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## versão impressa ISSN 0001-3765versão On-line ISSN 1678-2690

### An. Acad. Bras. Ciênc. vol.88 no.4 Rio de Janeiro out./dez. 2016

#### http://dx.doi.org/10.1590/0001-3765201620160013

Mathematical Sciences

On non-Kupka points of codimension one foliations on ℙ3

1Centro de Investigaciones en Matemáticas/CIMAT, A.C., Jalisco, s/n, Col. Valenciana CP 36023 Guanajuato, Gto, México

2Universidade Federal de Minas Gerais, Departamento de Matemática, Av. Antônio Carlos, 6627, 31270-901 Belo Horizonte, MG, Brazil

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.

Key words: 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,fα,ψαβ)} where

1. 𝔘={Uα} is an open cover of M .

2. fα:Uα is a holomorphic submersion for each α .

3. A family of biholomorphisms {ψαβ:fβ(Uαβ)fα(Uαβ)} such that

ψαβ=ψβα-1,fβ|UαUβ=ψβαfα|UαUβ and ψαγ=ψαβψβγ.

Since dfα(x)=ψαβ(fβ(x))dfβ(x), the set F=αKer(dfα)TM is a subbundle. Also [ψαβ(fβ)]Hˇ1(𝔘,𝒪) define a line bundle N=TM/F . The family of 1–forms {dfα} glue to a global section ωH0(M,Ω1(N)) . We have

0FTM{dfα}N0,0Θ{dfα}𝒩0,[,]

where =𝒪(F),Θ=𝒪(TM)and𝒩=𝒪(N) . We also obtain

nTM=det(F)N,ΩMn:=KM=det()𝒩,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 =(S,) , where SM is an analytic subset of codim(S)2 , and is a regular codimension one holomorphic foliation on MS .

2. A class of global sections [ω]H0(M,Ω1(L)) , where LPic(M) such that

1. the singular set Sω={pM|ωp=0} has codim(Sω)2 .

2. ωdω=0 in H0(M,Ω3(L2)) .

In this case, we denote by ω=(Sω,ω) the foliation represented by ω .

3. An exact sequence of sheaves

0Θ𝒩0,[,]

where is a reflexive sheaf of rank rk()=n-1 with torsion free quotient 𝒩𝒥SL, where 𝒥S is an ideal sheaf for some closed scheme S .

These three definitions are equivalents.

Remark 1.2. Let ωH0(M,Ω1(L)) be a section.

1. The section ω may be defined by a family of 1-forms

ωαΩ1(Uα),ωα=λαβωβinUαβ=UαUβ,L=[λαβ]Hˇ1(𝔘,𝒪).

2. The section ω is a morphism of sheaves Θ𝜔L . The kernel of ω is the tangent sheaf . The image of ω is a twisted ideal sheaf 𝒩=𝒥SωL . It is called the normal sheaf of .

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

KM=ΩMn=det()𝒩=KL-1,det(N)L

where KM,K=det() are the canonical sheaf of M and respectively.

We denote by

(M,L)={[ω]H0(M,Ω1(L))|codim(Sω)2,ωdω=0}(n,d)={[ω]H0(n,Ω1(d+2))|codim(Sω)2,ωdω=0}.

The number d0 is called the degree of the foliation represented by ω .

1-1 Statement of the results

In the sequel, M is a compact complex manifold with dim(M)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=k=2nSk where codim(Sk)=k.

For a foliation on M represented by ω(M,L), the Kupka set (Kupka 1964; De Medeiros 1977) is defined by

K(ω)={pM|ω(p)=0,dω(p)0}.

We recall that for points near K(ω) the foliation 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(ω)S2 .

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

NK(ω)={pM|ω(p)=0,dω(p)=0}S3Sn.

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

1. S2=K(ω) , then NK(ω)=S3Sn .

2. There is an irreducible component ZS2 such that ZK(ω)= .

3. For a foliation ω(3,d) . Let ZS2 be a connected component such that ZZK(ω) 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 ω(n,d) be a foliation with K(ω)=S2 and connected, then ω 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(ω)=Sn . In this note, we present a new and short proof of this fact when the transversal type of K(ω) is radial.

