On non-Kupka points of codimension one foliations on P 3
aa r X i v : . [ m a t h . AG ] A ug ON NON-KUPKA POINTS OF CODIMENSION ONEFOLIATIONS ON P OMEGAR CALVO-ANDRADE, MAUR´ICIO CORRˆEA,AND ARTURO FERN ´ANDEZ–P´EREZ
To Jos´e Seade in his 60 birthday
Abstract.
We study the singular set of a codimension one holomorphic foliation on P . We find a local normal form for these foliations near a codimension two component ofthe singular set that is not of Kupka type. We also determine the number of non-Kupkapoints immersed in a codimension two component of the singular set of a codimension onefoliation on P . Resumo.
Estudamos o conjunto singular de uma folhea¸c˜ao holomorfa de codimens˜aoum em P . Encontramos uma forma normal local para tais folhea¸c˜oes em torno deuma componente de codimens˜ao dois do seu conjunto singular que n˜ao ´e do tipo Kupka.Tamb´em, determinamos o n´umero de pontos n˜ao–Kupka imersos numa componente decodimens˜ao dois de uma folhea¸c˜ao de codimens˜ao um em P . Introduction A regular codimension one holomorphic foliation on a complex manifold M , canbe defined by a triple { ( U , f α , ψ αβ ) } where(i) U = { U α } is an open cover of M .(ii) f α : U α → C is a holomorphic submersion for each α .(iii) A family of biholomorphisms { ψ αβ : f β ( U αβ ) → f α ( U αβ ) } such that ψ αβ = ψ − βα , f β | U α ∩ U β = ψ βα ◦ f α | U α ∩ U β and ψ αγ = ψ αβ ◦ ψ βγ . Since df α ( x ) = ψ ′ αβ ( f β ( x )) · df β ( x ) , the set F = [ α Ker ( df α ) ⊂ T M is a subbundle.Also [ ψ ′ αβ ( f β )] ∈ ˇ H ( U , O ∗ ) define a line bundle N = T M/F . The family of 1–forms { df α } , glue to a section ω ∈ H ( M, Ω ( N )). We get0 → F → T M { df α } −−−→ N → , → F → Θ { df α } −−−→ N → , [ F , F ] ⊂ F where F = O ( F ) , Θ = O ( T M ) , and N = O ( N ). We also have ∧ n T M ∗ = det ( F ∗ ) ⊗ N ∗ , Ω nM := K M = det ( F ∗ ) ⊗ N ∗ , n = dim ( M ) . Mathematics Subject Classification.
Key words and phrases.
Holomorphic foliations, non-Kupka points, ample vector bundle. Fol-hea¸c˜oes holomorfas, pontos n˜ao-kupka, fibrados vetoriais amplos.Calvo-Andrade: FAPESP n o Definition 1.1.
Let M be a compact complex manifold with dim( M ) = n . A singular codimension one holomorphic foliation on M , may be defined by one ofthe following ways:(1) A pair F = ( S, F ), where S ⊂ M is an analytic subset of codim( S ) ≥ F is a regular codimension one holomorphic foliation on M \ S .(2) A class of sections [ ω ] ∈ P H ( M, Ω ( L )) where L ∈ P ic ( M ), such that(i) the singular set S ω = { p ∈ M | ω p = 0 } has codim( S ω ) ≥ ω ∧ dω = 0 in H ( M, Ω ( L ⊗ )).We denote by F ω = ( S ω , F ω ) the foliation represented by ω .(3) An exact sequence of sheaves0 → F → Θ → N → , [ F , F ] ⊂ F where F is a reflexive sheaf of rank rk ( F ) = n − N ≃ J S ⊗ L, where J S is an ideal sheaf for some closed scheme S .These three definitions are equivalents. Remark 1.2.
