On non-Kupka points of codimension one foliations on ℙ³

Abstract

We study the singular set of a codimension one holomorphic foliation on ℙ ³ . We find a local normal form for these foliations near a codimension two component of the singular set that is not of Kupka type. We also determine the number of non-Kupka points immersed in a codimension two component of the singular set of a codimension one foliation on ℙ ³.

holomorphic foliations; Kupka sets; non-Kupka points

1 - Introduction

A regular codimension one holomorphic foliation on a complex manifold $M$, can be defined by a triple$\{(𝔘,f_{\alpha },{\psi }_{\alpha \beta })\}$ where

1. $𝔘=\{U_{\alpha }\}$ is an open cover of $M$.

2. $f_{\alpha }:U_{\alpha }\to ℂ$ is a holomorphic submersion for each $\alpha$.

3. A family of biholomorphisms$\{{\psi }_{\alpha \beta }:f_{\beta }(U_{\alpha \beta })\to f_{\alpha }(U_{\alpha \beta })\}$ such that

   ${\psi }_{\alpha \beta }={\psi }_{\beta \alpha }^{-1},{f_{\beta }|}_{U_{\alpha }\cap U_{\beta }}={{\psi }_{\beta \alpha }\circ f_{\alpha }|}_{U_{\alpha }\cap U_{\beta }}\mathit{}\text{and}\mathit{}{\psi }_{\alpha \gamma }={\psi }_{\alpha \beta }\circ {\psi }_{\beta \gamma }.$

Since$d{f}_{\alpha }(x)={\psi }_{\alpha \beta }^{\prime }(f_{\beta }(x))\cdot d{f}_{\beta }(x),$ the set $F={\displaystyle \bigcup _{\alpha }}Ker(d{f}_{\alpha })\subset TM$ is a subbundle. Also$[{\psi }_{\alpha \beta }^{\prime }(f_{\beta })]\in {\check{H}}^1(𝔘,{𝒪}^{\ast })$define a line bundle $N=TM/F$. The family of 1–forms $\{d{f}_{\alpha }\}$ glue to a global section$\omega \in H^0(M,{\mathrm{\Omega }}^1(N))$. We have

$0\to F\to TM\stackrel{\{d{f}_{\alpha }\}}{\to }N\to 0,0\to ℱ\to \mathrm{\Theta }\stackrel{\{d{f}_{\alpha }\}}{\to }𝒩\to 0,[ℱ,ℱ]\subset ℱ$

where $ℱ=𝒪(F),\mathrm{\Theta }=𝒪(TM)\text{and}𝒩=𝒪(N)$. We also obtain

${\wedge }^{n}T{M}^{\ast }=det(F^{\ast })\otimes N^{\ast },{\mathrm{\Omega }}_M^n:=K_M=det({ℱ}^{\ast })\otimes {𝒩}^{\ast },n=dim(M).$

Definition 1.1. Let $M$ be a compact complex manifold of dimension $n$.A singular codimension one holomorphic foliation on $M$, may be defined by one of the following ways:

1. A pair $ℱ=(S,ℱ)$, where $S\subset M$ is an analytic subset of $\text{codim}(S)\ge 2$, and $ℱ$ is a regular codimension one holomorphic foliation on $M\setminus S$.

2. A class of global sections $[\omega ]\in \mathbb{P}{H}^0(M,{\mathrm{\Omega }}^1(L))$, where $L\in Pic(M)$ such that

   1. the singular set $S_{\omega }=\{p\in M|{\omega }_p=0\}$ has $\text{codim}(S_{\omega })\ge 2$.

   2. $\omega \wedge d\omega =0$ in $H^0(M,{\mathrm{\Omega }}^3(L^{\otimes 2}))$.

   In this case, we denote by ${ℱ}_{\omega }=(S_{\omega },{ℱ}_{\omega })$ the foliation represented by $\omega$.

3. An exact sequence of sheaves

   $0\to ℱ\to \mathrm{\Theta }\to 𝒩\to 0,[ℱ,ℱ]\subset ℱ$

   where $ℱ$ is a reflexive sheaf of rank $rk(ℱ)=n-1$with torsion free quotient $𝒩\simeq {𝒥}_S\otimes L,$ where ${𝒥}_S$ is an ideal sheaf for some closed scheme $S$.

These three definitions are equivalents.

Remark 1.2. Let $\omega \in H^0(M,{\mathrm{\Omega }}^1(L))$ be a section.

1. The section $\omega$ may be defined by a family of 1-forms

   ${\omega }_{\alpha }\in {\mathrm{\Omega }}^1(U_{\alpha }),{\omega }_{\alpha }={\lambda }_{\alpha \beta }{\omega }_{\beta }\text{in}{U}_{\alpha \beta }=U_{\alpha }\cap U_{\beta },L=[{\lambda }_{\alpha \beta }]\in {\check{H}}^1(𝔘,{𝒪}^{\ast }).$

2. The section $\omega$ is a morphism of sheaves $\mathrm{\Theta }\stackrel{𝜔}{\to }L$. The kernel of $\omega$ is the tangent sheaf$ℱ$. The image of $\omega$ is a twisted ideal sheaf $𝒩={𝒥}_{S_{\omega }}\otimes L$. It is called the normal sheaf of $ℱ$.

3. As in the non-singular case, the following equality of line bundles holds

   $K_M={\mathrm{\Omega }}_M^n=det({ℱ}^{\ast })\otimes {𝒩}^{\ast }=K_{ℱ}\otimes L^{-1},det(N)\simeq L$

   where $K_M,K_{ℱ}=det({ℱ}^{\ast })$ are the canonical sheaf of $M$ and $ℱ$ respectively.

We denote by

The number $d\ge 0$ is called the degree of the foliation represented by $\omega$.

1-1 Statement of the results

In the sequel, $M$ is a compact complex manifold with $dim(M)\ge 3$. We will use any of the above definitions for foliation. The singular set will be denoted by $S$. Observe that $S$ decomposes as

$S=\bigcup _{k=2}^n{S}_k\mathit{}\text{where}\mathit{}\text{codim}(S_k)=k.$

For a foliation $ℱ$ on $M$ represented by $\omega \in ℱ(M,L),$ the Kupka set (Kupka ¹⁹⁶⁴; De Medeiros ¹⁹⁷⁷) is defined by

$K(\omega )=\{p\in M|\omega (p)=0,d\omega (p)\ne 0\}.$

We recall that for points near $K(\omega )$ the foliation $ℱ$ is biholomorphic to a product of a dimension one foliation in a transversal section by a regular foliation of codimension two (Kupka ¹⁹⁶⁴) and in particular we have $K(\omega )\subset S_2$.