In the second section, we study the case of a non-Kupka irreducible component of S2 . These phenomenon arise naturally in the intersection of irreducible components of (M,L) . The following result is a local normal form for ω near the singular set and is a consequence of a result of Loray (2006).

Theorem 1. Let ωΩ1(n,0) , n3 , be a germ of integrable 1–form such that 𝑐𝑜𝑑𝑖𝑚(Sω)=2 , 0Sω is a smooth point and dω=0 on Sω . If j01ω0 , then or either

1. there exists a coordinate system (x1,,xn)n such that

j01(ω)=x1dx2+x2dx1

and ω 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 (x1,,xn)n such that

ω=x1dx1+g1(x2)(1+x1g2(x2))dx2,

such that g1,g2𝒪,0 with g1(0)=g2(0)=0 , or

3. ω has a non-constant holomorphic first integral in a neighborhood of 0n .

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 ω be a germ of a 1-form at 03 defined by

ω=xdx+(1+xf)df

where f(x,y,z)=y2z . We have

ω=xdx+2yz(1+xy2z)dy+y2(1+xy2z)dz.

The singular set of ω is {x=y=0} and {x=z=y2=0} , therefore the singular set has an embedding point {x=z=y2=0} and dω vanish along {x=y=0} . We will show that ω has a holomorphic first integral F in a neighborhood of 03 . In fact, let t=f(x,y,z)=y2z and set φ:(3,0)(2,0) defined by

φ(x,y,z)=(x,t).

Let η=xdx+(1+xt)dt be 1-form at 02 , note that ω=φ*(η) and moreover η(0,0)0 , this implies that η is non-singular at 02 and by Frobenius theorem η has a holomorphic first integral H(x,t) on (2,0) . Defining H1(x,y,z):=H(x,f(x,y,z))=H(x,y2z) , we get H1 is a holomorphic first integral for ω in a neighborhood of 03 .

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 ω(3,d) . Let Z be a connected component of S2 . We count the number |ZNK(ω)| of non-Kupka points of ω in ZS2 .

Theorem 2. Let ω(3,d) be a foliation and ZS2 a connected component of S2 . Suppose that Z is a local complete intersection and ZZK(ω) is a finite set of points, then dω|Z is a global section of KZ-1K|Z and the associated divisor Dω=pZordp(dω)p has degree

deg(Dω)=deg(K)-deg(KZ).

Note that the section dω|Z vanishes exactly in the non-Kupka points of ω in Z then the above theorem determine the number |ZNK(ω)| (counted with multiplicity) of non-Kupka points of ω in Z .

2 - The singular set

Let ω(M,L) be a codimension one holomorphic foliation then singular set of ω may be written as

S=j=2nSj where codim(Sj)=j.

The fact that K(ω)S2 implies that S3SnNK(ω) . To continue we focus in the components of singular set of ω 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 ω be a germ at 0n , n3 of an integrable 1–form singular at 0 , if 𝑐𝑜𝑑𝑖𝑚(Sω)3 , then there exist f𝒪n,0 and g𝒪n,0 such that

ω=gdf on a neighborhood of0n.

We have the following proposition.

Proposition 2.2. Let ω(M,L) be a foliation and let pSn an isolated singularity, then any germ of vector field tangent to the foliation vanishes at p .

Proof. Let ω=gdf,g𝒪p*,f𝒪p be a 1-form representing the foliation at p . Let 𝐗Θp be a vector field tangent to the foliation, i.e., ω(𝐗)=0 .If 𝐗(p)0 there exists a coordinate system with z(p)=0 and 𝐗=/zn , then

0=ω(𝐗)=g(i=1n(f/zi)dzi(/zn))=g(f/zn),thereforef/zn0,

and f=f(z1,,zn-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 ω . Note that, given a section ωH0(M,Ω1(L)) , along the singular set, the equation ωα=λαβωβ implies dωα|S=(λαβdωβ)|S. Then

{dωα}H0(S,(ΩM2L)|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 ω(M,L) . The Kupka set is

K(ω)={pM|ω(p)=0,dω(p)0}.

