---
abstract: 'Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name *persistent*, with the unfoldings of $\omega$, generalizing previous work done on foliations of codimension $1$ in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of $1$-forms defining the foliation. In the latter parts of the article we extend some of these results to toric varieties by making computations on the Cox ring and modules over this ring.'
author:
- 'César Massri[^1]'
- 'Ariel Molinuevo[^2]'
- 'Federico Quallbrunn$^*$'
title: Foliations with persistent singularities
---

Introduction {#introduction .unnumbered}
============

Overview of the subject and existing work {#overview-of-the-subject-and-existing-work .unnumbered}
-----------------------------------------

Foliations of arbitrary codimension over algebraic varieties have been considered at least since the seminal works of Malgrange [@malgrange; @malgrange2] in the local case, and Jouanolou [@jou] in a more global approach. Aside from the main result of [@malgrange2] and general definitions, most of the early theorems about foliations on projective algebraic varieties have been formulated for codimension $1$ foliations on the projective space ${\mathbb{P}}^n$. In those articles, codimension $q$ foliations were defined locally by $1$-forms $\omega_1,\dots,\omega_q$ satisfying Frobenius integrability equations: $d\omega_i\wedge\omega_1\wedge\dots\wedge\omega_q=0$ for $i=1,\dots,q$. Later, de Medeiros observed that this definition is not general enough for singular foliations of codimension $q$, as singular foliations by curves in dimension $n\geq 3$ cannot be given by $n-1$ forms even locally, see [@deMed1] and Example \[exampleCodimension2inA3\]. The correct definition is given by a $q$-form verifying the Plücker relations and Frobenius integrability (see below for definitions).

As for why many results were stated with ${\mathbb{P}}^n$ as ambient variety, notice that working in ${\mathbb{P}}^n$ allows the use of homogeneous coordinates and so one can define a codimension $1$ foliation with an integrable polynomial $1$-form $\omega=\sum_i f_i(x) dx_i$, that is a $1$-form verifying $\omega\wedge d\omega=0$ and $\sum_i x_i f_i(x)=0$. Such a setting can give concrete examples of foliations which may be hard to produce and study in more general contexts, see for instance the book [@omegarlibro].

Going beyond isolated examples there is the problem of establishing irreducible components for the space $\mathcal{F}ol(X)$ parameterizing all integrable forms on a variety $X$. The existence of this space for a projective variety $X$ follows from the existence of the $\mathrm{Quot}_X(\Omega^1_X)$ scheme and have been settled in [@quallbrunn]. In the case of codimension $1$ and $X={\mathbb{P}}^n$ there are several known examples of such components, the first known examples were established in [@jou] and up to the present is a very active research subject, in [@omegarlibro; @fj] one can find (non-exhaustive) lists of components. In the case of codimension $1$ and a general variety $X$ much less is known. In [@omegar] Calvo-Andrade proves that for a variety $X$ with $H^1(X,{\mathbb{C}})=0$ generic logarithmic $1$-forms give rise to integrable $1$-forms that are stable under small perturbations, *i.e.*: that there is an irreducible component of the space of integrable forms whose generic member defines a logarithmic foliation.

Besides stability, another important problem is the local and global characterization of the singularities of a foliation. Local results include the main theorems of [@malgrange] in codimension $1$ and of [@saito] and [@malgrange2] in higher codimension. Global studies have been made in the case of logarithmic foliations in [@fmi] and in the case of foliations defined by polynomial representations of affine lie algebras in [@ocgln] among others. An important type of singularity of a holomorphic foliation was discovered by Ivan Kupka in [@kupka]. A Kupka singularity for an integrable $1$-form $\omega$ is a point $p$ such that $\omega(p)=0$ and $d\omega(p)\neq 0$. Kupka showed that this type of singularity of codimension $1$ foliation is stable, meaning that if $\omega_t$ is a family of integrable $1$-forms parameterized by $t$ and $\omega_0$ has a Kupka singularity then $\omega_t$ also has a Kupka singularity for small enough $t$. Also if a foliation have a Kupka singularity then there is a codimension $2$ subvariety whose points are singular points of the foliation. Kupka singularities were generalized to arbitrary codimension by de Medeiros in [@deMedTesis], where stability for this singularities is proved in general. In codimension $q$ Kupka singularities come in subvarieties of codimension less or equal than $q+1$. In codimension $1$ there are many results relating the geometry of the variety of Kupka points with the global properties of the foliation, see *e.g.*: [@omegar-ivan; @omegar-kupka]. In higher codimension there is the work of Calvo-Andrade [@omegar2009].

A third subject we look upon in this work is the study of the unfoldings of a foliation. Unfoldings in the context of foliations were introduced independently by Suwa and Mattei in different contexts, see [@suwa-review] for a survey on the subject. Unfoldings of foliations where computed mostly in some codimension $1$ cases, locally by Suwa (see *loc. cit.*) and on ${\mathbb{P}}^n$ by Molinuevo in [@moli].

Recently we have related the study of unfoldings and singularities of a codimension $1$ foliation on ${\mathbb{P}}^n$. Indeed, in [@mmq] we define a homogeneous ideal $I(\omega)$ defining a subscheme of the singular scheme of $\omega$ (see below for precise definitions), the elements of degree equal to the degree of $\omega$ in $I(\omega)$ are in natural correspondence with the infinitesimal unfoldings of $\omega$. Under generic conditions we can prove that if $K(\omega)$ is the ideal defining the closure of the variety of Kupka points then $\sqrt{I(\omega)}=\sqrt{K(\omega)}$, using this result we were able to compute the unfoldings of foliations of codimension $1$ on ${\mathbb{P}}^n$ with split tangent sheaf and also prove the existence of Kupka points for every foliation in ${\mathbb{P}}^n$ with reduced singular scheme.

Main results {#main-results .unnumbered}
------------

Our aim in this article is to generalize previous results on the relation of unfoldings and singular points of a foliation to arbitrary codimension and to foliations on a non-singular projective variety. We also begin a study of this relation in toric varieties by making use of the Cox ring of the variety and modules over this ring. In codimension $1$ there is a direct relation between unfoldings and a certain type of singularities which we call persistent singularities. In this respect we prove Proposition \[JJcIIcKK\] relating Kupka and persistent singularities:

Let ${\mathscr{J}}$ be the ideal sheaf of the singular locus of $\omega$, ${\mathcal{K}}$ the ideal of the Kupka singularities of $\omega$ and ${\mathscr{I}}$ the ideal of persistent singularities. Then the following inclusions hold, $${\mathscr{J}}\subseteq {\mathscr{I}}\subseteq {\mathcal{K}}.$$

and Theorem \[teoKnotempty\] stating the existence of Kupka points under certain hypotheses:

Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals and such that $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$. Then $\omega$ has Kupka singularities.

In higher codimension the relation of persistent and Kupka singularities is not so clear, specially in the case where the foliation is not given locally by a complete intersection of $1$-forms, as in Example \[exampleCodimension2inA3\]. However, under suitable cohomological conditions the absence of persistent singularities impose very strong consequences on the foliation. If ${\mathcal{E}}$ is the sheaf of $1$-forms defining the foliation, and if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}}, \mathrm{Sym}^2{\mathcal{E}})=0$ then the absence of persistent singularities implies the existence of a connection on ${\mathcal{E}}$, see Theorem \[teoConnectionE\]:

Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega_X^q$ be an integrable $q$-form and ${\mathcal{E}}$ be the associated subsheaf of $1$-forms ${\mathcal{E}}\subseteq \Omega^1_X$. Let $\mathrm{Sym}^2({\mathcal{E}})$ denote the symmetric power of ${\mathcal{E}}$ and suppose ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=0$. If ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ then ${\mathcal{E}}$ admits a holomorphic connection, in particular is locally free and every Chern class of ${\mathcal{E}}$ vanishes.

In the case of toric varieties we can extend some of the results known for ${\mathbb{P}}^n$ by making use of the Cox ring, in this regard we obtain Theorem \[prop1notinI\]:

Let $X_\Sigma$ be a projective simplicial toric variety, $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ a codimension $q$ foliation and let $\mathcal{E}=\{\eta\in\widehat\Omega^1\,\colon\,\omega\wedge\eta=0\}$. Assume that $\mathcal{E}$ is locally free (*e.g.* $q=1$), $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element. Then $\omega$ has persistent singularities.

Furthermore, if instead ${\mathcal{E}}$ locally free, we require $\widehat\Omega^1/{\mathcal{E}}$ reflexive, the same conclusion holds.

Kupka scheme in the Projective space for codimension 1 foliations
=================================================================

Along this section we will revisit some definitions that we used in [@mmq], among them we will define the Kupka variety as a projective scheme ${\mathpzc{Kup}(\omega)}$ over ${\mathbb{P}}^n$ and $I=I(\omega)$ the *ideal of persistent singularities* (*a.k.a.* *unfoldings ideal*) of $\omega$. Then we will recall some results that we proved in *loc. cit.* that we will generalize later. The scheme ${\mathpzc{Kup}(\omega)}$ and the ideal $I$ were of central importance in those results. We refer the reader to [@mmq] for a full overview of this subjects.

With the exception of Theorem \[teodivision\] through this section we will restrict to the projective space ${\mathbb{P}}^n$. So let us denote $S={\mathbb{C}}[x_0,\dots,x_n]$ to the homogeneous coordinate ring of ${\mathbb{P}}^n$ and $\Omega^1_{{\mathbb{P}}^n}(e)$ the sheaf of twisted differential 1-forms in ${\mathbb{P}}^n$ of degree $e$. With ${\mathrm{Sing}(\omega)}_{set}$ we will denote the (set theoretic) singular set of $\omega\in H^0({\mathbb{P}}^n, \Omega^1_{{\mathbb{P}}^n}(e))$ in ${\mathbb{P}}^n$, $${\mathrm{Sing}(\omega)}_{set} = \{p\in {\mathbb{P}}^n: \omega(p) = 0 \}\ .$$

\[foliation\] Let ${\mathcal{L}}\simeq {\mathcal{O}}_{{\mathbb{P}}^n}(-e)$, $e\geq 2$, be a line bundle and $\omega:{\mathcal{L}}\to \Omega^1_{{\mathbb{P}}^n}$ be a morphism of sheaves, we will say that $\omega$ defines an *algebraic foliation of codimension 1* on ${\mathbb{P}}^n$, if $\Omega^1_{{\mathbb{P}}^n}/{\mathcal{L}}$ is torsion free and the morfism is generated by a non zero global section $\omega\in H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e))$ such that $\omega\wedge d\omega = 0$. We recall that such foliations have *geometric degree* $e-2$.

The condition of $\Omega^1_{{\mathbb{P}}^n}/{\mathcal{L}}$ to be torsion free in the definition of a foliation is equivalent to ask the singular set to have codimension greater than 2. Indeed, this is the same to ask that $\omega$ is not of the form $f.\omega'$, for some global section $f\in H^0({\mathbb{P}}^n,{\mathcal{O}}_{{\mathbb{P}}^n}(d))$ and a 1-form $\omega'\in H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e-d))$. Also, integrable differential 1-forms define the same foliation up to scalar multiplication. Then, we will denote the set of codimension 1 foliations of geometric degree $e-2$ as $$\label{torsion}
{\mathcal{F}}^1({\mathbb{P}}^n,e) := \left\{\omega\in{\mathbb{P}}\left(H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e))\right):\ \omega\wedge d\omega=0,\ {\mathrm{codim}}({\mathrm{Sing}(\omega)}_{set})\geq 2 \right\}.$$

\[IJ\] We define the graded ideals of $S$ associated to $\omega$ as $$\begin{aligned}
I(\omega) &:= \left\{ h\in S:\  h\ d\omega = \omega\wedge\eta\text{ for some } \eta\in\Omega^1_S\right\}\\
J(\omega) &:= \left\{ i_X(\omega)\in S:\ X\in T_S \right\}.\end{aligned}$$ We will name $I(\omega)$ the *ideal of persistent singularities* of $\omega$. We will also denote them $I=I(\omega)$ and $J=J(\omega)$ if no confusion arises.

\[1notinI\] Notice that $1\not\in I$, since the class of $d\omega$ in the Koszul complex of $\omega$, $H^2(\omega)$ is not zero, see Definition \[koszulcomplex\]. Also $J(\omega)$ equals the ideal defining the singular locus of $\omega$. This last thing, can be seen by contracting with the vector fields ${\partial}/{\partial x_i}$. The definition given for $J(\omega)$ is better suited for our schematic approach that we will develop next.

