On the geometry of Poincaré's problem for one-dimensional projective foliations

aabc Anais da Academia Brasileira de Ciências An. Acad. Bras. Ciênc. 0001-3765 1678-2690 Academia Brasileira de Ciências Rio de Janeiro, RJ, Brazil Consideramos o problema de relacionar carateres geométricos extrínsecos de uma variedade projetiva lisa e irredutível, que é invariante por uma folheação holomorfa de dimensão um de um espaço projetivo complexo, a objetos geométricos associados à folheação. On the geometry of Poincaré's problem for one-dimensional projective foliationsMARCIO G. SOARES*Departamento de Matemática, ICEx, UFMG - 31270-901 Belo Horizonte, Brazil Manuscript received on June 28, 2001; accepted for publication on July 18, 2001.ABSTRACTWe consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation. Key words: holomorphic foliations, invariant varieties, polar classes, degrees.1 INTRODUCTION H. Poincaré treated, in (1891), the question of bounding the degree of an algebraic curve, which is a solution of a foliation on with rational first integral, in terms of the degree of the foliation. This problem has been considered more recently in the following formulation: to bound the degree of an irreducible algebraic curve S, invariant by a foliation on , in terms of the degree of the foliation. Simple examples show that, when S is a dicritical separatrix of , the search for a positive solution to the problem is meaningless. The obstruction in this case was given by M. Brunella in (1997), and reads: the number c1(N) - S . S may be negative if S is a dicritical separatrix (here, N is the normal bundle of the foliation). More than that, A. Lins Neto constructs, in (2000), some remarkable families of foliations on providing counterexamples for this problem, all involving singular separatrices and dicritical singularities. However, as was shown in (Brunella 1997), when S is a non-dicritical separatrix, the number c1(N) - S . S is nonnegative and, in , this means d0() + 2 d0(S), where d0() and d0(S) are the degrees of the foliation and of the curve, respectively. Another solution to the problem, in the non-dicritical case, was given by M.M. Carnicer in (1994), using resolution of singularities. Let us now consider one-dimensional holomorphic foliations on , n 2, that is, morphisms : (m) T , m , m £ 1, with singular set of codimension at least 2. We write m = 1 - d0() and call d0() 0 the degree of . From now on we will consider d0() 2. This is the characteristic number associated to the foliation. On the other hand, if we consider -invariant algebraic varieties , it is natural to consider other characters associated to , not just its degree. This is the point of view we address. More precisely, we pose the question of relating extrinsic geometric characters of to geometric objects associated to . This approach produces some interesting results. Let us illustrate the two-dimensional situation. Suppose we have an -invariant irreducible plane curve S. We associate to a tangency divisor (depending on a pencil ), which is a curve of degree d0() + 1 and contains the first polar locus of S. Computing degrees we arrive at d0(S) £ d0() + 2 in case S is smooth, and at d0(S)(d0(S) - 1) - ( - 1) £ (d0() + 1)d0(S) in case S is singular, where is the Milnor number of S at p. This allows us to recover a result of D. Cerveau and A. Lins Neto (1991), which states that if S has only nodes as singularities, then d0(S) £ d0() + 2, regardless of the singularities of being dicritical or non-dicritical. In the higher dimensional situation, we obtain relations among polar classes of -invariant smooth varieties and the degree of the foliation. 2 THE TANGENCY DIVISOR OF WITH RESPECT TO A PENCIL Let be a one-dimensional holomorphic foliation on of degree d0() 2, with singular set of codimension at least 2. We associate a tangency divisor to as follows: Choose affine coordinates (z1,..., zn) such that the hyperplane at infinity, with respect to these, is not -invariant, and let be a vector field representing , where 0 is homogeneous of degree d0() and Yi(z1,..., zn) is a polynomial of degree £ d0(), 1 £ i £ n. Let H be a generic hyperplane in . Then, the set of points in H which are either singular points of or at which the leaves of are not transversal to H is an algebraic set, noted tang(H,), of dimension n - 2 and degree d0() (observe that g(z1,..., zn) = 0 is precisely tang(H,)). DEFINITION. Consider a pencil of hyperplanes = {Ht}t , with axis Ln - 2. The tangency divisor of with respect to is LEMMA 2.1. is a (possibly singular) hypersurface of degree d0() + 1. PROOF. Let p be a point in Ln - 2, the axis of the pencil. If p sing() then p is necessarily in , otherwise p is a regular point of . In this case, if is the leaf of through p, then either TpLn - 2 or, Tp together with Ln - 2 determine a hyperplane H, and hence we have p tang(H,) , so that Ln - 2. Now, let p Ln - 2 be a regular point of and choose a generic line , transverse to Ln - 2, passing through p and such that Ln - 2 and determine a hyperplane Hb, distinct from H. This line meets at p and at d0() further points, counting multiplicities, corresponding to the intersections of with tang(Hb,). Hence has degree d0() + 1. EXAMPLE. If we consider the two-dimensional Jouanolou's example and the pencil = {(at, bt) : t , (a : b) }, a straightforward manipulation shows that is given, in homogeneous coordinates (X : Y : Z) in , by 3. -INVARIANT SMOOTH IRREDUCIBLE VARIETIES Let us recall some facts about polar varieties and classes (Fulton 1984). If is a smooth irreducible algebraic subvariety of , of dimension n - k, and Lk + j - 2 is a linear subspace, then the j-th polar locus of is defined by for 0 £ j £ n - k. If Lk + j - 2 is a generic subspace, the codimension of j() in is precisely j. The j-th class, (), of is the degree of j() and, since the cycle associated to j() is we have LEMMA 3.1. Let be a smooth irreducible algebraic variety of dimension n - k, -invariant and not contained in sing(). Then PROOF. Let us first assume is a linear subspace of . In this case j = , for j 1, so the first assertion of the lemma is meaningless. Assume then is not a linear subspace and choose a pencil of hyperplanes = {Ht}t , with axis Ln - 2 generic, so that codim(n - k(),) = n - k. If q n - k(), then Tq meets Ln - 2 in a subspace W of dimension at least n - k - 1. If TqLn - 2 then any hyperplane Ht contains Tq, if not, a line Tq, Ln - 2, W consisting of a point determines, together with Ln - 2, a hyperplane Ht such that TqHt. Since is -invariant, we have TqTqHt, in case q is not a singular point of , where is the leaf of through q. This implies q tang(Ht,) , so that n - k() . Also, it follows from the definition of that is not contained in it. THEOREM I. Let be a one-dimensional holomorphic foliation on of degree d0() 2, with singular set of codimension at least 2, and let be an -invariant smooth irreducible algebraic variety, of dimension n - k, which is not a linear subspace of , and not contained in sing(). Suppose n - k - j() but n - k - j - 1() , for some 0 £ j £ n - k - 1. Then PROOF. Observe that we may assume n - k - j() n - k - j - 1() and hence Bézout's Theorem then gives COROLLARY 1. Let sing() be a smooth irreducible complete intersection in , which is not a linear subspace, defined by F1 = 0,..., Fk = 0 where F[z0,..., zn] is homogeneous of degree d, 1 £ £ k and -invariant, where is as in Theorem I. If n - k - j() but n - k - j - 1() then where is the Wronski (or complete symmetric) function of degree in k variables Observe that if is a smooth irreducible hypersurface, this reads d0() + 2 d0(). In (Soares 1997) we showed d0() + 1 d0(), but assumed to be a non-degenerate foliation on . Also, in (Soares 2000) the following estimate is obtained, provided n - k is odd and is non-degenerate: if 1 £ k £ n - 2 then We remark that this estimate is sharper than that given in Corollary 1. 4. THE TWO-DIMENSIONAL CASE As pointed out in Corollary 1, whenever we have a smooth irreducible -invariant plane curve S, the relation d0(S) £ d0() + 2 holds because (S) = d0(S)(d0(S) - 1), regardless of the nature of the singularities of , provided sing() has codimension two. In order to treat the case of arbitrary irreducible -invariant curves, let us recall the definition (see R. Piene 1978) of the class of a (possibly singular) irreducible curve S in . We let Sreg denote the regular part of S and, for a generic point p in , we consider the subset of Sreg consisting of the points q such that p TqSreg. The closure 1 of in S is the first polar locus of S, and the class (S) of S is its degree. 