Let ω ∈ H ( M, Ω ( L )) be a section.(i) The section ω may be defined by a family of 1-forms ω α ∈ Ω ( U α ) , ω α = λ αβ ω β in U αβ = U α ∩ U β , L = [ λ αβ ] ∈ ˇ H ( U , O ∗ ) . (ii) The section ω is a morphism of sheaves Θ ω −→ L . The kernel of ω is thetangent sheaf F . The image of ω is a twisted ideal sheaf N = J S ω ⊗ L . Itis called the normal sheaf .(iii) As in the non singular case, the following equality of line bundles holds K M = Ω nM = det ( F ∗ ) ⊗ N ∗ = K F ⊗ L − , det ( N ) ≃ L where K M , K F = det ( F ∗ ) are the canonical sheaf of M and F respectively.We denote by F ( M, L ) = { [ ω ] ∈ P H ( M, Ω ( L )) | codim( S ω ) ≥ , ω ∧ dω = 0 }F ( n, d ) = { [ ω ] ∈ P H ( P n , Ω ( d + 2)) | codim( S ω ) ≥ , ω ∧ dω = 0 } . The number d ≥ degree of the foliation represented by ω .1.1. Statement of the results.
In the sequel, M is a compact complex manifoldwith dim ( M ) ≥
3. We will use any of the above definitions for foliation. Thesingular set will be denoted by S . Observe that S decomposes as S = n [ k =2 S k where codim( S k ) = k. For a foliation F on M represented by ω ∈ F ( M, L ) , the Kupka set [15, 17] isdefined by K ( ω ) = { p ∈ M | ω ( p ) = 0 , dω ( p ) = 0 } . We recall that for points near K ( ω ) the foliation F is biholomorphic to a prod-uct of a dimension one foliation in a transversal section by a regular foliation ofcodimension two [15] and in particular we have K ( ω ) ⊂ S .In this note, we focus our attention on the set of non-Kupka points NK ( ω ) of ω .The first remark is NK ( ω ) = { p ∈ M | ω ( p ) = 0 , dω ( p ) = 0 } ⊃ S ∪ · · · ∪ S n . We analyze three cases, one in each section, the last two being the core of the work.
N NON-KUPKA POINTS OF CODIMENSION ONE FOLIATIONS 3 (1) S = K ( ω ), then NK ( ω ) = S ∪ · · · ∪ S n .(2) There is an irreducible component Z ⊂ S such that Z ∩ K ( ω ) = ∅ .(3) For a foliation ω ∈ F (3 , d ). Let Z ⊂ S be a connected component suchthat Z \ Z ∩ K ( ω ) is a finite set of points.The first case has been considered in [3, 5, 6, 7, 10]. Let ω ∈ F ( n, d ) be a foliationwith K ( ω ) = S and connected, then ω has a meromorphic first integral. In thegeneric case, the leaves define a Lefschetz or a Branched Lefschetz Pencil . Thenon-Kupka points are isolated singularities NK ( ω ) = S n . In this note, we presenta 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 S . These phenomenon arise naturally in the intersection of irreducible componentsof F ( M, L ). The following result is a local normal form for ω near the singular setand is a consequence of a result of F. Loray [16]. Theorem 1.
Let ω ∈ Ω ( C n , , n ≥ , be a germ of integrable 1–form such thatcodim ( S ω ) = 2 , ∈ S ω is a smooth point and dω = 0 on S ω . If j ω = 0 , then oreither (1) there exists a coordinate system ( x , . . . , x n ) ∈ C n such that j ( ω ) = x dx + x dx and F ω is biholomorphic to the product of a dimension one foliation in atransversal section by a regular foliation of codimension two, or (2) there exists a coordinate system ( x , . . . , x n ) ∈ C n such that ω = x dx + g ( x )(1 + x g ( x )) dx , such that g , g ∈ O C , with g (0) = g (0) = 0 , or (3) ω has a non-constant holomorphic first integral in a neighborhood of ∈ C n . The alternatives are not exclusives. The following example was suggest by thereferee 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 0 ∈ C defined by ω = xdx + (1 + xf ) df where f ( x, y, z ) = y z . We have ω = xdx + 2 yz (1 + xy z ) dy + y (1 + xy z ) dz. The singular set of ω is { x = y = 0 } and { x = z = y = 0 } , therefore the singularset has an embedding point { x = z = y = 0 } and dω vanish along { x = y = 0 } .We will show that ω has a holomorphic first integral F in a neighborhood of 0 ∈ C .In fact, let t = f ( x, y, z ) = y z and set ϕ : ( C , → ( C ,
0) defined by ϕ ( x, y, z ) = ( x, t ) . Let η = xdx + (1 + xt ) dt be 1-form at 0 ∈ C , note that ω = ϕ ∗ ( η ) and moreover η (0 , = 0, this implies that η is non-singular at 0 ∈ C and by Frobenius theo-rem η has a holomorphic first integral H ( x, t ) on ( C , H ( x, y, z ) := H ( x, f ( x, y, z )) = H ( x, y z ), we get H is a holomorphic first integral for ω in aneighborhood of 0 ∈ C . CALVO-ANDRARE, C ˆORREA, FERN´ANDEZ–P´EREZ
We apply Theorem 1 to a codimension one holomorphic foliation of the projectivespace with empty Kupka set.About the third case, consider a foliation ω ∈ F (3 , d ). Let Z be a connectedcomponent of S . We count the number | Z ∩ N K ( ω ) | of non-Kupka points of ω in Z ⊂ S . Theorem 2.
