AUTOMORPHISMS AND NON-INTEGRABILITY

On this note we prove that a holomorphic foliation of the projective plane with rich, but finite, automorphism group does not have invariant algebraic curves.


INTRODUCTION
On the monograph (Jouanolou 1979) it is proved that for every d ≥ 2, there exists a residual subset of the space of degree d holomorphic foliations of the projective plane whose elements do not have any invariant algebraic curve.The hard part of the proof is to exhibit an explicit example having such property.Jouanolou shows that the foliations induced by the homogeneous 1-forms In (Zoladek 1998) new examples of foliations of the projective plane without invariant algebraic curves are constructed.These examples also have rich automorphism groups.
In (Maciejewski et al. 2000), the four authors prove that some higher dimensional analogues of Jouanolou's examples do not have invariant algebraic hypersurfaces.Again the automorphism group of these foliations play a major role on the course of the proof.
Besides taking advantage of a rich automorphism group the above mentioned works make use of the explicit form of the equations investigated and impose some restrictions on the singularities of the equations.
The purpose of this note is to settle a general result for foliations of the projective plane with rich automorphism group.Unfortunately we cannot completely get ride of the hypothesis on the singular set of the foliation, although our assumptions are on the eigenvalues of just one of the singularities.More precisely we prove the following Theorem 1.Let F be a holomorphic foliations foliations of P 2 .Suppose that F has at least one singularity with non-singular linear part and whose quotient of eigenvalues is not rational nor a root of a unity.If the automorphism group of F is finite and acts without nontrivial fixed points on the space of cofactors then F does not admit any invariant algebraic curve.
For a definition of the space of cofactors and the action of automorphism group of the foliation on it the reader should consult section 2.

COFACTOR REPRESENTATION
On this paper a degree d singular holomorphic foliation of P 2 will be given by a 1-form ω on C 3 annihilated by the radial vector field with homogeneous coefficients of degree d + 1.More explicitly if ω = Adx + Bdy + Cdz then A, B and C are homogenous polynomials of degree d + 1 and xA + yB + zC = 0. We will denote the foliation induced by ω by F ω .
Note that if φ ∈ Aut(ω) then there exists a non zero complex number λ(φ) such that φ * ω = λ(φ) • ω.Moreover, where F is a 3-form satisfying the following Lemma 1.There exists a unique 3-form β F such that The existence and uniqueness of β F follow from the De Rham's division Theorem.
From the above Lemma it follows that for every automorphism φ of ω the following identity holds If F is a foliation of degree d then β F has degree d − 1. Denote by 3 d−1 the space of homogeneous 3-forms on C 3 of degree d − 1. Hence we have a representation ω : is the natural projection and is, by definition, the cofactor representation.We will say that the automorphism group of F acts without nontrivial fixed points on the space of cofactors if Proposition 1.Let F be a foliation on P 2 .If the automorphism group of F is finite and acts without nontrivial fixed points on the space of cofactors then either F admits liouvillian first integral or F does not admit an invariant algebraic curve.
Proof.Suppose that F admits an invariant algebraic curve C given by homogeneous polynomial F .If we consider the homogeneous polynomial we see that G is invariant by both F and Aut(F).Consequently the cofactor associated to G is a fixed point for F ω and therefore, by hypothesis, is equal to zero.Consequently, it follows from Lemma 1 that To conclude one has to observe that the multivalued 1-form • ω is closed and induces a closed multivalued one-form on the projective plane.In particular F admits a liouvillian first integral.
Example 1.Let H be the subgroup of Aut(P 2 ) generated by φ 1 [x : y : z] = [−x : −y : z] and φ 2 [x : y : z] = [−x : y : −z].Among the degree 2 foliations invariant by G there exists a P 2 ⊂ Fol(2) = PH 0 (P 2 , 1 (4)) whose elements are induced by the 1-forms The cofactor representation is given by In particular H acts without nontrivial fixed points.The lines {ay 2 +bz 2 = 0} and {−ax 2 −cz 2 = 0} are invariant by both F (a,b,c) and Aut(F a,b,c ).Consequently and F (a,b,c) has a liouvillian first integral as predicted by Proposition 1.In fact one can easily verify that F (a,b,c) has a rational first integral.