In this note, we focus our attention on the set of non-Kupka points $NK(\omega )$ of $\omega$. The first remark is

$NK(\omega )=\{p\in M|\omega (p)=0,d\omega (p)=0\}\supset S_3\cup \mathrm{\cdots }\cup S_n.$

We analyze three cases, one in each section, the last two being the core of the work.

1. $S_2=K(\omega )$, then $NK(\omega )=S_3\cup \mathrm{\dots }\cup S_n$.

2. There is an irreducible component $Z\subset S_2$ such that $Z\cap K(\omega )=\mathrm{\varnothing }$.

3. For a foliation $\omega \in ℱ(3,d)$. Let $Z\subset S_2$ be a connected component such that $Z\setminus Z\cap K(\omega )$ is a finite set of points.

The first case has been considered in Brunella (²⁰⁰⁹); Calvo-Andrade (¹⁹⁹⁹, ²⁰¹⁶); Calvo-Andrade andSoares (¹⁹⁹⁴); Cerveau andLins Neto (¹⁹⁹⁴). Let $\omega \in ℱ(n,d)$ be a foliation with $K(\omega )=S_2$ and connected, then $\omega$ has a meromorphic first integral. In the generic case, the leaves define a Lefschetz or a Branched Lefschetz Pencil. The non-Kupka points are isolated singularities $NK(\omega )=S_n$. In this note, we present a new and short proof of this fact when the transversal type of $K(\omega )$ is radial.

In the second section, we study the case of a non-Kupka irreducible component of $S_2$. These phenomenon arise naturally in the intersection of irreducible components of $ℱ(M,L)$. The following result is a local normal form for $\omega$ near the singular set and is a consequence of a result of Loray (²⁰⁰⁶).

Theorem 1. Let $\omega \mathrm{\in }{\mathrm{\Omega }}^{\mathrm{1}}\mathit{}\mathrm{(}{ℂ}^n\mathrm{,}\mathrm{0}\mathrm{)}$, $n\mathrm{\ge }\mathrm{3}$, be a germ of integrable 1–form such that $\text{𝑐𝑜𝑑𝑖𝑚}\mathit{}\mathrm{(}{S}_{\omega }\mathrm{)}\mathrm{=}\mathrm{2}$, $\mathrm{0}\mathrm{\in }{S}_{\omega }$ is a smooth point and $d\mathit{}\omega \mathrm{=}\mathrm{0}$ on $S_{\omega }$. If $j_{\mathrm{0}}^{\mathrm{1}}\mathit{}\omega \mathrm{\ne }\mathrm{0}$, then or either

1. there exists a coordinate system $(x_1,\mathrm{\dots },x_n)\in {ℂ}^n$ such that

   $j_0^1(\omega )=x_{1}d{x}_2+x_{2}d{x}_1$

   and ${ℱ}_{\omega }$ is biholomorphic to the product of a dimension one foliation in a transversal section by a regular foliation of codimension two, or

2. there exists a coordinate system $(x_1,\mathrm{\dots },x_n)\in {ℂ}^n$ such that

   $\omega =x_{1}d{x}_1+g_1(x_2)(1+x_1{g}_2(x_2))d{x}_2,$

   such that $g_1,g_2\in {𝒪}_{ℂ,0}$ with $g_1(0)=g_2(0)=0$ , or

3. $\omega$ has a non-constant holomorphic first integral in a neighborhood of $0\in {ℂ}^n$ .

The alternatives are not exclusives. The following example was suggest by the referee and show that the case (3) of Theorem 1 cannot be avoid.

Example 1.3. Let $\omega$ be a germ of a 1-form at $0\in {ℂ}^3$ defined by

$\omega =xdx+(1+xf)df$

where $f(x,y,z)=y^{2}z$. We have

$\omega =xdx+2yz(1+x{y}^{2}z)dy+y^2(1+x{y}^{2}z)dz.$

The singular set of$\omega$ is $\{x=y=0\}$ and $\{x=z=y^2=0\}$, therefore the singular set has an embedding point $\{x=z=y^2=0\}$ and $d\omega$ vanish along $\{x=y=0\}$. We will show that $\omega$ has a holomorphic first integral $F$ in a neighborhood of $0\in {ℂ}^3$. In fact, let $t=f(x,y,z)=y^{2}z$ and set $\phi :({ℂ}^3,0)\to ({ℂ}^2,0)$ defined by

$\phi (x,y,z)=(x,t).$

Let $\eta =xdx+(1+xt)dt$ be 1-form at $0\in {ℂ}^2$, note that $\omega ={\phi }^{*}(\eta )$ and moreover $\eta (0,0)\ne 0$, this implies that $\eta$ is non-singular at $0\in {ℂ}^2$ and by Frobenius theorem $\eta$ has a holomorphic first integral $H(x,t)$ on $({ℂ}^2,0)$. Defining $H_1(x,y,z):=H(x,f(x,y,z))=H(x,y^{2}z)$, we get $H_1$ is a holomorphic first integral for $\omega$ in a neighborhood of $0\in {ℂ}^3$.

We apply Theorem 1 to a codimension one holomorphic foliation of the projective space with empty Kupka set.

About the third case, consider a foliation $\omega \in ℱ(3,d)$. Let $Z$ be a connected component of $S_2$. We count the number $|Z\cap NK(\omega )|$ of non-Kupka points of $\omega$ in $Z\subset S_2$.