Let ω ∈ F (3 , d ) be a foliation and Z ⊂ S a connected component of S . Suppose that Z is a local complete intersection and Z \ Z ∩ K ( ω ) is a finite setof points, then dω | Z is a global section of K − Z ⊗ K F | Z and the associated divisor D ω = X p ∈ Z ord p ( dω ) · p has degree deg( D ω ) = deg( K F ) − deg( K Z ) . Note that the section dω | Z vanishes exactly in the non-Kupka points of ω in Z then the above theorem determine the number | Z ∩ N K ( ω ) | (counted with multi-plicity) of non-Kupka points of ω in Z .2. The singular set
Let ω ∈ F ( M, L ) be a codimension one holomorphic foliation then singular setof ω may be written as S = n [ j =2 S j where codim( S j ) = j. The fact that K ( ω ) ⊂ S implies that S ∪ . . . ∪ S n ⊂ N K ( ω ). To continue wefocus 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 resultdue to Malgrange [18].
Theorem 2.1 (Malgrange) . Let ω be a germ at ∈ C n , n ≥ of an integrable1–form singular at , if codim ( S ω ) ≥ , then there exist f ∈ O C n , and g ∈ O ∗ C n , such that ω = gdf on a neighborhood of ∈ C n . We have the following proposition.
Proposition 2.2.
Let ω ∈ F ( M, L ) be a foliation and let p ∈ S n an isolatedsingularity, then any germ of vector field tangent to the foliation vanishes at p .Proof. Let ω = gdf, g ∈ O ∗ p , f ∈ O p be a 1–form representing the foliation at p .Let X ∈ Θ p be a vector field tangent to the foliation, i.e., ω ( X ) = 0. If X ( p ) = 0there exists a coordinate system with z ( p ) = 0 and X = ∂/∂z n , then0 = ω ( X ) = g · n X i =1 ( ∂f /∂z i ) dz i ( ∂/∂z n ) ! = g · ( ∂f /∂z n ) , therefore ∂f /∂z n ≡ , and f = f ( z , . . . , z n − ), but this function does not have an isolated singularity. (cid:3) Now, we begin our study of the irreducible components of codimension two ofthe singular set of ω . Note that, given a section ω ∈ H ( M, Ω ( L )), along thesingular set, the equation ω α = λ αβ ω β implies dω α | S = ( λ αβ dω β ) | S . Then(2.1) { dω α } ∈ H ( S, (Ω M ⊗ L ) | S ) . N NON-KUPKA POINTS OF CODIMENSION ONE FOLIATIONS 5
The Kupka set.
These singularities has bee extensively studied and the mainproperties have been established in [15, 17].
Definition 2.3.
