Brieskorn module and Center conditions: pull-back of differential equations in projective space
BBrieskorn module and Center conditions:pull-back of differential equations in pro jectivespace Yadollah Zare , Susumu Tanab´e
Abstract
The moduli space of algebraic foliations on P of a fixed degree and with a centersingularity has many irreducible components. We find a basis of the Brieskorn moduledefined for a rational function and prove that pull-back foliations forms an irreduciblecomponent of the moduli space. The main tools are Picard-Lefschetz theory of arational function in two variables, period integrals and Brieskorn module. A holomorphic foliation F ( α ) in P is defined by a 1-form α = AdX + BdY + CdZ ,where
A, B and C are homogeneous polynomials of degree d + 1 satisfying the identity α ( υ ) = XA + Y B + ZC = 0 for υ = X ∂∂X + Y ∂∂Y + Z ∂∂Z , the Euler vector field. Forgeneric A, B and C the degree of the foliation F ( α ) is defined as d = deg ( A ) − LS ]and [ CL ]). The space of algebraic foliations F ( α ) , α ∈ Ω d +1 , where(0.1) Ω d +1 := { α = AdX + BdY + CdZ | A, B, C ∈ C [ X, Y, Z ] hd +1 , α ( υ ) ≡ } , is the projectivization of the vector space Ω d +1 and is denoted by F (2 , d ). From here onwe use the notation C [ x, y, z ] h D to denote the ring of homogeneous polynomials of degree D . The space F (2 , d ) is a rational variety C N for some N , the space of coefficients ofpolynomials A, B, C . A singularity of F ( α ) is a common zero of A, B and C . We denotethe set of singularities of F ( α ) by Sing ( F ( α )). For an isolated singularity p ∈ Sing ( F ( α )),if there is a holomorphic coordinate system (˜ x, ˜ y ) in a neighborhood of the point p with˜ x ( p ) = 0, ˜ y ( p ) = 0 such that in this coordinate system α | ∧ d (˜ x + ˜ y ) = 0 , holds then the point p is called a center singularity . The closure of the set of algebraicfoliations of fixed degree d with at least one center in F (2 , d ), which is denoted by M (2 , d ),is an algebraic subset of F (2 , d ) (see for instance, [ Mo1 ] and [ LS ]). Identifying irreduciblecomponents of M (2 , d ) is the center condition problem in the context of polynomial differ-ential equations on the real plane. The complete classification of irreducible componentsof M (2 ,
2) is done by H. Dulac in [ D ] (see also [ CL , p.601]). For the classification ofpolynomial 1-forms of degree 3 the reader can consult ˙Zo(cid:32)l¸adek’s articles [ Z1 ], [ Z2 ]. This Keywords: Holomorphic foliations, Picard-Lefschetz theoryAMS Classification: 32L30, 14D05. This work is partially supported by T ¨UB˙ITAK project 116F130”Period integrals associated to algebraic varieties. ” a r X i v : . [ m a t h . D S ] O c t lassification gives applications on the number of limit cycles in the context of polynomialdifferential equations on the real plane.Consider the morphism(0.2) F : P → P [ x, y, z ] → [ R, S, T ]where
R, S, T ∈ C [ x, y, z ] hs . Let P (2 , a, s ) be the set of foliations(0.3) F ( F ∗ ( α )) where α ∈ Ω a +1 . Let us denote by D the zero locus of the determinant of Jacobian matrix J F of F i.e.(0.4) J F ( x, y, z ) = R x R y R z S x S y S z T x T y T z (0.5) D = { [ x : y : z ] | det ( J F ( x, y, z )) = 0 } ⊂ P . Definition 0.1.
For a generic morphism F and foliation F , there exists local chart ( φ, U ) (resp. ( ψ, V ) ) of the critical point p ∈ D (resp. F ( p ) ) such that F | U = ( x , y ) and theleaves of F | ( ψ, V ) are given by X + Y = t . Therefore, F ∗ ( F ) with local expression x + y = t has a center singularity. We call this singularity of the foliation F ∗ ( F )tangency center singularity . We define the space P (2 , a, s ) of pull-back foliations F ( ω ) defined by(0.6) ω = F ∗ ( A ) dR + F ∗ ( B ) dS + F ∗ ( C ) dT, where R, S, T ∈ C [ x, y, z ] hs , A, B, C ∈ C [ X, Y, Z ] ha +1 ,RF ∗ ( A ) + SF ∗ ( B ) + T F ∗ ( C ) = 0 . Theorem 0.1.