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

1. K(ω) is smooth of codimension two.

2. K(ω) has local product structure and the tangent sheaf is locally free near K(ω) .

3. K(ω) is subcanonically embedded and

2NK(ω)=L|K(ω),KK(ω)=(KML)|K(ω)=K|K(ω).

Let ω(n,d) be a foliation with S2=K(ω) .By Calvo-Andrade and Soares (1994), there exists a pair (V,σ) , where V is a rank two holomorphic vector bundle and σH0(n,V) , such that

0𝒪𝜎V𝒥K(d+2)0 with {σ=0}=K

and the total Chern class

c(V)=1+(d+2)𝐡+deg(K(ω))𝐡2H(n,)[𝐡]/𝐡n+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 ω(n,d) be a foliation with S2=K(ω) , (connected if n=3 ) and of radial transversal type. Then K(ω) is a complete intersection and ω 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 2 with c1(F)=0 and c2(F)=0 . Then F𝒪𝒪 , is holomorphically trivial.

Now, we prove Proposition 2.4.

Proof of Proposition 2.4. Let (V,σ) 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)𝐡+(d+2)24𝐡2=(1+(d+2)𝐡2)2H(n,)[𝐡]/𝐡n+1.

The vector bundle E=V(-d+22) , has c1(E)=0 and c2(E)=0 . Let ξ:2n be a linear embedding. By the preceding lemma we have

ξE𝒪2𝒪2

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

E𝒪n𝒪n

is trivial and hence V splits as 𝒪n(d+22)𝒪n(d+22) and K is a complete intersection. The existence of the meromorphic first integral follows from Theorem A of (Cerveau and Lins Neto 1994). ∎

If ω is such that K(ω)=S2 and connected, the set of non-Kupka points of ω is

NK(ω)=S3Sn.

A generic rational map, that means, a Lefschetz or a Branched Lefschetz Pencil φ:n1 , has only isolated singularities away its base locus. The singular set of the foliation defined by the fibers of φ is SnS2 . The Kupka set corresponds away from its base locus and Sn=NK(ω) are the singularities as a map. Sn is empty if and only if the degree of the foliation is 0 . The number (Sn) of isolated singularities counted with multiplicities can be calculated by (Cukierman et al. 2006). If ωp is a germ of form that defines the foliation at pSn , we have

(Sn)=pSnμ(ωp,p),μ(ω,p)=dim𝒪p(ω1,,ωn),ωp=i=1nωidzi.

We have that cn()=(Sn) .

3 - Foliations with a non-Kupka component

It is well known that K(ω){pM|jp1ω0} , 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 S2 .

Proof of Theorem 1. By hypotheses, dω(p)=0 for any pSω . Since

ω=ω1+,dω=dω1+=0,

we get dω1(p)=0 for any pSω . Now, as ω10 and codim(Sω)=2 , we have 1codim(Sω1)2 . We distinguish two cases.

1. codim(Sω1)=2 : there is a coordinate system (x1,,xn)n such that

ω1=x1dx2+x2dx1.

2. codim(Sω1)=1 : there is a coordinate system (x,ζ)×n-1 such that x(p)=0 and ω1=xdx .

The first case is known, the foliation ω is equivalent in a neighborhood of 0n 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,ζ)×n-1 , a germ f𝒪n-1,0 with f(0)=0 , and germs g,h𝒪,0 such that the foliation is defined by the 1–form

ω=xdx+[g(f(ζ))+xh(f(ζ))]df(ζ). (5.1)

Since Sω1={x=0} and 0Sω is a smooth point, we can assume that Sω,p={x=ζ1=0} , where Sω,p is the germ of Sω at p=0 . Therefore,

Sω,p={x=ζ1=0}={x=g(f(ζ))=0}{x=fζ1==fζn-1=0}.

