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A contact distribution on projective three-space is defined by the 1-form $x_2dx_1-x_1dx_2+x_4dx_3-x_3dx_4$, up to a change of projective coordinates. The family of contact distributions is parameterized by the complement of the Pfaff-Pl\"ucker quadric in the projective 5-space of antisymmetric $4\times4$ matrices. A foliation of dimension 1 and degree $d$ is specified by a polynomial vector field $\sum{}p_i\partial_{x_i}, p_i$ homogeneous of degree $d$. The foliation is called Legendrian if tangent to some distribution of contact. Our goal is to give formulas for the dimensions and degrees of the varieties of Legendrian foliations, and of the varieties of foliations tangent to a pencil of planes., Comment: 20 pages; To appear on Perspectives on four decades: Algebraic Geometry 1980-2020. In memory of Alberto Collino. Trends in Mathematics; source.tex contains scripts for singular

Let $\mathcal{F}$ be a holomorphic foliation of degree $2$ on $\mathbb{P}^n$ with dimension $k\geq 2$. We prove that either $\mathcal{F}$ is algebraically integrable, or $\mathcal{F}$ is the linear pull-back of a degree-2 foliation by curves on $\mathbb{P}^{n-k+1}$, or $\mathcal{F}$ is a logarithmic foliation of type $(1^{n-k +1},2)$, or $\mathcal{F}$ is a logarithmic foliation of type $(1^{n-k+3})$, or $\mathcal{F}$ is the linear pull-back of a degree-2 foliation of dimension 2 on $\mathbb{P}^{n-k+2}$ tangent to an action of the Lie algebra $\mathfrak{aff}(\mathbb{C})$. As a byproduct, we describe the geometry of Poisson structures on $\mathbb{P}^n$ with generic rank two., Comment: 19 pages; major changes with several improvements and corrections. Comments welcome!

We study two-dimensional holomorphic distributions on $\mathbb{P}^4$. We classify dimension two distributions, of degree at most $2$, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non-integrable to integrable distributions. In the maximally non-integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in $\mathbb{P}^4$ whose the conormal sheaves are the Horrocks-Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non-integrable, uniquely determined by their singular schemes and invariant by a group $H_5 \rtimes SL(2,\mathbb{Z}_5) \subset Sp(4, \mathbb{Q})$, where $H_5$ is the Heisenberg group of level $5$. We prove that the moduli spaces of Horrocks-Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree $d\geq 6$, on $\mathbb{P}^4$ have a family of degenerated flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by $H_5 \rtimes SL(2,\mathbb{Z}_5)$ and singular along a degenerate abelian surface with $(1,5)$-polarization and level-$5$-structure., Comment: Improved exposition

The current state of electronic component miniaturization coupled with the increasing efficiency in hardware and software allow the development of smaller and compact robotic systems. The convenience of using these small, simple, yet capable robots has gathered the research community's attention towards practical applications of swarm robotics. This paper presents the design of a novel platform for swarm robotics applications that is low cost, easy to assemble using off-the-shelf components, and deeply integrated with the most used robotic framework available today: ROS (Robot Operating System). The robotic platform is entirely open, composed of a 3D printed body and open-source software. We describe its architecture, present its main features, and evaluate its functionalities executing experiments using a couple of robots. Results demonstrate that the proposed mobile robot is very effective given its small size and reduced cost, being suitable for swarm robotics research and education., Comment: Submitted to Autonomous Robots - S.I. 208: Robot Swarms in the Real World: from Design to Deployment

In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which have been motivated by classical works of Darboux, Poincar\'e and Painlev\'e on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincar\'e and Painlev\'e problem of bounding the degree and the genus of analytic varieties invariant by holomorphic foliations and Pfaff systems. We shall discuss the general ideas of the theory of integrability characterizing the existence of meromorphic first integrals for complex analytic Pfaff equations., Comment: This is intended as a chapter of the forthcoming "Handbook on the Geometry and Topology of Singularities"

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists., Comment: 17 pages, To appear in in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