The space P (2 , a, s ) constitutes an irreducible component of M (2 , d ) for d = s ( a + 2) − , s ≥ . This article treats the question of the vector tangent to M (2 , d ) at a foliation withrational first integral f. This question has been studied for the case of pull-back of poly-nomial differential equations in [ Za ].A sketch of contents of the article is as follows. In section 1 we prepare notations andnotions that will be used in the course of exposition. Since the calculation of the tangentspace of M (2 , d ) at its generic point requires more information, we adopt a strategy tochoose a special point F in the intersection of P (2 , a, s ) with the space of projectivelogarithmic foliations (1.5). Let F (cid:15) be a deformation of F such that for any (cid:15) , F (cid:15) has acenter singularity closed to a tangency center singularity of F . We assume that F (resp. F (cid:15) ) is defined by F ∗ ( α ) (resp. ω (cid:15) = F ∗ ( α )+ (cid:15) ω + · · · ) for α (1.6). By using the resultsconcerning vanishing cycles from section 2 and relatively exact 1-forms from section 3, weestablish that there are 1-forms α and ω e such that ω be expressed as ω = F ∗ ( α ) + ω e .In section 4 by using the first order Melnikov function and period integral we calculate2n explicit form of the tangent vector to the irreducible component X ( a, s ) of M (2 , d )containing P (2 , a, s ). This calculation gives a proof of Theorem 1.1. In section 5 we finda basis for the Brieskon module H f with f = P q Q p for P, Q generic polynomials and p, q co-prime. This is used to prove that F ∗ ( B ), for B a basis of H f , is extendable to a basisof H F ∗ ( f ) . This result furnishes us with the proof of Theorem 5.1 that is necessary toestablish Corollary 3.2 that in its turn plays an essential rˆole during the proof of the keyLemma 4.1.Authors are grateful to Hossein Movasati for useful remarks. Let F be the morphism defined in (0.2). If the coefficients of the pull-back F ∗ ( α ) of α = AdX + BdY + CdZ for
A, B, C ∈ C [ X, Y, Z ] ha +1 do not have a common factor thenthe degree of the foliation F ∗ ( F ( α )) is s ( a + 2) − P (2 , a, s ) is an irreducible algebraicsubset of M (2 , d ). We take a point F ( F ∗ ( α )) in P (2 , a, s ) for a 1-form α like in (0.1),make a deformation F (cid:15) ∈ P (2 , a, s ) and calculate the tangent vector space of P (2 , a, s ) at F ( F ∗ ( α )): F ∗ (cid:15) ( α (cid:15) ) = F ∗ (cid:15) ( α + (cid:15)α ) + O ( (cid:15) )(1.1)for α (cid:15) = α + (cid:80) j ≥ (cid:15) j α j with α j ∈ Ω ≤ d +1 , ∀ j ≥ . The pull-back by a morphism F (cid:15) = F + (cid:15)F + O ( (cid:15) ) with F = ( R, S, T ), F = ( R , S , T ) of the form (1.1)(1.2) F ∗ (cid:15) ( α + (cid:15)α + · · · ) = F ∗ ( α ) + (cid:15)ω W + O ( (cid:15) ) . Here the 1-form ω W has the following form: ω W = F ∗ ( A ) dR + F ∗ ( B ) dS + F ∗ ( C ) dT + ( R .F ∗ ( A X ) + S .F ∗ ( A Y ) + T .F ∗ ( A Z )) dR + ( R .F ∗ ( B X ) + S .F ∗ ( B Y ) + T .F ∗ ( B Z )) dS + ( R .F ∗ ( C X ) + S .F ∗ ( C Y ) + T .F ∗ ( C Z )) dT + F ∗ ( α ) . (1.3)Consider a foliation in P with the first integral f : P \ ( { P = 0 } ∩ { Q = 0 } ) → ¯ C f ( X, Y, Z ) = P ( X, Y, Z ) q Q ( X, Y, Z ) p (1.4)with deg ( P ) deg ( Q ) = pq , g.c.d ( p, q ) = 1 . We denote by(1.5) I ( deg ( P ) − , deg ( Q ) − F ( α ) with the first integrals of type (1.4) where(1.6) α = qQdP − pP dQ. In [
Mo1 , Theorem 5.1] it is shown that the space of foliations with a first integral of type(1.4) is an irreducible component of M (2 , deg ( P )+ deg ( Q ) −
2) for deg ( P )+ deg ( Q ) − ≥ I ] established for the polynomial first integral to thecase of foliations in P with a generic rational first integral of type (1.4).3 emark 1.1. By calculating the tangent vector ω W to P (2 , a, n ) at the point F = F ∗ ( α ) for α (1.6) , we have A = qQP X − pP Q X , B = qQP Y − pP Q Y , C = qQP Z − pP Q Z in (1.3) and obtain (1.7) ω W = ω pl + F ∗ ( α ) , where ω pl = qF ∗ ( Q ) dP − pP dF ∗ ( Q ) − qQ dF ∗ ( P ) + pF ∗ ( P ) dQ ,P = R F ∗ ( P X ) + S F ∗ ( P Y ) + T F ∗ ( P Z ) ,Q = R F ∗ ( Q X ) + S F ∗ ( Q Y ) + T F ∗ ( Q Z ) . See the calculation of (4.14) . Let X ( a, s ) be the irreducible component of M (2 , d ) containing P (2 , a, s ). To calculatetangent cone of X ( a, s ) at a special point in I ( ms − , ns − ∩ P (2 , a, s ). Let F be ageneric morphism of P into itself such that each component of F and f be a rationalfunction f = P q Q p where P, Q are two homogeneous polynomials degrees m, n with(1.8) a = n + m − mq = np with the condition (2.1). Theorem 1.1.