For ω ∈ F ( M, L ). The Kupka set is K ( ω ) = { p ∈ M | ω ( p ) = 0 , dω ( p ) = 0 } . The following properties of Kupka sets, are well known [17].(1) K ( ω ) is smooth of codimension-two.(2) K ( ω ) has local product structure and the tangent sheaf F is locally freenear K ( ω ).(3) K ( ω ) is subcanonically embedded and ∧ N K ( ω ) = L | K ( ω ) , K K ( ω ) = ( K M ⊗ L ) | K ( ω ) = K F | K ( ω ) . Let ω ∈ F ( n, d ) be a foliation with S = K ( ω ). By [7], there exists a pair ( V, σ ),where V is a rank two holomorphic vector bundle and σ ∈ H ( P n , V ), such that0 −→ O σ −−→ V −→ J K ( d + 2) → { σ = 0 } = K and the total Chern class c ( V ) = 1 + ( d + 2) · h + deg( K ( ω )) h ∈ H ∗ ( P n , Z ) ≃ Z [ h ] / h n +1 . In 2009, Marco Brunella [3] proved that following result, which in a certain sensesay that the local transversal type of the singular set of foliation determines itsbehavior globally. Here we present a new proof of this fact. The techniques usedin the proof could be of independent interest.
Proposition 2.4.
Let ω ∈ F ( n, d ) be a foliation with S = 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 requires the following lemma. This result may bewell 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 P with c ( F ) = 0 and c ( F ) = 0 . Then F ≃ O ⊕ O , is holomorphically trivial. Now, we prove Proposition 2.4.
Proof of Proposition 2.4.
Let (
V, σ ) be the vector bundle with a section definingthe Kupka set as scheme. The radial transversal type implies [7] c ( V ) = 1+( d +2) · h + ( d + 2) · h = (cid:18) d + 2) · h (cid:19) ∈ H ∗ ( P n , Z ) ≃ Z [ h ] / h n +1 . The vector bundle E = V ( − d +22 ), has c ( E ) = 0 and c ( E ) = 0. Let ξ : P ֒ → P n be a linear embedding. By the preceding lemma we have ξ ∗ E ≃ O P ⊕ O P and by the Horrocks’ criterion [19, Theorem 2.3.2 pg. 22], E ≃ O P n ⊕ O P n is trivialand hence V splits as O P n ( d +22 ) ⊕ O P n ( d +22 ) and K is a complete intersection. Theexistence of the meromorphic first integral follows from [10, Th. A]. (cid:3) CALVO-ANDRARE, C ˆORREA, FERN´ANDEZ–P´EREZ If ω is such that K ( ω ) = S and connected, the set of non-Kupka points of ω is NK ( ω ) = S ∪ · · · ∪ S n . A generic rational map, that means, a
Lefschetz or a Branched Lefschetz Pencil ϕ : P n P , has only isolated singularities away its base locus. The singular setof the foliation defined by the fibers of ϕ is S n ∪ S . The Kupka set correspondsaway from its base locus and S n = NK ( ω ) are the singularities as a map. S n isempty if and only if the degree of the foliation is 0. The number ℓ ( S n ) of isolatedsingularities counted with multiplicities can be calculated by[12, Th. 3]. If ω p is agerm of form that defines the foliation at p ∈ S n , we have ℓ ( S n ) = X p ∈ S n µ ( ω p , p ) , µ ( ω, p ) = dim C O p ( ω , . . . , ω n ) , ω p = n X i =1 ω i dz i . We have that c n ( F ) = ℓ ( S n ).3. Foliations with a non-Kupka component
It is well known that K ( ω ) ⊂ { p ∈ M | j p ω = 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 irreduciblenon-Kupka component of S . Proof of Theorem 1.