For $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$, we define the *Kupka set* as the subset of the singular set $$\mathpzc{K}_{set} = \overline{\{p\in {\mathrm{Sing}(\omega)}_{set} :  d\omega(p)\neq 0 \}}\ .$$

Notice that the definition above it is not the standard definition of the Kupka set. Usually it is defined just as the set of points in ${\mathrm{Sing}(\omega)}_{set}$ such that $d\omega(p)\neq 0$. Instead, we consider the closure of that set.

\[KK\] For $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$, we define the *Kupka scheme* ${\mathpzc{Kup}(\omega)}$ as the scheme theoretic support of $d\omega$ at $\Omega^2_{S}\otimes_S 
(S\big/J)$. Then, ${\mathpzc{Kup}(\omega)}={\mathrm{Proj}}(S/K(\omega))$ where $K(\omega)$ is the homogeneous ideal defined as $$K(\omega)={\mathrm{ann}}(\overline{d\omega})+J(\omega)\subseteq S,\quad \overline{d\omega}\in \Omega^2_{S}\otimes_S \left(S\big/J(\omega)\right).$$ We will denote $K=K(\omega)$ if no confusion arises.

We recall the notion of *ideal quotient* of two $S$-modules $M$ and $N$ as $$(N:M) := \left\{a\in S: a.M\subseteq N\right\},$$ then, one could also define $K(\omega)$ as $K(\omega)=(J\cdot \Omega^2_S: d\omega)$. Also, given that $\Omega^2_S$ is free, we can also write $$\label{Kbis}
K(\omega)=(J(\omega):J(d\omega)),$$ where $J(d\omega)$ denotes the ideal generated by the polynomial coefficients of $d\omega$.

From the properties of ideal quotient, it follows that if $J$ is radical, then $K$ is radical as well. With the Example 4.5 in [@mmq]\[p. 1034\] we showed that the algebraic geometric approach is indeed necessary, since the reduced structure associated to the Kupka scheme ${\mathcal{K}}$ differs from the reduced variety associated to $\mathpzc{K}_{set}$. With the following lemma we show that the Kupka scheme and the Kupka set coincide when the singular locus it is radical.

([@mmq]\[Lemma 4.6, p.1034\])\[K=Kset\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$ such that $J=\sqrt{J}$. Then $${\mathpzc{Kup}(\omega)}= \mathpzc{K}_{set}.$$

We have the following chain of inclusions, see Proposition \[JJcIIcKK\] and Proposition \[JJcIIcKK2\] for a generalization, in the codimension one and codimension $q$ case, respectively:

([@mmq]\[Proposition 4.7, p. 1035\])\[incl2\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$. Then, we have the following relations $$J\subseteq I\subseteq K\ .$$

Let $\mathfrak{p}$ be a point in ${\mathbb{P}}^n$, *e.g.*, an homogeneous prime ideal in $S$ different from the *irrelevant ideal* $(x_0.\ldots,x_n)$, and let $\omega$ be an integrable differential 1-form. We will denote with a subscript $\mathfrak{p}$ the localization at the point $\mathfrak{p}$ and with $\widehat{S}_\mathfrak{p}$ the completion of the local ring $S_\mathfrak{p}$ with respect to the maximal ideal defined by $\mathfrak{p}$.

We say that $\mathfrak{p}\in{\mathbb{P}}^n$ is a *division point of $\omega$* if $1\in I(\omega)_\mathfrak{p}$.

We now define a subset of the moduli space of foliations on which we are going to state our next result.

\[generic\] We define the set ${\mathcal{U}}\subseteq{\mathcal{F}}^1({\mathbb{P}}^n,e)$ as $${\mathcal{U}}= \left\{\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e): \ \forall \mathfrak{p}\not\in{\mathpzc{Kup}(\omega)},\,\mathfrak{p}\text{ is a division point of }\omega\right\}.$$

See Theorem \[propP=K\] for a generalization of the following:

([@mmq]\[Theorem 4.12, p. 1036\])\[teo1\] Let $\omega\in\mathcal{U}\subseteq{\mathcal{F}}^1({\mathbb{P}}^n,e)$. Then, $$\sqrt{I}=\sqrt{K}.$$ Furthermore, if $\sqrt{I}=\sqrt{K}$ then $\omega\in{\mathcal{U}}$.

See Theorem \[teoKnotempty\] for a generalization of the following:

([@mmq]\[Theorem 4.24, p. 1041\])\[teo3\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$ such that $J=\sqrt{J}$. Then $${\mathpzc{Kup}(\omega)}=\mathpzc{K}_{set}\neq \emptyset.$$

The following statement is valid in a non-singular variety $X$ and we will use it later. We will consider a $1$-form $\omega$ on $X$ with singular set of codimension equal or greater than 2. And we will denote with $\mathcal{J}$ the ideal sheaf of ${\mathrm{Sing}(\omega)}$.

([@mmq]\[Theorem 2.7, p. 1030\])\[teodivision\] Let $\omega$ be an integrable $1$-form in a non-singular variety $X$ and let ${\mathfrak{p}}\in{\mathrm{Sing}(\omega)}$ be such that ${\mathscr{J}}_{\mathfrak{p}}$ is radical and such that $d\omega_{\mathfrak{p}}\in {\mathscr{J}}_{\mathfrak{p}}\cdot \Omega^2_{X,{\mathfrak{p}}}$. Then there is a formal $1$-form $\eta$ such that $d\omega=\omega\wedge\eta$.

Unfoldings over schemes {#unf-schemes}
=======================

Along this section we will give the definition of *codimension $q$ foliation* on a smooth variety $X$. Then we will redefine the *singular locus* with a scheme theoretic approach. Finally we define an *unfolding* of a codimension $q$ foliation.

For the rest of the article, until Section \[section\_toric\], let us consider $X$ as a non-singular projective variety unless stated otherwise.

If $\Xi \in \Gamma(U, \bigwedge^p TX)$ is a multivector and $\varpi\in \Gamma(U,\Omega^q_X)$ a $q$-form we will denote by $i_\Xi \varpi \in \Gamma(U,\Omega^{q-p}_X)$ the contraction. Recall that the *Plücker relations* for $\varpi$ are given by $$i_\Xi \varpi\wedge \varpi=0$$ for any $\Xi\in\bigwedge^{q-1} TX$.

When $\varpi({\mathfrak{p}})\neq 0$ for some closed point ${\mathfrak{p}}\in X$ then $\varpi$ is *locally decomposable* as a product $\varpi=\varpi_1\wedge\dots\wedge\varpi_q$ of $q$ $1$-forms.

Let ${\mathcal{L}}$ be a line bundle and $\omega:{\mathcal{L}}\to \Omega^q_X$ be a morphism of sheaves, we will say that the morphism is *integrable* if

-   $\Omega^q_X/{\mathcal{L}}$ is torsion free.

-   The map $$i_\Xi \omega\wedge \omega:{\mathcal{L}}\to\Omega^{q+1}_X\otimes{\mathcal{L}}^{-1}$$ is zero for every local section $\Xi$ of $\bigwedge^{q-1} TX$.

-   For every local section $s$ of ${\mathcal{L}}$ and $\Xi$ of $\bigwedge^{q-1} TX$, $\omega(s)$ verifies $$\label{frobenius2}
     d (i_\Xi \omega(s))\wedge \omega(s)=0.$$

We also say that $\omega$ determines a *codimension $q$ foliation*.

By using Equation (\[frobenius2\]) with $q=1$ we recover the definition of codimension one foliation as in Definition \[foliation\].

The fact that $\omega(s)$ is locally decomposable as a product of $q$ $1$-forms $\varpi_1,\dots,\varpi_q$ implies that there exist a rank $q$ vector bundle $\mathcal{E}\hookrightarrow \Omega^1_X$, locally generated by $\omega_1(s),\dots,\omega_q(s)$ and such that ${\mathcal{L}}\simeq \bigwedge^q\mathcal{E}$. Reciprocally, given a locally free sheaf of rank $q$, $\mathcal{E}$ and a map $\mathcal{E}\hookrightarrow \Omega^1_X$, we have that $\bigwedge^q\mathcal{E}$ is a line bundle ${\mathcal{L}}$ and a map ${\mathcal{L}}\to\Omega^q_X$. The condition that $\Omega^q_X/{\mathcal{L}}$ is torsion free is equivalent to $\Omega^1_X/\mathcal{E}$ being torsion free. Example \[exampleCodimension2inA3\] shows that the condition *locally free* is necessary for this equivalence.

Let $\omega:{\mathcal{L}}\to\Omega^q$ be a integrable $q$-form. Then, we can consider two maps, $$\xymatrix{
\bigwedge^{q-1} TX\otimes {\mathcal{L}}\ar[r]^<<<<<{i_{(\cdot)}\omega}
&\Omega^1_X\ar[r]^<<<<<{(\cdot)\wedge\omega}&\Omega_X^{q+1}\otimes{\mathcal{L}}^{-1}
}$$ The integrability condition on $\omega$ implies that this diagram is a complex and it is easy to check that its homology is supported over the points where $\omega$ is not decomposable. We define the sheaf associated to $\omega$, denoted ${\mathcal{E}}={\mathcal{E}}(\omega)$, as the kernel of $(\cdot)\wedge\omega$. By definition, ${\mathcal{E}}$ is a reflexive sheaf.

\[exampleCodimension2inA3\] Let $X=\mathbb{A}^3$ or, in the holomorphic case, a polydisk of dimension $3$. We take $v\in \Gamma(X,TX)$ a vector field, generic in the sense that in a coordinate system $(x_1,x_2,x_3)$ we can write $v=f_1\frac{\partial}{\partial x_1}-f_2\frac{\partial}{\partial x_2}+f_3\frac{\partial}{\partial x_3}$ with $f_1, f_2, f_3 \in k[x_1,x_2,x_3]$ and such that the ideal $(f_1, f_2,f_3)\subseteq k[x_1,x_2,x_3]$ is a complete intersection, that is, there are no nontrivial relations among the $f_i$’s.

The vector field $v$ generates a codimension $2$ foliation in $X$, this foliation is determined by a $2$-form $\omega$ such that $i_v\omega=0$. One such $\omega$ is given by $$\omega= f_3 dx_1\wedge dx_2 + f_2 dx_1\wedge dx_3 + f_1 dx_2\wedge dx_3.$$ It can be verified that this $\omega$ satisfies Plücker relations, is integrable, $i_v\omega=0$ and that $\Omega^2_X/(\omega)$ is torsion free. Therefore $\omega$ determines the same foliation of codimension $2$ as $v$. If we now look at the $1$-forms annihilated by $v$ we get the subsheaf generated by the forms $$\omega_1= f_3 dx_2+f_2dx_3,\quad \omega_2=f_3 dx_1- f_1 dx_3\ \text{ and } 
\omega_3=f_2 dx_1 + f_1 dx_2.$$ These generators satisfy the relation $f_1\omega_1 + f_2 \omega_2= f_3\omega_3$. The subsheaf $\mathcal{E}=(\omega_1,\omega_2,\omega_3)$ is generically of rank $2$ outside the zeros of the ideal $(f_1,f_2,f_3)$ but $\mathcal{E}\otimes k({\mathfrak{p}})$ is of rank $3$ when ${\mathfrak{p}}$ is in the zeros of this ideal. Therefore $\mathcal{E}$ is *not* locally free. Moreover when we compute the determinant of $\mathcal{E}$ we get $\wedge^2\mathcal{E}= (f_1,f_2,f_3)\cdot(\omega)\subseteq \Omega^2_X$, $$\omega_1\wedge\omega_2=f_3\omega,\quad
\omega_3\wedge\omega_1=f_2\omega,\quad
\omega_2\wedge\omega_3=f_1\omega.$$ In particular $\omega$ is not in $\wedge^2\mathcal{E}$. But by [@GH Lemma, p. 210], if $\omega$ is locally decomposable, then $\omega\in \wedge^2\mathcal{E}$. Then $\omega$ is *not* locally decomposable around the zeros of the ideal $(f_1,f_2,f_3)$.

Composing a morphism $\omega:{\mathcal{L}}\to \Omega^q$ with the contraction of forms with vector fields give us a morphism $$\bigwedge^q TX\otimes {\mathcal{L}}\to{\mathcal{O}}_X.$$

The ideal sheaf ${\mathscr{J}}(\omega)$ is defined to be the sheaf-theoretic image of the morphism $\bigwedge^q TX\otimes {\mathcal{L}}\to{\mathcal{O}}_X$. The subscheme it defines is called *the singular set* of $\omega$ and denoted ${\mathrm{Sing}(\omega)}\subseteq X$. We will denote it just as ${\mathscr{J}}$ if no confusion arises.

This definition agrees with Remark \[1notinI\], where we said that the ideal $J(\omega)$ gives the ideal defining the singular locus of $\omega$.

From [@suwa-review]\[(4.6) Definition, p. 192\] we get the following definition for a codimension $q$ foliation:

Let $S$ be a scheme, $p\in S$ a closed point, and ${\mathcal{L}}\xrightarrow{\omega} \Omega^q_X$ a codimension $q$ foliation on $X$. An *unfolding* of $\omega$ is a codimension $q$ foliation $\widetilde{{\mathcal{L}}}\xrightarrow{\widetilde{\omega}}\Omega^q_{X\times S}$ on $X\times S$ such that $\widetilde{\omega}|_{X\times\{p\}}\cong \omega$. In the case $S=\mathrm{Spec}(k[x]/(x^2))$ we will call $\widetilde{\omega}$ a *first order infinitesimal unfolding*.