Tangent cone of X ( a, s ) at the point F := F ( F ∗ ( α )) for α (1.6) is equalto tangent cone of P (2 , a, s ) at this point T C F X ( a, s ) = T C F P (2 , a, s ) . Let us consider a deformation F ( ω (cid:15) ) ∈ X ( a, s ) defined by the following 1-form:(1.9) ω (cid:15) = F ∗ ( α ) + (cid:15)ω + (cid:15) ω + · · · , deg ( ω j ) ≤ d + 1 ∀ j. The hypothesis F ( ω (cid:15) ) ∈ X ( a, s ) implies that it always has a tangency center singularity(Definition 0.1) near the tangency center singularity p of F ( ω ) defined by ω = F ∗ ( α ) . We call it also tangent critical point of the rational mapping F ∗ ( f ) : P → P in view of thecircumstance that requires analysis of vanishing cycles and their monodromy associatedto F ∗ ( f ) in § E , and let δ t be a continuous family of vanishingcycles in ( F ∗ ( f )) − ( t ) around tangent critical point p and Σ ∼ = C be a transverse sectionof F at some points of δ t . We write Taylor expansion of the deformed holonomy h (cid:15) ( t ) h (cid:15) ( t ) − t = M ( t ) (cid:15) + M ( t ) (cid:15) + · · · + M i ( t ) (cid:15) i + · · · where M i ( t ) is i − th Melnikov function of the deformation (see [ F , Theorem 1.1], [ Mo1 ,Section 3]). If Σ is parametrized by the image of F ∗ ( f ) i.e. t = F ∗ ( f )( σ ) for σ ∈ Σ then(1.10) M ( t ) = − (cid:90) δ t F ∗ ( f ) ω F ∗ ( P Q ) = − t (cid:90) δ t ω F ∗ ( P Q ) . for δ t ∈ H (( F ∗ ( f ) − ( t ) , Z ) , the vanishing cycle associated to the tangent critical point p ∈ C F ∗ ( f ) . In fact M j ( t ) , ∀ j ≥ t near to zero as F ∗ ( F ( ω η )) has atangency center singularity ∀ η ∈ [0 , (cid:15) ] . The condition M ( t ) = 0 plays a central rˆole in the proof of Theorem 1.1. See Lemma4.1, Lemma 4.2. 4 Monodromy action on tangency cycles
In this section we formulate Theorem 2.1 that establishes a relation between vanishingcycles associated to f and F ∗ ( f ) . Let F be a foliation in P with the first integral f = P q Q p like in (1.4). In addition to (1.4) we impose on F satisfies the following conditions: Conditions 2.1. (1) The curves V ( P ) ⊂ P and V ( Q ) ⊂ P are smooth and intersecteach other transversally, and also each one has transversal intersection with line at infinity.We denote the set V ( P ) ∩ V ( Q ) by R . (2) All critical points of f = P q Q p are distinct andbelong to the affine plane C . (3) deg ( P ) > deg ( Q ) ≥ , i.e.m > n ≥ . Let us denote by M f : π ( C \ C f , B ) −→ GL ( H ( f − ( B ) , Z ) , Z )the monodromy representation of the fundamental group of the complement to the crit-ical value set C f of f acting on the first homology group of a smooth generic fiber H ( f − ( B ) , Z ) . In a similar manner we consider the following monodromy representation for the firsthomology group of a smooth generic fiber H (( F ∗ ( f )) − ( b ) , Z ) : M F ∗ ( f ) : π ( C \ C F ∗ ( f ) , b ) −→ GL ( H (( F ∗ ( f )) − ( b ) , Z ) , Z ) . By virtue of the assumption made on the critical points of f and the genericity of F themonodromy representations M f and M F ∗ ( f ) can be realized with integer coefficients. Here C F ∗ ( f ) denotes the critical value set of F ∗ ( f ) . Let us call vanishing cycle δ t around a tangent critical point tangency vanishing cycle (see (2.3) below). Theorem 2.1.
The morphism (2.1) F ∗ : H (( F ∗ ( f )) − ( b ) , Z ) → H ( f − ( b ) , Z ) is surjective and Ker ( F ∗ ) is generated by the result of the monodromy group action M F ∗ ( f ) ( π ( C \ C F ∗ ( f ) , b )) on a tangency cycle δ t around a tangent critical point.Proof. For D defined in (0.5) the morphism F | : C \ D → C \ F ( D ) is a covering map.Inverse image of each vanishing cycle ∆ ∈ H ( f − ( B ) , Z ) via F − | contains s disjointsvanishing cycles. Moreover, H (( F ∗ ( f )) − ( b ) , Z ) contains a subgroup consisiting of s copies of a group isomorphic to H ( f − ( B ) , Z ).Let us take a pull-back vanishing cycle δ such that F ∗ ( δ ) = ∆ . According to [ La ,(7.3.5)], [ Mo5 , Theorem 2.3] the action of the monodromy group M F ∗ ( f ) (cid:0) π ( C \ C F ∗ ( f ) , b ) (cid:1) on the vanishing cycle δ generates a subgroup of H (( F ∗ ( f )) − ( b ) , Z ) which is isomorphicto H ( f − ( B ) , Z ).It is well-known that Dynkin diagram of F ∗ ( f ) is connected, see for instance [ AGV ].If we remove the vertices which correspond to vanishing cycles around the tangent criticalpoints in D (we call them tangency vertices ) the Dynkin diagram becomes s disjointgraphs P i , 1 ≤ i ≤ s each of which is isomorphic to the Dynkin diagram of f . See [ Za ,Figure 9].There exists a local chart around the tangent critical point p ∈ D such that in thischart F = ( x , y ). Therefore the graph P i is connected to a graph P j by tangency vertices.5ndeed, a tangency vertex corresponding to a tangency cycle around a tangent criticalpoint p ∈ D is connected to some P i or to another vertex corresponding to a vanishingcycle around p ∈ D . Therefore we conclude that the monodromy group M F ∗ ( f ) ( π ( C \ C F ∗ ( f ) , b )) actions on the tangency cycle δ t generate cycles of the following two types. Thedifference between two vanishing cycles associated to different critical values c i (cid:54) = c j of F ∗ ( f ) satisfying F ( c i ) = F ( c j ) = c, c ∈ C F ∗ ( f ) . (2.2) δ ic − δ jc , ≤ i, j ≤ s , . and vanishing cycles that are associated to p ∈ D, (2.3) δ p , p ∈ D. In [ Za , Theorem 4.9, Figure 9] cycles (2.3) are divided into tangency and exceptionalvanishing cycles .We denote by H the free Abelian group generated by cycles (2.2), (2.3) that is asubgroup of ker ( F ∗ ) . The space of cycles of type (2.2) has rank ( s − µ f for µ f = rank H ( f − ( b ) , Z ) . Moreprecisely we calculate µ f = ( m + n − − mn as | V ( P ) ∩ V ( Q ) | = mn (see Proposition 5.1).In a similar manner the equality µ = rank H (( F ∗ ( f )) − ( b ) , Z ) = ( s ( m + n ) − − s mn holds.Thus the rank of the space H is equal to(2.4) ( s − µ f + ρ D for ρ D : the rank of cycles (2.3).Here we remark that(2.5) µ = rank H (( F ∗ ( f )) − ( b ) , Z ) = s µ f + ρ D thanks to [ Za , Theorem 4.9].From the definition of the morphism F ∗ (2.1) we see that rank ( ker ( F ∗ )) = µ − µ f . The combination of (2.4), (2.5) shows the equality rank ( H ) = rank ( ker ( F ∗ )) . Thustogether with H ⊆ ker ( F ∗ ) we conclude H = ker ( F ∗ ) . A foliation F = F ( ω ) defined by a holomorphic 1-form ω is called integrable if there existsa meromorphic function f on P such that df ∧ ω | F = 0. In this case the meromorphicfunction f is said to be the first integral of F .The concept of relatively exact forms has been investigated by many authors, e.g.[ Mu ], [ Mo1 , Section 4].
