upon defining the notion of generalized critical value for such a family we show that the functions $s \to |K(s)|$ and $s\to K(s)$, respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of $T_s$, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.

math.AG↗
arXiv2015

Resolution of singularities of the cotangent sheaf of a singular variety

The main problem studied is resolution of singularities of the cotangent sheaf of a complex- or real-analytic variety Y (or of an algebraic variety Y over a field of characteristic zero). Given Y, we ask whether there is a global resolution of singularities s: X -> Y such that the pulled-back cotangent sheaf of Y is generated by differential monomials in suitable coordinates at every point of X ("Hsiang-Pati coordinates''). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of s. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety Y; there have been important applications of the latter to L2-cohomology.

math.AG↗
arXiv2015

Thin-thick decomposition for real definable isolated singularities

Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thin-thick decomposition of an isolated singularity germ - which happens to be a natural blow-spherical invariant. This decomposition is a generalization of the thin-thick decomposition of normal complex surface singularity germs introduced in [7]

math.MG↗
arXiv2015

The exponential map at a cuspidal singularity

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant -- essentially the only boundary invariant -- of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric. Revision: restructured many parts of the paper completely; stated main theorems for cusps of any order; replaced 'cusp' by 'cuspidal', also in title, added references

math.DG↗
arXiv2014

Lipschitz contact equivalence of function germs in $\mathbb{R}^2$

In this paper we study Lipschitz contact equivalence of continuous function germs in the plane definable in a polynomially bounded o-minimal structure, such as semialgebraic and subanalytic functions. We partition the germ of the plane at the origin into zones where the function has explicit asymptotic behavior. Such a partition is called a pizza. We show that each function germ admits a minimal pizza, unique up to combinatorial equivalence. We show then that two definable continuous function germs are definably Lipschitz contact equivalent if and only if their corresponding minimal pizzas are equivalent.

math.AG↗