Kupka scheme in general for codimension 1 foliations
====================================================

Over this section we will restate the definition of *persistent singularities* and of the *Kupka scheme*, through its ideal sheaf, in a more general setting, see Definition \[II2\] and Definition \[KK2\], respectively. In [@mmq] we showed that persistent singularities are related to unfoldings in codimension one. We want to extend this relation to higher codimension.

First we prove Proposition \[JJcIIcKK\], generalizing Proposition \[incl2\] in the codimension one case. Then we define the Kupka scheme and we prove Theorem \[propP=K\] and Theorem \[teoKnotempty\], generalizing Theorems \[teo1\] and \[teo3\].

Given a line bundle ${\mathcal{L}}$ and a global section $\omega\in H^0\left(X,\Omega^1_X\otimes {\mathcal{L}}^{-1}\right)$ we will consider the Koszul complex associated with $\omega$, $$\label{koszulcomplex}
\xymatrix{K(\omega): & {\mathcal{O}}_X\ar[r]^-{\wedge\omega}& \Omega^1_X\otimes{\mathcal{L}}^{-1} \ar[r]^-{\wedge\omega} & \dots\ar[r] & \Omega^i\otimes{\mathcal{L}}^{-i}\ar[r]& \dots}$$ where we are following [@GKZ Chapter 2, B, p. 51] and using the identification $\bigwedge^k\left(\Omega^1_X\otimes {\mathcal{L}}^{-1}\right)\simeq \left(\bigwedge^k \Omega^1_X \right)\otimes \left({\mathcal{L}}^{-k}\right)$. We will denote the cohomology sheaves of this complex by $H^\bullet(\omega)$, the *Koszul cohomology sheaves* of $\omega$.

We can use $K(\omega)$ to compute the codimension of ${\mathrm{Sing}(\omega)}$ by the well known result, see [@eisenbud Theorem 17.4, p. 424]:

\[koszul\] Let $\omega\in H^0\left(X,\Omega^1_X\otimes {\mathcal{L}}^{-1}\right)$. The following statements are equivalent:

-   $codim({\mathrm{Sing}(\omega)})\geq k$

-   $H^\ell(\omega)=0$ for all $\ell <k$

\[notation\] Suppose now the morphism $\omega:{\mathcal{L}}\to \Omega^1_X$ defines a foliation on $X$. Given a trivializing open set $U$ and a choice of a trivialization ${\mathcal{O}}_X|_U\cong {\mathcal{L}}|_U$, we take a local generator $\varpi$ of ${\mathcal{L}}(U)$ (we think about it as a $1$-form through the morphism ${\mathcal{L}}\to \Omega^1_X$) and take the differential $d \varpi$. This defines a ${\mathbb{C}}$-linear morphism ${\mathcal{L}}(U)\to \Omega^2_X(U)$, which in turn we can compose with the projection $\Omega^2_X\to H^2(\omega)\otimes {\mathcal{L}}^{\otimes 2}$. Note that the submodule ${\mathcal{O}}_X(U)\cdot (d\varpi)$ of $\left(H^2(\omega)\otimes{\mathcal{L}}^{\otimes 2}\right)(U)$ is independent of the choice of the trivialization. In this way one gets a morphism of coherent sheaves, $${\mathcal{L}}\to H^2(\omega)\otimes{\mathcal{L}}^{\otimes 2}.$$ Or, equivalently, a (non trivial) global section of $H^2(\omega)\otimes{\mathcal{L}}$. We will denote the global section or the morphism indistinctly by $[d\omega]$. By Theorem \[koszul\] above, we conclude that $codim({\mathrm{Sing}(\omega)})\geq 2$.

\[II2\] The subscheme of *persistent singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathscr{I}}(\omega):={\mathrm{ann}}([d\omega])$, for $[d\omega]\in H^0\left(X,H^2(\omega)\otimes{\mathcal{L}}\right)$. We will denote it just as ${\mathscr{I}}$ if no confusion arises.

\[remarkI\] Let $\varpi\in \Omega^1_X(U)$ be a local generator of the image of ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$, then the local sections of ${\mathscr{I}}(\omega)$ in $U$ are given by $${\mathscr{I}}(U)=\{h\in {\mathcal{O}}_X(U): \text{there is a section } \eta\in\Gamma(U,\Omega^1_X/{\mathcal{L}})\text{ s.t. }  hd \varpi=\varpi\wedge\eta\}.$$

\[formalremark\] For a regular local ring $(R,\mathfrak{m})$, an $R$-module $M$ and an element $m\in M$, let us denote $\widehat{R}$ the $\mathfrak{m}$-adic completion of $R$ and $\widehat{M}=M\otimes\widehat{R}$. The element $m\otimes 1\in\widehat{M}$ has as annihilator the ideal ${\mathrm{ann}}(m)\otimes \widehat{R}$. Setting $R={\mathcal{O}}_{X,{\mathfrak{p}}}$, $M= \left(H^2(\omega)\otimes{\mathcal{L}}\right)_{\mathfrak{p}}$ and $m=[d\omega]_{\mathfrak{p}}$, and following the notation of Remark \[notation\], we have that $${\mathrm{ann}}([d\omega]_{\mathfrak{p}}\otimes 1)=\{h\in \widehat{{\mathcal{O}}_{X,{\mathfrak{p}}}}: \text{there is a formal $1$-form } \eta \ \text{ s.t. } hd \varpi=\varpi\wedge\eta\}.$$

Let ${\mathfrak{p}}\in X$ be a point in ${\mathrm{Sing}(\omega)}$, ${\mathcal{O}}_{X,{\mathfrak{p}}}$ the local ring around ${\mathfrak{p}}$, and $X_{\mathfrak{p}}=\mathrm{Spec}({\mathcal{O}}_{X,{\mathfrak{p}}})$. Then ${\mathfrak{p}}$ is in the subscheme of persistent singularities if and only if for any infinitesimal first order unfolding $\widetilde{\omega}$ of $\omega$ in $X_{\mathfrak{p}}$, the point $({\mathfrak{p}}, 0)\in X_{\mathfrak{p}}\times \mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$ is a singular point of $\widetilde{\omega}$.

Let $S=\mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$, $0\in S$ be its closed point, $p:X\times S\to S$ be the projection and $\iota:X\cong X\times\{0\}\hookrightarrow X\times S$ be the inclusion. Then the sheaf $\Omega^1_{X\times S}$ can be decomposed as direct sum of $\iota_*({\mathcal{O}}_X)$-modules as $$\Omega^1_{X\times S}\cong \iota_*(\Omega^1_X)\oplus \epsilon\cdot \iota_*(\Omega^1_X)\oplus \iota_*({\mathcal{O}}_X)d\epsilon.$$ A point ${\mathfrak{p}}\in{\mathrm{Sing}(\omega)}$ is *not* a persistent singularity if and only if $1\in {\mathscr{I}}_{\mathfrak{p}}\subseteq {\mathcal{O}}_{X,{\mathfrak{p}}}$ which, by Remark \[remarkI\], means that there is an open neighborhood $U\subseteq X$ of ${\mathfrak{p}}$, a local generator $\varpi$ of the image of ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$, and a section $\eta\in\Gamma(U,\Omega^1_X/{\mathcal{L}})$ such that $d\omega=\omega\wedge \eta$. By shrinking $U$ if necessary we can take a lifting of $\eta$ in $\Omega^1_X$ which by abuse of notation we also call $\eta$ and define $$\widetilde{\omega}= \omega + \varepsilon \eta + d\varepsilon.$$ Thus $\widetilde{\omega}$ is a form in $\Omega^1_{X\times S}$ and $\widetilde{\omega}({\mathfrak{p}},0)=
d\varepsilon\neq 0$, so ${\mathfrak{p}}\times \{0\}$ is *not* a singular point of $\widetilde{\omega}$. Reciprocally, if there is an unfolding $\widetilde{\omega}$ of $\omega|_U$, then $$\widetilde{\omega}({\mathfrak{p}}, 0)=\omega({\mathfrak{p}}) + h(0) d\varepsilon.$$ As ${\mathfrak{p}}$ is a singular point of $\omega$, we have $\omega({\mathfrak{p}})$, so if $({\mathfrak{p}},0)$ is not a singular point of $\widetilde{\omega}$, then $h(0)\neq 0$, so again shrinking $U$ if necessary we have that $h$ is a unit, hence $1\in {\mathscr{I}}_{\mathfrak{p}}$.

Most of the known families of foliations on algebraic varieties present persistent singularities, see [@gmln; @omegar; @celn; @pullback; @fji; @fmi; @fj; @mmq]. As it happens the absence of persistent singularities impose some restrictions on the line bundle ${\mathcal{L}}$. To explain this we have to make explicit use of a result that is implied in the proof of Lefschetz Theorem on $(1,1)$ classes as is proved in [@GH Chapter 1.1 p.:  141].

Let ${\mathcal{L}}$ be a line bundle. Choose a trivialization $(U_i, \phi_i)_{i\in I}$ of ${\mathcal{L}}$ with gluing data $g_{ij}\in {\mathcal{O}}^*_X(U_{ij})$. The Čech cocycle $\frac{1}{2\pi i }[d\log g_{ij}]\in Z^1(\Omega^1_X)$ represents the Chern class $c_1({\mathcal{L}})$ of ${\mathcal{L}}$ in $H^1(X,\Omega^1_X)$.

The claim follows from a careful reading of the proof of the Proposition in page 141 of [@GH Chern classes of line bundles, Chapter 1.1, p. 141], as we will show next. There is shown that de Rham’s Theorem for $\mathcal{C}^\infty$-forms with complex coefficients give us exact sequence of sheaves $$0\to {\mathbb{C}}\to \mathcal{A}^0\to \mathcal{Z}^1_d\to 0,\qquad 0\to\mathcal{Z}^1_d\to\mathcal{A}^1\to \mathcal{Z}^2_d\to 0,$$ where $\mathcal{A}^i$ are $\mathcal{C}^\infty$ $i$-forms with complex coefficients and $\mathcal{Z}^i_d$ are the cycles of the de Rham complex. These exact sequences give us boundary isomorphisms $$\frac{H^0(\mathcal{Z}^2)}{dH^0(\mathcal{A}^1)}\xrightarrow{\delta_1}H^1(\mathcal{Z}^1_d),\qquad
H^1(\mathcal{Z}^1_d) \xrightarrow{\delta_2} H^2({\mathbb{C}}).$$ As explained in *loc. cit.* the multiple of the Chern class $-(2\pi i)c_1({\mathcal{L}})$ can be calculated as $\delta_2\delta_1(\Theta)$, where $\Theta$ is the curvature form of a connection on ${\mathcal{L}}$. It also follows from *loc. cit.* that $\delta_1(\Theta)$ is represented by the cocycle $-d\log g_{ij}$.

\[propc1L=0\] Let $X$ be a smooth projective variety over ${\mathbb{C}}$. If ${\mathcal{L}}$ is a line bundle such that $H^1(X,{\mathcal{L}})=0$ and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ is a foliation without persistent singularities then $c_1({\mathcal{L}})=0$, where $c_1({\mathcal{L}})$ is the Chern class of the line bundle viewed in $H^2(X,{\mathbb{C}})$.

Let $(U_i,\phi_i)$ be a trivialization of ${\mathcal{L}}$ with gluing data $g_{ij}\in {\mathcal{O}}^*_X(U_{ij})$. On each $U_i$ we have a local generator of ${\mathcal{L}}(U_i)$, namely $\phi_i^{-1}(1)$, we denote by $\omega_i$ the image under $\omega$ of this generator. The fact that the foliation defined by $\omega$ has no persistent singularities means that on each $U_i$ there is a local section $\eta_i$ of $\Omega^1_X/{\mathcal{L}}(U_i)$ such that $d\omega_i=\omega_i\wedge\eta_i$. On $U_{ij}$ the restriction of the local $1$-form $\omega_i$ satisfies $$\omega_i=g_{ij}\omega_j.$$ So computing the de Rham differential of this forms on $U_{ij}$ gives us, $$\begin{aligned}
\omega_i\wedge\eta_i=d\omega_i=d(g_{ij}\omega_j)=\\
=g_{ij}d\omega_j+dg_{ij}\wedge \omega_j=\\
=g_{ij} \omega_j\wedge\eta_j + dg_{ij}\wedge \omega_j=\\
=g_{ij}\omega_j\wedge\left(\eta_j-\frac{dg_{ij}}{g_{ij}}\right).\end{aligned}$$ Subtracting both sides of the equality we get that, on $U_{ij}$, $$\eta_i-\eta_j=\frac{dg_{ij}}{g_{ij}},$$ as sections of $\Gamma(U_{ij},\Omega^1_X /{\mathcal{L}})$. Therefore we get a Čech cochain $(\eta_i)_{i\in I}$ of $C^0(\Omega^1_X/{\mathcal{L}})$ whose border is $$\partial (\eta)_{ij}=d\log g_{ij}\in B^1(\Omega^1/{\mathcal{L}}).$$ As the cocycle $(d\log g_{ij})\in Z^1(\Omega^1_X)$ represents $(2\pi i )c_1({\mathcal{L}})$, the existence of the cochain $(\eta_i)$ implies $c_1({\mathcal{L}})$ is in the kernel of the map $H^1(\Omega^1_X)\to H^1(\Omega^1_X/{\mathcal{L}})$ induced by the short exact sequence of sheaves $$0\to{\mathcal{L}}\xrightarrow{\omega}\Omega^1_X\to \Omega^1_X/{\mathcal{L}}\to 0.$$ The hypothesis $H^1({\mathcal{L}})=0$ then implies $c_1({\mathcal{L}})=0$.

