Abelian Relations are Liouvillian
We prove that every germ of an abelian relation of a codimension-one web on a complex projective manifold is defined over a Liouvillian extension of the function field.
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arXiv subjects
Jorge Vitório Pereira
Publications and source records attributed to Jorge Vitório Pereira.
The space of foliations on projective spaces in positive characteristic
This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces varies with the characteristic of the base field in very low degrees and establishes an arbitrary characteristic version of Calvo-Andrade's stability of generic logarithmic $1$-forms under deformation.
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Bondal's conjecture in dimension five
Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves--Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.
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Transversely projective structures on smooth foliations on surfaces
Brunella's classification implies that every smooth foliation on a compact complex surface admits a singular transversely projective structure. However, Biswas and Dumitrescu's recent work shows that certain foliations on compact complex surfaces, despite possessing a singular transversely projective structure, do not admit a regular transversely projective structure. Here, we describe the smooth foliations on compact complex surfaces which fail to possess a regular transversely projective structure.
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Unlikely intersections of codimension one foliations
We study families of singular holomorphic foliations on complex projective manifolds whose total intersection defines a foliation of unexpectedly low codimension.
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Numerically flat foliations and holomorphic Poisson geometry
We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact Kähler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact Kähler manifolds.
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Codimension one foliations in positive characteristic
We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds.
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Closed meromorphic 1-forms
We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities under the lenses of the theory of closed meromorphic $1$-forms and flat meromorphic connections. We apply the theory to investigate the algebraicity separatrices in a semi-global setting (neighborhood of a compact curve contained in the singular set of the foliation), and the geometry of smooth hypersurfaces with numerically trivial normal bundle on compact Kähler manifolds.
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Submanifolds with ample normal bundle
We construct germs of complex manifolds of dimension $m$ along projective submanifolds of dimension $n$ with ample normal bundle and without non-constant meromorphic functions whenever $m \geq 2n$. We also show that our methods do not allow the construction of similar examples when $m < 2n$ by establishing an algebraicity criterion for foliations on projective spaces which generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.
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Primitive Lie algebras of rational vector fields
A transitive Lie algebra g of rational vector fields on a projective manifold which do not preserve any foliation determines a rational map to an algebraic homogenous space G/H which maps g to lie(G).
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Codimension one foliations of degree three on projective spaces
We establish a structure theorem for degree three codimension one foliations on projective spaces of dimension $n\ge 3$, extending a result by Loray, Pereira, and Touzet for degree three foliations on $\mathbb P^3$. We show that the space of codimension one foliations of degree three on $\mathbb{P}^n$, $n\ge 3$, has exactly $18$ distinct irreducible components parameterizing foliations without rational first integrals, and at least $6$ distinct irreducible components parameterizing foliations with rational first integrals.
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Failure of the Brauer-Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
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A global Weinstein splitting theorem for holomorphic Poisson manifolds
We prove that if a compact Kähler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact Kähler manifolds with split tangent bundle.
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On the formal principle for curves on projective surfaces
We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.
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Toward Effective Liouvillian Integration
We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree bounded by a function of the degree of the foliation. We establish, for the same class of foliations, the existence of a bound for the degree of the simplest integrating factor depending only on the degree of the foliation and on the nature of its singularities. We also prove the existence of invariant algebraic curves of small degree for foliations with rational first integral and intermediate Kodaira dimension.
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Transversely affine foliations on projective manifolds
We describe the structure of singular transversely affine foliations of codimension one on projective manifolds X with zero first Betti number. Our result can be rephrased as a theorem on rank two reducible flat meromorphic connections.
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An invitation to web geometry
This book was intended to serve as supporting material for a mini-course on web geometry delivered at the 27th Brazilian Mathematical Colloquium which took place at IMPA in the last week of July 2009.
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On the degree of Polar Transformations -- An approach through Logarithmic Foliations
We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation associated to a homogeneous polynomial $F$ is determined by the zero locus of $F$. For zero dimensional-dimensional linear spaces this was conjecture by Dolgachev and proved by Dimca-Papadima using topological arguments. Our methods are algebro-geometric and rely on the study of the Gauss map of naturally associated logarithmic foliations.
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