Non-degenerate Singularities Research Articles

The blow-up method applied to monodromic singularities

The blow-up method proves its effectiveness to characterize the integrability of the resonant saddles giving the necessary conditions to have formal integrability and the sufficiency doing the resolution of the associated recurrence differential equation using induction. In this work we apply the blow-up method to monodromic singularities in order to solve the center-focus problem. The case of nondegenerate monodromic singularities is straightforward since any real nondegenerate monodromy singularity can be embedded into a complex system with a resonant saddle. Here we apply the method to nilpotent and degenerate monodromic singularities solving the center problem when the center conditions are algebraic.

Heat Diffusions on Holomorphic Foliations With Non-Degenerate Singularities

Heat Diffusions on Holomorphic Foliations With Non-Degenerate Singularities

On real center singularities of complex vector fields on surfaces

One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98–99 (1982), pp. 216–223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with ‘many’ periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.

Poincaré metric of holomorphic foliations with non-degenerate singularities

Consider a Brody hyperbolic foliation [Formula: see text] with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincaré metric is transversally Hölder continuous with a logarithmic slope towards the singular set of [Formula: see text].

Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson–Lie groups

In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson–Lie groups and the unimodularity of the Poisson–Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson–Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Poisson–Lie structure is necessarily unimodular. Furthermore, we illustrate our theory with different interesting examples, both on semisimple and unimodular Poisson–Lie groups.

Integrability and linearizability of symmetric three-dimensional quadratic systems

<p style='text-indent:20px;'>We study local integrability and linearizability of polynomial and analytic systems of ODEs. It is proven that in the case of non-degenerate singularity if an analytic system is completely integrable and one equation is linearizable, then the system is linearizable in a neighborhood of the singularity. Some integrable and linearizable systems in a family of three-dimensional quadratic autonomous systems of ODEs depending on ten parameters are found.</p>

Directed harmonic currents near non-hyperbolic linearizable singularities

AbstractLet$(\mathbb {D}^2,\mathscr {F},\{0\})$be a singular holomorphic foliation on the unit bidisc$\mathbb {D}^2$defined by the linear vector field$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$where$\unicode{x3bb} \in \mathbb {C}^*$. Such a foliation has a non-degenerate singularity at the origin${0:=(0,0) \in \mathbb {C}^2}$. LetTbe a harmonic current directed by$\mathscr {F}$which does not give mass to any of the two separatrices$(z=0)$and$(w=0)$. Assume$T\neq 0$. The Lelong number ofTat$0$describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when$\unicode{x3bb} \notin \mathbb {R}$, that is, when$0$is a hyperbolic singularity, the Lelong number at$0$vanishes. Suppose the trivial extension$\tilde {T}$across$0$is$dd^c$-closed. For the non-hyperbolic case$\unicode{x3bb} \in \mathbb {R}^*$, we prove that the Lelong number at$0$:(1)is strictly positive if$\unicode{x3bb}>0$;(2)vanishes if$\unicode{x3bb} \in \mathbb {Q}_{<0}$;(3)vanishes if$\unicode{x3bb} <0$andTis invariant under the action of some cofinite subgroup of the monodromy group.

Limit spectral distribution for non-degenerate hypersurface singularities

We establish Kyoji Saito's continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito's notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.

On the monodromy conjecture for non-degenerate hypersurfaces

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein–Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations – the Denef– Loeser conjecture on topological zeta functions – is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting. We develop new tools to deal with such multidimensional phenomena, and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.

On non-projective small resolutions

We construct a large class of projective threefolds with one node (aka non-degenerate quadratic singularity) such that their small resolutions are not projective.

