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Maurício Corrêa

Publications and source records attributed to Maurício Corrêa.

At least 37 records · Page 2Linked to original sources
arXiv2026

Holomorphic foliations of degree two and arbitrary dimension

We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a degree-$2$ foliation by curves on $\mathbb{P}^{n-k+1}$, or a logarithmic foliation of type $(1^{n-k+1},2)$, or a logarithmic foliation of type $(1^{n-k+3})$, or 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 $\mathfrak{aff}(\mathbb{C})$. Meanwhile, we prove that any $2$-dimensional foliation tangent to a global vector field must satisfy that its tangent sheaf is either not locally free or has a direct summand isomorphic to $\mathcal{O}_{\mathbb{P}^{n}}(a)$, with $a\in\{0,1\}$. As a byproduct of our classification, we describe the geometry of Poisson structures on $\mathbb{P}^{4}$ with generic rank two.

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arXiv2025

A higher-dimensional Van den Essen type formula for projective foliations and applications

Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ μ(F, W)$ under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of $W$. This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on $\mathbb{P}^n$ that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve.

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arXiv2024

Global residue formula for logarithmic indices of foliations

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 Poincaré-Hopf type formula for singular normal projective varieties.

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arXiv2024

Dimension two holomorphic distributions on four-dimensional projective space

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.

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arXiv2023

Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

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.

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arXiv2022

Moduli of Distributions via Singular Schemes

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 $\mathbb{P}^3$.

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arXiv2021

Families and unfoldings of singular holomorphic Lie Algebroids

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.

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arXiv2021

Analytic varieties invariant by foliations and Pfaff systems

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, Poincaré and Painlevé on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincaré and Painlevé 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.

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arXiv2020

Classification of holomorphic Pfaff systems on Hopf manifolds

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.

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arXiv2020

On Gauss-Bonnet and Poincaré-Hopf type theorems for complex $\partial$-manifolds

We prove a Gauss-Bonnet and Poincaré-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.

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arXiv2020

Rational Morita equivalence for holomorphic Poisson modules

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.

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arXiv2020

GSV-index for holomorphic Pfaff Systems

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 Poincaré problem for Pfaff systems on projectives spaces.

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arXiv2020

Codimension one distributions and stable rank 2 reflexive sheaves on threefolds

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.

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arXiv2018

Codimension one holomorphic distributions on the projective three-space

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.

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arXiv2018

Residue Formulas for logarithmic Foliations and applications

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 Poincaré-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 $Sing(\mathscr F) \subsetneq \mathcal{D}.$

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