Definition 3.1.
A meromorphic 1-form ω on P is called relatively exact modulo a folia-tion F in P if the restriction of ω to each leaf L of F is exact, i.e. there is a meromorphicfunction g on L such that ω |L = dg . Let us call f - fiber the set { u ∈ P \ R : f ( u ) = t } defined for some t ∈ C . SeeConditions 2.1. 6 roposition 3.1. [ Mu , §
2] A 1-form ω is relatively exact modulo the foliation F ( df ) with rational first integral f if and only if (cid:90) δ ω = 0 for every closed curve δ in a f − fiber.Proof. Let L be a line in P which is not F -invariant and does not pass through the pointin R as in Conditions 2.1. For any point u ∈ U := P \ R let f − ( f ( u )) ∩ L = { p , p , · · · , p r } where the intersection multiplicity of p i might be greater than 1.Define g : P \ R → C g ( u ) = 1 r ( r (cid:88) i =1 (cid:90) p i u ω )(3.1)where (cid:82) p i u is an integral over a path in f − ( f ( u )) which connects u to p i . The function g is well-defined and does not depend on the choice of the paths connecting u to p i because (cid:82) δ ω = 0 on any close curve δ in each level set f − ( f ( u )). Furthermore it is clear thata monodromy action on the line L leaves the set L ∩ f − ( f ( u )) invariant as it inducesmerely a permutation among its points. Thus we conclude that g ( u ) is a meromorphicfunction on P \ ( V ( P ) ∪ V ( Q )) in taking Levi extension theorem and Hartogs theoreminto account.A function f is called non-composite if every generic f -fiber is irreducible. It is easyto see that f is non-composite if and only if f can not be factored as a composite(3.2) P f (cid:48) → ¯ C i → ¯ C where i is a non-constant holomorphic map. In fact, if the composite factorization like(3.2) does not take place then the generic f - fiber cannot be reducible. From (3.2) thereducibility of generic f -fiber follows.Let f = P q Q p be a rational function satisfying Conditions 2.1 and suppose that for every t ∈ C the fiber f − ( t ) is connected. Let F ( ω ) be a foliation on P with the non-compositefirst integral f not satisfying (3.2) for(3.3) ω = dff . Let ω be a rational 1-form with the pole divisor(3.4) ˜ D = n D + n D , where D := V ( P ) , D := V ( Q ) . Theorem 3.1.
Every relatively exact rational 1-form ω modulo F ( ω ) with pole divisor (3.4) ˜ D has the form (3.5) ω = dg + T ω where g and T are rational functions with the pole divisor ˜ D . Mo1 , T heorem .
1] adapted to our situation.
Proof.
The function g in (3.1) is a holomorphic function in P \ ( D ∪ D ). For a point u ∈ U \ ( D ∪ D ), by the hypothesis q, p are the multiplicities of f along D , D , respectively.The function f n q is an univalent function in a small neighborhood of the path connecting u to p i and we have (cid:90) p i u ω = f ( − n q ) (cid:90) p i u f ( n q ) ωf n q ω is a holomorphic 1-form along U ∩ D therefore the above integral has poles of orderat most n along D . By using the chart around infinity and applying the above argumentto D once again, one can check that each component integral in (3.1) has poles of orderat most n along D . The equalities dg ∧ ω = ω ∧ ω ⇒ ( ω − dg ) ∧ ω = 0imply that there is a rational function T with pole divisor ˜ D such that ω = dg + T ω . Corollary 3.1.
Suppose that ω is a polynomial homogeneous − f orm on P with deg( F ( ω )) = deg ( F ( ω )) and ωF ∗ ( P Q ) is relatively exact modulo F ( ω ) . Then there are polynomials ( P , Q ) ∈ P ms × P ns such that ω has the form (3.6) ω = qF ∗ ( Q ) dP − pP dF ∗ ( Q ) − qQ dF ∗ ( P ) + pF ∗ ( P ) dQ . Proof.
By the Theorem 3.1 there are polynomials
B, A of degree at most ( m + n ) s suchthat ωP Q = d ( BP Q ) − ( AP Q )( q.QdP − p.P dQP Q )(3.7) ω = QP.dB − Bd ( P Q ) − A ( qQdP − p.P dQ ) P Q
This implies that P | B + qA, Q | B − pA ⇒ B + qA = ( p + q ) P.Q , B − pA = ( p + q ) QP ⇒ B = pP Q + qQP , A = − QP + P.Q where P , Q are two polynomials of respective degrees at most ms, ns. Substituting thesein (3.7) we get the result.In the sequel, for a form ω defined on P we shall use the notation ω | of its restrictionon the affine variety P . Corollary 3.2.
