Isotrivial unfoldings and structural theorems for foliations on Projective spaces

Specialness and Isotriviality for Regular Algebraic Foliations

We show that an everywhere regular foliation $\mathcal F$ with compact canonically polarized leaves on a quasi-projective manifold $X$ has isotrivial family of leaves when the orbifold base of this family is special. By a recent work of Berndtsson, Paun and Wang, the same proof works in the case where the leaves have trivial canonical bundle. The specialness condition means that the $p$-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal Kodaira dimension $p$, for any $p>0$. This condition is satisfied, for example, in the ver…

Unfoldings and Deformations of Rational and Logarithmic Foliations

We study codimension one foliations in projective space \PP^n over \CC by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type. For a differential form ωdefining a codimension one foliation, we present a graded module \UU(ω), related to the first order unfoldings of ω. If ωis a generic form of rational or logarithmic type, as a first application of the construction of \UU(ω), we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in t…

导出叶状结构的横向展开

Transversal unfoldings of derived foliations

本文发展了相对导出叶状结构横向展开的CE-表示、同伦不变形式，将展开视为可积横向形变，通过横向控制器（基本零次混合导出的商）控制提升，并计算了严格仿射CE模型下的导出空间。

The Kupka Scheme and Unfoldings

Let $ω$ be a differential 1-form defining an algebraic foliation of codimension 1 in projective space. In this article we use commutative algebra to study the singular locus of $ω$ through its ideal of definition. Then, we expose the relation between the ideal defining the Kupka components of the singular set of $ω$ and the first order unfoldings of $ω$. Exploiting this relation, we show that the set of Kupka points of $ω$ is generically not empty. As an application of this results, we can compute the ideal of first order unfoldings for some known components of the space of foliations.

On the Camacho-Lins Neto regularity

We work with codimension one foliations in the projective space $\mathbb{P}^{n}$, given a differential one form $ω\in H^0(\mathbb{P}^n,Ω^1_{\mathbb{P}^n}(e))$, such differential form verifies the Frobenius integrability condition $ω\wedge dω=0$. In this work we show that the Camacho-Lins Neto regularity, applied for $ω$, is equivalent to the fact that every first order unfolding of $ω$ is trivial up to isomorphism. We do this by computing the Castelnuovo-Mumford regularity of the ideal $I(ω)$ of first order unfoldings. With this result, we are also showing that the only regular projective foli…

Rational pullbacks of toric foliations

This article is dedicated to the study of singular codimension $1$ foliations $\mathcal{F}$ on a simplicial complete toric variety $X$ and their pullbacks by dominant rational maps $φ:\mathbb{P}^n\dashrightarrow X$. First, we describe the singularities of $\mathcal{F}$ and $φ^*\mathcal{F}$ for a generic pair $(φ,\mathcal{F})$. Then we show that the first order deformations of $φ^*\mathcal{F}$ arising from first order unfoldings are the families of the form $φ_\varepsilon^*\mathcal{F}$, where $φ_\varepsilon$ is a perturbation of $φ$. We also prove that the deformations of the form $φ^*\mathcal{…