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Bruno Scardua

Publications and source records attributed to Bruno Scardua.

At least 19 recordsLinked to original sources
arXiv2026

On meromorphic pencils, cusp singularities and holomorphic foliations in the complex plane

We study polynomial holomorphic $1$-forms in $\mathbb{C}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $ \phi = \frac{f^p}{g^q}, $ where $f,g$ are holomorphic functions (possibly polynomials) in general position and $(p,q)=1$. We first establish a homological characterization of relative exactness: if a polynomial $1$-form $\Omega$ has vanishing periods along every closed path contained in the fibers $\phi_c$, then $\Omega$ decomposes as $\Omega = a\omega_0 + dh,$ where $\omega_0 = p gdf - q f dg,$ for suitable polynomials $a$ and $h$. In the homogeneous case, degree constraints force $a$ to be constant. We then apply this integration principle to foliations leaving invariant plane curve singularities of cusp type \[ f^p + g^q = 0. \] Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that $\Omega$ is a polynomial cusp basic form, \[ \Omega = d(f^p + g^q) + \lambda (p gdf - q fdg), \qquad \lambda \in \mathbb{C}. \] In particular, such foliations admit Liouvillian first integrals of hypergeometric type. Our results provide a bridge between relative cohomology, the geometry of rational pencils, and the analytic structure of cusp foliations, yielding explicit normal forms and first integrals under homological hypotheses.

math.CV↗
arXiv2019

A swift introduction to holomorphic foliations with singularities

These are the notes of a series of lectures delivered by the author at the Graduate School of Mathematics of the University of Tokyo, during the month of October 2015. They were meant to be as self-contained as possible, taking into account time and space. The basic idea was to introduce the concepts and some of the basic results in the theory of holomorphic foliations with singularities. Another goal is to guide the reader to some of the recent questions and problems in the field, providing in this way a motivating introduction to those who are interested in studying a new subject.

math.CV↗
arXiv2016

Integrable deformations of local analytic fibrations with singularities

We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of integrable germs writing as $ω= df + f η$ where $f$ is quasi-homogeneous. Under the same hypotheses for $X_f : (f=0)$ we prove that $ω$ also admits a holomorphic first integral. Finally, we conclude that an integrable germ $ω= adf + f η$ admits a holomorphic first integral provided that: (i) $X_f: (f=0)$ is irreducible with an isolated singularity at the origin $0 \in \mathbb C^n, n \geq 3$; \, (ii) the algebraic multiplicities of $ω$ and $f$ at the origin satisfy $ν(ω) = ν(df)$. In the case of an isolated singularity for $(f=0)$ the writing $ω= adf + f η$ is always assured so that we conclude the existence of a holomorphic first integral. Some questions related to Relative Cohomology are naturally considered and not all of them answered.

math.CV↗
arXiv2016

Stable singularities of holomorphic vector fields

We consider germs of holomorphic vector fields with an isolated singularity at the origin $0\in\mathbb{C}^2$. We introduce a notion of stability, similar to "Lyapunov stability". For such a germ, called $L$-stable singularity, either the corresponding foliation admits a holomorphic first integral, or it is a real logarithmic foliation singularity. A notion of $L$-stability is also naturally introduced for a leaf of a foliation. In the complex codimension one case, for holomorphic foliations, the holonomy groups of $L$-stable leaves are proved to be abelian, of a suitable type. This implies the existence of local closed meromorphic one-forms defining the foliation, in a neighborhood of $L$-stable leaves.

math.DS↗
arXiv2014

Transversely affine holomorphic foliations of arbitrary codimension - I

We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theorem is then proved in the generic singularities framework.

math.GT↗
arXiv2014

Closed Orbits and Integrability for singularities of complex vector fields in dimension three

This paper is about the integrability of complex vector fields in dimension three in a neighborhood of a singular point. More precisely, we study the existence of holomorphic first integrals for isolated singularities of holomorphic vector fields in complex dimension three, pursuing the discussion started in \cite{CaSc2009}. Under generic conditions, we prove a topological criteria for the existence of a holomorphic first integral. Our result may be seen as a kind of Reeb stability result for the framework of vector fields singularities in complex dimension three. As a consequence, we prove that, for the class of singularities we consider, the existence of a holomorphic first integral is invariant under topological equivalence.

math.DS↗
arXiv2013

Complex polynomial vector fields with many algebraic orbits

We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits infinitely many algebraic orbits. In this paper, we give an interpretation of this result in terms of the classical Reeb stability theorems, for compact leaves of (non-singular) smooth foliations. Then we give versions of Darboux's theorem, assuring, for a (non-singular) holomorphic foliation of any codimension, the existence of an open set of compact leaves provided that the measure of the set of compact leaves is not zero. As for the case of polynomial vector fields in the complex affine space of dimenion $m\geq2$, we prove suitable versions of the above results, based also on the very special geometry of the complex projective space of dimension $m$, and on the nature of the singularities of such vector fields we consider.

math.DS↗
arXiv2012

A measurable stability theorem for holomorphic foliations transverse to fibrations

We prove that a transversely holomorphic foliation which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not of zero measure. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold, is finite provided that the set of periodic orbits is not of zero measure.

math.CV↗
arXiv2012

Integrating factors for groups of formal complex diffeomorphisms

We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable {\it integrating factors}, i.e., invariant vector fields and one-forms associated to the group. In this paper we search the basic lines of such dictionary for the case of several complex variables groups. For abelian, metabelian, solvable or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension.

math.CV↗
arXiv2011

Groups of formal diffeomorphisms in several complex variables and closed one-forms

We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension.

math.CV↗
arXiv2011

Codimension one foliations with Bott-Morse singularities II

We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among other things, a good topological understanding of the leaves one may have around a center-type component of the singular set, and also of the topology of its basin. The stability theorem further allows the description of the topology of the boundary of the basin and how the topology of the leaves changes when passing from inside to outside the basin. This is described via fiberwise Milnor-Wallace surgery. A key-point for this is to show that if the boundary of the basin of a center is non-empty, then it contains a saddle; in this case we say that the center and the saddle are paired. We then describe the possible pairings one may have in dimension three and use a construction motivated by the classical saddle-node bifurcation, that we call foliated surgery, that allows the reduction of certain pairings of singularities of a foliation. This is used together with our previous work on the topic to prove an extension for 3-manifolds of Reeb's sphere recognition theorem.

math.GT↗
arXiv2010

Transversely projective holomorphic foliations with singularities

We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic subset. The basic motivation is the characterization of pull-backs of Riccati foliations on projective spaces. Our techniques apply to give a description of the generic models of codimension one foliations on compact manifolds of dimension $\ge 3$.

math.DS↗
arXiv2007

On the integrability of holomorphic vector fields

We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal F(X)$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F \colon(\mathbb{C}^{3},0)\longrightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal F(X)$ are contained in the level curves of $F$.

math.CV↗