The morphism F ∗ : H f → H F ∗ ( f ) is injective and the image of the basisof H f is can be extended to a basis of H F ∗ ( f ) .Proof. We consider the pull-back by F of the projectivized 1-form ω for ω | ∈ H f . Werestrict F ∗ ( ω ) on C = { z = 1 } in H F ∗ ( f ) . It is well known that F ∗ is injective (see[ H , Proposition 1.1]). For each element ˜ ω j | = m j η | of the basis H f , ˜ ω j = m j ( X, Y, Z ) η
8s a polynomial 1-form on P and F ∗ (˜ ω j ) | = m j ( R, S, T ) F ∗ ( η ) | where the form η | = axdy − bydx such that dη | = dx ∧ dy . According to Theorem 5.1, F ∗ ( η ) | can be written as F ∗ ( η ) | = µ (cid:88) (cid:96) =1 g (cid:96) ( F ∗ ( f ))˜ ω (cid:96) | , where ˜ ω (cid:96) | is a basis of H F ∗ ( f ) . The inequality (5.6) entails deg ( g (cid:96) ) ≤ deg ( F ∗ ( η )) + ns − deg ( Z (˜ ω (cid:96) )) − ms.q < . We recall here that deg ( F ∗ ( η )) ≤ s and m > n ≥ g (cid:96) is in fact a constant for every (cid:96) . Therefore the1-form F ∗ (˜ ω j ) | = m j ( R, S, T ) F ∗ ( η ) | is free of F ∗ ( f ). This terminates the proof. We begin our discussion on the tangent vector of M (2 , d ) ( d = s ( a + 2) −
2) at the point F ( F ∗ ( α )) ∈ I ( ms − , ns − ∩ P (2 , a, s ), for α defined in (1.6). First of all, we show thefollowing lemma on a decomposition (4.1) valid for ω used to define the first Melnikovfunction (1.10). Lemma 4.1.
For ω in (1.10) we find a homogeneous 1-form α on P of degree a + 1 = m + n − and two homogeneous polynomials P , Q of respective degrees ms, ns such that (4.1) ω = F ∗ ( α ) + ω e , where (4.2) ω e = qF ∗ ( Q ) dP − pP dF ∗ ( Q ) − qQ dF ∗ ( P ) + pF ∗ ( P ) dQ . Proof.
In the affine coordinate the polynomial map F introduces a morphism F ∗ : H f → H F ∗ ( f ) between two C [ τ ]-module H f and C [ t ]-module H F ∗ ( f ) . The linear map H ( f − ( b ) , Z ) → C given by ∆ → (cid:90) F − ∗ (∆) ω F ∗ ( P Q )is well-defined because (cid:82) δ ω F ∗ ( P Q ) = 0, ∀ δ ∈ ker ( F ∗ ) by virtue of Theorem 2.1.By the duality between de Rham cohomology and singular homology there is a C ∞ differential form α b in regular fiber f − ( b ) such that (cid:90) F − ∗ (∆) ω F ∗ ( P Q ) = (cid:90) ∆ α b P Q .
According to Atiyah-Hodge theorem (See [ AH , Theorem 4], [ M-VL , Chapter 4]) α b canbe taken holomorphic thus polynomial. The analytic continuation of α b with respect to9he parameter b ∈ C \ C f gives rise to a holomorphic global section α of cohomology bundleof f . Thus in the affine coordinate C ⊂ P we have the following decomposition in H f (4.3) α | = µ f (cid:88) (cid:96) =1 h (cid:96) ( f | ) η (cid:96) | where h (cid:96) ( τ ) is holomorphic in τ ∈ C \ C f . The coefficients h (cid:96) ( τ ) in (4.3) are rational functions in τ because of the followingrelation h ( τ )... h µ f ( τ ) = (cid:104)(cid:82) δ k η (cid:96) | (cid:105) − µ f × µ f (cid:82) δ α | ... (cid:82) δ µf α | All the elements of the matrices in the right side of the equality have finite growth atcritical values. This is an analogy of the argument used to show (5.11) with the aid ofCramer’s rule.Pull-back of forms η (cid:96) | , ∀ (cid:96) are independent in H f under the map F ∗ and can be extendedto a basis for H F ∗ ( f ) in view of Corollary 3.2.There is a polynomial K ( τ ) ∈ C [ τ ] such that K ( f | ) .α | be a holomorphic form. We canwrite K ( f | ) .α | = (cid:80) (cid:96) h (cid:48) (cid:96) ( f | ) η (cid:96) | then F ∗ ( K ) ω | − F ∗ ( K.α | ) = 0 in H F ∗ ( f ) .Now we shall show the Claim: ω | − F ∗ ( α | ) = 0 in H F ∗ ( f ) . The set { F ∗ ( η (cid:96) | ) } µ f (cid:96) =1 canbe extended to a basis of H F ∗ ( f ) by Corollary 3.2. So we have in H F ∗ ( f ) F ∗ ( K ) ω | = µ f (cid:88) (cid:96) =1 F ∗ ( K ) .h (cid:48) (cid:96) ( F ∗ ( f | )) F ∗ ( η (cid:96) | ) + µ (cid:88) σ = µ f +1 F ∗ ( K ) a σ ˜ η σ | (4.4)Here { ˜ η σ } µσ = µ f +1 is a basis of H F ∗ ( f ) alien to F ∗ ( H f ) . Since each element of H F ∗ ( f ) can be uniquely written as a linear combination of theelements in this basis we get the vanishing coefficients a σ = 0 for all σ . In other words, inview of (4.3), we have F ∗ ( K ) .h (cid:96) = F ∗ ( h (cid:48) (cid:96) ) hence K | h (cid:48) (cid:96) . This means that ω | − F ∗ ( α | ) = 0in H F ∗ ( f ) . This is nothing but the Claim in question.To find the degree of α , we write ω = (cid:88) l F ∗ ( h l η l ) = (cid:88) (cid:96) h l ( F ( f )) (cid:88) β g lβ ( F ( f )) η β = (cid:88) (cid:96),β F ∗ ( g lβ h l ) η β , and we conclude that deg ( h l ) = 0 by virtue of Theorem 5.1. Therefore deg ( α ) = a + 1.Thanks to the Claim, we have(4.5) (cid:90) δ ω − F ∗ ( α ) F ∗ ( P Q ) = 0 , ∀ δ ∈ H (( F ∗ ( f )) − ( b ) , Z ) , which implies that the integrand rational form of (4.5) is a relatively exact 1-form modulothe foliation F ∗ ( ω ) for (3.3). By Corollary 3.1 there is a 1-form ω e of the form (3.6) suchthat ω = F ∗ ( α ) + ω e . In fact there are polynomials P and Q with degree ms and ns respectively such that ω e = qF ∗ ( Q ) dP − pP dF ∗ ( Q ) − qQ dF ∗ ( P ) + pF ∗ ( P ) dQ .We know that there are rational function ˜ h and a 1 − f orm β on P such that(4.6) ω e F ∗ ( P Q ) = β + ˜ h F ∗ ( α ) F ∗ ( P Q ) . emma 4.2. The 1-form ω e in the equality (4.1) is of the form (4.7) ω e = qF ∗ ( Q ) dP − pP dF ∗ ( Q ) − qQ dF ∗ ( P ) + pF ∗ ( P ) dQ with P = q < F , F ∗ ( gradP ) >, Q = p < F , F ∗ ( gradQ ) > for a vector (4.8) F = ( R , S , T ) , defined by some homogeneous polynomials R , S , T ∈ C [ x, y, z ] hs . Proof.