Let $X$ be a smooth projective variety over ${\mathbb{C}}$ such that every line bundle ${\mathcal{L}}$ verifies $H^1(X,{\mathcal{L}})=0$ and such that $Pic(X)$ is torsion-free (*e.g.*: $X$ smooth complete intersection). Then every foliation on $X$ have persistent singularities.

From the exponential sequence and the hypothesis $H^1(X,{\mathcal{O}}_X)=0$ it follows that $c_1:Pic(X)\to H^2(X,{\mathbb{Z}})$ is injective. Assume that $\omega:{\mathcal{L}}\to\Omega^1_X$ is a foliation without persistent singularities. Then the above Proposition imply that $c_1({\mathcal{L}})$ is a torsion element in $H^2(X,{\mathbb{Z}})$. But given that $Pic(X)$ is torsion-free, we get ${\mathcal{L}}\cong{\mathcal{O}}_X$.

In particular, $\omega$ is a global differential $1$-form which contradicts the fact that $H^0(X,\Omega^1_X)=H^1(X,{\mathcal{O}}_X)=0$.

\[[domega]{}\] Given a trivializing open set $U$, a choice of a trivialization ${\mathcal{O}}_X|_U\cong {\mathcal{L}}|_U$ and a local generator $\varpi$ of ${\mathcal{L}}(U)$, the mapping $\varpi\mapsto d\varpi$ defines a ${\mathcal{O}}_X$-linear morphism ${\mathcal{L}}\to \Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}$. We will denote by $\{d\omega\}$ this morphism or equivalently the global section of $\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes{\mathcal{L}}^{-1}$ it defines.

\[KK2\]The subscheme of *Kupka singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathcal{K}}(\omega):={\mathrm{ann}}(\{d\omega\})\in\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes{\mathcal{L}}^{-1}$. We will denote it just as ${\mathcal{K}}$ if no confusion arises.

\[JJcIIcKK\] Let ${\mathscr{J}}$ be the ideal sheaf of the singular set of $\omega$, ${\mathcal{K}}$ the ideal of the Kupka singularities of $\omega$ and ${\mathscr{I}}$ the ideal of persistent singularities. Then the following inclusions hold, $${\mathscr{J}}\subseteq {\mathscr{I}}\subseteq {\mathcal{K}}.$$

Let $U\subseteq X$ be an open subscheme such that ${\mathcal{L}}|_U\simeq{\mathcal{O}}_X$, and $\varpi$ a local generator of ${\mathcal{L}}(U)$.

Suppose $h\in{\mathscr{J}}(U)\subseteq{\mathcal{O}}_X(U)$ is a local section. By shrinking $U$ if necessary we may assume that there is a vector field $v\in T_X(U)$ such that $h= i_v(\omega)$. Then we have $$0=i_v(\varpi\wedge d\varpi)=i_v(\varpi)d\varpi - \varpi \wedge i_v(d\varpi).$$ So, calling $\eta=i_v(d\varpi)$, we get $hd\varpi=\varpi\wedge \eta$. Hence $h$ is in ${\mathscr{I}}(U)$, which proves the first inclusion.

Now assume $h\in {\mathscr{I}}(U)$, then again by shrinking $U$ if necessary, we may assume that there is a $\eta\in \Omega^1_X/{\mathcal{L}}(U)$ such that $hd\varpi= \varpi\wedge\eta$. By definition we have $\varpi \in {\mathscr{J}}(U)\cdot \Omega^1_X(U)$, then $hd\varpi\in {\mathscr{J}}(U)\cdot \Omega^2_X(U)$ so $h$ is in the annihilator of $\{d\omega\}$ in $\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}$. Then $h\in {\mathcal{K}}(U)$, which proves the second inclusion.

With the following results we can generalize Theorem \[teo1\] and Theorem \[teo3\] giving conditions for the existence of Kupka singularities:

Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$, we are going to call ${\mathpzc{Per}(\omega)}\subseteq X$ the subschemes of persistent singularities.

\[propP=K\] Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals. Let ${\mathpzc{Per}(\omega)}\subseteq X$ and ${\mathpzc{Kup}(\omega)}\subseteq X$ be the subschemes of persistent and Kupka singularities respectively. Then ${\mathpzc{Per}(\omega)}_{\mbox{red}}={\mathpzc{Kup}(\omega)}_{\mbox{red}}$.

We are going to prove that $X\setminus{\mathpzc{Per}(\omega)}=X\setminus{\mathpzc{Kup}(\omega)}$. By Proposition \[JJcIIcKK\] we have ${\mathpzc{Kup}(\omega)}\subseteq{\mathpzc{Per}(\omega)}$, so $X\setminus {\mathpzc{Per}(\omega)}\subseteq X\setminus {\mathpzc{Kup}(\omega)}$. Now suppose ${\mathfrak{p}}$ is a point *not* in ${\mathpzc{Kup}(\omega)}$, by abuse of notation we will call $\omega$ a local generator of ${\mathcal{L}}_{\mathfrak{p}}$ viewed as a $1$-form. As ${\mathfrak{p}}$ is not in ${\mathpzc{Kup}(\omega)}$ then $d\omega \in {\mathscr{J}}_{\mathfrak{p}}\cdot\Omega^2_{X,{\mathfrak{p}}}$. By hypothesis ${\mathscr{J}}_{\mathfrak{p}}$ is radical and so by Theorem \[teodivision\] we have that $d\omega$ decomposes as $\omega\wedge\eta$ for some formal $1$-form $\eta$, this implies $1\in {\mathscr{I}}_{\mathfrak{p}}$, so ${\mathfrak{p}}$ is not in ${\mathpzc{Per}(\omega)}$ (see Remark \[formalremark\]).

\[teoKnotempty\] Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals and such that $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$. Then $\omega$ has Kupka singularities.

This follows from Proposition \[propc1L=0\] and Theorem \[propP=K\], as a foliation with $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$ has persistent singularities on one hand, and having radical singular ideal implies the reduced scheme defined by persistent singularities is equal to the reduced scheme of Kupka singularities, in particular this last scheme is not empty.

Infinitesimal unfoldings in codimension $q$
===========================================

Along this section we review the definition of unfolding of a codimension $q$ foliation on a variety $X$. We will also generalize the definitions of persistent singularities and of Kupka singularities for codimension $q$ foliations, see Definition \[defII\] and Definition \[defKK\], respectively. We classify which singular points of $\omega$ are such that they extend to singular points of every unfolding $\widetilde{\omega}$ (Proposition \[prop\]) and then, we generalize Proposition \[incl2\] and Proposition \[JJcIIcKK\] to the codimension $q$ case (Proposition \[JJcIIcKK2\]). Finally, with Theorem \[teoConnectionE\] we establish that the absence of persistent singularities implies the existence of a connection on ${\mathcal{E}}$, the sheaf of 1-forms defining the foliation.

Let $S=\mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$, $0\in S$ be its closed point, $p:X\times S\to S$ be the projection and $\iota:X\cong X\times\{0\}\hookrightarrow X\times S$ be the inclusion. Then the sheaf $\Omega^q_{X\times S}$ can be decomposed as direct sum of $\iota_*({\mathcal{O}}_X)$-modules as $$\Omega^q_{X\times S}\cong \iota_*\Omega^q_X\oplus \varepsilon\cdot (\iota_*\Omega^q_X)\oplus \iota_*\Omega^{q-1}_X\wedge d\varepsilon.$$ Given a codimension $q$ foliation determined by a morphism ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$, and a first order infinitesimal unfolding $\widetilde{\omega}: \widetilde{{\mathcal{L}}}\to \Omega^q_{X\times S}$ of $\omega$, we take local generators $\varpi$ of ${\mathcal{L}}(U)$ and $\widetilde{\varpi}$ of $\widetilde{{\mathcal{L}}}(U\times S)$. Suppose $\omega$ and $\widetilde{\omega}$ are locally decomposable, then we may take $U$ small enough such that $\varpi$ and $\widetilde{\varpi}$ decompose as products $$\varpi=\varpi_1\wedge\dots\wedge\varpi_q, \qquad \widetilde{\varpi}=\widetilde{\varpi}_1\wedge\dots\wedge \widetilde{\varpi}_q.$$ Then we can write $\widetilde{\varpi}_i=\varpi_i + \varepsilon \eta_i + h_i d\varepsilon$ and the equations $d\widetilde{\varpi}_i\wedge \widetilde{\varpi}=0$ for $i=1,\dots, q$ are equivalent to the equations

$$\left\{
\begin{aligned}
&d\eta_i\wedge \varpi+ d\varpi_i\wedge \left(\sum_{j=1}^q (-1)^j\eta_j \varpi_{\widehat{j}}\right)=0,\qquad (i=1,\dots,q), \\
&(dh_i-\eta_i)\wedge \varpi+ d\varpi_i\wedge \left(\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\right)=0,\qquad (i=1,\dots,q), 
\end{aligned}
\right.$$

where $ \varpi_{\widehat{j}}=\varpi_1\wedge\dots\wedge\varpi_{j-1}\wedge\varpi_{j+1}\wedge\dots\wedge\varpi_q\in \Omega^{q-1}_X(U)$.

As is shown in [@suwa-review proof of (6.1) Theorem, p. 199] the second equation implies the first. So we finally get that the equations $d\widetilde{\varpi}_i\wedge \widetilde{\varpi}=0$ for $i=1,\dots, q$ are equivalent to

$$\label{equnfcodq}
\begin{aligned}
 \left\{
 (dh_i-\eta_i)\wedge \varpi+ d\varpi_i\wedge \left(\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\right)=0,\qquad (i=1,\dots,q)\ .
\right. 
\end{aligned}$$

\[prop\] Suppose ${\mathfrak{p}}$ is a singular point of $\omega$. Then there exist an infinitesimal unfolding $\widetilde{\omega}$ of $\omega$ in $X_{\mathfrak{p}}$ such that $({\mathfrak{p}},0)$ is *not* a singular point of $\widetilde{\omega}$ if and only if $\omega$ is decomposable locally around ${\mathfrak{p}}$, not all $\varpi_{\widehat{j}}({\mathfrak{p}})$ vanish and there are $1$-forms $\alpha_{ij}$ such that $$d\varpi_i=\sum_{j=1}^q \alpha_{ij}\wedge \varpi_j,\qquad (i=1,\dots,q).$$

Given local forms $\alpha_{ij}\in \Omega^1_{X,{\mathfrak{p}}}$ such that $d\varpi_i=\sum_{j=1}^q \alpha_{ij}\wedge \varpi_j, (i=1,\dots,q)$ we may take local sections $h_i\in{\mathcal{O}}_{X,{\mathfrak{p}}}$ such that $\sum_{i=1}^q(-1)^ih_i({\mathfrak{p}})\varpi_{\widehat{i}}({\mathfrak{p}})\neq 0$. With that choice of $h_i$’s we take $\eta_i:=dh_i+\sum_{j=1}^q (-1)^j h_j\alpha_{ij}$. We will see that the $\eta_i$’s and $h_i$’s determine an unfolding of $\omega$ locally around ${\mathfrak{p}}$. For that we need to verify the Equation (\[equnfcodq\]) above. Indeed we have $$\begin{aligned}
  (dh_i-\eta_i)&\wedge\varpi+d\varpi_i\wedge\left(\sum_{j=1}^q (-1)^jh_j\varpi_{\widehat{j}}\right)=\\
 &= (dh_i-\eta_i)\wedge\varpi+\left( \sum_{k=1}^q\alpha_{ik}\wedge \varpi_k \right)\wedge\sum_{j=1}^q (-1)^j\varpi_{\widehat{j}}=\\
 &=(dh_i-\eta_i)\wedge\varpi + \left(\sum_{j,k=1}^1(-1)^j h_j\alpha_{ik}\wedge\varpi_k\wedge\varpi_{\widehat{j}}\right)=\\
 &=(dh_i-\eta_i)\wedge \varpi + \left(\sum_{j=1}^q (-1)^j\alpha_{ij}\wedge \varpi\right) =\\
 &=\left((dh_i-\eta_i)+\sum_{j=1}^q (-1)^j h_j\alpha_{ij} \right)\wedge \varpi .
 \end{aligned}$$ And from the definition of the $\eta_i$ we have that $$\begin{aligned}
 &\left((dh_i-\eta_i)+\sum_{j=1}^q (-1)^j h_j\alpha_{ij} \right)\wedge \varpi =\\
 &= \left(-\sum_{j=1}^q (-1)^j h_j\alpha_{ij} +\sum_{j=1}^q (-1)^j h_j\alpha_{ij} \right)\wedge \varpi= 0\\ 
 \end{aligned}$$

Then we have an unfolding $\widetilde{\omega}$ given locally around ${\mathfrak{p}}$ by $$\bigwedge_{i=1}^q (\varpi_i+\varepsilon \eta_i + h_i d\varepsilon)= \varpi+\varepsilon \left(\sum_{i=1}^q \eta_i\wedge \varpi_{\widehat{j}}\right)+ \left(\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\right)\wedge d\varepsilon.$$ As $\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\neq 0$ then $\widetilde{\omega}$ does not vanishes on $({\mathfrak{p}},0)$.