Hence, either g(0)=0 and ζ1|f , or g(0)0 and ζ1|fζj for all j=1,,n-1 . In any case, we have ζ1|f and then f(ζ)=ζ1kψ(ζ) , where ψ is a germ of holomorphic function in the variable ζ ; k and ζ1 does not divide ψ . We have two possibilities:

1st case.– ψ(0)0 . In this case, we consider the biholomorphism

G(x,ζ)=(x,ζ1ψ1/k(ζ),ζ2,,ζn)=(x,y,ζ2,,ζn)

where ψ1/k is a branch of the kth root of ψ , we get fG-1(x,y,ζ2,,ζn)=yk and

G*(ω)=xdx+(g(yk)+xh(yk))kyk-1dy=xdx+(g1(y)+xh1(y))dy,

where g1(y)=kyk-1g(yk) , h1(y)=kyk-1h(yk) . Therefore, ω~:=G*(ω) is equivalent to ω and moreover ω~ is given by

ω~=xdx+(g1(y)+xh1(y))dy with Sω~={x=g1(y)=0}. (5.2)

Since dω~=h1(y)dxdy is zero identically on {x=g1(y)=0} ,we get g1|h1 , so that h1(y)=(g1(y))mH(y) , for some m and such that H(y) does not divided g1(y) . Using the above expression for h1 in (3.2), we have

ω~=xdx+g1(y)(1+x(g1(y))m-1H(y))dy=xdx+g1(y)(1+xg2(y))dy,

where g2(y)=(g1(y))m-1H(y) . Consider φ:(,0)×(n-1,0)(2,0) defined by φ(x,ζ)=(x,y) , then

ω=φ(xdx+g1(y)(1+xg2(y))dy). (5.3)

2nd case.– ψ(0)=0 . We have Sω,p={x=ζ1=0} and

ω=xdx+(g(ζ1kψ)+xh(ζ1kψ))d(ζ1kψ), (5.4)

therefore

ω=xdx+(g(ζ1kψ)+xh(ζ1kψ))ζ1k-1(kψdζ1+ζ1dψ). (5.5)

Note that g(0)0 , otherwise {x=ζ1ψ(ζ)=0} would be contained in Sω,p , but it is contradiction because Sω,p={x=ζ1=0}{x=ζ1ψ(ζ)=0} . Furthermore k2 , because otherwise ζ1|ψ .

Let φ:(,0)×(n-1,0)(2,0) be defined by

φ(x,ζ)=(x,ζ1kψ(ζ))=(x,t),

then from (3.4), we get that

ω=φ(η),

where η=xdx+(g(t)+xh(t))dt . Since η(0,0)=g(0)dt0 , we deduce that η has a non-constant holomorphic first integral F𝒪2,0 such that dF(0,0)0 . Therefore, F1(x,ζ)=F(x,ζ1kψ(ζ)) is a non-constant holomorphic first integral for ω in a neighborhood of 0n .∎

3.2 - Applications to foliations on 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 𝒢ω=(S,𝒢) be a codimension one holomorphic foliation represented by ωH0(M,Ω1(L)) . We have the exact sequence

0𝒢ΘM𝜔𝒩𝒢0,𝒩𝒢𝒥SL.

Set M0=MS and take p0M0 . Then in a neighborhood Uα of p0 the foliation 𝒢 is induced by a holomorphic 1 –form ωα and there exists a differentiable 1 –form θα such that

dωα=θαωα

Let Z be an irreducible component of S2 . Take a generic point pZ ,that is, p is a point where Z is smooth and disjoint from the other singular components. Pick Bp a ball centered at p sufficiently small, so that S(Bp) is a sub-ball of Bp of codimension 2 . Then the De Rhamclass can be integrated over an oriented 3 -sphere LpBp* positively linked with S(Bp) :

BB(𝒢,Z)=1(2πi)2Lpθdθ.

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

c1(L)2=c12(𝒩𝒢)=ZS2𝐵𝐵(𝒢,Z)[Z],

where 𝒩𝒢=𝒥SL is the normal sheaf of 𝒢 on M and the sum is done over all irreducible components of S2 .

In particular, if 𝒢 is a codimension one foliation on n of degree d , then the normal sheaf 𝒩𝒢=𝒥S(d+2) and the Baum-Bott Theorem looks as follows

ZS2BB(𝒢,Z)deg[Z]=(d+2)2.

Remark 3.2. If there exist a coordinates system (U,(x,y,z3,,zn)) around pZS2 such that x(p)=y(p)=0 and S(𝒢)U=ZU={x=y=0} . Moreover, if we assume that

ω|U=P(x,y)dy-Q(x,y)dx

is a holomorphic 1-form representing 𝒢|U . Then we can consider the 𝒞 (1,0)-form θ on UZ given by

θ=(Px+Qy)|P|2+|Q|2(P¯dx+Q¯dy).