By hypotheses, dω ( p ) = 0 for any p ∈ S ω . Since ω = ω + · · · , dω = dω + · · · = 0 , we get dω ( p ) = 0 for any p ∈ S ω . Now, as ω = 0 and codim( S ω ) = 2, we have1 ≤ codim( S ω ) ≤
2. We distinguish two cases.(1) codim( S ω ) = 2: there is a coordinate system ( x , . . . , x n ) ∈ C n such that ω = x dx + x dx . (2) codim( S ω ) = 1: there is a coordinate system ( x, ζ ) ∈ C × C n − such that x ( p ) = 0 and ω = xdx .The first case is known, the foliation F ω is equivalent in a neighborhood of0 ∈ C n to a product of a dimension one foliation in a transversal section by aregular foliation of codimension two [9, p. 31].In the second case, Loray’s preparation theorem [16], shows that there exists acoordinate system ( x, ζ ) ∈ C × C n − , a germ f ∈ O C n − , with f (0) = 0, and germs g, h ∈ O C , such that the foliation is defined by the 1–form(3.1) ω = xdx + [ g ( f ( ζ )) + xh ( f ( ζ ))] df ( ζ ) . Since S ω = { x = 0 } and 0 ∈ S ω is a smooth point, we can assume that S ω,p = { x = ζ = 0 } , where S ω,p is the germ of S ω at p = 0. Therefore, S ω,p = { x = ζ = 0 } = { x = g ( f ( ζ )) = 0 } ∪ (cid:26) x = ∂f∂ζ = · · · = ∂f∂ζ n − = 0 (cid:27) . Hence, either g (0) = 0 and ζ | f , or g (0) = 0 and ζ | ∂f∂ζ j for all j = 1 , . . . , n −
1. Inany case, we have ζ | f and then f ( ζ ) = ζ k ψ ( ζ ), where ψ is a germ of holomorphicfunction in the variable ζ ; k ∈ N and ζ does not divide ψ . We have two possibilities: N NON-KUPKA POINTS OF CODIMENSION ONE FOLIATIONS 7 st case.– ψ (0) = 0. In this case, we consider the biholomorphism G ( x, ζ ) = ( x, ζ ψ /k ( ζ ) , ζ , . . . , ζ n ) = ( x, y, ζ , . . . , ζ n )where ψ /k is a branch of the k th root of ψ , we get f ◦ G − ( x, y, ζ , . . . , ζ n ) = y k and G ∗ ( ω ) = xdx + ( g ( y k ) + xh ( y k )) ky k − dy = xdx + ( g ( y ) + xh ( y )) dy, where g ( y ) = ky k − g ( y k ), h ( y ) = ky k − h ( y k ). Therefore, ˜ ω := G ∗ ( ω ) is equiva-lent to ω and moreover ˜ ω is given by(3.2) ˜ ω = xdx + ( g ( y ) + xh ( y )) dy with S ˜ ω = { x = g ( y ) = 0 } . Since d ˜ ω = h ( y ) dx ∧ dy is zero identically on { x = g ( y ) = 0 } , we get g | h , sothat h ( y ) = ( g ( y )) m H ( y ), for some m ∈ N and such that H ( y ) does not divided g ( y ). Using the above expression for h in (3.2), we have˜ ω = xdx + g ( y )(1 + x ( g ( y )) m − H ( y )) dy = xdx + g ( y )(1 + xg ( y )) dy, where g ( y ) = ( g ( y )) m − H ( y ). Consider ϕ : ( C , × ( C n − , → ( C ,
0) definedby ϕ ( x, ζ ) = ( x, y ), then(3.3) ω = ϕ ∗ ( xdx + g ( y )(1 + xg ( y )) dy ) . nd case.– ψ (0) = 0. We have S ω,p = { x = ζ = 0 } and(3.4) ω = xdx + ( g ( ζ k ψ ) + xh ( ζ k ψ )) d ( ζ k ψ ) , therefore(3.5) ω = xdx + ( g ( ζ k ψ ) + xh ( ζ k ψ )) ζ k − ( kψdζ + ζ dψ ) . Note that g (0) = 0, otherwise { x = ζ ψ ( ζ ) = 0 } would be contained in S ω,p , butit is contradiction because S ω,p = { x = ζ = 0 } ( { x = ζ ψ ( ζ ) = 0 } . Furthermore k ≥
2, because otherwise ζ | ψ . Let ϕ : ( C , × ( C n − , → ( C ,
0) be defined by ϕ ( x, ζ ) = ( x, ζ k ψ ( 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) dt = 0, we deduce that η has a non-constant holomorphic first integral F ∈ O C , such that dF (0 , = 0.Therefore, F ( x, ζ ) = F ( x, ζ k ψ ( ζ )) is a non-constant holomorphic first integral for ω in a neighborhood of 0 ∈ C n . (cid:3) Applications to foliations on P n . In order to give some applications ofTheorem 1, we need the Baum-Bott index associated to singularities of foliationsof codimension one.Let M be a complex manifold and let G