Reciprocally, let us suppose there is an unfolding $\widetilde{\omega}$ such that $\widetilde{\omega}({\mathfrak{p}},0)\neq 0$. As $\widetilde{\omega}$ satisfies Plücker relations and does not vanish in ${\mathfrak{p}}$, then it decomposes as a product of $1$-forms $\varpi_i+\varepsilon\eta_i+h_id\varepsilon$, $i=1,\dots,q$. As $\widetilde{\omega}|_{X\times\{0\}}=\omega$ then any local generator $\varpi$ of the image of $\omega$ is locally decomposable as $\varpi_1\wedge\dots\wedge\varpi_q$. We want to prove that the class $[d\varpi_i]$ of $d\varpi_i$ in $\Omega^2_{X,{\mathfrak{p}}}/((\varpi_1,\dots,\varpi_q)\wedge\Omega^1_{X,{\mathfrak{p}}})$ is zero for $i=1,\dots, q$. Let $\mathfrak{q}$ be a point in the support of $[d\varpi]$, then $\omega$ is singular in $\mathfrak{q}$, for otherwise $[d\varpi]=0$ because of the Frobenius condition $d\varpi_i\wedge \varpi = 0$, for $i=1,\ldots,q$. By Equation (\[equnfcodq\]), we have $\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}({\mathfrak{p}})\neq 0$, in particular not all of the $\varpi_{\widehat{j}}({\mathfrak{p}})$ vanishes. Without any loss of generality, we may assume $\varpi_{\widehat{1}}({\mathfrak{p}})$ does not vanish. Then also $\varpi_{\widehat{1}}(\mathfrak{q})\neq 0$. But $\varpi(\mathfrak{q})=0$, therefore $\varpi_2(\mathfrak{q}),\dots,\varpi_q(\mathfrak{q})$ are linearly independent and $\varpi_1(\mathfrak{q})$ is a linear combination of them. Hence $\varpi_{\widehat{j}}(\mathfrak{q})=f_j \varpi_{\widehat{1}}(\mathfrak{q})$. Then evaluating Equation (\[equnfcodq\]) in $\mathfrak{q}$, and adding the term $h_{1}(\mathfrak{q}) d\varpi_i(\mathfrak{q}) \wedge \varpi_{\widehat{1}}(\mathfrak{q})$, gives $$h_{1}(\mathfrak{q}) d\varpi_i(\mathfrak{q}) \wedge \varpi_{\widehat{1}}(\mathfrak{q})=(dh_i-\eta_i)\wedge\varpi(\mathfrak{q}) +\left(\sum_{j=2}^q h_j(\mathfrak{q}) f_j(\mathfrak{q})\right)d\varpi_i(\mathfrak{q})\wedge\varpi_{\widehat{1}}(\mathfrak{q}) \ .$$ So, after clearing $h_1(\mathfrak{q})\neq 0$, there is a $1$-form $\alpha_{i1}$ such that $$d\varpi_i\wedge \varpi_{\widehat{1}}(\mathfrak{q})= \alpha_{i1}\wedge \varpi_1\wedge \varpi_{\widehat{1}}(\mathfrak{q}) ,$$ then we have $(d\varpi_i-\alpha_{i1}\wedge\varpi_1)\wedge \varpi_{\widehat{1}}(\mathfrak{q})=0$, but as $\varpi_{\widehat{1}}(\mathfrak{q})\neq 0$ , this implies that there are forms $\alpha_{ij}$ such that $$(d\varpi_i-\alpha_{i1}\wedge\varpi_1)(\mathfrak{q})=\sum_{j\neq 1}\alpha_{ij}\wedge \varpi_j(\mathfrak{q}).$$ Hence $[d\varpi_i]=0$ in any point of its support, a contradiction, so $[d\varpi_i]=0$ in $\Omega^2_{X,{\mathfrak{p}}}/((\varpi_1,\dots,\varpi_q)\wedge\Omega^1_{X,{\mathfrak{p}}})$.

Let ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ be an integrable morphism determining a subsheaf $\mathcal{E}\to\Omega^1_X$. Composing $\omega$ with wedge product gives a morphism ${\mathcal{L}}\otimes\Omega^2_X\xrightarrow{\omega\wedge -} \Omega^{q+2}_X$. As $\omega$ is integrable the sheaf $\mathcal{E}\otimes\Omega^1_X$ is in the kernel of $\omega\wedge -$. We define the sheaf $H^2(\omega)$ as $$H^2(\omega):= \ker(\omega\wedge -) /\mathcal{E}\otimes \Omega^1_X.$$

The restriction of the de Rham differential to $\mathcal{E}$ gives a sheaf map $\mathcal{E}\to \Omega^2_X$ which is not ${\mathcal{O}}_X$-linear but whose image is in $ \ker(\omega\wedge -) $ as $\omega$ is integrable. The projection of this map to $H^2(\omega)$ is however ${\mathcal{O}}_X$-linear as $d g\varpi\cong g d\varpi \mod \mathcal{E}\otimes\Omega^1_X$ for every local section $\varpi$ of $\mathcal{E}$.

Let us fix ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ be an integrable morphism determining a subsheaf $\mathcal{E}\to\Omega^1_X$. Then we have the following definitions:

\[defII\] The subscheme of *persistent singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathscr{I}}(\omega)$ to be the annihilator of $d(\mathcal{E})$ in $H^2(\omega)$. In other words the local sections of ${\mathscr{I}}(\omega)$ in an open set $U\subseteq X$ are given by $${\mathscr{I}}(\omega)(U)=\{ h\in {\mathcal{O}}_X(U): \forall \varpi\in\mathcal{E}(U),\ hd\varpi=\sum_j \alpha_j\wedge\omega_j \},$$ for some local $1$-forms $\alpha_j$ and forms $\omega_j$ in $\mathcal{E}(U)$. We will denote it just as ${\mathscr{I}}$ if no confusion arises.

With the following example we are showing that the ideal ${\mathscr{I}}(\omega)$ can have codimension greater than $2$. Let us consider the $2$-form in ${\mathbb{P}}^3$: $$\begin{aligned}
\omega&=(-288 x_2+2880 x_3) dx_0 dx_1+(288 x_1-96 x_3) dx_0 dx_2+\\
&\hspace{.3cm}+(-288 x_0-1152 x_3) dx_1 dx_2+(-2880 x_1+96 x_2) dx_0 dx_3+\\
&\hspace{.3cm}+(2880 x_0+1152 x_2) dx_1 dx_3+(-96 x_0-1152 x_1) dx_2 dx_3
 \end{aligned}$$ Suche a differential form it is locally decomposable and locally integrable and has singular locus of codimension 3. The ideal of persistent singularities has also codimension 3. We did the computations using the software [DiffAlg]{}, see [@diffalg].

We can consider an extension of Remark \[[domega]{}\] for $\omega\in\Omega^q_X$. Then:

\[defKK\] The subscheme of *Kupka singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathcal{K}}(\omega):= {\mathrm{ann}}(\{d\omega\})\in \Omega^{q+1}_X\otimes{\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes {\mathcal{L}}^{-1}$. We will denote it just as ${\mathcal{K}}$ if no confusion arises.

We would like to notice that both definitions above coincide to the ones given in the codimension 1 case, as the reader can see by comparing them to Definition \[II2\] and Remark \[remarkI\] and to Defintion \[KK2\], respectively.

\[pluckerFiltration\] Given a short exact sequence of modules $$0\to M\to P\to N \to 0,$$ there is a filtration in $\bigwedge^q P$. $$\bigwedge^q P=F^0 \supseteq F^1\supseteq \dots \supseteq F^{q+1}=(0),$$ such that $$F^i/F^{i+1}\cong \bigwedge^{q-i}N\otimes \bigwedge^{i}M.$$

The result follows from defining $F^i\subseteq \bigwedge^q P$ to be the submodule generated by the elements of the form $(m_1\wedge\dots\wedge m_i\wedge a_{i+1}\wedge\dots\wedge a_q)$ where $m_j\in M$.

\[JJcIIcKK2\] Given an integrable morphism ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ we have the inclusions ${\mathscr{J}}(\omega)\subseteq{\mathscr{I}}(\omega)$ and ${\mathscr{J}}(\omega) \subseteq {\mathcal{K}}(\omega)$. If moreover $\omega$ is locally decomposable (*i.e.*: if ${\mathcal{E}}$ is locally free) then we have ${\mathscr{J}}(\omega)\subseteq{\mathscr{I}}(\omega)\subseteq {\mathcal{K}}(\omega)$.

To ease the notation let us set ${\mathscr{J}}={\mathscr{J}}(\omega)$, and likewise with ${\mathscr{I}}$ and ${\mathcal{K}}$. Let $h$ be a local section of ${\mathscr{J}}$, and by abuse of notation we will call $\omega$ a local generator of the image of the morphism $\omega:{\mathcal{L}}\to\Omega^q_X$, then by definition of ${\mathscr{J}}$ there is a local $q-1$-vector $v\in \bigwedge^{q}T_X$ such that $h=i_v\omega$. Then taking the filtration $\Omega^2_X=F^0\supseteq F^1\supseteq F^2\supseteq F^3=0$ of lemma \[pluckerFiltration\], we can say that $h$ is in ${\mathscr{I}}$ if and only if for every local section $\varpi \in {\mathcal{E}}$ we have $d\varpi\in F^1$. To establish this we recall that for every local section $\varpi$ of ${\mathcal{E}}$ the equation $d\varpi\wedge\omega=0$ holds. Then contracting with $v$ we get $$\begin{aligned}
  0= i_v(d\varpi\wedge\omega)=\\
  =d\varpi\wedge i_v\omega + \sum_{\substack{a_j\in T_X,\ b_j\in \bigwedge^{q-1}T_X\\ a_1\wedge b_1+\dots +a_r\wedge b_r=v}} i_a d\varpi\wedge i_b \omega+ \sum_{\substack{c_j\in \bigwedge^2 T_X,\ d_j\in \bigwedge^{q-2}T_X\\ c_1\wedge d_1+\dots +c_r\wedge d_s=v}} i_c d\varpi \wedge i_d\omega.
 \end{aligned}$$ By definition of ${\mathcal{E}}$ we have that $i_b \omega$ is a local section of ${\mathcal{E}}$, so every summand of the form $i_a d\varpi \wedge i_b \omega$ is in $\Omega^1_X\otimes{\mathcal{E}}$. Hence, to see that $hd\omega\in F^1$ it suffices to show that $i_d\omega$ is in $F^1$ for every $d\in \bigwedge^{q-2}T_X$. To see this we can calculate the class of $i_d\omega$ in $\Omega^2_X/F^1=F^0/F^1\cong \bigwedge^2(\Omega^1_X/{\mathcal{E}})$. The dual sheaf $(\Omega^1_X/{\mathcal{E}})^\vee \subseteq T_X$ is the distribution defined by $\omega$, that is, is the sheaf of vector fields $w$ such that $i_w\omega=0$. Then, when we evaluate $i_d \omega$ in a section $w_1\wedge w_2\in \bigwedge^2(\Omega^1_X/{\mathcal{E}})^\vee$ we get $0$. As $\bigwedge^2(\Omega^1_X/{\mathcal{E}})$ is torsion-free then the class of $i_d \omega$ in $\Omega^2_X/F^1$ is zero, then $i_d\omega\in F^1$, which means $h d\omega$ is in $F^1$ as we wanted to show.

The second assertion is clear by definition, as ${\mathcal{K}}$ is the annihilator of a section whose support is contained in ${\mathrm{Sing}(\omega)}$.