Since dω=θω , we get

BB(𝒢,Z)=1(2πi)2Lpθdθ=Res0{Tr(D𝐗)dxdyPQ}, (5.6)

where Res0 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

BB(𝒢,Z)=Tr(D𝐗(p))2det(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)x+Q(x,y)y,z3,,zn

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 ω(M,L) be a foliation and ZS2K(ω) . Suppose that Z is smooth and jp1ω0 for all pZ , then 𝐵𝐵(ω,Z)=0 .

Proof. We work in a small neighborhood U of pZM . According to Theorem 1 there exist a coordinate system (x,y,z3,,zn) at p such that ZU={x=y=0} and one has three cases. In the first case, ω is the product of a dimension one foliation in a section transversal to Z by a regular foliation of codimension two and jp1(ω)=xdy+ydx . In this case, it follows from (3.6) that BB(ω,Z)=0 . In the second case

ω=xdx+g1(y)(1+xg2(y))dy,

where g1,g2𝒪,0 and it follows from Lemma 3.9 of Cerveau and Lins Neto (2013) that

BB(ω,Z)=Rest=0[(g1(t)g2(t))2dtg1(t)]=Rest=0[g1(t)(g2(t))2].

Since g1(y)(g2(y))2 is holomorphic at y=0 , we get BB(ω,Z)=0. In the third case ω has a holomorhic first integral in neighborhood of p and is known that BB(ω,Z)=0.

The Baum-Bott formula implies the following result.

Corollary 3.4. Let ω(n,d) , n3 , be a foliation with K(ω)= . Then there exists a smooth point pS2 such that jp1ω=0 .

Proof. If for all smooth point pS2 one has jp1ω0 , the above theorem shows that BB(ω,Z)=0 for all irreducible components ZS2 . By Baum–Bott’s theorem, we get

0<(d+2)2=ZS2BB(ω,Z)deg[Z]=0

which is a contradiction. Therefore there exists a smooth point pS2 such that jp1ω=0 . ∎

In particular, if ω(n,d) , n3 , is a foliation with jp1ω0 for any pn , then its Kupka set is not empty.

4 - The number of non-Kupka points

Through this section, we consider codimension one foliations on 3 , but some results remain valid to codimension one foliations on others manifolds of dimension three.

4.1 - Simple singularities

Let ω be a germ of 1–form at 03 . We define the rotational of ω as the unique vector field 𝐗 such that

rot(ω)=𝐗dω=ı𝐗dxdydz,

moreover ω is integrable if and only if ω(rot(ω))=0 .

Let ω be a germ of an integrable 1–form at 03 . We say that 0 is a simple singularity of ω if ω(0)=0 and either dω(0)0 or dω has an isolated singularity at 0 . In the second case, these kind of singularities, are classified as follows

1. Logarithmic. The second jet j02(ω)0 and the linear part of 𝐗=rot(ω) 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 λ1+λ2=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 pS is a simple singularity and dω(p)=0 , then p is a singular point of S .

Theorem 4.1. Let ωΩ1(3,0) , n3 , be a germ of integrable 1-form such that ω has a simple singularity at 0 then the tangent sheaf =Ker(ω) is locally free at 0 and it is generated by rot(ω),𝐒 , where 𝐒 has non zero linear part.

Proof.Let ω be a germ at 03 of an integrable 1–form and 0 a simple non-Kupka singularity. Then 03 is an isolated singularity of 𝐗=rot(ω) . Consider the Koszul complex of the vector field 𝐗 at 0

𝕂(𝐗)0:0Ω3,03ı𝐗Ω3,02ı𝐗Ω3,01ı𝐗𝒪3,00

Since ω(𝐗)=0 , then ωH1(𝕂(𝐗)0) that vanishes because 𝐗 has an isolated singularity at 0 . Therefore, there exists θΩ3,02 such that ı𝐗θ=ω . The map Θ3,0𝐙ı𝐙dxdydzΩ3,02 is an isomorphism, hence