First of all we introduce the following polynomials { λ j ( X, Y, Z ) } j =1 defined by therelation(4.9) α = qQdP − pP dQ = λ dX + λ dY + λ dZ. Secondly, we define polynomials { ρ j ( x, y, z ) } j =1 by means of (0.4), (4.9),(4.10) ( F ∗ ( λ ) , F ∗ ( λ ) , F ∗ ( λ )) .J F ( x, y, z ) = ( ρ , ρ , ρ ) . In other words, F ∗ ( α ) = ρ dx + ρ dy + ρ dz. Now we pass to the investigation of polynomials P , Q present in (4.2). By multiplythe relation (4.6) with F ∗ ( α ) we get the following equality:( − Q F ∗ ( P ) + P F ∗ ( Q )) F ∗ ( dP ∧ dQ ) = ( pq ) − ( β − qF ∗ ( Q ) dP + pF ∗ ( P ) dQ ) ∧ F ∗ ( α ) . (4.11)Now let us consider the ideals in C [ x, y, z ] I = < F ∗ ( λ ) , F ∗ ( λ ) , F ∗ ( λ ) >, I = < ρ , ρ , ρ > for polynomials from (4.9), (4.10). We also consider the ideal J = < J F ( j, (cid:96) ) > ≤ j,(cid:96) ≤ ⊂ C [ x, y, z ] generated by 2 × J F ( x, y, z ) (0.4).Since V ( J ) ∩ V ( I ) = ∅ we see that I + J = C [ x, y, z ] and I ∩ J = I .J. The equality (4.11) entails that ( − Q F ∗ ( P ) + P F ∗ ( Q )) .J ∈ I ⊂ I . This means that( − Q F ∗ ( P ) + P F ∗ ( Q )) .J ∈ I ∩ J = I .J thus ( − Q F ∗ ( P ) + P F ∗ ( Q )) ∈ I . In otherwords, there exist polynomials R , S , T of degree s such that − Q F ∗ ( P ) + P F ∗ ( Q ) = R F ∗ ( λ ) + S F ∗ ( λ ) + T F ∗ ( λ ) . (4.12)Since P, Q are co-prime we have the required expressions for P , Q (4.7): P = q ( R .F ∗ ( P X ) + S .F ∗ ( P Y ) + T .F ∗ ( P Z )) ,Q = p ( R .F ∗ ( Q X ) + S .F ∗ ( Q Y ) + T .F ∗ ( Q Z )) . (4.13)Now we proceed to the proof of Theorem 1.1.11 roof. Let us introduce the notation P = qP, Q = pQ. We see that the 1-form F ∗ ( qQ dP − pP dQ ) defines the foliation F ( α ) . With this notation and (1.1), (4.13) wehave P = < F , F ∗ ( grad P ) >, Q = < F , F ∗ ( grad Q ) > .The (cid:15) part of the numerator of the rational form d (cid:18) ( P + (cid:15)P ) q ( Q − (cid:15)Q ) p (cid:19) gives rise to pq ( qQ dP − pP dQ − qQ dP + pP dQ ) = qQd ( < F , F ∗ ( grad P ) > ) − p < F , F ∗ ( grad P ) > dQ − q < F , F ∗ ( grad Q ) > dP + pP d ( < F , F ∗ ( grad Q ) > ) . (4.14)If we consider the 1-form (4.7) in replacing ( P, Q ) by ( P , Q ) and divide it by pq , thenthe result will concide with (4.14). This implies that every tangent vector ω to X ( a, s )at F from (1.9) can be interpreted as a tangent vector ω W (1.7) to P (2 , a, s ) at the samepoint. In this section we establish a new formulation of results concerning the Brieskorn/Petrovmodule defined for a rational function of type (1.4). This generalization furnishes us withthe proof of Theorem 5.1 that is necessary to establish Corollary 3.2 that in its turn playsan essential rˆole during the proof of the key Lemma 4.1.Let us consider a rational function f : P \ R → C as in (1.4) satisfying Conditions2.1. Further in this section, we regard f = P q Q p with P, Q ∈ C [ X, Y ] defined on C . Inother words in the sequel P = P ( X, Y, Q = Q ( X, Y,
1) in terms of polynomials in (1.4).This reduction is possible due to the fact that f is transversal to Z = 0.We recall that D := f − ( ∞ ) = V ( Q ) is smooth due to the Condition 2.1, (1). Let Ω i ( ∗ D ) be the set ofrational i -forms on P with poles of arbitrary order along D . Let t be an affine coordinateof C = P \ {∞} . The set Ω i ( ∗ D ) can be regarded as a C [ t ]-module according to thefollowing identification: p ( t ) .ω = p ( f ) ω, ω ∈ Ω i ( ∗ D )for p ( t ) ∈ C [ t ] . Any i -form ω ∈ Ω i ( ∗ D ) can be considered as a polynomial i -form in threevariables X, Y, ζ with dζ = − ζ dQ . The C [ t ]-module of relative rational 2-forms withpoles of arbitrary order is defined as follows:Ω P / P ( ∗ D ) = Ω ( ∗ D ) df ∧ Ω ( ∗ D ) . One can find a C [ t ]-module injective and surjective homomorphism from Ω P / P ( ∗ D ) | C to the following C [ t ]-module with quotient ring structure(5.1) M ( ∗ D ) := C [ X, Y, ζ ] I defined for the ideal(5.2) I = < ζ.Q − , qP X − pζ.Q X .P, qP Y − pζ.Q Y .P > . We remark here that the element Q is invertible in the quotient ring M ( ∗ D ) . roposition 5.1. Ω P / P ( ∗ D ) has