Holomorphic Linearization of Commuting Germs of Holomorphic Maps

Let f ₁,…,f _h be h≥2 germs of biholomorphisms of ? ⁿ fixing the origin. We investigate the shape that a (formal) simultaneous linearization of the given germs can have, and we prove that if f ₁,…,f _h commute and their linear parts are almost simultaneously Jordanizable, then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commute and our Brjuno-type condition holds, then they are holomorphically simultaneously linearizable. This answers a multi-dimensional version of a problem raised by Moser.     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

C-GV triviality of function germs and Newton polyhedra

We provide estimates on the degree of C l GV determinacy ( G is one of Mather’s groups R or K ) of function germs which are defined on analytic variety V and satisfies a non-degeneracy condition with respect to some Newton polyhedron. The result gives an explicit order such that the C l geometrical structure of a function germ is preserved after higher order perturbations, which generalizes the result on C l G triviality of function germs given by M.A.S.Ruas.     

On the Hypoellipticity of Differential Forms with Isolated Singularities

Hypoellipticity for a class of germs of ?-valued, real-analytic 1-forms with an isolated singularity is studied. The main result states that in dimension ≥3 such forms are real-analytic hypoelliptic over L ^p functions, but are not real-analytic hypoelliptic in dimension 2. The case of C ^∞-hypoellipticity for a class of rotation invariant forms in ?² is also considered.     

Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ? ⁿ . More precisely, we are interested in the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms ??, ?? such that every formal conjugation $\hat \sigma$ (i.e. $\eta \circ \hat \sigma = \hat \sigma \circ \phi$ ) does not extend to the fixed points set Fix(??) of ??, meaning that it is not transversally formal (or semi-convergent) along Fix(??). We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.     

A forgotten theorem on Z k × R m -action germs and related questions

Two weakly hyperbolic smooth Z ^k × R ^m -action germs are smoothly conjugate if and only if they are formally conjugate, and such.     

Sur la topologie d'une famille de pinceaux de germes d'hypersurfaces complexes

We study the pencils of germs of complex hypersurfaces defined by the couples (ƒ, g): (C^{n+1, 0}) → (C², 0) without blowing-up in codimension 0. We approximate the equisingularity set of such a pencil in terms of the tangent directions to the discriminant germ of the morphism (ƒ, g). We next compare the monodromy modules of the generic member of this pencil with those of the atypical ones.     

Rigidity of Poisson structures

We study germs of analytic Poisson structures which are suitable perturbations of a quasihomogeneous Poisson structure in a neighborhood of the origin of ℝ ⁿ or ℂ ⁿ , a fixed point of the Poisson structures. We define a “diophantine condition” relative to the quasihomogeneous initial part ie256-1 which ensures that such a good perturbation of 256-2 which is formally conjugate to 256-3 is also analytically conjugate to it.     

Équation homologique et cycles asymptotiques d'une singularité nœud-col

Let a germ of holomorphic vector field Z be given and assume that is an isolated degenerate-resonnant singular point for Z (one and only one non-zero eigenvalue). Such a vector field acts as a derivative over the space of holomorphic germs at the origin of the complex plane. We obtain the solutions of the homological equation Z·F=G by integrating G along some asymptotic paths tangent to the complex trajectories of Z and ending at the singularity; this locate the obstructions to solve such an equation in the period of G along asymptotic cycles. The Borel transform is thus extended to the foliated setting and this geometrical approach helps us in the study of the conjugacy problem. For instance we find without expense of computation the obstructions obtained previously by P.M. Elizarov for the Poincaré-Dulac models. This approach of the caracteristics method in the singular setting will lead us, in a further print, to describe the analytical classification of germs of degenerate-resonnant vector fields.     

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.     

Topological {\mathcal{K}} -classification of finitely determined map germs

We consider smooth finitely C ⁰- ${\mathcal{K}}$ -determined map germs ${f : (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)}$ and we look at the classification under C ⁰- ${\mathcal{K}}$ -equivalence. The main tool is the homotopy type of the link, which is obtained by intersecting the image of f with a small enough sphere centered at the origin. When f ^?1(0) = {0}, the link is a smooth map between spheres and f is C ⁰- ${\mathcal{K}}$ -equivalent to the cone of its link. When f ^?1(0) ≠ {0}, we consider a link diagram, which contains some extra information, but again f is C ⁰- ${\mathcal{K}}$ -equivalent to the generalized cone. As a consequence, we deduce some known results due to Nishimura (for n = p) or the first named author (for n < p). We also prove some new results of the same nature.     

The proper L-S category of Whitehead manifolds

It is shown that the proper L-S category of an eventually end-irreducible, R²-irreducible Whitehead 3-manifold is 4. For this we prove, in the category of germs at infinity of proper maps, a partial analogue of the characterization by Eilenberg and Ganea of the L-S category of an aspherical space.     

The Index of a 1-Form on a Real Quotient Singularity

Let G be a finite Abelian group acting (linearly) on space ?ⁿ and, therefore, on its complexification ?ⁿ, and let W be the real part of the quotient ?ⁿ/G (in the general case, W ≠ ?ⁿ/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the residue bilinear form on the G-invariant part of the quotient of the space of germs of n-forms on (?ⁿ, 0) by the subspace of forms divisible by the 1-form under consideration.     

Commutative cochains of germs of differential forms for Euclidean compacta

The purpose of the paper is for any compactum K⊂Rn to construct a space Cp(K) of commutative germs of C^∞-differential p-forms, which is a reflexive locally convex topological vector space with the Cp(K) as its dual.     

On derivative-dependent infinitesimal deformations of differentiable maps

We study, by means of flows in jet bundles, infinitesimal deformations of germs of $\text{b}$^∞ maps $\text{?:R}{}^{\mathrm{n}}\text{→R}{}^{\mathrm{m}}$ that depend on a finite number of derivatives of these maps. We show that for m = 1 such deformations cannot essentially depend on derivatives higher than the first and that the deformations generalize the notion of an infinitesimal contact transformation. Analogous results hold for m > 1. Some applications to differential equations are given.     

Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

Let V, $\tilde{V}$ be hypersurface germs in ? ^m , each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for V, $\tilde{V}$ reduces to the linear equivalence problem for certain polynomials P, $\tilde{P}$ arising from the moduli algebras of V, $\tilde{V}$ . The polynomials P, $\tilde{P}$ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for V, $\tilde{V}$ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.     

The duality theorem for min-max functions

The set of min-max functions F : ℝⁿ → ℝⁿ is the least set containing coordinate substitutions and translations and closed under pointwise max, min, and function composition. The Duality Conjecture asserts that the trajectories of a min-max function, considered as a dynamical system, have a linear growth rate (cycle time) and shows how this can be calculated through a representation of F as an infimum of max-plus linear functions. We prove the conjecture using an analogue of Howard's policy improvement scheme, carried out in a lattice ordered group of germs of affine functions at infinity. The methods yield an efficient algorithm for computing cycle times.     

On the local classification of smooth maps induced by Newton's method

The classification of germs of smooth maps f:Rⁿ→Rⁿ induced by the continuous Newton method is addressed in this paper. This classification problem is shown to rely on an equivalence notion located between right and contact equivalences, driving the classification problem from the setting of quasilinear ODEs to the singularity theory framework. One-dimensional problems and regular, n-dimensional cases are easily characterized, and normal forms for them are given. Folded zeros in Rⁿ display a much richer behavior: an invariant and a preliminary normal form are derived for these cases.