Theorem 2. Let $\omega \mathrm{\in }ℱ\mathit{}\mathrm{(}\mathrm{3}\mathrm{,}d\mathrm{)}$ be a foliation and$Z\mathrm{\subset }{S}_{\mathrm{2}}$ a connected component of $S_{\mathrm{2}}$. Suppose that $Z$ is a local complete intersection and $Z\mathrm{\setminus }Z\mathrm{\cap }K\mathit{}\mathrm{(}\omega \mathrm{)}$ is a finite set of points, then${d\mathit{}\omega \mathit{|}}_Z$ is a global section of ${K_Z^{\mathrm{-}\mathrm{1}}\mathrm{\otimes }{K}_{ℱ}\mathit{|}}_Z$ and the associated divisor $D_{\omega }\mathrm{=}{\displaystyle \mathrm{\sum }_{p\mathrm{\in }Z}}o\mathit{}r\mathit{}{d}_p\mathit{}\mathrm{(}d\mathit{}\omega \mathrm{)}\mathrm{\cdot }p$ has degree

$\mathrm{deg}(D_{\omega })=\mathrm{deg}(K_{ℱ})-\mathrm{deg}(K_Z).$

Note that the section ${d\omega |}_Z$ vanishes exactly in the non-Kupka points of $\omega$ in $Z$ then the above theorem determine the number $|Z\cap NK(\omega )|$ (counted with multiplicity) of non-Kupka points of $\omega$ in $Z$.

2 - The singular set

Let $\omega \in ℱ(M,L)$ be a codimension one holomorphic foliation then singular set of $\omega$ may be written as

$S=\bigcup _{j=2}^n{S}_j\mathit{}\text{where}\mathit{}\text{codim}(S_j)=j.$

The fact that $K(\omega )\subset S_2$ implies that $S_3\cup \mathrm{\dots }\cup S_n\subset NK(\omega )$. To continue we focus in the components of singular set of $\omega$ of dimension at least three.

2.1 - Singular set of codimension at least three

We recall the following result due to B. Malgrange.

Theorem 2.1.(Malgrange ¹⁹⁷⁶)Let $\omega$ be a germ at $\mathrm{0}\mathrm{\in }{ℂ}^n$, $n\mathrm{\ge }\mathrm{3}$ of an integrable 1–form singular at $\mathrm{0}$, if $\text{𝑐𝑜𝑑𝑖𝑚}\mathit{}\mathrm{(}{S}_{\omega }\mathrm{)}\mathrm{\ge }\mathrm{3}$, then there exist $f\mathrm{\in }{𝒪}_{{ℂ}^n\mathrm{,}\mathrm{0}}$ and $g\mathrm{\in }{𝒪}_{{ℂ}^n\mathrm{,}\mathrm{0}}^{\mathrm{\ast }}$ such that

$\omega =gdf\mathit{}\mathit{\text{on a neighborhood of}}0\in {ℂ}^n.$

We have the following proposition.

Proposition 2.2. Let $\omega \mathrm{\in }ℱ\mathit{}\mathrm{(}M\mathrm{,}L\mathrm{)}$ be a foliation and let $p\mathrm{\in }{S}_n$ an isolated singularity, then any germ of vector field tangent to the foliation vanishes at $p$.

Proof. Let $\omega =gdf,g\in {𝒪}_p^{*},f\in {𝒪}_p$ be a 1-form representing the foliation at $p$. Let $𝐗\in {\mathrm{\Theta }}_p$ be a vector field tangent to the foliation, i.e., $\omega (𝐗)=0$.If $𝐗(p)\ne 0$ there exists a coordinate system with $z(p)=0$ and $𝐗=\partial /\partial z_n$, then

$0=\omega (𝐗)=g\cdot \left(\sum _{i=1}^n(\partial f/\partial z_i)d{z}_i(\partial /\partial z_n)\right)=g\cdot (\partial f/\partial z_n),\text{therefore}\partial f/\partial z_n\equiv 0,$

and $f=f(z_1,\mathrm{\dots },z_{n-1})$, but this function does not have an isolated singularity.∎

Now, we begin our study of the irreducible components of codimension two of the singular set of $\omega$. Note that, given a section $\omega \in H^0(M,{\mathrm{\Omega }}^1(L))$, along the singular set, the equation ${\omega }_{\alpha }={\lambda }_{\alpha \beta }{\omega }_{\beta }$implies ${d{\omega }_{\alpha }|}_S={({\lambda }_{\alpha \beta }d{\omega }_{\beta })|}_S.$ Then

$\{d{\omega }_{\alpha }\}\in H^0(S,{({\mathrm{\Omega }}_M^2\otimes L)|}_S).$ (4.1)

2.2 - The Kupka set

These singularities has bee extensively studied and the main properties have been established in (Kupka ¹⁹⁶⁴; De Medeiros ¹⁹⁷⁷).

Definition 2.3. For $\omega \in ℱ(M,L)$. The Kupka set is

$K(\omega )=\{p\in M|\omega (p)=0,d\omega (p)\ne 0\}.$

The following properties of Kupka sets, are well known (De Medeiros ¹⁹⁷⁷).

1. $K(\omega )$ is smooth of codimension two.

2. $K(\omega )$ has local product structure and the tangent sheaf $ℱ$ is locally free near $K(\omega )$.

3. $K(\omega )$ is subcanonically embedded and

   ${\wedge }^2{N}_{K(\omega )}={L|}_{K(\omega )},K_{K(\omega )}={(K_M\otimes L)|}_{K(\omega )}={K_{ℱ}|}_{K(\omega )}.$

Let $\omega \in ℱ(n,d)$ be a foliation with $S_2=K(\omega )$.By Calvo-Andrade and Soares (¹⁹⁹⁴), there exists a pair $(V,\sigma )$, where $V$ is a rank two holomorphic vector bundle and $\sigma \in H^0({\mathbb{P}}^n,V)$, such that

$0⟶𝒪\stackrel{𝜎}{\to }V⟶{𝒥}_K(d+2)\to 0\mathit{}\text{with}\mathit{}\{\sigma =0\}=K$

and the total Chern class

$c(V)=1+(d+2)\cdot 𝐡+\mathrm{deg}(K(\omega )){𝐡}^2\in H^{\ast }({\mathbb{P}}^n,\mathbb{Z})\simeq \mathbb{Z}[𝐡]/{𝐡}^{n+1}.$

In 2009, Marco Brunella proved that following result, which in a certain sense say that the local transversal type of the singular set of foliation determines its behavior globally. Here we present a new proof of this fact. The techniques used in the proof could be of independent interest.

Proposition 2.4. Let $\omega \mathrm{\in }ℱ\mathit{}\mathrm{(}n\mathrm{,}d\mathrm{)}$ be a foliation with $S_{\mathrm{2}}\mathrm{=}K\mathit{}\mathrm{(}\omega \mathrm{)}$, (connected if $n\mathrm{=}\mathrm{3}$) and of radial transversal type. Then $K\mathit{}\mathrm{(}\omega \mathrm{)}$ is a complete intersection and $\omega$ has a meromorphic first integral.