1 is a subvariety of codimension 1 whose degree is given by Teissier's formula (Teissier 1973): where the summation is over all singular points q of S, mq denotes the Milnor number of S at q and mq denotes the multiplicity of S at q. Because 1 is a finite set of regular points in S, revisiting Lemma 3.1 we conclude: 1S. Also, sing(S) Í sing(), so that sing(S) S and hence 1sing(S) S. It follows from Bézout's theorem that (S) + mq (d0() + 1)d0(S) Therefore we obtain the THEOREM II. Let S be an irreducible curve, of degree d0(S) > 1, invariant by a foliation on , of degree d0() 2 with sing() of codimension 2. Then where the summation extends over all singular points q of S.This gives at once the following result, first obtained by Cerveau and Lins Neto (1991); COROLLARY 2. If all the singularities of S are ordinary double points (so that mq= 1) then d0(S) d0() + 2. Theorem II illustrates one obstruction to solving Poincaré's problem in general, since we cannot estimate the sum åq(mq- 1) when dicritical singularities are present. However, if S is an irreducible -invariant algebraic curve, which is a non-dicritical separatrix, then it follows from (Brunella 1997) that where the sum is over all singular points q of S, B1q,..., Brqq are the analytic branches of S at q, and GSV denotes the Gomez-Mont/Seade/Verjovsky index. REMARK. Let S be a non-dicritical separatrix of , so that d0(S) £ d0() + 2. Assume equality holds in the expression in Theorem II, which amounts to Hence we conclude d0(S) = d0() + 2 and S has only ordinary double points as singularities. 5. -INVARIANT SMOOTH IRREDUCIBLE CURVES We have the following immediate consequence of Corollary 1: if we consider an -invariant smooth one-dimensional complete intersection S = sing(), then d1 + ... + dn - 1d0() + n so that d0(S) provided codim sing() 2. In the general case we have: COROLLARY 3. Let S sing() be an -invariant smooth irreducible curve of degree d0(S) > 1, where is a one-dimensional holomorphic foliation on of degree d0() 2, with singular set of codimension at least 2. Then the first class (S) of S satisfies (S) (d0() + 1)d0(S), the geometric genus g of S satisfies g + 1. Also, if N(, S) is the number of singularities of along S, then N(, S) (d0() + 1)d0(S). PROOF. Since S is a curve which is not a line, we have to consider only (S) = d0(S) and (S). The first inequality follows immediately from Theorem I. To bound the genus we observe that Lefschetz' theorem on hyperplane sections (Lamotke 1981) gives (S) = 2d0(S) + 2g - 2 and the second inequality follows. On the other hand, since S is irreducible and not contained in sing(), Whitney's finiteness theorem for algebraic sets (Milnor 1968) implies that S sing() is connected, and hence N(, S) is necessarily finite. Also, sing() S Ì S and Bézout's theorem implies N(, S) (d0() + 1)d0(S). The first class of a smooth irreducible curve S in was calculated by R. Piene (1976), and is as follows: (S) = 2(d0(S) + g - 1) - where g is the genus of S and 0 is an integer, called the 0 - th stationary index. It follows from Theorem I that: COROLLARY 4. With the same hypothesis of Corollary 3 2d0(S) - (S) - (d0() + 1)d0(S). REMARK ON EXTREMAL CURVES. We can obtain an estimate for d0(S) in terms of d0() and n 3, provided S is non-degenerate (that is, is not contained in a hyperplane) and extremal (that is, the genus of S attains Castelnuovo's bound). Recall that, for S a smooth non-degenerate curve in of degree d0(S) 2n, Castenuovo's bound is (Arbarello et al. 1985): g(n - 1) + m, where d0(S) - 1 = m(n - 1) + . The inequality g + 1 together with S extremal give, performing a straightforward manipulation: d0(S) 2(d0() - 1)(n - 1) + . ACKNOWLEGMENTS I'm grateful to M. Brunella for useful conversations, to PRONEX-Dynamical Systems (Brazil) for support and to Laboratoire de Topologie, Univ. de Bourgogne (France) for hospitality. RESUMO Consideramos o problema de relacionar carateres geométricos extrínsecos de uma variedade projetiva lisa e irredutível, que é invariante por uma folheação holomorfa de dimensão um de um espaço projetivo complexo, a objetos geométricos associados à folheação. Palavras-chave: folheações holomorfas, variedades invariantes, classes polares, graus. LINS NETO A. 2000. Some examples for Poincaré and Painlevé problems. Pre-print IMPA. * Member of Academia Brasileira de Ciências E-mail: [e-mail] ARBARELLO E, CORNALBA M, GRIFFITHS PA AND HARRIS J. 1985. Geometry of Algebraic Curves, volume I. Grundlehren der mathematischen Wissenschaften 267, Springer-Verlag. Geometry of Algebraic Curves 1985 I ARBARELLO E CORNALBA M GRIFFITHS PA HARRIS J BRUNELLA M. 1997. Some remarks on indices of holomorphic vector fields. 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