arXiv subjects
Publications and source records attributed to Víctor León.
We study third-order linear differential equations through their associated Pfaffian and projective foliations. In the regular-singular setting, we introduce a Fuchs-adapted Pfaffian model in which the indicial polynomial appears directly in the special-fiber singular scheme, and Frobenius resonance acquires a projective geometric interpretation.
We study Lyapunov-type stability for invariant algebraic divisors of codimension-one holomorphic foliations on complex projective spaces. Motivated by the notion of $L$-stability introduced in \cite{LeonScardua2018} for plane singularities, we define a projective stability condition which is compatible with the logarithmic setting and controls leafwise holonomy along the regular part of the invariant divisor, while discarding any stability requirement inside prescribed neighborhoods of its singular locus. Our main result then is a global projective counterpart of the local classification in \cite{LeonScardua2018}: for a maximal invariant algebraic divisor, projective $L$-stability together with non-dicritical non-nodal generalized-curve singularities on a generic plane section forces the ambient foliation to be globally logarithmic. As an application we obtain a topological rigidity consequence on $\mathbb P^2$ for logarithmic foliations under mild generic conditions. The proof of the main theorem is based on propagating the consequences of the stability hypothesis through the reduction tree by means of an explicit Dulac transport argument, together with the classification of $L$-stable groups of germs of one-dimensional complex diffeomorphisms.
Let $\F$ be a codimension-one holomorphic foliation of degree $d$ on $\PP^n$, $n\geq3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\cap\Sing(\F))_{\rm red}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homogeneous coordinates with $H=(t=0)$, \[ Ω=Q\,dt-\frac{t}{d+1}\,dQ , \] where $Q$ is homogeneous of degree $d+1$. Thus $Q/t^{d+1}$ is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface $D\subset\PP^n$: if the reduced singular divisor on $D$ is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every $d+1$ which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.
We study relative cohomology for logarithmic differential forms and meromorphic forms that are relatively closed with respect to the associated logarithmic foliation. Under suitable Diophantine, geometric, and topological hypotheses, we obtain decompositions into a meromorphic multiple of the defining logarithmic form, an exact meromorphic form, and a logarithmic form with constant residues. We establish a meromorphic extension theorem from suitable two-dimensional sections and prove local, polynomial, and homogeneous versions of the relative-cohomology decomposition; the global polynomial result is proved in arbitrary dimension by ambient leafwise continuation and meromorphic extension. Resolution of singularities, non-nodal saturation, and holonomy gluing are used to treat singular logarithmic foliations. As an application, we derive formal normal forms for analytic integrable deformations, including a several-component result in dimension two and a two-component extension in higher dimensions.
In this paper, we study weakly orthogonally invariant Finsler metrics and derive explicit expressions for their Berwald and Landsberg curvatures. We then obtain the system of partial differential equations characterizing generalized Finsler warped product metrics on $I \times \mathbb{R}^n$, which form a subclass of weakly orthogonally invariant Finsler metrics, under the conditions of vanishing Berwald and Landsberg curvatures. As an application, we construct examples of non-regular Landsberg metrics that are not of Berwald type.
We investigate the travel time in a navigation problem from a geometric perspective. The setting involves an open subset of the Euclidean plane, representing a lake perturbed by a symmetric wind flow proportional to the distance from the origin. The Randers metric derived from this physical problem generalizes the well-known Euclidean metric on the Cartesian plane and the Funk metric on the unit disk. We obtain formulas for distances, or travel times, from point to point, from point to line, and vice-versa
In this paper, we consider the cylindrically symmetric Finsler metrics and we obtain their Douglas curvature. Furthermore, we obtain the differential equation system of the cylindrically symmetric Finsler metrics with vanishing Douglas curvature. Many examples are included.
We introduce the notion of the Bruce-Roberts Tjurina number for holomorphic 1-forms relative to a pair $(X,V)$ of complex analytic subvarieties. When the pair $(X,V)$ consists of isolated complex analytic hypersurfaces, we prove that the Bruce-Roberts Tjurina number is related to the Bruce-Roberts number, the Tjurina number of the 1-form with respect to $V$ and the Tjurina number of $X$, among other invariants. As an application, we present a quasihomogeneity result for germs of holomorphic foliations in complex dimension two.
We introduce the notion of the \textit{Bruce-Roberts number} for holomorphic 1-forms relative to complex analytic varieties. Our main result shows that the Bruce-Roberts number of a 1-form $ω$ with respect to a complex analytic hypersurface $X$ with an isolated singularity can be expressed in terms of the \textit{Ebeling--Gusein-Zade index} of $ω$ along $X$, the \textit{Milnor number} of $ω$ and the \textit{Tjurina number} of $X$. This result allows us to recover known formulas for the Bruce-Roberts number of a holomorphic function along $X$ and to establish connections between this number, the radial index, and the local Euler obstruction of $ω$ along $X$. Moreover, we present applications to both global and local holomorphic foliations in complex dimension two.
In this paper we study holomorphic actions of the complex multiplicative group on complex manifolds around a singular (fixed) point. We prove linearization results for the germ of action and also for the whole action under some conditions on the manifold. This can be seen as a follow-up to previous works of M. Suzuki and other authors.
We study parabolas in the two dimensional unit disk equipped with a Funk metric. Four types of parabolas are obtained, due to the non-reversibility of the Funk metric, each one with applications to physics in the Zermelo navigation problem. We show that two of the four parabolas obtained are well known conics, and the remaining two are characterized by irreducible quartics. Explicit examples are given.
We study the cylindrical symmetric Finsler metrics. We obtain the system of differential equations of such metrics which are projectively flat. We give a family of solutions of this system. Examples are included.