To prove Proposition 2.4, we require the following lemma. This result may be well known but for lack of a suitable reference we include the proof in an appendix.

Lemma 2.5. Let $F$ be a rank two holomorphic vector bundle over ${\mathbb{P}}^{\mathrm{2}}$ with $c_{\mathrm{1}}\mathit{}\mathrm{(}F\mathrm{)}\mathrm{=}\mathrm{0}$ and $c_{\mathrm{2}}\mathit{}\mathrm{(}F\mathrm{)}\mathrm{=}\mathrm{0}$. Then $F\mathrm{\simeq }𝒪\mathrm{\oplus }𝒪$, is holomorphically trivial.

Now, we prove Proposition 2.4.

Proof of Proposition 2.4. Let $(V,\sigma )$ be the vector bundle with a section defining the Kupka set as scheme. The radial transversal type implies (Calvo-Andrade and Soares ¹⁹⁹⁴)

$c(V)=1+(d+2)\cdot 𝐡+\frac{{(d+2)}^2}{4}\cdot {𝐡}^2={\left(1+\frac{(d+2)\cdot 𝐡}{2}\right)}^2\in H^{\ast }({\mathbb{P}}^n,\mathbb{Z})\simeq \mathbb{Z}[𝐡]/{𝐡}^{n+1}.$

The vector bundle$E=V(-\frac{d+2}{2})$, has $c_1(E)=0$ and $c_2(E)=0$. Let$\xi :{\mathbb{P}}^2\hookrightarrow {\mathbb{P}}^n$ be a linear embedding. By the preceding lemma we have

${\xi }^{\ast }E\simeq {𝒪}_{{\mathbb{P}}^2}\oplus {𝒪}_{{\mathbb{P}}^2}$

and by the Horrocks’ criterion (Okonek et al. ¹⁹⁸⁰),

$E\simeq {𝒪}_{{\mathbb{P}}^n}\oplus {𝒪}_{{\mathbb{P}}^n}$

is trivial and hence $V$ splits as ${𝒪}_{{\mathbb{P}}^n}(\frac{d+2}{2})\oplus {𝒪}_{{\mathbb{P}}^n}(\frac{d+2}{2})$ and $K$ is a complete intersection. The existence of the meromorphic first integral follows from Theorem A of (Cerveau and Lins Neto ¹⁹⁹⁴). ∎

If $\omega$ is such that $K(\omega )=S_2$ and connected, the set of non-Kupka points of $\omega$ is

$NK(\omega )=S_3\cup \mathrm{\cdots }\cup S_n.$

A generic rational map, that means, a Lefschetz or a Branched Lefschetz Pencil$\phi :{\mathbb{P}}^n⇢{\mathbb{P}}^1$, has only isolated singularities away its base locus. The singular set of the foliation defined by the fibers of $\phi$ is $S_n\cup S_2$. The Kupka set corresponds away from its base locus and $S_n=NK(\omega )$ are the singularities as a map. $S_n$ is empty if and only if the degree of the foliation is $0$. The number$\mathrm{\ell }(S_n)$ of isolated singularities counted with multiplicities can be calculated by (Cukierman et al. ²⁰⁰⁶). If ${\omega }_p$ is a germ of form that defines the foliation at $p\in S_n$, we have

$\mathrm{\ell }(S_n)=\sum _{p\in S_n}\mu ({\omega }_p,p),\mu (\omega ,p)=di{m}_{ℂ}\frac{{𝒪}_p}{({\omega }_1,\mathrm{\dots },{\omega }_n)},{\omega }_p=\sum _{i=1}^n{\omega }_{i}d{z}_i.$

We have that $c_n(ℱ)=\mathrm{\ell }(S_n)$.

3 - Foliations with a non-Kupka component

It is well known that $K(\omega )\subset \{p\in M|j_p^1\omega \ne 0\}$, but the converse is not true. Our first result describes the singular points with this property.

3.1 - A normal form

Now, we analyze the situation when there is an irreducible non-Kupka component of $S_2$.

Proof of Theorem 1. By hypotheses, $d\omega (p)=0$ for any $p\in S_{\omega }$. Since

$\omega ={\omega }_1+\mathrm{\cdots },d\omega =d{\omega }_1+\mathrm{\cdots }=0,$

we get $d{\omega }_1(p)=0$ for any $p\in S_{\omega }$. Now, as ${\omega }_1\ne 0$ and $\text{codim}(S_{\omega })=2$, we have $1\le \text{codim}(S_{{\omega }_1})\le 2$. We distinguish two cases.

1. $\text{codim}(S_{{\omega }_1})=2$: there is a coordinate system $(x_1,\mathrm{\dots },x_n)\in {ℂ}^n$ such that

   ${\omega }_1=x_{1}d{x}_2+x_{2}d{x}_1.$

2. $\text{codim}(S_{{\omega }_1})=1$: there is a coordinate system $(x,\zeta )\in ℂ\times {ℂ}^{n-1}$ such that$x(p)=0$ and ${\omega }_1=xdx$.

The first case is known, the foliation ${ℱ}_{\omega }$ is equivalent in a neighborhood of $0\in {ℂ}^n$ to a product of a dimension one foliation in a transversal section by a regular foliation of codimension two (Cerveau and Mattei ¹⁹⁸²).

In the second case, Loray’s preparation theorem (Loray ²⁰⁰⁶), shows that there exists a coordinate system $(x,\zeta )\in ℂ\times {ℂ}^{n-1}$, a germ $f\in {𝒪}_{{ℂ}^{n-1},0}$ with $f(0)=0$, and germs $g,h\in {𝒪}_{ℂ,0}$ such that the foliation is defined by the 1–form

$\omega =xdx+[g(f(\zeta ))+xh(f(\zeta ))]df(\zeta ).$ (5.1)

Since $S_{{\omega }_1}=\{x=0\}$ and $0\in S_{\omega }$ is a smooth point, we can assume that $S_{\omega }{}_{,p}=\{x={\zeta }_1=0\}$, where $S_{\omega ,p}$ is the germ of $S_{\omega }$ at $p=0$. Therefore,

$S_{\omega }{}_{,p}=\{x={\zeta }_1=0\}=\{x=g(f(\zeta ))=0\}\cup \{x=\frac{\partial f}{\partial {\zeta }_1}=\mathrm{\cdots }=\frac{\partial f}{\partial {\zeta }_{n-1}}=0\}.$