Hidden toric symmetry and structural stability of singularities in integrable systems

The goal of the paper is to develop a systematic approach to the study of (perhaps degenerate) singularities of integrable systems and their structural stability. As the main tool, we use "hidden" system-preserving torus actions near singular orbits. We give sufficient conditions for the existence of such actions and show that they are persistent under integrable perturbations. We find toric symmetries for several infinite series of singularities and prove, as an application, structural stability of Kalashnikov's parabolic orbits with resonances in the real-analytic case. We also classify all Hamiltonian $k$-torus actions near a singular orbit on a symplectic manifold $M^{2n}$ (or on its complexification) and prove that the normal forms of these actions are persistent under small perturbations. As a by-product, we prove an equivariant version of the Vey theorem (1978) about local symplectic normal form of nondegenerate singularities.

Conjectures on Stably Newton Degenerate Singularities

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, x^p+x^q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

Geometric quantization via cotangent models

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on \(T^*(SO(3))\) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.

On the integrability problem for the Hopf-zero singularity and its relation with the inverse Jacobi multiplier

In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields.

Realization of focal singularities of integrable systems using billiard books with a Hooke potential field

Systems of particle motion in the Hooke central potential field on a billiard book glued from flat circular billiard domains are considered. An important class of nondegenerate focal singularities of the rank 0 of integrable systems with 2 degrees of freedom is completely realized by this class of billiards. Namely, for each semi-local focal singularity the constructed billiard system has a singularity fiberwise homeomorphic to the given one.

Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows

In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows $\varphi$, GS-flows, on singular surfaces $M$. This theory generalizes the classical theory of Morse complexes of smooth dynamical systems together with the corresponding cancellation theory for non-degenerate singularities. This is accomplished by defining a GS-chain complex for $(M,\varphi)$ and computing its spectral sequence $(E^r,d^r)$. As $r$ increases, algebraic cancellations occur, causing modules in $E^r$ to become trivial. The main theorems herein relate these algebraic cancellations within the spectral sequence to a family $\{M_r,\varphi_r\}$ of GS-flows $\varphi_r $ on singular surfaces $M_r$, all of which have the same homotopy type as $M$. The surprising element in these results is that the dynamical homotopical cancellation of GS-singularities of the flows $\varphi_r$ are in consonance with the algebraic cancellation of the modules in $E^r$ of its associated spectral sequence. Also, the convergence of the spectral sequence corresponds to a GS-flow $\varphi_{\bar{r}}$ on $M_{\bar{r}}$, for some $\bar{r}$, with the property that $\varphi_{\bar{r}}$ admits no further dynamical homotopical cancellation of GS-singularities.

Hodge filtration and Hodge ideals for $\mathbb{Q}$-divisors with weighted homogeneous isolated singularities or convenient non-degenerate singularities

Hodge filtration and Hodge ideals for $\mathbb{Q}$-divisors with weighted homogeneous isolated singularities or convenient non-degenerate singularities

Complex planar Hamiltonian systems: Linearization and dynamics

<p style='text-indent:20px;'>Global dynamics of complex planar Hamiltonian polynomial systems is difficult to be characterized. In this paper, for general complex quadratic Hamiltonian systems of one degree of freedom, we obtain some sufficient conditions on the existence of family of invariant tori. We also complete characterization on locally analytic linearizability of complex planar Hamiltonian systems with homogeneous nonlinearity of degrees either 2 or 3 at a nondegenerate singularity, and present their global dynamics. For these classes of systems we also prove existence of families of invariant tori, together with isochronous periodic orbits.

Boundary of the Milnor fiber of a Newton non-degenerate surface singularity

We give an algorithm for the description as a graph manifold of the boundary of the Milnor fiber of a Newton non-degenerate surface singularity embedded in a germ of toric three-dimensional variety, starting from its local Newton polyhedron. This algorithm is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any smoothing of a reduced non-isolated singularity of complex surface.

On a Generic Dimension of the Critical Locus

Let f:({mathbb {C}}^n,0)rightarrow left( {mathbb {C}},0right) ,nle 3, be a nondegenerate singularity. In this article we give a combinatorial characterization of the dimension of the critical locus of f in terms of its support. We also show that this dimension can be read off from the Newton diagram of f, which solves one of Arnold’s problems in this case.