Now suppose ${\mathcal{E}}$ is locally free. So we can take local generators $\varpi_1,\dots,\varpi_q$ of ${\mathcal{E}}$, this sections verify that $\omega=\varpi_1\wedge\dots\wedge\varpi_q$. Then for every section $h$ of ${\mathscr{I}}$ there are local $1$-forms $\alpha_{ij}$ such that $$hd\varpi_i=\sum_{j} \alpha_{ij}\wedge\varpi_j.$$ Therefore we have $$\begin{aligned}
  hd\omega&=d(\varpi_1\wedge\dots\wedge\varpi_q)=\sum_i (-1)^i \varpi_1\wedge\dots\wedge d\varpi_i \wedge \varpi_{i+1}\wedge\dots\wedge\varpi_q=\\
  &=\sum_{i} (-1)^i \varpi_1\wedge\dots\wedge (\sum_j \alpha_{ij})\wedge\varpi_j \wedge \varpi_{i+1}\wedge\dots\wedge\varpi_q=\\
&=\sum_i \alpha_{ii}\wedge \varpi_1\wedge\dots\wedge\varpi_q=(\sum_i \alpha_{ii})\wedge\omega.
 \end{aligned}$$ In particular $hd\omega$ vanishes in ${\mathrm{Sing}(\omega)}$ so $h$ is in ${\mathcal{K}}$.

Let $\omega\in \Omega^2_{\mathbb{A}^3}$ be like in Example \[exampleCodimension2inA3\] so we write $$\omega= f_3 dx_1\wedge dx_2+f_2 dx_1\wedge dx_3+f_1 dx_2\wedge dx_3.$$ So we have $$d\omega = \left(\frac{\partial f_3 }{\partial x_3}- \frac{\partial f_2 }{\partial x_2}+ \frac{\partial f_1 }{\partial x_1}\right) dx_1\wedge dx_2\wedge dx_3.$$ For a general choice of the $f_i$’s the restriction $d\omega|_{{\mathrm{Sing}(\omega)}}$ does not vanish, so ${\mathscr{J}}={\mathcal{K}}$.

However, by setting for instance $f_3=f_3(x_1,x_2)$, $f_2=f_2(x_1,x_3)$ and $f_1=f_1(x_2,x_3)$, we get a form $\omega$ such that $d\omega=0$. With this choice of $\omega$ we have ${\mathcal{K}}={\mathcal{O}}_X$. When computing the ideal ${\mathscr{I}}$ for this case we need to check that $h d\omega_i= \alpha_{i1}\wedge\omega_1+\alpha_{i2}\wedge\omega_2+\alpha_{i3}\wedge\omega_3$ for $i=1,2,3$, where the $\omega_i$’s are the generators of ${\mathcal{E}}$ of Example \[exampleCodimension2inA3\] and $h\in {\mathcal{O}}_X$. Further specializing our choice of $\omega$ we can take $f_3=x_2$ and $f_2=-x_1$, in order to get $d\omega_1=dx_1\wedge dx_2 + dx_1\wedge dx_3$, so clearly $1\notin {\mathscr{I}}(\omega)$.

So we see that there are cases where ${\mathcal{K}}={\mathcal{O}}_X$ and $1\notin {\mathscr{I}}$. This is in stark contrast to the situation in codimension $1$ where, from Theorem \[teodivision\], follows that the condition ${\mathscr{J}}=\sqrt{{\mathscr{J}}}$ implies $\sqrt{{\mathscr{I}}}=\sqrt{{\mathcal{K}}}$.

\[teoConnectionE\] Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega_X^q$ be an integrable $q$-form and ${\mathcal{E}}$ be the associated subsheaf of $1$-forms ${\mathcal{E}}\subseteq \Omega^1_X$. Let $\mathrm{Sym}^2({\mathcal{E}})$ denote the symmetric power of ${\mathcal{E}}$ and suppose ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=0$. If ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ then ${\mathcal{E}}$ admits a holomorphic connection, in particular is locally free (in other words the foliation is locally decomposable) and every Chern class of ${\mathcal{E}}$ vanishes.

In order to prove the vanishing of the Chern classes of ${\mathcal{E}}$ we are going to use Atiyah’s classical result [@atiyah Theorem 4, p. 192] which states that if a holomorphic vector bundle on a compact Kähler manifold admits a holomorphic connection, then its Chern classes are all zero. We will then produce a holomorphic connection for ${\mathcal{E}}$ in this case. The condition ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ implies that for every local section $\varpi$ of ${\mathcal{E}}$ we have $d\varpi=\sum_i\alpha_i\wedge\omega_i$ for some local $1$-forms $\alpha_i$ and $\omega_i \in {\mathcal{E}}$. In other words, de Rham differential applied to sections of ${\mathcal{E}}$ give us a map $d:{\mathcal{E}}\to F^1\subseteq \Omega^2_X$ such that $d(f\varpi)=df\wedge\varpi+fd\varpi$, that is a differential operator of order $1$ between ${\mathcal{E}}$ and $F^1$. We will call $\mathrm{Diff}^{\leq 1}(A, B)$ the set of differential operators of order $\leq 1$ between two sheaves $A$ and $B$. Let us denote with $\mathcal{PE}$ the sheaf of principal parts of ${\mathcal{E}}$ of order $1$, this sheaf is defined by the universal property ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},M)=\mathrm{Diff}^{\leq 1}({\mathcal{E}}, M)$ for every coherent sheaf $M$. So the de Rham differential defines a coherent sheaves morphism $[\nabla]:\mathcal{PE}\to F^1$. To see if we can lift $[\nabla]$ to a morphism $\nabla:\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ defining a connection, we observe that the short exact sequence $0\to\mathrm{Sym}^2({\mathcal{E}})\to\Omega^1_X\otimes {\mathcal{E}}\to F^1\to 0$ gives an exact sequence of modules $$\begin{aligned}
  0 &\to {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},\Omega^1_X\otimes{\mathcal{E}})\to {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )\xrightarrow{\delta} \\
  &\xrightarrow{\delta} {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\Omega^1_X\otimes{\mathcal{E}})\to\cdots
 \end{aligned}$$ So $[\nabla]$ lifts to a morphism $\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ if and only if is in the kernel of $ {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1)\xrightarrow{\delta} {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$.

In order to compute ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$ recall the short exact sequence of sheaves $$0\to \Omega^1_X\otimes{\mathcal{E}}\to \mathcal{PE} \to {\mathcal{E}}\to 0,$$ which give rise to an exact sequence $$\begin{aligned}
   \cdots\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))&\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to \\
  &\to{\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\Omega^1_X\otimes{\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))\to \cdots\end{aligned}$$ Recall that the group ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$ can be regarded as the group of isomorphism classes of extensions of $\mathcal{PE}$ by $\mathrm{Sym}^2({\mathcal{E}})$. Viewed like this, the morphism $\delta: {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )\to  {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$ evaluated at an element $a\in {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1)$ returns the isomorphism class of the extension $0 \to \mathrm{Sym}^2({\mathcal{E}})\to A \to \mathcal{PE}\to 0$ where $A$ is the pull-back of the diagram $$\xymatrix{
  A \ar[d] \ar[r] & \mathcal{PE}\ar[d]^a \\
  \Omega^1_X\otimes{\mathcal{E}}\ar[r] & F^1
 }$$ In particular the composition $${\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )\xrightarrow{\delta} {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\Omega^1_X\otimes{\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$$ evaluated at the element $[\nabla] \in {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )$ returns the isomorphism class of the extension $0\to \mathrm{Sym}^2({\mathcal{E}}) \to B \to \Omega^1_X\otimes{\mathcal{E}}$ where $B$ is the pull-back of the diagram $$\xymatrix{
  B \ar[d] \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[d]^{[\nabla]\circ i} \\
  \Omega^1_X\otimes{\mathcal{E}}\ar[r] & F^1
 }$$ where $i:\Omega^1_X\otimes{\mathcal{E}}\to\mathcal{PE}$ is the canonical immersion.

Now to compute $[\nabla]\circ i:\Omega^1_X\otimes{\mathcal{E}}\to F^1$ recall that $[\nabla]$ is defined by being the unique ${\mathcal{O}}_X$-linear morphism making the following diagram commute, $$\xymatrix{
  {\mathcal{E}}\ar[d]^{d^{(1)}} \ar[r]^d & F^1 \\
  \mathcal{PE} \ar[ru]_{[\nabla]} & 
 }$$ where $d^{(1)}:{\mathcal{E}}\to \mathcal{PE}$ is the universal differential operator of order $1$. Then, as follows from the formulas of [@atiyah p. 193] explicitly describing the ${\mathcal{O}}_X$-module structure of $\mathcal{PE}$, given local sections $f$ of ${\mathcal{O}}_X$ and $\varpi$ of ${\mathcal{E}}$ we have $$[\nabla](df\otimes \varpi) = d(f\varpi)-fd(\varpi).$$ So, $[\nabla]\circ i$ is just the exterior product of forms, hence the sequence $0\to \mathrm{Sym}^2({\mathcal{E}}) \to B \to \Omega^1_X\otimes{\mathcal{E}}\to 0$ splits, then the class of $\delta([\nabla])$ in ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\Omega^1_X\otimes{\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$ is zero. Therefore $\delta([\nabla])$ is in the image of ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$. Hence if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$ then $\delta([\nabla])=0$. So, if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$, then there is a morphism $\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ lifting $[\nabla]$.

What we need to prove to conclude is that among the morphisms $\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$, there is one $\nabla:\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ such that $\nabla|_{\Omega^1_X\otimes {\mathcal{E}}}$ is the identity. To do this we consider the short exact sequence $0\to \Omega^1_X\otimes{\mathcal{E}}\to \mathcal{PE}\to {\mathcal{E}}\to 0$ and the exact sequence of ${\mathrm{Hom}}$ groups $$\begin{aligned}
  {\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{E}},\Omega_X\otimes{\mathcal{E}})\to{\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},\Omega_X\otimes{\mathcal{E}})&\to{\mathrm{Hom}}_{{\mathcal{O}}_X}(\Omega_X\otimes{\mathcal{E}},\Omega_X\otimes{\mathcal{E}})\xrightarrow{\delta} \\
  &\xrightarrow{\delta}{\mathrm{Ext}}^1({\mathcal{E}},\Omega_X\otimes{\mathcal{E}})\to\cdots
 \end{aligned}$$ The identity is an element $\mathrm{id}\in {\mathrm{Hom}}_{{\mathcal{O}}_X}(\Omega_X\otimes{\mathcal{E}},\Omega_X\otimes{\mathcal{E}})$ and we want to show that it is the restriction of some morphism $\nabla:\mathcal{PE}\to  \Omega_X\otimes{\mathcal{E}}$, which is equivalent to the condition $\delta(\mathrm{id})=0$. We already know that there is a morphism $\tilde{\nabla}:\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ lifting $[\nabla]$, so the restriction of $\tilde{\nabla}$ to $\Omega^1_X\otimes{\mathcal{E}}$, which we also denote $\tilde{\nabla}$, makes the following diagram commute. $$\xymatrix{
  0 \ar[r] &  \mathrm{Sym}^2({\mathcal{E}}) \ar[d] \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[d]^{\tilde{\nabla}} \ar[r] & F^1 \ar[r] \ar@{=}[d] & 0 \\
  0 \ar[r] &  \mathrm{Sym}^2({\mathcal{E}})  \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[r] & F^1 \ar[r] & 0
 }$$ Being a restriction we have $\delta(\tilde{\nabla})=0$. If we can prove that $\delta(\tilde{\nabla}-\mathrm{id})=0$ then $\delta(\mathrm{id})=0$ and we are set. The image of $\tilde{\nabla}-\mathrm{id}$ is in $\mathrm{Sym}^2({\mathcal{E}})$ so the element $\delta(\tilde{\nabla}-\mathrm{id})\in {\mathrm{Ext}}^1({\mathcal{E}},\Omega_X\otimes{\mathcal{E}})$ is the class of the extension in the last row of the diagram: $$\xymatrix{
  0 \ar[r] &  \Omega^1_X\otimes{\mathcal{E}}\ar[d]^{\tilde{\nabla}-\mathrm{id}} \ar[r] & \mathcal{PE} \ar[d] \ar[r] & {\mathcal{E}}\ar[r] \ar[d] & 0 \\
  0 \ar[r] &  \mathrm{Sym}^2({\mathcal{E}}) \ar[d] \ar[r] & I \ar[d] \ar[r] & {\mathcal{E}}\ar[d] \ar[r] & 0 \\
  0 \ar[r] &  \Omega^1_X\otimes{\mathcal{E}}\ar[r] & II \ar[r] & {\mathcal{E}}\ar[r] & 0
 }$$ Where $I$ and $II$ are push-forwards. If ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$ then $[I]=0\in {\mathrm{Ext}}^1({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$ and so $\delta(\tilde{\nabla}-\mathrm{id})=[II]=0\in {\mathrm{Ext}}^1({\mathcal{E}},\Omega^1_X\otimes{\mathcal{E}})$. So the condition ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$ implies that there is a connection on ${\mathcal{E}}$.

Let $X$ be a smooth projective variety over ${\mathbb{C}}$ such that every line bundle ${\mathcal{L}}$ verifies $H^1(X,{\mathcal{L}})=0$ and such that $Pic(X)$ is torsion-free (*e.g.*: $X$ smooth complete intersection). And let ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ be a foliation such that ${\mathcal{E}}\cong\bigoplus_i {\mathcal{L}}_i$ for some line bundles ${\mathcal{L}}_i$. Then ${\mathcal{L}}$ has persistent singularities.