Hence, either $g(0)=0$ and ${\zeta }_1|f$, or $g(0)\ne 0$ and ${\zeta }_1|\frac{\partial f}{\partial {\zeta }_j}$ for all $j=1,\mathrm{\dots },n-1$. In any case, we have ${\zeta }_1|f$ and then $f(\zeta )={\zeta }_1^k\psi (\zeta )$, where $\psi$ is a germ of holomorphic function in the variable $\zeta$; $k\in \mathbb{N}$ and ${\zeta }_1$ does not divide $\psi$. We have two possibilities:

$1^{st}$case.–$\psi (0)\ne 0$. In this case, we consider the biholomorphism

$G(x,\zeta )=(x,{\zeta }_1{\psi }^{1/k}(\zeta ),{\zeta }_2,\mathrm{\dots },{\zeta }_n)=(x,y,{\zeta }_2,\mathrm{\dots },{\zeta }_n)$

where ${\psi }^{1/k}$ is a branch of the $k^{th}$ root of $\psi$, we get $f\circ G^{-1}(x,y,{\zeta }_2,\mathrm{\dots },{\zeta }_n)=y^k$ and

$G_{*}(\omega )=xdx+(g(y^k)+xh(y^k))k{y}^{k-1}dy=xdx+(g_1(y)+x{h}_1(y))dy,$

where $g_1(y)=k{y}^{k-1}g(y^k)$, $h_1(y)=k{y}^{k-1}h(y^k)$. Therefore, $\tilde{\omega }:=G_{*}(\omega )$ is equivalent to $\omega$ and moreover $\tilde{\omega }$ is given by

$\tilde{\omega }=xdx+(g_1(y)+x{h}_1(y))dy\mathit{}\text{with}\mathit{}{S}_{\tilde{\omega }}=\{x=g_1(y)=0\}.$ (5.2)

Since $d\tilde{\omega }=h_1(y)dx\wedge dy$ is zero identically on $\{x=g_1(y)=0\}$,we get $g_1|h_1$, so that $h_1(y)={(g_1(y))}^{m}H(y)$, for some $m\in \mathbb{N}$ and such that $H(y)$ does not divided $g_1(y)$. Using the above expression for $h_1$ in (3.2), we have

$\tilde{\omega }=xdx+g_1(y)(1+x{(g_1(y))}^{m-1}H(y))dy=xdx+g_1(y)(1+x{g}_2(y))dy,$

where$g_2(y)={(g_1(y))}^{m-1}H(y)$. Consider $\phi :(ℂ,0)\times ({ℂ}^{n-1},0)\to ({ℂ}^2,0)$ defined by $\phi (x,\zeta )=(x,y)$, then

$\omega ={\phi }^{\ast }(xdx+g_1(y)(1+x{g}_2(y))dy).$ (5.3)

$2^{nd}$case.–$\psi (0)=0$. We have $S_{\omega ,p}=\{x={\zeta }_1=0\}$ and

$\omega =xdx+(g({\zeta }_1^k\psi )+xh({\zeta }_1^k\psi ))d({\zeta }_1^k\psi ),$ (5.4)

therefore

$\omega =xdx+(g({\zeta }_1^k\psi )+xh({\zeta }_1^k\psi )){\zeta }_1^{k-1}(k\psi d{\zeta }_1+{\zeta }_{1}d\psi ).$ (5.5)

Note that $g(0)\ne 0$, otherwise $\{x={\zeta }_1\psi (\zeta )=0\}$ would be contained in $S_{\omega }{}_{,p}$, but it is contradiction because $S_{\omega ,p}=\{x={\zeta }_1=0\}⊊\{x={\zeta }_1\psi (\zeta )=0\}$. Furthermore $k\ge 2$, because otherwise ${\zeta }_1|\psi$.

Let $\phi :(ℂ,0)\times ({ℂ}^{n-1},0)\to ({ℂ}^2,0)$ be defined by

$\phi (x,\zeta )=(x,{\zeta }_1^k\psi (\zeta ))=(x,t),$

then from (3.4), we get that

$\omega ={\phi }^{\ast }(\eta ),$

where $\eta =xdx+(g(t)+xh(t))dt$. Since $\eta (0,0)=g(0)dt\ne 0$, we deduce that $\eta$ has a non-constant holomorphic first integral $F\in {𝒪}_{{ℂ}^2,0}$ such that $dF(0,0)\ne 0$. Therefore, $F_1(x,\zeta )=F(x,{\zeta }_1^k\psi (\zeta ))$ is a non-constant holomorphic first integral for $\omega$ in a neighborhood of $0\in {ℂ}^n$.∎

3.2 - Applications to foliations on ${\mathbb{P}}^n$

In order to give some applications of Theorem 1, we need the Baum-Bott index associated to singularities of foliations of codimension one.

Let $M$ be a complex manifold and let ${𝒢}_{\omega }=(S,𝒢)$ be a codimension one holomorphic foliation represented by $\omega \in H^0(M,{\mathrm{\Omega }}^1(L))$. We have the exact sequence

$0\to 𝒢\to {\mathrm{\Theta }}_M\stackrel{𝜔}{\to }{𝒩}_{𝒢}\to 0,{𝒩}_{𝒢}\simeq {𝒥}_S\otimes L.$

Set $M^0=M\setminus S$ and take $p_0\in M^0$. Then in a neighborhood $U_{\alpha }$ of $p_0$the foliation $𝒢$ is induced by a holomorphic $1$–form ${\omega }_{\alpha }$ and there exists a differentiable $1$–form ${\theta }_{\alpha }$ such that

$d{\omega }_{\alpha }={\theta }_{\alpha }\wedge {\omega }_{\alpha }$

Let $Z$ be an irreducible component of $S_2$. Take a generic point $p\in Z$,that is, $p$ is a point where $Z$ is smooth and disjoint from the other singular components. Pick $B_p$ a ball centered at $p$ sufficiently small, so that$S(B_p)$ is a sub-ball of $B_p$ of codimension $2$. Then the De Rhamclass can be integrated over an oriented $3$-sphere$L_p\subset B_p^{*}$ positively linked with $S(B_p)$:

$\text{BB}(𝒢,Z)=\frac{1}{{(2\pi i)}^2}{\int }_{L_p}\theta \wedge d\theta .$

This complex number is the Baum-Bott residue of $𝒢$ along Z.We have a particular case of the general Baum-Bott residues Theorem (Baum and Bott ¹⁹⁷²), reproved by Brunella and Perrone (²⁰¹¹).