ω = ( S, G ) be a codimension one holo-morphic foliation represented by ω ∈ H ( M, Ω ( L )). We have the exact sequence0 → G → Θ M ω −→ N G → , N G ≃ J S ⊗ L. Set M = M \ S and take p ∈ M . Then in a neighborhood U α of p the foliation G is induced by a holomorphic 1–form ω α and there exists a differentiable 1–form θ α such that dω α = θ α ∧ ω α Let Z be an irreducible component of S . Take a generic point p ∈ Z , that is, p is a point where Z is smooth and disjoint from the other singular components. CALVO-ANDRARE, C ˆORREA, FERN´ANDEZ–P´EREZ
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 Rham class can be integrated over an oriented3-sphere L p ⊂ B ∗ p positively linked with S ( B p ):BB( G , Z ) = 1(2 πi ) Z L p θ ∧ dθ. This complex number is the
Baum-Bott residue of G along Z . We have a particularcase of the general Baum-Bott residues Theorem [1] reproved by Brunella andPerrone in [4]. Theorem 3.1 (Baum-Bott [1]) . Let G be a codimension one holomorphic foliationon a complex manifold M . Then c ( L ) = c ( N G ) = X Z ⊂ S BB ( G , Z )[ Z ] , where N G = J S ⊗ L is the normal sheaf of G on M and the sum is done over allirreducible components of S . In particular, if G is a codimension one foliation on P n of degree d , then thenormal sheaf N G = J S ( d + 2) and the Baum-Bott Theorem looks as follows X Z BB( G , Z ) deg[ Z ] = ( d + 2) . Now, if there exist a coordinates system ( U, ( x, y, z , . . . , z n )) around p ∈ Z ⊂ S such that x ( p ) = y ( p ) = 0 and S ( G ) ∩ U = Z ∩ U = { x = y = 0 } . Assume that ω | U = P ( x, y ) dy − Q ( x, y ) dx is a holomorphic 1-form representing G| U . Let θ be the C ∞ (1,0)-form on U \ Z given by θ = ( ∂P∂x + ∂Q∂y ) | P | + | Q | ( ¯ P dx + ¯
Qdy ) . Since dω = θ ∧ ω , then(3.6) BB( G , Z ) = 1(2 πi ) Z L p θ ∧ dθ = Res ( Tr (cid:0) D X (cid:1) dx ∧ dyP Q ) , where Res denotes the Grothendieck residue, D X is the Jacobian of the holomor-phic map X = ( P, Q ). It follows from of Grothendieck residues [14, Chapter 5] thatif D X ( p ) is non-singular, thenBB( G , Z ) = Tr( D X ( p )) det( D X ( p )) . In the situation explained above, the tangent sheaf G ( U ) is locally free andgenerated by the holomorphic vector fields G ( U ) = (cid:28) X = P ( x, y ) ∂∂x + Q ( x, y ) ∂∂y , ∂∂z , . . . , ∂∂z n (cid:29) and the vector field X carries the information of the Baum–Bott residues.The next result, in an application of Theorem 1 Theorem 3.2.
Let ω ∈ F ( M, L ) be a foliation and Z ⊂ S \ K ( ω ) . Suppose that Z is smooth and j p ω = 0 for all p ∈ Z , then BB ( F ω , Z ) = 0 . N NON-KUPKA POINTS OF CODIMENSION ONE FOLIATIONS 9
Proof.
We work in a small neighborhood U of p ∈ Z ⊂ M . According to Theorem 1there exists a coordinate system ( x, y, z , . . . , z n ) at p such that Z ∩ U = { x = y = 0 } and one has three cases. In the first case F ω is the product of a dimension onefoliation in a section transversal to Z by a regular foliation of codimension two and j p ( ω ) = xdy + ydx . In this case, it follows from (3.6) that BB( F ω , Z ) = 0. In thesecond case ω = xdx + g ( y )(1 + xg ( y )) dy, where g , g ∈ O C , and it follows from [11, Lemma 3.9] thatBB( F ω , Z ) = Res t =0 (cid:20) ( g ( t ) g ( t )) dtg ( t ) (cid:21) = Res t =0 (cid:2) g ( t )( g ( t )) (cid:3) . Since g ( y )( g ( y )) is holomorphic at y = 0, we get BB( F ω , Z ) = 0 . In the thirdcase F ω has a holomorhic first integral in neighborhood of p and is known thatBB( F ω , Z ) = 0 . (cid:3) The Baum-Bott formula implies the following result.