As ${\mathcal{E}}$ is a direct sum of line bundles the group ${\mathrm{Ext}}^1_X({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$ decomposes as $${\mathrm{Ext}}^1_X({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=\bigoplus_{i}\bigoplus_{j\le k}
  H^1(X,{\mathcal{L}}_i^{-1}\otimes {\mathcal{L}}_j\otimes{\mathcal{L}}_k)=0.$$ So, as $\omega$ is decomposable, if it does not posses persistent singularities then ${\mathscr{I}}={\mathcal{O}}_X$ so by Theorem \[teoConnectionE\] there is a connection on ${\mathcal{E}}$. This implies that the Chern classes of the line bundle ${\mathcal{L}}_1,\dots,{\mathcal{L}}_q$ are all zero. Then ${\mathcal{L}}_i\cong{\mathcal{O}}_X$ for $i=1,\dots,q$, giving global sections of $\Omega^1_X$, contradicting the fact that $H^0(X,\Omega^1_X)=H^1(X,{\mathcal{O}}_X)=0$.

Applications to toric varieties {#section_toric}
===============================

In this section we try to generalize some of our results to foliations on varieties that may have singularities. We focus our attention in normal toric varieties and make use of the Cox ring of such a variety to generalize what is known in the case of projective space.

Let $X_{\Sigma}$ be a toric variety with no torus factors and let $S$ be its Cox ring graded by $\text{Cl}(X_{\Sigma})$, $$S={\mathbb{C}}[x_\rho\,\colon\,\rho\in\Sigma(1)].$$ The graded pieces $S_\beta$ for $\beta\in \text{Cl}(X_{\Sigma})$ consist of toric-homogeneous polynomials. Given $\phi\in{\mathrm{Hom}}_{{\mathbb{Z}}}(\text{Cl}(X_{\Sigma}),{\mathbb{Z}})$, let us define the vector field $R_{\phi}$, $$R_\phi:=\sum_{\rho\in\Sigma(1)}\phi(\rho)x_\rho\frac{\partial}{\partial x_\rho}.$$ Denote $Lie_\phi$ to the Lie derivative with respect to $R_\phi$, $$Lie_\phi=di_{R_{\phi}}+i_{R_{\phi}}d.$$

Let $X_{\Sigma}$ be a toric variety with no torus factors and let $S$ be its Cox ring. A $q$-form $\omega=\sum a_I dx_I$ with coefficients in $S$ has (homogeneous) *degree* $\beta\in\text{Cl}(X_{\Sigma})$ if $a_Ix_I\in S_{\beta}$ for all multi-index $|I|=q$.

Clearly if $\omega'$ is a $q'$-form of degree $\beta'$, then $\omega\wedge\omega'$ has degree $\beta+\beta'$. Furthermore, the degree of $d\omega$ and of $i_{R_{\phi}}(\omega)$ is also $\beta$.

Let $X_{\Sigma}$ be a toric variety with no torus factors and free class group $\text{Cl}(X_{\Sigma})$. The following are equivalent for a $q$-form $\omega$,

-   $\omega$ has degree $\beta$.

-   $Lie_\phi(\omega)=\phi(\beta)\omega$ for all $\phi\in{\mathrm{Hom}}_{{\mathbb{Z}}}(\text{Cl}(X_{\Sigma}),{\mathbb{Z}})$

Given that $Lie_\phi$ is a derivation and commutes with $d$, the result follows from the $0$-form case, see [@COX Exercise 8.1.8, p. 357], $$Lie_{\phi}(f)=\phi(\beta)f,\quad \forall f\in S_\beta.$$

Let $X_{\Sigma}$ be a toric variety with no torus factors and let $S$ be its Cox ring. A $q$-form $\omega$ with coefficients in $S$ is said to *descend to $X_{\Sigma}$* if $i_{R_\phi}\omega=0$ for all $\phi\in{\mathrm{Hom}}_{{\mathbb{Z}}}(\text{Cl}(X_{\Sigma}),{\mathbb{Z}})$.

Let $X_{\Sigma}$ be a simplicial toric variety with no torus factors and let $\omega$ be a $q$-form of degree $\beta$. Let $\widehat{\Omega}^q=(\Omega^q)^{\vee\vee}$ be the sheaf of Zariski $q$-forms, [@COX Equation 8.0.5, p. 347]. The following are equivalent,

-   $\omega$ descends to $X_{\Sigma}$.

-   $\omega\in H^0(X_{\Sigma},\widehat{\Omega}^q(\beta))$.

If $\omega$ is a $1$-form, the result follows from [@COX Corollary 8.1.5, p. 354] and [@COX Theorem 8.1.6, p. 355]. The general case follows from [@COX Corollary 8.2.17, p. 368] and [@COX Theorem 8.2.16, p. 367].

\[prop1notinI\] Let $X_\Sigma$ be a projective simplicial toric variety (hence with no torus factors) and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$. If $d\omega=\omega\wedge\eta$ for some $1$-form $\eta$, then $\beta$ is a torsion element and $d\omega=0$.

Let us apply the contraction with respect to the radial field $R_\phi$, where $\phi:\text{Cl}(X_\Sigma)\to{\mathbb{Z}}$ is some linear function, $$i_{R_\phi}(d\omega)=i_{R_\phi}(\omega\wedge\eta)\Longrightarrow
\phi(\beta)\omega+\omega i_{R_\phi}(\eta)=0\Longrightarrow
\phi(\beta)+i_{R_\phi}(\eta)=0.$$ Given that the degree of $d\omega$ is equal to the degree of $\omega$, it follows that the degree of $\eta$ is $0\in\text{Cl}(X_\Sigma)$, $$\eta=\sum_{\rho\in\Sigma(1)} a_{\rho}dx_{\rho}\Longrightarrow
-\phi(\beta)=\sum_{\rho\in\Sigma(1)} \phi(\rho)a_{\rho}x_{\rho}\Longrightarrow
\phi(\beta)\in\langle x_\rho\,\colon\,\rho\in\Sigma(1)\rangle.$$ Since $X_{\Sigma}$ has no torus factors, it follows $\phi(\beta)=0$ for all $\phi$ and this implies $\beta$ a torsion element and $\eta\in H^0(X_\Sigma,\widehat{\Omega}^1)$, but from [@materov Theorem 2.14], $H^0(X_\Sigma,\widehat{\Omega}^1)=0$. Hence, $\eta=0$ and then $d\omega=0$.

It is a priori unclear what a foliation is when the ambient space is singular. At the very least, we should ask a foliation on a singular space $X$ to restrict to a foliation on its maximal non-singular subscheme $U$. When $X$ is a normal variety, which is the case we will consider here, $U$ is an open subscheme whose complement $X\setminus U$ is a closed subset of codimension at least $2$. So whatever a codimension $q$ foliation on such an $X$ is, it should determine a morphism $L\xrightarrow{\omega}\Omega^q_{U}$ for some line bundle on $U$ verifying the Plücker relations and integrability. As $X$ is normal and the complement of $U$ is of codimension at least $2$ then restriction of line bundles defines an isomorphism $\mathrm{Pic}(X)\simeq\mathrm{Pic}(U)$, so there is a line bundle ${\mathcal{L}}$ on $X$ such that ${\mathcal{L}}|_U\simeq L$. So the foliation on $U$ defines a section (which we also call $\omega$ by abuse of notation) $\omega \in \Gamma(U, \Omega^q_X\otimes {\mathcal{L}}^{-1}|_U)$. Would $\Omega^q_X$ be a reflexive sheaf, a section on $U$ would extend in a unique way to a section on the whole $X$, this needs not to be the case if $X$ has singularities. To remedy this we can take the reflexive hull of $\Omega^q_X$, this sheaf is known as the *sheaf of Zariski $q$-forms on $X$*, we denote it by $\widehat{\Omega}^q_X$ and is defined as the double dual $\widehat{\Omega}^q_X:=({\Omega^q_X})^{\vee\vee}$. As $U$ is by definition a non-singular open subcheme we have identifications $\Omega^q_U=\Omega^q_X|_U=\widehat{\Omega}^q_X|_U$. Therefore a foliation on $X$ defines a section $\omega\in\Gamma(U, \widehat{\Omega}^q_X\otimes{\mathcal{L}}^{-1})$, as $X$ is normal, $\widehat{\Omega}^q_X\otimes{\mathcal{L}}^{-1}$ is a reflexive sheaf on $X$ and $\omega$ is a section defined outside a codimension $2$ subset, there is a unique way in which $\omega$ extends to a global section of $\widehat{\Omega}^q_X\otimes{\mathcal{L}}^{-1}$. As $\omega|_U$ verify the Plücker relations and the integrability condition, so does its global extension. We see in this way that considering twisted Zariski forms which are integrable is general enough and, as we will see bellow, is also manageable enough.

Let $X_\Sigma$ be a toric variety, let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$. We say that $\omega$ defines a *codimension $q$ foliation* if ${\mathrm{codim}}({\mathrm{Sing}(\omega)})\ge 2$ and for all $\eta\in E$, $$\begin{array}{rcl}
i_\Xi(\omega)\wedge\omega=0,&\forall\,\Xi\in\bigwedge^{q-1}T_S&\text{(Pl\"ucker relations),}\\
i_\Xi(\omega)\wedge d\omega=,0&\forall\,\Xi\in\bigwedge^{q-1}T_S&\text{(locally integrable).}
\end{array}$$ Notice that if the first condition is true, then the second condition is equivalent to $d(i_\Xi(\omega))\wedge\omega=0$.

The ideal of *singularities* of $\omega$ is the ideal in the Cox ring $S$ generated by the coefficients of $\omega$, $$J(\omega)=\{i_\Xi(\omega)\,\colon\, 
\Xi\in\bigwedge^{q}T_S\}\subseteq S.$$ Analogously, define $J(d\omega)$ as the ideal of coefficients of $d\omega\in\widehat{\Omega}^{q+1}_S$. Notice that if $\beta$ is not a torsion element, there exists $\phi:\text{Cl}(X_\Sigma)\to{\mathbb{Z}}$ such that $\phi(\beta)\neq 0$ and then contracting with $R_{\phi}$, the equation $i_{R_{\phi}}d\omega=\phi(\beta)\omega$ implies $J(\omega)\subseteq J(d\omega)$.

Let $$E:=\{\eta\,\colon\,\eta\wedge\omega=0\}
\subseteq\widehat\Omega^1_S.$$ The ideal of *persistent singularities* is $$I(\omega):=
(E\wedge\widehat{\Omega}^1_S\,:dE)=
\bigcap_{\eta\in E}(E\wedge\widehat{\Omega}^1_S\,:d\eta)$$ and the Kupka ideal is defined as the colon ideal, $$K(\omega):=(J\cdot \widehat{\Omega}^{q+1}_S:d\omega). $$ The *subscheme of persistent singularities* ${\mathpzc{Per}(\omega)}$ is defined by the homogeneous ideal $I(\omega)\subseteq S$, the *subscheme of singularities* ${\mathrm{Sing}(\omega)}$ by $J(\omega)$ and the *Kupka subscheme* ${\mathpzc{Kup}(\omega)}$ by $K(\omega)$.

In this simple example we show that the ideals $J(\omega)$ and $I(\omega)$ might be nontrivial ideals of the Cox ring although they may define empty subschemes. Let $\omega$ be the following $1$-form $$\omega=-x_3dx_2+x_2dx_3$$ and let $R_1$ and $R_2$ be the vector fields, $$R_1= x_0\partial_{x_0}+x_1\partial_{x_1}
,\quad
R_2=
x_2\partial_{x_2}+x_3\partial_{x_3}.$$ Notice that, $$d\omega=
2dx_2\wedge dx_3
,\quad
i_{R_1}d\omega= 0 =0\omega
,\quad
i_{R_2}d\omega=
-2x_3dx_2+2x_2dx_3
=2\omega.$$ So clearly $\omega\in H^0({\mathbb{P}}^1\times{\mathbb{P}}^1,\Omega^1(0,2))$. Indeed, $\omega$ is the pull-back $\pi_2^*\eta$, where $\pi_2:{\mathbb{P}}^1\times{\mathbb{P}}^1\to{\mathbb{P}}^1$ is the projection on the second factor and $\eta\in H^0({\mathbb{P}}^1,\Omega^1_{{\mathbb{P}}^1}(2))$ is the unique (up to multiplication by a constant) global $1$-form of degree $2$ in ${\mathbb{P}}^1$. Then the foliation defined by $\omega$ is non-singular, its leaves being the subvarieties ${\mathbb{P}}^1\times\{p\}$ with $p\in{\mathbb{P}}^1$. Also note that in this case $I(\omega)=J(\omega)=(x_3,x_2)$ is a non-trivial ideal of the Cox ring although it contains the irrelevant ideal of ${\mathbb{P}}^1\times{\mathbb{P}}^1$ (so they define the empty subscheme in $X_{\Sigma}$) in accordance to the foliation defined by $\omega$ being non-singular.