Theorem 3.1. Let $𝒢$ be a codimension one holomorphic foliation on a complex manifold $M$. Then

$c_1{(L)}^2=c_1^2({𝒩}_{𝒢})=\sum _{Z\subset S_2}\text{𝐵𝐵}(𝒢,Z)[Z],$

where ${𝒩}_{𝒢}\mathrm{=}{𝒥}_S\mathrm{\otimes }L$ is the normal sheaf of $𝒢$ on $M$ and the sum is done over all irreducible components of $S_{\mathrm{2}}$.

In particular, if $𝒢$ is a codimension one foliation on ${\mathbb{P}}^n$ of degree $d$, then the normal sheaf ${𝒩}_{𝒢}={𝒥}_S(d+2)$ and the Baum-Bott Theorem looks as follows

$\sum _{Z\subset S_2}\text{BB}(𝒢,Z)\mathrm{deg}[Z]={(d+2)}^2.$

Remark 3.2. If there exist a coordinates system $(U,(x,y,z_3,\mathrm{\dots },z_n))$ around $p\in Z\subset S_2$ such that $x(p)=y(p)=0$ and $S(𝒢)\cap U=Z\cap U=\{x=y=0\}$. Moreover, if we assume that

${\omega |}_U=P(x,y)dy-Q(x,y)dx$

is a holomorphic 1-form representing ${𝒢|}_U$. Then we can consider the ${𝒞}^{\mathrm{\infty }}$ (1,0)-form $\theta$ on $U\setminus Z$ given by

$\theta =\frac{(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y})}{{|P|}^2+{|Q|}^2}(\overline{P}dx+\overline{Q}dy).$

Since$d\omega =\theta \wedge \omega$, we get

$\text{BB}(𝒢,Z)=\frac{1}{{(2\pi i)}^2}{\int }_{L_p}\theta \wedge d\theta ={\text{Res}}_0\left\{\frac{\text{Tr}\left(D𝐗\right)dx\wedge dy}{PQ}\right\},$ (5.6)

where ${\text{Res}}_0$ denotes the Grothendieck residue, $D𝐗$ is the Jacobian of the holomorphic map $𝐗=(P,Q)$. It follows from Griffiths andHarris (¹⁹⁷⁸) that if $D𝐗(p)$ is non-singular, then

$\text{BB}(𝒢,Z)=\frac{\text{Tr}{(D𝐗(p))}^2}{det(D𝐗(p))}.$

In the situation explained above, the tangent sheaf $𝒢(U)$ is locally free and generated by the holomorphic vector fields

$𝒢(U)=⟨𝐗=P(x,y)\frac{\partial }{\partial x}+Q(x,y)\frac{\partial }{\partial y},\frac{\partial }{\partial z_3},\mathrm{\dots },\frac{\partial }{\partial z_n}⟩$

and the vector field $𝐗$ carries the information of the Baum–Bott residues.

The next result, in an application of Theorem 1

Theorem 3.3. Let $\omega \mathrm{\in }ℱ\mathit{}\mathrm{(}M\mathrm{,}L\mathrm{)}$ be a foliation and $Z\mathrm{\subset }{S}_{\mathrm{2}}\mathrm{\setminus }K\mathit{}\mathrm{(}\omega \mathrm{)}$. Suppose that $Z$ is smooth and $j_p^{\mathrm{1}}\mathit{}\omega \mathrm{\ne }\mathrm{0}$ for all $p\mathrm{\in }Z$, then $\text{𝐵𝐵}\mathit{}\mathrm{(}{ℱ}_{\omega }\mathrm{,}Z\mathrm{)}\mathrm{=}\mathrm{0}$.

Proof. We work in a small neighborhood $U$ of $p\in Z\subset M$. According to Theorem 1 there exist a coordinate system $(x,y,z_3,\mathrm{\dots },z_n)$ at $p$ such that $Z\cap U=\{x=y=0\}$ and one has three cases. In the first case, ${ℱ}_{\omega }$ is the product of a dimension one foliation in a section transversal to $Z$ by a regular foliation of codimension two and $j_p^1(\omega )=xdy+ydx$. In this case, it follows from (3.6) that $\text{BB}({ℱ}_{\omega },Z)=0$. In the second case

$\omega =xdx+g_1(y)(1+x{g}_2(y))dy,$

where $g_1,g_2\in {𝒪}_{ℂ,0}$ and it follows from Lemma 3.9 of Cerveau and Lins Neto (²⁰¹³) that

$\text{BB}({ℱ}_{\omega },Z)={\text{Res}}_{t=0}\left[\frac{{(g_1(t)g_2(t))}^{2}dt}{g_1(t)}\right]={\text{Res}}_{t=0}\left[g_1(t){(g_2(t))}^2\right].$

Since $g_1(y){(g_2(y))}^2$ is holomorphic at $y=0$, we get $\text{BB}({ℱ}_{\omega },Z)=0.$ In the third case ${ℱ}_{\omega }$ has a holomorhic first integral in neighborhood of $p$ and is known that $\text{BB}({ℱ}_{\omega },Z)=0.$∎

The Baum-Bott formula implies the following result.

Corollary 3.4. Let $\omega \mathrm{\in }ℱ\mathit{}\mathrm{(}n\mathrm{,}d\mathrm{)}$, $n\mathrm{\ge }\mathrm{3}$, be a foliation with $K\mathit{}\mathrm{(}\omega \mathrm{)}\mathrm{=}\mathrm{\varnothing }$. Then there exists a smooth point $p\mathrm{\in }{S}_{\mathrm{2}}$ such that $j_p^{\mathrm{1}}\mathit{}\omega \mathrm{=}\mathrm{0}$.

Proof. If for all smooth point $p\in S_2$ one has $j_p^1\omega \ne 0$, the above theorem shows that$\text{BB}({ℱ}_{\omega },Z)=0$ for all irreducible components $Z\subset S_2$. By Baum–Bott’s theorem, we get

which is a contradiction. Therefore there exists a smooth point $p\in S_2$ such that $j_p^1\omega =0$. ∎

In particular, if $\omega \in ℱ(n,d)$, $n\ge 3$, is a foliation with $j_p^1\omega \ne 0$ for any $p\in {\mathbb{P}}^n$, then its Kupka set is not empty.