a structure of vector space of dimension µ f = ( n + m − − nm where µ f is the global Milnor number of f .Proof. According to [ B ], [ Mo3 , Corollary 1.1], Ω P / P ( ∗ D ) is a vector space with dimen-sion µ f : the sum of local Milnor numbers of f . We remark here that the cardinality ofthe set R = V ( P ) ∩ V ( Q ) is equal to nm. We can prove the Proposition 5.1 by the aid of the Gr¨obner basis. In fact by usingthe graded lex order on C [ X, Y, ζ ] one can find a Gr¨obner basis ˜ I for the ideal I (5.2) andthen by considering the leading part of ˜ I the basis of M ( ∗ D ) is obtained. Indeed thisbasis depends on ζ but by changing the basis we can write ζ as a polynomial in variables X, Y because ζ is an invertible.By the Condition 2.1, (1) we have(5.3) M ( ∗ D ) ∼ = µ f (cid:88) j =1 C .m j ( X, Y )for m j ( X, Y ) ∈ C [ X, Y ] . We see that the concrete calculation of the monomials m j ( X, Y ) in (5.3) can be donewith the following example that essentially covers all necessary cases (1.4) under theConditions 2.1. Namely these conditions mean that the Newton polyhedron of the poly-nomial P ( X, Y ) (resp. Q ( X, Y )) is a triangle with vertices { (0 , , ( m, , (0 , m ) } (resp. { (0 , , ( n, , (0 , n ) } ) with non-degenerate condition on the edge s ( m, − s )(0 , m ) , s ∈ [0 ,
1] (resp. s ( n,
0) + (1 − s )(0 , n ) , s ∈ [0 , n, m of the form P = p X m + p Y m and Q = q X n + q Y n where P, Q are co-prime. The module M ( ∗ D ) is generatedby X i Y (cid:96) , s.t X m − Y n − (cid:54) | X i Y (cid:96) , ≤ i, (cid:96) ≤ ( m + n − , Figure 1: ( i, j ) ↔ X i Y (cid:96) Thus we get monomials m j ( X, Y ) for j = 1 , · · · , µ f = ( n + m − − nm. See Figure 1. 13ow let us define the relative cohomology associated to f that is endowed with C [ t ]-module structure(5.4) H f := Ω ( ∗ D ) df ∧ Ω ( ∗ D ) + d Ω ( ∗ D ) . where t = f in the affine coordinate C . This module is called Brieskorn C [ t ] -module (or Petrov module according to [ G1 ]). It is worthy noticing that the denominator of (5.4)represents the space of relatively exact rational 1-forms modulo F ( ω ) for (3.3).We define the degree of the zero divisor Z ( η ) of a rational form η = P η ( X, Y ) dX + P η ( X, Y ) dYQ (cid:96) , (cid:96) ≥ Q (cid:54) | P η , P η as follows(5.5) degZ ( η ) = max { degP η ( X, Y ) , degP η ( X, Y ) } Now we state the following (see [
Mo6 , Theorem 10.12.1]):
Theorem 5.1.
The C [ t ] -module H f is free and finitely generated by 1-forms α j , j =1 , · · · , µ f and each 1-form α j can be defined by the condition dα j = m j ( x, y ) dx ∧ dy, where m j ( x, y ) is an element of the monomial basis (5.3) for M ( ∗ D ) . Furthermore, arational 1-form α in Ω ( ∗ D ) can be written as follows α = µ (cid:88) j =1 C j ( f ) α j + df ∧ ζ + dζ where ζ , ζ ∈ Ω ( ∗ D ) and C j is a polynomial with degree that admits an evaluation, (5.6) deg ( C j ) ≤ deg ( Z ( α )) + deg ( Q ) − deg ( Z ( α j )) − qdeg ( P ) = deg ( Z ( α )) + n − deg ( Z ( α j )) − mq . Here Z ( η ) is zero divisor of η . In order to give a proof of this theorem, we introduce period integrals associated tothe rational function f as in (1.4).Let δ i ( t ) , i = 1 , · · · , µ f be a continuous family of vanishing cycles around the criticalpoints of f which form a basis of H ( f − ( t ) , Z ) for any regular value t ∈ C . Suppose thatthe 1-forms α , · · · , α µ f ∈ H f such that dα j = m j dx ∧ dy where m j is an element of thebasis (5.3) for M ( ∗ D ) . With these homology and cohomology bases one can associate the period matrix (5.7) Y ( t ) := (cid:82) δ ( t ) α · · · (cid:82) δ µ ( t ) α ... . . . ... (cid:82) δ ( t ) α µ · · · (cid:82) δ µ ( t ) α µ which is an analytic multi-valued matrix-function ramified over the critical values C f of f . Also, W ( t ) := det ( Y ( t )) is called Wronskian function.14 roposition 5.2. For f under the Conditions 2.1, the determinant W ( t ) of any periodmatrix is a polynomial in t with zeros at C f . This fact holds regardless of the choice of theforms which constitute a basis of H f . Proof.