Let $X_\Sigma$ be a toric variety and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$. The sheaves associated to $J(\omega)$, $K(\omega)$ and $I(\omega)$ are $\mathscr{J}(\omega)$, $\mathscr{K}(\omega)$ and $\mathscr{I}(\omega)$ respectively.

First of all, if $\omega=\sum a_Idx_I$, then $J(\omega)$ is the ideal generated by $\{a_I\}$. Hence, $\Gamma_{*}(\mathscr{J}(\omega))=J(\omega)$. Now, let ${\mathfrak{p}}\in X_\Sigma$ and consider the stalk $K(\omega)_{{\mathfrak{p}}}$. Then, $K(\omega)_{{\mathfrak{p}}}=(J_{\mathfrak{p}}\cdot \widehat{\Omega}^{q+1}_{\mathfrak{p}}:d\omega)=
\{f\in S_{\mathfrak{p}}\,\colon\,fd\omega\in J_{\mathfrak{p}}\cdot \widehat{\Omega}^{q+1}_{\mathfrak{p}}\}$ and this is equal to the annihilator of $d\omega$ in $\widehat{\Omega}^{q+1}_{\mathfrak{p}}/(J_{\mathfrak{p}}\cdot \widehat{\Omega}^{q+1}_{\mathfrak{p}})\cong
\widehat{\Omega}^{q+1}_{\mathfrak{p}}\otimes (S/J)_{\mathfrak{p}}$. Hence, $\Gamma_{*}(\mathscr{K}(\omega))=K(\omega)$. Finally, $I(\omega)_{\mathfrak{p}}=\{f\in S_p\,\colon\,fd\varpi=\sum \alpha_j\wedge\varpi_j
\text{ where }
\varpi,\varpi_j\in E_{\mathfrak{p}}\}=\mathscr{I}(\omega)_{\mathfrak{p}}$. Then, $\Gamma_{*}(\mathscr{I}(\omega))=I(\omega)$.

Let $X_\Sigma$ be a projective simplicial toric variety and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ be a codimension $q$ foliation such that $\omega$ is locally decomposable on $X_\Sigma$ (*e.g.* $q=1$). If $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element, then ${\mathpzc{Per}(\omega)}\neq\emptyset$, where $\mathcal{E}=\{\eta\in\widehat\Omega^1\,\colon\,\omega\wedge\eta=0\}$.

Let $\{U_r\}$ be an open cover trivializing ${\mathcal{O}}_X(\beta)$ and let $(g_{rs},U_{rs})$ be a 1-cocycle in $\check{H}^1(X_\Sigma,{\mathcal{O}}_{X_\Sigma}^*)$ such that $\omega_{r}=g_{rs}\omega_s$ in $\Gamma(U_{rs},\widehat{\Omega}^q)$, where $U_{rs}:=U_r\cap U_s$.

Assume ${\mathpzc{Per}(\omega)}=\emptyset$. On $U_r$, the $q$-form $\omega_r$ is decomposable and there exists $\eta_r\in\Gamma(U_r,\widehat\Omega^1)$ such that $d\omega_r=\eta_r\wedge\omega_r$. Indeed, if $\omega_r=\varpi_1\wedge\ldots\wedge\varpi_q$, then $d\varpi_i=\sum\theta_{ij}\wedge\varpi_j$ because ${\mathpzc{Per}(\omega)}=\emptyset$. Then $d\omega_r=(\sum\theta_{ii})\wedge\omega_r$.

Now, on $U_{rs}$ we have, $$\eta_r\wedge\omega_r=d\omega_r=d(g_{rs}\omega_s)=dg_{rs}\wedge\omega_s+g_{rs}d\omega_s=
dg_{rs}\wedge\omega_s+g_{rs}\eta_s\wedge\omega_s\Longrightarrow$$ $$(\eta_r-dg_{rs}/g_{rs}-\eta_s)\wedge\omega_s=0\Longrightarrow
\overline\eta_r-\overline\eta_s=\overline{dg_{rs}/g_{rs}}\in \Gamma(U_{rs},\widehat\Omega^1/\mathcal{E}).$$ Hence, the 1-cocycle $(dg_{rs}/g_{rs},U_{rs})$ is in the kernel of the map $$\check{H}^1(X_{\Sigma},\widehat\Omega^1)\to 
\check{H}^1(X_{\Sigma},\widehat\Omega^1/\mathcal{E})$$ which is $0=\check{H}^1(X_\Sigma,\mathcal{E})$. Then, $0=[(dg_{\alpha\beta}/g_{\alpha\beta},U_{\alpha\beta})]
\in \check{H}^1(X_{\Sigma},\widehat\Omega^1)$.

Now, let us recall several results from complete simplicial toric varieties. By [@COX proof of Theorem 12.3.11, p. 588] $H^2(X_\Sigma,\mathbb{C})\cong H^1(X_{\Sigma},\widehat\Omega^1)$, by [@COX Theorem 12.3.2, p. 577] $H^2(X_\Sigma,\mathbb{C})\cong
\text{Pic}(X_\Sigma)_{\mathbb{C}}$ and by [@COX Proposition 4.2.7, p. 180] $\text{Pic}(X_\Sigma)_{\mathbb{C}}\cong\text{Cl}(X_\Sigma)_{\mathbb{C}}$. Then, $$H^1(X_{\Sigma},\widehat\Omega^1)\cong\text{Cl}(X_\Sigma)_{\mathbb{C}}.$$ Hence, $[\beta]=0$ in $\text{Cl}(X_\Sigma)_{\mathbb{C}}$ and this implies $\beta$ is a torsion element.

Let $X_\Sigma$ be a projective simplicial toric variety and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ be a codimension $q$ foliation such that $\widehat{\Omega}^1/{\mathcal{E}}$ is reflexive. If $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element, then ${\mathpzc{Per}(\omega)}\neq\emptyset$.

Let $\mathrm{Sing}(X_\Sigma)$ be the singular locus of $X_\Sigma$ and let $Z:=\text{Sing}(X_\Sigma)\cup{\mathrm{Sing}(\omega)}$ which has codimension $\ge 2$.

Let $\{U_r\}$ be an open cover trivializing ${\mathcal{O}}_X(\beta)$ and let $(g_{rs},U_{rs})$ be a 1-cocycle in $\check{H}^1(X_\Sigma,{\mathcal{O}}_{X_\Sigma}^*)$ such that $\omega_{r}=g_{rs}\omega_s$ in $\Gamma(U_{rs},\widehat{\Omega}^q)$, where $U_{rs}:=U_r\cap U_s$.

Assume ${\mathpzc{Per}(\omega)}=\emptyset$. Take an open cover $\{V_a\}$ of $U_r\setminus Z$ small enough such that the $q$-form $\omega_r$ is decomposable. Hence, as before, there exists $\zeta_a\in\Gamma(V_a,\widehat\Omega^1/{\mathcal{E}})$ such that $d\omega_r=\zeta_a\wedge\omega_r$ on $V_a$. Notice that $\zeta_a-\zeta_b=0$ in $\Gamma(V_a\cap V_b,\widehat\Omega^1/{\mathcal{E}})$. Then, $\zeta_r:=\{(\zeta_a,V_a)\}$ defines a section in $\check H^0(U_r\setminus Z,\widehat\Omega^1/{\mathcal{E}})$. Being $\widehat\Omega^1/{\mathcal{E}}$ reflexive, there exists a unique extension $\eta_r\in\check H^0(U_r,\widehat\Omega^1/{\mathcal{E}})$ and it satisfies $d\omega_r-\eta_r\wedge\omega_r\in \Gamma_Z(U_r,\widehat\Omega^{q+1})$. From the reflexivity of $\widehat\Omega^{q+1}$, we get $\Gamma_Z(U_r,\widehat\Omega^{q+1})=0$. Then, $d\omega_r=\eta_r\wedge\omega_r$ in $\Gamma(U_r,\widehat\Omega^{q+1})$.

The result follows by repeating the arguments of the previous proof.

Let $\omega$ be the following $1$-form $$\omega=
x_2^2x_0dx_1-x_2^2x_1dx_0+x_0^2x_2dx_3-x_0^2x_3dx_2$$ and let $R_1$ and $R_2$ be the vector fields, $$R_1= x_0\partial_{x_0}+x_1\partial_{x_1}
,\quad
R_2=
x_2\partial_{x_2}+x_3\partial_{x_3}.$$ Clearly $\omega\in H^0({\mathbb{P}}^1\times{\mathbb{P}}^1,\Omega^1(2,2))$ and, as $H^1({\mathbb{P}}^1\times{\mathbb{P}}^1,{\mathcal{O}}(-2,-2))=0$ it follows from the previous corollary that $\omega$ must have persistent singularities. Indeed $$\begin{aligned}
d\omega&=
2x_2^2dx_0\wedge dx_1+(2x_1x_2-2x_0x_3)dx_0\wedge dx_2\\
&-2x_0x_2dx_1\wedge dx_2+
2x_0x_2dx_0\wedge dx_3+2x_0^2dx_2\wedge dx_3,\\
i_{R_1}d\omega&=
-2x_1x_2^2dx_0+2x_0x_2^2dx_1-2x_0^2x_3dx_2+2x_0^2x_2dx_3
=2\omega,\\
i_{R_2}d\omega&=
-2x_1x_2^2dx_0+2x_0x_2^2dx_1-2x_0^2x_3dx_2+2x_0^2x_2dx_3
=2\omega.\end{aligned}$$ And from this one can compute the ideal $I(\omega)$, it turns out to be $$I(\omega)= (x_0^2, x_3 x_0 + x_4 x_1, x_4 x_0, x_4^2),\quad \sqrt{I(\omega)}=(x_0,x_4).$$ This computations were done with [DiffAlg]{}, see [@diffalg]. The singular ideal in this case is $J(\omega)=(x_0^2, x_4^2)$, so ${\mathpzc{Per}(\omega)}$ and ${\mathrm{Sing}(\omega)}$ have equal reduced structure although their scheme structure is not the same.

Here we provide an example were the subscheme ${\mathpzc{Per}(\omega)}$ is supported on a proper closed subset of ${\mathrm{Sing}(\omega)}$. This example was done with [DiffAlg]{}, see [@diffalg]. Let $\omega$ be the following $1$-form $$\omega = x_1x_2x_3dx_0+x_0x_2x_3dx_1+2x_0x_1x_3dx_2-2x_0x_1x_2dx_3$$ and let $R_1$ and $R_2$ be the vector fields, $$R_1= x_0\partial_{x_0}+x_1\partial_{x_1}+x_3\partial_{x_3}
,\quad
R_2=
x_2\partial_{x_2}+x_3\partial_{x_3}.$$ Then, it is easy to see that $i_{R_1}\omega=i_{R_2}\omega=0$, hence $\omega$ defines a $1$-form over the Hirzebruch surface $\mathcal{H}_1\cong \text{Bl}_{p}({\mathbb{P}}^2)$ of bi-degree $(2,3)$. From $$\begin{aligned}
d\omega&=
x_1x_3dx_0\wedge dx_2+
x_0x_3dx_1\wedge dx_2 -3x_1x_2dx_0\wedge dx_3\\
&-3x_0x_2dx_1\wedge dx_3-4x_0x_1dx_2\wedge dx_3,\\
i_{R_1}d\omega&=
3x_1x_2x_3dx_0+3x_0x_2x_3dx_1+6x_0x_1x_3dx_2-6x_0x_1x_2dx_3
=3\omega,\\
i_{R_2}d\omega&=
2x_1x_2x_3dx_0+2x_0x_2x_3dx_1+4x_0x_1x_3dx_2-4x_0x_1x_2dx_3
=2\omega,\end{aligned}$$ follows that $$I(\omega)=(x_2,x_1)\cap (x_1,x_0)\cap (x_2,x_3),\quad J(\omega)=(x_2,x_1)\cap (x_1,x_0)\cap (x_2,x_3)\cap (x_0, x_3).$$

[CACGLN04]{}

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  ----------------------------- --
  César Massri$^*$              
  Ariel Molinuevo$^\dag$        
  Federico Quallbrunn$^\ddag$   
  ----------------------------- --

  ---------------------------------------- --
                                           
  $^*$[Departamento de Matemática]{}       
  [Pabellón I]{}                           
  [Ciudad Universitaria]{}                 
  [CP C1428EGA]{}                          
  [Buenos Aires]{}                         
  [Argentina]{}                            
                                           
  $^\dag$Instituto de Matemática           
  Universidade Federal do Rio de Janeiro   
  Caixa Postal 68530                       
  CEP. 21945-970 Rio de Janeiro - RJ       
  BRASIL                                   
                                           
  ---------------------------------------- --

  ------------------------------------- --
  $^\ddag$ Departamento de Matemática   
  Universidad CAECE                     
  Av. de Mayo 866                       
  CP C1084AAQ                           
  [Buenos Aires]{}                      
  [Argentina]{}                         
  ------------------------------------- --

[^1]: The author was fully supported by CONICET, Argentina.

[^2]: The author was fully supported by Universidade Federal do Rio de Janeiro, Brasil.