4 - The number of non-Kupka points

Through this section, we consider codimension one foliations on ${\mathbb{P}}^3$, but some results remain valid to codimension one foliations on others manifolds of dimension three.

4.1 - Simple singularities

Let $\omega$ be a germ of 1–form at $0\in {ℂ}^3$. We define the rotational of $\omega$ as the unique vector field $𝐗$ such that

$rot(\omega )=𝐗⟺d\omega ={ı}_{𝐗}dx\wedge dy\wedge dz,$

moreover $\omega$ is integrable if and only if $\omega (rot(\omega ))=0$.

Let $\omega$ be a germ of an integrable 1–form at $0\in {ℂ}^3$. We say that $0$ is a simple singularity of $\omega$ if $\omega (0)=0$ and either $d\omega (0)\ne 0$ or $d\omega$ has an isolated singularity at $0$. In the second case, these kind of singularities, are classified as follows

1. Logarithmic. The second jet $j_0^2(\omega )\ne 0$ and the linear part of $𝐗=rot(\omega )$ at $0$ has non zero eigenvalues.

2. Degenerated. The rotational has a zero eigenvalue, the other two are non zero and necessarily satisfies the relation ${\lambda }_1+{\lambda }_2=0$.

3. Nilpotent. The rotational vector field $𝐗,$ is nilpotent as a derivation.

The structure near simple singularity is known (Calvo-Andrade et al. ²⁰⁰⁴). If $p\in S$ is a simple singularity and $d\omega (p)=0$, then $p$ is a singular point of $S$.

Theorem 4.1. Let $\omega \mathrm{\in }{\mathrm{\Omega }}^{\mathrm{1}}\mathit{}\mathrm{(}{ℂ}^{\mathrm{3}}\mathrm{,}\mathrm{0}\mathrm{)}$, $n\mathrm{\ge }\mathrm{3}$, be a germ of integrable 1-form such that $\omega$ has a simple singularity at $\mathrm{0}$ then the tangent sheaf $ℱ\mathrm{=}K\mathit{}e\mathit{}r\mathit{}\mathrm{(}\omega \mathrm{)}$ is locally free at $\mathrm{0}$ and it is generated by$\mathrm{⟨}r\mathit{}o\mathit{}t\mathit{}\mathrm{(}\omega \mathrm{)}\mathrm{,}𝐒\mathrm{⟩}$, where $𝐒$ has non zero linear part.

Proof.Let $\omega$ be a germ at $0\in {ℂ}^3$ of an integrable 1–form and $0$ a simple non-Kupka singularity. Then $0\in {ℂ}^3$ is an isolated singularity of $𝐗=rot(\omega )$. Consider the Koszul complex of the vector field $𝐗$ at $0$

$𝕂{(𝐗)}_0:0\to {\mathrm{\Omega }}_{{ℂ}^3,0}^3\stackrel{{ı}_{𝐗}}{\to }{\mathrm{\Omega }}_{{ℂ}^3,0}^2\stackrel{{ı}_{𝐗}}{\to }{\mathrm{\Omega }}_{{ℂ}^3,0}^1\stackrel{{ı}_{𝐗}}{\to }{𝒪}_{{ℂ}^3,0}\to 0$

Since $\omega (𝐗)=0$, then $\omega \in H^1(𝕂{(𝐗)}_0)$ that vanishes because $𝐗$ has an isolated singularity at $0$. Therefore, there exists $\theta \in {\mathrm{\Omega }}_{{ℂ}^3,0}^2$ such that ${ı}_{𝐗}\theta =\omega$. The map${\mathrm{\Theta }}_{{ℂ}^3,0}\ni 𝐙\mapsto {ı}_{𝐙}dx\wedge dy\wedge dz\in {\mathrm{\Omega }}_{{ℂ}^3,0}^2$is an isomorphism, hence

$\omega ={ı}_{𝐗}\theta ,\text{and}\mathit{}\theta ={ı}_{𝐒}dx\wedge dy\wedge dz,\text{implies}\mathit{}\omega ={ı}_{𝐗}\theta ={ı}_{𝐗}{ı}_{𝐒}dx\wedge dy\wedge dz$

and then, the vector fields $\{𝐗,𝐒\}$ generate the sheaf $ℱ$ in a neighborhood of $0$. ∎

Let $\omega \in ℱ(3,d)$ be a foliation and $Z\subset S_2$ be a connected component of $S_2$. Assume that $Z$ is a local complete intersection and has only simple singularities. We will calculate the number $|NK(\omega )\cap Z|$ of non-Kupka points in $Z$.

Proof of Theorem 2. Let $𝒥$ be the ideal sheaf of $Z$. Since $Z$ is a local complete intersection, consider the exact sequence

$0\to 𝒥/{𝒥}^2\to {\mathrm{\Omega }}^1\otimes {𝒪}_Z\to {\mathrm{\Omega }}_Z^1\to 0$

Taking ${\wedge }^2$ and twisting by $L=K_{{\mathbb{P}}^3}^{-1}\otimes K_Z=K_Z(4)$ we get

$0\to {\wedge }^2𝒥/{𝒥}^2\otimes L\to {{\mathrm{\Omega }}_{{\mathbb{P}}^3}^2|}_Z\otimes L\to \mathrm{\cdots }$

Since $Z\subset S$, the singular set, we have seen before that

${d\omega |}_Z\in H^0(Z,{\wedge }^2(𝒥/{𝒥}^2)\otimes L)$

Now, from the equalities of sheaves

$K_Z^{-1}\otimes K_{{\mathbb{P}}^3}\simeq {\wedge }^2(𝒥/{𝒥}^2),\text{and}\mathit{}L\simeq K_{{\mathbb{P}}^3}^{-1}\otimes K_{ℱ}$

we have

$H^0(Z,{\wedge }^2(𝒥/{𝒥}^2)\otimes L)=H^0(Z,{K_Z^{-1}\otimes K_{ℱ}|}_Z),$

the non-Kupka points of $\omega$ in $Z$ satisfies ${d\omega |}_Z=0$, denoting

$D_{\omega }=\sum _{p\in Z}or{d}_p(d\omega )$

the associated divisor to ${d\omega |}_Z$, one has

$\mathrm{deg}(D_{\omega })=\mathrm{deg}(K_{ℱ})-\mathrm{deg}(K_Z),$

as claimed. ∎

Remark 4.2. The method of the proof works also in projective manifolds, and does not depends on the integrability condition.