Let $X$ be a smooth projective variety. We show that the map that sends a codimension one distribution on $X$ to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when $X = \mathbb{P}^n$, compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on $\mathbb{P}^3$. We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on $\mathbb{P}^3$., Comment: 21 pages. Some results and exposition were improved. Final version to appear in Math. Zeitschrift

We classify holomorphic Pfaff systems (possibly non locally decomposable) on certain Hopf manifolds. As consequence, we prove some integrability results. We also prove that any holomorphic distribution on a general (non-resonance) Hopf manifold is integrable., Comment: To appear in European Journal of Mathematics

In this work we provide a localization formulae for odd holomorphic super vector fields on compact complex supermanifolds with fermionic dimension equal to the bosonic dimension. We prove a residue theorem for holomorphic super vector fields with non-degenerated and isolated singularities and we determine the residues under certain generic local conditions., Comment: 40 pages

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces., Comment: 40 pages; to appear in Revista Matem\'atica Iberoamericana; major changes following suggestions of the referees; the title has been slightly modified; new explicit examples of degree 3 foliations are presented, which includes a foliation whose conormal bundle is an instanton bundle of charge 5 and the singular scheme is not reduced

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foliations. Moreover, we describe the geometry of the symplectic foliation induced by the Poisson connection on the projectivization of the Poisson module., Comment: 22 pages; To appear in Advances in Mathematics

We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\mathbb{P}^3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\mathscr{G}$ is a subfoliation of a codimension one distribution $\mathscr{F}$ with isolated singularities, then $Sing(\mathscr{G})$ is a curve. As a consequence, we give a criterion to decide whether $\mathscr{G}$ is globally given as the intersection of $\mathscr{F}$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme., Comment: 12 pages; To appear in Annals of the Brazilian Academy of Sciences

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable., Comment: 18 pages; Improved exposition; To Israel Vainsencher on the occasion of his 70th birthday

We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invariant divisor. As an application, we provide a formula for the number of singularities in the complement of the invariant divisor on complex projective spaces. Finally, we obtain a Poincar\'e-Hopf type formula for singular normal projective varieties., Comment: 19 pages. To appear in Communications in Analysis and Geometry. arXiv admin note: text overlap with arXiv:1611.01203

We prove a Gauss-Bonnet and Poincar\'e-Hopf type theorems for complex $\partial$-manifold $\tilde{X} = X - D$, where $X$ is a complex compact manifold and $D$ is a reduced divisor. We will consider the cases such that $D$ has isolated singularities and also if $D$ has a (not necessarily irreducible) decomposition $D=D_1\cup D_2$ such that $D_1$, $D_2$ have isolated singularities and $C=D_1\cap D_2$ is a codimension $2$ variety with isolated singularities., Comment: To appear in Moscow Mathematical Journal; To Omegar Calvo-Andrade on the occasion of his 60th birthday

We show a residues formula for maps generically transversal to regular holomorphic distributions., Comment: To appear in Proceedings of the American Mathematical Society

We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on projective spaces., Comment: To Omegar Calvo-Andrade on the occasion of his 60th birthday; Major changes following suggestions of the referee

In this work we prove a Brunella-Khanedani-Suwa variational type residue theorem for currents invariant by holomorphic foliations. As a consequence, we give conditions for the leaves of a singular holomorphic foliation to accumulate in the intersection of the singular set of the foliation with the support of an invariant current., Comment: To Israel Vainsencher on the occasion of his 70th birthday

In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. As a consequence, we obtain a classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one., Comment: 23 pages

Let $\mathscr{F}$ be a singular holomorphic foliation, of codimension $k$, on a complex compact manifold such that its singular set has codimension $\geq k+1$. In this work we determinate Baum-Bott residues for $\mathscr{F}$ with respect to homogeneous symmetric polynomials of degree $k+1$. We drop the Baum-Bott's generic hypothesis and we show that the residues can be expressed in terms of the Grothendieck residue of an one-dimensional foliation on a $(k+1)$-dimensional disc transversal to a $(k+1)$-codimensional component of the singular set of $\mathscr{F}$. Also, we show that Cenkl's algorithm for non-expected dimensional singularities holds dropping the Cenkl's regularity assumption., Comment: 13 pages. Improved exposition. To appear in Asian Journal of Mathematics