PROOF OF THE MAIN RESULT
Before starting the proof of the main result let us state a simple lemma Lemma 2. Let X be a holomorphic vector field on a neighborhood of 0 ∈ C 2 and ω = i X dx ∧ dy a 1-form dual to X. Assume that 0 is a singularity of X.If there exists a 1-form η such that dω = η ∧ ω then the trace of DX(0) is zero.
Proof.Since ω = i X dx ∧ dy we have dω = div(X)dx ∧ dy = (Tr(DX(0)) + h.t.o)dx ∧ dy.Now as 0 is a singularity of X we have: Let p be the reduced singularity of F whose quotient of eigenvalues is not rational nor a root of unity.If F admits an invariant algebraic curve C we can suppose without loss of generality that C is reduced and invariant by Aut(F).Moreover we can also assume that Aut(F) acts transitively on the set of irreducible components of C.
It follows from Lemma 1 that where F is a reduced homogeneous polynomial defining C.
From Lemma 2 we have p ∈ C. Since p is reduced, we can choose local coordinates (x, y) where p = (0, 0) and F is defined in these coordinates by An Acad Bras Cienc (2005) 77 (3) Suppose that C is smooth at p and locally defined by either {x = 0} or {y = 0}.From ( 1) and ( 2) we obtain in both cases that λ is rational contradicting our hypothesis.Therefore C will have two branches passing through p.In this case Comparing this last equality with (1) we deduce that d + 2 = deg(F ) and that the foliation is given by the closed 1-form ω F with simple poles along C. By Theorem 2.1 of (Cerveau and Mattei 1982) it follows that F is defined by a closed meromorphic 1-form where Writing the closed 1-form above in local coordinates (x, y) this implies that x and y divide locally two distinct polynomials F i , since otherwise the quotient of eigenvalues at p would be 1.
Without loss of generality we may suppose that x divides F 1 and y divides F 2 .Thus we obtain that Consequently we have λ = − α 2 α 1 .We can suppose without loss of generality that there exists φ ∈ Aut(F) such that φ * dF 1 F 1 = dF 2 F 2 .Applying φ to ω F we see that there exists a root of unity β such that Comparing the coefficients of dF 2 F 2 in both sides of the equation we infer that α 1 = βα 2 .Therefore λ = −β −1 contrary to our hypothesis.The cofactor representation is given by d (φ 1 ) (F (x, y, z) Lemma 3. If d ≡ 1 mod 3 then the automorphism group of J d acts without nontrivial fixed points on the space of cofactors.
Proof.The eigenvalue of d (φ 2 ) : 3 d−1 → 3 d−1 corresponding to the eigenvector x i y j z k dx ∧ dy ∧ dz, i + j + k = d − 1, is given by δ i+jd+kd 2 .To prove the lemma it is sufficient to show that for d ≡ 1 mod 3 the system has no solutions with i, j and k nonnegative integers.
Subtracting the first equation from the second we obtain that From the lemma above and Theorem 1 we obtain a proof that J d has non algebraic solutions when d = 1 mod 3. Note that this result also holds for every integer d greater than or equal to 2, see (Jouanolou 1979).
d ≥ 2, have such properties.His proof explores the richness of the automorphism groups of such examples.
examples of Jouanolou As stated in the introduction the examples of Jouanolou are the foliations of P 2 induced by the 1-forms ω d = i R i X d dx ∧ dy ∧ dz, where d ≥ 2 and X d = y d ∂ ∂x + z d ∂ ∂y + x d ∂ ∂z .We will denote the foliation induced by ω d by J d .The automorphisms group of J d is isomorphic to the semidirect product of Z 3 and Z d 2 +d+1 .In particular it has order 3 • (d 2 + d + 1).It is generated by φ 1 [x : y : z] = [z : x : y] and φ 2 [x : y : z] = [δx : δ d y : δ d 2 ] where δ is a d 2 + d + 1-root of the unity.