Corollary 3.3.
Let ω ∈ F ( n, d ) , n ≥ , be a foliation with K ( ω ) = ∅ . Then thereexists a smooth point p ∈ S such that j p ω = 0 .Proof. If for all smooth point p ∈ S one has j p ω = 0, the above theorem shows thatBB( F ω , Z ) = 0 for all irreducible components Z ⊂ S . By Baum–Bott’s theorem,we get 0 < ( d + 2) = X Z ⊂ S BB( F ω , Z ) = 0which is a contradiction. Therefore there exists a smooth point p ∈ S such that j p ω = 0. (cid:3) In particular, if ω ∈ F ( n, d ), n ≥
3, is a foliation with j p ω = 0 for any p ∈ 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 P , but someresults remain valid to codimension one foliations on others manifolds of dimensionthree.4.1. Simple singularities.
Let ω be a germ of 1–form at 0 ∈ C . We define the rotational of ω as the unique vector field X such that rot ( ω ) = X ⇐⇒ dω = ı X dx ∧ dy ∧ dz, moreover ω is integrable if and only if ω ( rot ( ω )) = 0.Let ω be a germ of an integrable 1–form at 0 ∈ C . We say that 0 is a simplesingularity of ω if ω (0) = 0 and either dω (0) = 0 or dω has an isolated singularityat 0. In the second case, these kind of singularities, are classified as follows(1) Logarithmic . The second jet j ( 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 nonzero and necessarily satisfies the relation λ + λ = 0.(3) Nilpotent . The rotational vector field X , is nilpotent as a derivation. The structure near simple singularity is known [8]. If p ∈ S is a simple singularityand dω ( p ) = 0, then p is a singular point of S . Theorem 4.1.
Let ω ∈ Ω ( C , , n ≥ , be a germ of integrable 1-form such that ω has a simple singularity at then the tangent sheaf F = Ker ( ω ) is locally free at and it is generated by h rot ( ω ) , S i , where S has non zero linear part.Proof. Let ω be a germ at 0 ∈ C of an integrable 1–form and 0 a simple non-Kupkasingularity. Then 0 ∈ C is an isolated singularity of X = rot ( ω ). Consider theKoszul complex of the vector field X at 0 K ( X ) : 0 → Ω C , ı X −→ Ω C , ı X −→ Ω C , ı X −→ O C , → ω ( X ) = 0, then ω ∈ H ( K ( X ) ) that vanishes because X has an isolatedsingularity at 0. Therefore, there exists θ ∈ Ω C , such that ı X θ = ω . The mapΘ C , ∋ Z ı Z dx ∧ dy ∧ dz ∈ Ω C , is an isomorphism, hence ω = ı X θ, and θ = ı S dx ∧ dy ∧ dz, implies ω = ı X θ = ı X ı S dx ∧ dy ∧ dz and then, the vector fields { X , S } generate the sheaf F in a neighborhood of 0. (cid:3) Let ω ∈ F (3 , d ) be a foliation and Z ⊂ S be a connected component of S .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 completeintersection, consider the exact sequence0 → J / J → Ω ⊗ O Z → Ω Z → ∧ and twisting by L = K − P ⊗ K Z = K Z (4) we get0 → ∧ J / J ⊗ L → Ω P | Z ⊗ L → · · · Since Z ⊂ S , the singular set, we have seen before that dω | Z ∈ H ( Z, ∧ ( J / J ) ⊗ L )Now, from the equalities of sheaves K − Z ⊗ K P ≃ ∧ ( J / J ) , and L ≃ K − P ⊗ K F we have H ( Z, ∧ ( J / J ) ⊗ L ) = H ( Z, K − Z ⊗ K F | Z ) , the non-Kupka points of ω in Z satisfies dω | Z = 0, denoting D ω = X p ∈ Z ord p ( dω )the associated divisor to dω | Z , one hasdeg( D ω ) = deg( K F ) − deg( K Z ) , as claimed. (cid:3) Remark 4.2.
The method of the proof works also in projective manifolds, anddoes not depends on the integrability condition.