In this work we introduce a GSV type index for varieties invariant by holomorphic Pfaff systems (possibly non locally decomposables) on projective manifolds. We prove a non-negativity property for the index. As an application, we prove that the non-negativity of the GSV-index gives us an obstruction to the solution of the Poincar\'e problem for Pfaff systems on projectives spaces., Comment: 15 pages; to appear in Documenta Mathematica. Major changes following suggestions of the referees

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves., Comment: 43 pages, to appear in International Mathematics Research Notices. Major changes following suggestions of the referees

In this work we prove a Baum-Bott type formula for non-compact complex manifold of the form $\tilde{X}=X- \mathcal{D}$, where $X$ is a complex compact manifold and $\mathcal{D}$ is a normal crossing divisor on $X$. As applications, we provide a Poincar\'e-Hopf type Theorem and an optimal description for a smooth hypersurface $\mathcal{D}$ invariant by an one-dimensional foliation $\mathscr F$ on $\mathbb{P}^n$ satisfying $Sing(\mathscr F) \subsetneq \mathcal{D}.$, Comment: 20 pages, to appear in Transactions of the American Mathematical Society. Major changes following suggestions of the referees

In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this context ., Comment: 22 pages, To appear in Advances in Mathematics

Let $(X, \mathcal{F})$ be a foliated surface and $G$ a finite group of automorphisms of $X$ that preserves $\mathcal{F}$. We investigate invariant loci for $G$ and obtain upper bounds for its order that depends polynomially on the Chern numbers of $X$ and $\mathcal{F}$. As a consequence, we estimate the order of the automorphism group of some foliations under mild restrictions. We obtain an optimal bound for foliations on the projective plane which is attained by the automorphism groups of the Jouanolou's foliations., Comment: To appear in Revista Matem\'atica Iberoamericana

In this work we prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues., Comment: To appear in Comptes Rendus Mathematique

We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular holomorphic distributions on Hopf manifolds., Comment: 13 pages, to appear Mathematische Annalen

Let $(\mathcal{E}, \phi)$ be a rank two co-Higgs vector bundles on a K\"ahler compact surface $X$ with $\phi\in H^0(X,End(\mathcal{E})\otimes T_X)$ nilpotent. If $(\mathcal{E}, \phi)$ is semi-stable, then one of the following holds up to finite \' etale cover: $i)$ $X$ is uniruled. $ii)$ $X$ is a torus and $(\mathcal{E}, \phi)$ is strictly semi-stable. $iii)$ $X$ is a properly elliptic surface and $(\mathcal{E}, \phi)$ is strictly semi-stable., Comment: Major changes, following suggestions of the referee

We prove residual formulas for vector fields defined on compact complex orbifolds with isolated singularities and give some applications of these on weighted projective spaces., Comment: To appear in International Mathematics Research Notices. To J. Seade on the occasion of his 60th birthday

In this work we study some problems related with algebraic hypersurfaces invariant by foliations on weighted projective spaces $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ generalizing some results known for $\p$, as for example: the number of singularities, with multiplicities, contained in the invariant quasi-smooth hypersurfaces; Poincare problem on weighted projective plane and the number of the hypersurfaces, of a degree fixed, invariant by a foliation on $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ which does not admit a rational first integral.

In this paper we consider the question of bounding the degree of an divisor $D$ invariant by a $\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety $X$ in terms of degree of $\F$ and some invariants of $D$ and $X$. Particularly, if $\F$ is a foliation of degree $d$ on $\mathbb{P}_{\mathbb{C}}^2$, whose the number of invariants curves is greater that ${k+2\choose k}$, we show that there exist a number $\mathcal{M}(d,k)$ such that if $k>\mathcal{M}(d,k),$ then $\F$ admits a rational first integral of degree $\leq k$. Moreover, there exist a number $\mathscr{G}(d,k)$, such that if $\F$ has an algebraic solution of degree $k$ and genus smaller than $\mathscr{G}(d,k)$, then it has a rational first integral of degree $\leq k$.