Analytic Germs Research Articles

Jet scheme and simple isolated hypersurface singularities

Let m≥2 be a positive integer. Let (H(f),o) be the analytic germ of hypersurface attached to a weighted-homogeneous polynomial f∈C[x1,…,xm]. In this article, we mainly characterize the simpleness of any isolated analytic quasi-homogeneous complex hypersurface singularity (H,o) in Cm in terms of the associated jet schemes. We also compute the dimension of the jet scheme Ln(H)o, formed by the n-jets centered at o, for every positive integer n>>0, which is the crucial technical part of our simpleness criterion.

Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them

AbstractThe first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.

Gorenstein curve singularities of genus three

We classify the analytic germs of isolated Gorenstein curve singularities of genus three, and relate them to the connected components of strata of abelian differentials.

Topology of Fold Map Germs from mathbb R^3 to mathbb R^5

Let f:(mathbb R^3,0)rightarrow (mathbb R^5,0) be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of f with a small enough sphere S^4_epsilon centered at the origin in mathbb R^5. If f is of fold type, we define a labeled tree associated to its link and prove that is a complete topological invariant for it. As an application we obtain the complete topological classification of map germs contained in the mathcal {A}^2-class (x,y,z^2,xz,0).

A criterion for singular complex analytic germs not to be Lipschitz normally embedded

AbstractRecently, the notion of Lipschitz normally embedded sets introduced by Birbrair and Mostowski has been extensively studied. In the present paper we give a simple ‘effective’ criterion for a pure dimensional complex analytic germ not to be normally embedded.

A Note on Bernstein–Sato Varieties for Tame Divisors and Arrangements

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs, we consider general types of multivariate Bernstein–Sato ideals associated to arbitrary factorizations of our germ. We show that these ideals are principal, and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we obtain nice estimates for the zero locus of its Bernstein–Sato ideal for arbitrary factorizations and show that the Bernstein–Sato ideal attached to a factorization into linear forms is reduced. As an application, we independently verify and improve upon an estimate of Maisonobe regarding standard Bernstein–Sato ideals for reduced, generic arrangements: we compute the Bernstein–Sato ideal for a factorization into linear forms, and we compute its zero locus for other factorizations.

Almost-formality and deformations of representations of the fundamental groups of Sasakian manifolds

For a \(2n+1\)-dimensional compact Sasakian manifold, if \(n\ge 2\), we prove that the analytic germ of the variety of representations of the fundamental group at every semi-simple representation is quadratic. To prove this result, we prove the almost-formality of de Rham complex of a Sasakian manifold with values in a semi-simple flat vector bundle. By the almost-formality, we also prove the vanishing theorem on the cup product of the cohomology of semi-simple flat vector bundles over a compact Sasakian manifold.

Finite determinacy and approximation of flat maps

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real (resp. complex) analytic spaces to R m \mathbb {R}^m (resp. C m \mathbb {C}^m ). It is shown that the flatness of analytic maps from germs of real (resp. complex) analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection (resp. Cohen-Macaulay) analytic germs can be approximated arbitrarily well by polynomial (resp. Nash) maps in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney’s tangent cone.

On the normal sheaf of Gorenstein curves

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective 3-fold rational normal scroll S, implying that the normal sheaf on C embedded in S, and in Pg−1 as well, is unstable for g≥5, provided that S is smooth. We also compute the degree of the normal sheaf of any singular reduced curve in terms of the Tjurina and Deligne numbers, providing a semicontinuity of the degree of the normal sheaf over suitable deformations, revisiting classical results of the local theory of analytic germs.

Variation of local fundamental group and local first homology group

In this paper, we discuss various results about variation of the local fundamental group of normal complex spaces. It is proved that the finite Galois descent of upper semicontinuity of the local fundamental group holds at a factorial complex analytic germ. We also show by an example that finite Galois descent of upper semicontinuity of the local first homology group at a smooth germ is not true in general. We prove that the local fundamental groups of a normal complex algebraic variety are finite in number upto isomorphism.

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

Firstly, we see that the bases of the miniversal deformations of isolated Q \mathbb {Q} -Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension ≤ 2 \leq 2 which are the bases of the miniversal deformations of isolated Q \mathbb {Q} -Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric 3 3 -fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension ≥ 3 \geq 3 . As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension ≥ 3 \geq 3 .

The dual tree of a fold map germ from to

Let $f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_\epsilon$ centred at the origin in $\mathbb {R}^{4}$. If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

Inner geometry of complex surfaces: a valuative approach

Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.

Formal Gevrey solutions: in analytic germs—for higher order holomorphic PDEs

We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map P with P(0)=0. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to determine their divergence rate. In fact, we prove that the solution is Gevrey in P, giving new information on divergency while compared to the classical Gevrey classes. If P is not singular at 0, we also provide Poincaré conditions to recover convergent solutions. Our strategy is to extend the dimension and lift the given PDE to a problem where results of singular PDEs can be applied. Finally, examples where the Gevrey class in P is optimal are included.

Nonlinear Conditions for Ultradifferentiability: A Uniform Approach

Recent work showed that a theorem of Joris (that a function $f$ is smooth if two coprime powers of $f$ are smooth) is valid in a wide variety of ultradifferentiable classes $\mathcal C$. The core of the proof was essentially $1$-dimensional. In certain cases a multidimensional version resulted from subtle reduction arguments, but general validity, notably in the quasianalytic setting, remained open. In this paper we give a uniform proof which works in all cases and dimensions. It yields the result even on infinite dimensional Banach spaces and convenient vector spaces. We also consider more general nonlinear conditions, namely general analytic germs $\Phi$ instead of the powers, and characterize when $\Phi \circ f \in \mathcal C$ implies $f \in \mathcal C$.

On the Moduli Space of Quasi-Homogeneous Functions

We relate the moduli space of analytic equivalent germs of reduced quasi-homogeneous functions at $$({\mathbb {C}}^{2},0)$$ with their bi-Lipschitz equivalence classes. We show that any non-degenerate continuous family of (reduced) quasi-homogeneous (but not homogeneous) functions with constant Henry–Parusiński invariant is analytically trivial. Further, we show that there are only a finite number of distinct bi-Lipschitz classes among quasi-homogeneous functions with the same Henry–Parusiński invariant providing a maximum quota for this number. Finally, we conclude that the moduli space of bi-Lipschitz equivalent quasi-homogeneous function-germs admits an analytic structure.

Pluriharmonic Definable Functions in Some O-Minimal Expansions of the Real Field

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

A proof of A. Gabrielov’s rank theorem

This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.

Moderately Discontinuous Homology

AbstractWe introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow “moderately discontinuous” chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group for any and homomorphisms for any . Here is a “discontinuity rate”. The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms are essential. For Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for ‐Homology recovers the homology of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi‐Lipschitz subanalytic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer‐Vietoris long exact sequences. Moreover, fixed a discontinuity rate b we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are b‐moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, and recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs. © 2020 Wiley Periodicals LLC.

Topological Classification and Finite Determinacy of Knotted Maps

We show that the knot type of the link of a real analytic map germ with isolated singularity f : ( R 2 , 0 ) → ( R 4 , 0 ) is a complete invariant for C 0 - A -equivalence. Moreover, we also prove that isolated singularity implies finite C 0 -determinacy, giving an explicit estimate for its degree. For the general case of real analytic map germs, f : ( R n , 0 ) → ( R p , 0 ) ( n ≤ p ), we use the Lojasiewicz exponent associated with Mond’s double point ideal I 2 ( f ) to obtain some criteria of Lipschitz and analytic regularity.