N NON-KUPKA POINTS OF CODIMENSION ONE FOLIATIONS 11
Examples.
We apply Theorem 2 for some codimension one holomorphic foli-ations on P and determine the number of non-Kupka points. Example 4.3 (Degree two Logarithmic foliations) . Recall that the canonical bun-dle of a degree two foliation of P is trivial. There are two irreducible componentsof logarithmic foliations in the space of foliations of P of degree two: L (1 , ,
2) and L (1 , , , L (1 , , ω be a generic element of L (1 , ,
2) and consider its singular scheme S = S ∪ S . By [12, Theorem 3] ℓ ( S ) = 2. On the other hand, S has threeirreducible components, two quadratics and a line, the arithmetic genus is p a ( S ) =2. Note that Theorem 2, implies that the number | NK ( ω ) ∩ S | , of non-Kupka pointsin S is | NK ( ω ) ∩ S | = deg ( D ω ) = deg( K F ) − deg( K S ) = − χ ( S ) = 2 . The non-Kupka points of the foliation F ω are | NK ( ω ) | = ℓ ( S ) + | N K ∩ S | = 4. L (1 , , , ω be a generic element of L (1 , , ,
1) then the tangent sheafis
O ⊕ O and the singular scheme S = S [13] moreover consists of 6 lines giventhe edges of a tetrahedron, obtained by intersecting any two of the four invarianthyperplanes H i . The arithmetic genus is p a ( S ) = 3, by Theorem 2, | NK ( ω ) | = | NK ( ω ) ∩ S | = 4, corresponding to the vertices of the tetrahedron where there aresimple singularities of logarithmic type. Example 4.4 (The exceptional component E (3)) . The leaves of a generic foliation ω ∈ E (3) ⊂ F (3 , , are the orbits of an action of Aff ( C ) × P → P and its tangentsheaf is O⊕O [8, 13]. Its singular locus S = S has deg ( S ) = 6 and three irreduciblecomponents: a line L, a conic C tangent to L at a point p , and a twisted cubic Γwith L as an inflection line at p . The point NK ( ω ) = L ∩ C ∩ Γ = { p } ⊂ S is theonly non-Kupka point.The arithmetic genus is p a ( S ) = 3 and the canonical bundle of the foliation againis trivial, by Theorem 2, the number of non-Kupka points | NK ( ω ) | = 4. Thereforethe non-Kupka divisor NK ( ω ) ∩ S = 4 p . If ω represents the foliation at p , then µ ( dω, p ) = µ ( rot ( ω ) , p ) = 4 . Acknowledgements.
The first author thanks the Federal University of Minas GeraisUFMG, IMPA, and IMECC–UNICAMP for the hospitality during the elaborationof this work. The third author thanks the IMCA-Per´u for the hospitality. Finally,we would like to thank the referee by the suggestions, comments and improvementsto the exposition. 5.
Appendix
We prove Lemma 2.5.
Proof.
First, we see that h ( F ) ≥
1. By Riemann–Roch–Hirzebruch, we have χ ( F ) = h ( F ) − h ( F ) + h ( F ) = [ ch ( F ) · T d ( P )] = 2 , then h ( F ) + h ( F ) = [ ch ( F ) · T d ( P )] + h ( F ) ≥ [ ch ( F ) · T d ( P )] = 2 By Serre duality [14, 19], we get h ( F ) = h ( F ( − h ( F ) ≥ h ( F ( − k ))for all k >
0, hence h ( F ) ≥
1. Let τ ∈ H ( F ) be a non zero section, consider theexact sequence(5.1) 0 −→ O · τ −→ F −→ Q −→ Q = F/ O . The sheaf Q is torsion free, therefore Q ≃ J Σ for some Σ ⊂ P . The sequence (5.1),is a free resolution of the sheaf Q with vector bundles with zero Chern classes.From the definition of Chern classes for coherent sheaves [1], we get c ( Q ) = 1, inparticular deg(Σ) = c ( Q ) = 0, we conclude that Σ = ∅ and Q ≃ O . Then F is anextension of holomorphic line bundles, hence it splits [19, p. 15]. (cid:3) References [1] Baum, P., Bott, R.
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