arXiv subjects

Bruno Scárdua

Publications and source records attributed to Bruno Scárdua.

10 recordsLinked to original sources

arXiv2026

Stability and Logarithmic Foliations on Complex Projective Spaces

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.

math.DS↗

arXiv2026

An arithmetic integrability result for codimension-one foliations on complex projective spaces

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.

math.AG↗

arXiv2026

Relative Cohomology and Deformations of Logarithmic Foliations on Complex Spaces

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.

math.CV↗

arXiv2026

Holomorphic and Formal First Integrals for Foliations of Codimension One on Complex Analytic Space Germs

We study holomorphic and formal first integrals for germs of codimension-one holomorphic foliations on normal complex analytic spaces. In dimension two, under the assumption that the dual graph of the exceptional divisor of a resolution is a tree, we prove that the foliation admits a holomorphic first integral if and only if its leaves are closed outside the singular point and only finitely many leaves accumulate at that point. This extends a classical integrability theorem of Mattei and Moussu to singular ambient spaces. We also prove a holomorphic prolongation theorem for normal quotient germs admitting a smooth quasi-étale cover and a smooth connected lift of a generic two-dimensional section. We record, in addition, a conditional formal prolongation statement under depth assumptions on the conormal powers and an injectivity condition for the corresponding differential-form obstruction modules. Under the quotient-prolongation hypothesis, and with a reduced tangent cone where formal restriction must be detected, the higher-dimensional integrability results follow from their surface counterparts. We give a reduced nonnormal example satisfying both dynamical conditions but admitting no holomorphic first integral, showing that normality is essential. Our arguments combine resolution of singularities, holonomy techniques, formal completion, and extension properties of holomorphic functions on normal analytic spaces.

math.CV↗

arXiv2024

On holomorphic $\mathbb{C}^*$-actions

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.

math.CV↗

arXiv2019

On singular Frobenius for second order linear partial differential equations

The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the method of Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the paper is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

math.DS↗

arXiv2019

On the integrability of Hill's equation of the motion of the moon

We study under the standpoint of integrable complex analytic 1-forms (complex analytic foliations), a class of second order ordinary differential equations with periodic coefficients. More precisely, we study Hill's equations of motion of the moon, which are related to the dynamics of the system Sun-Earth-Moon. We associate to the {\em complex Hill equation} an integrable complex analytic one-form in dimension three. This defines a {\it Hill foliation}. The existence of first integral for a Hill foliation is then studied. The simple cases correspond to the existence of rational or Liouvillian first integrals. We then prove the existence of a {\it Bessel type} first integral in a more general case. We construct a standard two dimensional model for the foliation which we call {\it Hill fundamental form}. This plane foliation is then studied also under the standpoint of reduction of singularities and existence of first integral. For the more general case of the Hill equation, we prove for the corresponding Hill foliation, the existence of a Laurent-Fourier type formal first integral. Our approach suggests that there may be a class of plane foliations admitting Bessel type first integrals, in connection with the classification of (holonomy) groups of germs of complex diffeomorphisms associate to a certain class of second order ODEs.

math.DS↗

arXiv2018

Integrable deformations of foliations: a generalization of Ilyashenko's result

We study analytic deformations of holomorphic differential 1-forms. The initial 1-form is exact homogeneous and the deformation is by polynomial integrable 1-forms. We investigate under which conditions the elements of the deformation are still exact or, more generally, exhibit a first integral. Our results are related to natural extensions of classical results of Ilyashenko on limit cycles of perturbations of hamiltonian systems in two complex variables.

math.AG↗

arXiv2018

On first order deformations of homogeneous foliations

We study analytic deformations of holomorphic foliations given by homogeneous integrable one-forms in the complex affine space $\mathbb C^n$. The deformation is supposed to be of first order (order one in the parameter). We also assume that the deformation is given by homogeneous polynomial one-forms. The deformation takes place in the affine space since we are not assuming that the foliations descent to the projective space. We describe the space of such deformations in three main situations: (1) the given foliation is given by the level hypersurfaces of a homogeneous polynomial. (2) the foliation is rational, ie., has a first integral of type $P^r/Q^s$ for some homogeneous polynomials $P,Q$. (3) the foliation is logarithmic of a generic type. We prove that, for each class above, the first order homogeneous deformations of same degree are in the very same class. We also investigate the existence of such deformations with different degree.

math.AG↗

arXiv2010

Extension theorems for analytic objects associated to foliations

In this paper we will establish a structure theorem concerning the extension of analytic objects associated to germs of dimension one foliations on surfaces, through one-dimensional barriers. As an application, an extension theorem for projective transverse structures is obtained.

math.CV↗