From the Picard-Lefschetz theorem we see that the monodromy action around each t = t i ∈ C f induces a monodromy transformation T i Y ( t ) on Y ( t ) (5.7) and det ( T i Y ( t )) = det ( Y ( t )) . This means that the determinant W ( t ) is a single-valued function on C . As t tends to infinity the integrals occurring in the entries of Y ( t ) grow no faster than apolynomial in | t | in any sector. This implies that W ( t ) is a polynomial. When t tends toa point c ∈ C f , at least one of vanishing cycles δ i ( t ) vanishes, hence one row of Y ( t ) iszero and the determinant of W ( t ) tends to zero.Let us define the function(5.8) ∆( t ) = ( t − t ) µ ( t − t ) µ · · · ( t − t s ) µ s where µ i is the summation of local Milnor numbers of critical points located on the fiber f − ( t i ) and (cid:80) si =1 µ i = µ . When the f is a polynomial ∆ is called determinant functionof f (see e.g. [ G1 ]). Lemma 5.1.
There exists a non-zero constant c such that W ( t ) = c ∆( t ) .Proof. If t tends to the critical value t i then the vanishing cycles δ i ( t ) correspond to t i tend to points therefore (cid:82) δ i ( t ) α j tends to zero. This implies that the matrix Y ( t ) at point t i is not of full rank, in other words t i is a root of Wronskian function W . If all criticalvalues C f of f are distinct then we W ( t ) = c · ∆( t ) as the cardinality of C f is µ f .If not, consider a deformation f A,t ( X, Y ) = ( P + a X + b Y ) q ( Q + a X + b Y ) p − t where A = ( a , b , a , b ). There is an open subset U of ( A, t ) ∈ C such that the function f A,t has distinct critical values and all the critical points belongs to affine coordinate C .Let Σ A,t = { ( A, t ) | D ( A, t ) = 0 } . As the Milnor number of f a,t is equal to the Milnornumber f then D ( A, t ) is also has degree µ therefore if A = 0 then D (0 , t ) = ∆( t ). Let { δ j ( t, A ) } µ f j =1 be a continuous family of cycles which forms a basis of H ( f − A,t (0) , Z ) for any( A, t ) / ∈ Σ A,t . For the polynomial f A,t consider the Wronskian function˜ W ( A, t ) = det (cid:32)(cid:90) δ i ( t,A ) α j (cid:33) . This function is polynomial in (
A, t ) and vanishes along Σ
A,t , this implies that ˜ W ( A, t ) = C A · D ( A, t ). The polynomial C A does not depend on t because deg ( ˜ W ( A, t )) respect to t is equal to deg ( D ( A, t )) = µ f . To finish the proof it is enough to recall that ˜ W (0 , t ) = W ( t ).Now we pass to the proof of Theorem 5.1. Proof.
First of all we see that { α j } µ f j =1 are linearly independent in H f . The form dα j df of α j coincides with the Gel’fand-Leray form of m j dx ∧ dydf . ZO , Thoerem 5.25, b] we have(5.9) det (cid:32)(cid:90) δ i ( t ) dα j df (cid:33) = det (cid:32) ddt (cid:90) δ i ( t ) α j (cid:33) = c = const (cid:54) = 0 . This means the required linear independence. We remark also that this relation can bededuced from a refinement of [ T , Theorem 2.7] established for the Gauss-Manin systemsatisfied by period integrals (cid:82) δ i ( t ) α j .Let us consider a system of equations for a column vector of unknown functions C ( t ) =( C ( t ) , · · · , C µ f ( t ))(5.10) Y ( t ) . C ( t ) = A δ ( t )with Y ( t ) (5.7) and a µ f column vector: A δ ( t ) = (cid:82) δ ( t ) α ... (cid:82) δ µ ( t ) α . Cramer’s rule solves (5.10) with solutions C j ( t ) = det Y A ,j ( t ) /det Y ( t ) j = 1 , · · · , µ f where Y A ,j ( t ) is a µ f × µ f matrix obtained by replacing the j − th column of Y ( t ) with A δ ( t ). The function det Y A ,j ( t ) has at most polynomial growth as | t | → ∞ or as t approaches to a zero of ∆( t ). By Lemma 5.1 ∀ j, C j ( t ) ∈ C [ t ] as Y A ,j ( t ) vanishes at zerosof ∆( t ).In other words, for a fixed i we have (cid:90) δ i ( t ) α = µ (cid:88) j =1 C j ( t ) (cid:90) δ i ( t ) α j . This implies that(5.11) α = µ (cid:88) j =1 C j ( f ) α j in the C [ t ]-module H f . Assume that α = Q − l ( P α dx + P α dy ) has poles of order l along V ( Q ) therefore dα = hQ − l − dx ∧ dy where h is a polynomial of degree deg ( Z ( α )) + n − d ( C j ( f ) α j ) = C (cid:48) j ( f ) f ( qQdP − pP dQ ) ∧ α j P Q + C j ( f ) dα j . Each term of the above expression has degree ( deg ( C j ) − mq + ( mq −
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Y. Zare , S. Tanab´eDepartment of Mathematics,Galatasaray University,C¸ ıra˘gan cad. 36,Be¸sikta¸s, Istanbul, 34357, Turkey.
E-mails : [email protected],[email protected]@gsu.edu.tr: [email protected],[email protected]@gsu.edu.tr