4.2 - Examples

We apply Theorem 2 for some codimension one holomorphic foliations on ${\mathbb{P}}^3$ and determine the number of non-Kupka points.

Example 4.3 (Degree two logarithmic foliations). Recall that the canonical bundle of a degree two foliation of ${\mathbb{P}}^3$ is trivial. There are two irreducible components of logarithmic foliations in the space of foliations of ${\mathbb{P}}^3$ of degree two:$ℒ(1,1,2)$ and $ℒ(1,1,1,1)$. We analyze generic foliations on each component.

Component $ℒ\mathbf{}\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{)}$: let $\omega$ be a generic element of $ℒ(1,1,2)$ and consider its singular scheme $S=S_2\cup S_3$. By Theorem 3 of Cukierman et al. (²⁰⁰⁶), we have $\mathrm{\ell }(S_3)=2$. On the other hand, $S_2$ has three irreducible components, two quadratics and a line, the arithmetic genus is $p_a(S_2)=2$. Note that Theorem 2, implies that the number $|NK(\omega )\cap S_2|$, of non-Kupka points in $S_2$ is

$|NK(\omega )\cap S_2|=deg(D_{\omega })=\mathrm{deg}(K_{ℱ})-\mathrm{deg}(K_{S_2})=-\chi (S_2)=2.$

The non-Kupka points of the foliation ${ℱ}_{\omega }$ are $|NK(\omega )|=\mathrm{\ell }(S_3)+|NK\cap S_2|=4$.

Component $ℒ\mathbf{}\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{)}$: let $\omega$ be a generic element of $ℒ(1,1,1,1)$ then the tangent sheaf is $𝒪\oplus 𝒪$ and the singular scheme $S=S_2$(Giraldo and Pan-Collantes ²⁰¹⁰), moreover consists of 6 lines given the edges of a tetrahedron, obtained by intersecting any two of the four invariant hyperplanes $H_i$. The arithmetic genus is $p_a(S_2)=3$, by Theorem 2, $|NK(\omega )|=|NK(\omega )\cap S_2|=4$, corresponding to the vertices of the tetrahedron where there are simple singularities of logarithmic type.

Example 4.4 (The exceptional component $ℰ\mathbf{}\mathrm{(}\mathrm{3}\mathrm{)}$). The leaves of a generic foliation$\omega \in ℰ(3)\subset ℱ(3,2),$ are the orbits of an action of $\mathrm{𝐀𝐟𝐟}(ℂ)\times {\mathbb{P}}^3\to {\mathbb{P}}^3$ and its tangent sheaf is $𝒪\oplus 𝒪$ (see Calvo-Andrade et al. ²⁰⁰⁴; Giraldo and Pan-Collantes ²⁰¹⁰). Its singular locus $S=S_2$ has $deg(S)=6$ and three irreducible components: a line $L,$ a conic $C$ tangent to $L$ at a point $p$, and a twisted cubic $\mathrm{\Gamma }$ with $L$ as an inflection line at $p$. Then $NK(\omega )=L\cap C\cap \mathrm{\Gamma }=\{p\}\subset S$.

The arithmetic genus is $p_a(S)=3$ and the canonical bundle of the foliation again is trivial, by Theorem 2, the number of non-Kupka points $|NK(\omega )|=4$. Therefore the non-Kupka divisor $NK(\omega )\cap S=4p$.If $\omega$ represents the foliation at $p$, then $\mu (d\omega ,p)=\mu (rot(\omega ),p)=4.$

ACKNOWLEDGMENTS

The first author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP Nº 2014/23594-6), CONACYT 262121 and thanks the Federal University of Minas Gerais (UFMG), Instituto de Matemática Pura e Aplicada (IMPA), and Instituto de Matemática, Estatística e Computação Científica da UNICAMP for the hospitality during the elaboration of this work. The second author was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-DGU 247/11), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq 300351/2012-3 and PPM-00169-13). The third author was partially supported by Bolsista/CAPES and thanks the Instituto de Matemática y Ciencias Afines (IMCA) for the hospitality. Finally, we would like to thank the referee by the suggestions, comments and improvements to the exposition.

APPENDIX

We prove Lemma 2.5.

Proof.First, we see that $h^0(F)\ge 1$. By Riemann–Roch–Hirzebruch, we have

$\chi (F)=h^0(F)-h^1(F)+h^2(F)={[ch(F)\cdot Td({\mathbb{P}}^2)]}_2=2,$

then

$h^0(F)+h^2(F)={[ch(F)\cdot Td({\mathbb{P}}^2)]}_2+h^1(F)\ge {[ch(F)\cdot Td({\mathbb{P}}^2)]}_2=2$

By Serre duality (Griffiths and Harris ¹⁹⁷⁸; Okonek et al. ¹⁹⁸⁰), we get $h^2(F)=h^0(F(-3))$. Moreover $h^0(F)\ge h^0(F(-k))$ for all $k>0$, hence $h^0(F)\ge 1$.Let $\tau \in H^0(F)$ be a non zero section, consider the exact sequence

$0⟶𝒪\stackrel{\cdot \tau }{⟶}F⟶𝒬⟶0\mathit{}\text{with}\mathit{}𝒬=F/𝒪.$ (A.1)

The sheaf $𝒬$ is torsion free, therefore $𝒬\simeq {𝒥}_{\mathrm{\Sigma }}$ for some $\mathrm{\Sigma }\subset {\mathbb{P}}^2$. The sequence (A.1), is a free resolution of the sheaf $𝒬$ with vector bundles with zero Chern classes. From the definition of Chern classes for coherent sheaves (Baum and Bott ¹⁹⁷²), we get $c(𝒬)=1$, in particular $\mathrm{deg}(\mathrm{\Sigma })=c_2(𝒬)=0$, we conclude that $\mathrm{\Sigma }=\mathrm{\varnothing }$ and$𝒬\simeq 𝒪$. Then $F$ is an extension of holomorphic line bundles, hence it splits (Okonek et al. ¹⁹⁸⁰, p. 15).∎

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Publication Dates

• Publication in this collection
  Dec 2016

History

• Received
  13 Jan 2016
• Accepted
  15 Aug 2016