ω=ı𝐗θ,and θ=ı𝐒dxdydz,implies ω=ı𝐗θ=ı𝐗ı𝐒dxdydz

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

Let ω(3,d) be a foliation and ZS2 be a connected component of S2 . Assume that Z is a local complete intersection and has only simple singularities. We will calculate the number |NK(ω)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

0𝒥/𝒥2Ω1𝒪ZΩZ10

Taking 2 and twisting by L=K3-1KZ=KZ(4) we get

02𝒥/𝒥2LΩ32|ZL

Since ZS , the singular set, we have seen before that

dω|ZH0(Z,2(𝒥/𝒥2)L)

Now, from the equalities of sheaves

KZ-1K32(𝒥/𝒥2),and LK3-1K

we have

H0(Z,2(𝒥/𝒥2)L)=H0(Z,KZ-1K|Z),

the non-Kupka points of ω in Z satisfies dω|Z=0 , denoting

Dω=pZordp(dω)

the associated divisor to dω|Z , one has

deg(Dω)=deg(K)-deg(KZ),

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 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 3 is trivial. There are two irreducible components of logarithmic foliations in the space of foliations of 3 of degree two:(1,1,2) and (1,1,1,1). We analyze generic foliations on each component.

Component (1,1,2): let ω be a generic element of (1,1,2) and consider its singular scheme S=S2S3. By Theorem 3 of Cukierman et al. (2006), we have (S3)=2. On the other hand, S2 has three irreducible components, two quadratics and a line, the arithmetic genus is pa(S2)=2. Note that Theorem 2, implies that the number |NK(ω)S2|, of non-Kupka points in S2 is

|NK(ω)S2|=deg(Dω)=deg(K)-deg(KS2)=-χ(S2)=2.

The non-Kupka points of the foliation ω are |NK(ω)|=(S3)+|NKS2|=4.

Component (1,1,1,1): let ω be a generic element of (1,1,1,1) then the tangent sheaf is 𝒪𝒪 and the singular scheme S=S2(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 Hi. The arithmetic genus is pa(S2)=3, by Theorem 2, |NK(ω)|=|NK(ω)S2|=4, corresponding to the vertices of the tetrahedron where there are simple singularities of logarithmic type.

Example 4.4 (The exceptional component (3)). The leaves of a generic foliationω(3)(3,2), are the orbits of an action of 𝐀𝐟𝐟()×33 and its tangent sheaf is 𝒪𝒪 (see Calvo-Andrade et al. 2004; Giraldo and Pan-Collantes 2010). Its singular locus S=S2 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 Γ with L as an inflection line at p. Then NK(ω)=LCΓ={p}S.

The arithmetic genus is pa(S)=3 and the canonical bundle of the foliation again is trivial, by Theorem 2, the number of non-Kupka points |NK(ω)|=4. Therefore the non-Kupka divisor NK(ω)S=4p.If ω represents the foliation at p, then μ(dω,p)=μ(rot(ω),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 h0(F)1 . By Riemann–Roch–Hirzebruch, we have

χ(F)=h0(F)-h1(F)+h2(F)=[ch(F)Td(2)]2=2,

then

h0(F)+h2(F)=[ch(F)Td(2)]2+h1(F)[ch(F)Td(2)]2=2

By Serre duality (Griffiths and Harris 1978; Okonek et al. 1980), we get h2(F)=h0(F(-3)) . Moreover h0(F)h0(F(-k)) for all k>0 , hence h0(F)1 .Let τH0(F) be a non zero section, consider the exact sequence

0𝒪τF𝒬0 with 𝒬=F/𝒪. (A.1)

The sheaf 𝒬 is torsion free, therefore 𝒬𝒥Σ for some Σ2 . 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 deg(Σ)=c2(𝒬)=0 , we conclude that Σ= and 𝒬𝒪 . Then F is an extension of holomorphic line bundles, hence it splits (Okonek et al. 1980, p. 15).∎

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Received: January 13, 2016; Accepted: August 15, 2016

*Correspondence to: Maurício Côrrea E-mail: mauriciomatufmg@gmail.com

Dedicated to José Seade in his 60 birthday.