Abstracts
We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a onedimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.
holomorphic foliations; invariant varieties; polar classes; degrees
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.
folheações holomorfas; variedades invariantes; classes polares; graus
On the geometry of Poincaré's problem for onedimensional projective foliations
MARCIO G. SOARES^{*}* Member of Academia Brasileira de Ciências Email: msoares@math.ufmg.br
Departamento de Matemática, ICEx, UFMG  31270901 Belo Horizonte, Brazil
Manuscript received on June 28, 2001; accepted for publication on July 18, 2001.
ABSTRACT
We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a onedimensional 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
SSimple 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 c_{1}(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 nondicritical separatrix, the number c_{1}(N)  S^{ . }S is nonnegative and, in , this means d^{0}() + 2 d^{0}(S), where d^{0}() and d^{0}(S) are the degrees of the foliation and of the curve, respectively. Another solution to the problem, in the nondicritical case, was given by M.M. Carnicer in (1994), using resolution of singularities.
Let us now consider onedimensional 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  d^{0}() and call d^{0}() 0 the degree of . From now on we will consider d^{0}() 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 twodimensional 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 d^{0}() + 1 and contains the first polar locus of S. Computing degrees we arrive at d^{0}(S) £ d^{0}() + 2 in case S is smooth, and at d^{0}(S)(d^{0}(S)  1)  (  1) £ (d^{0}() + 1)d^{0}(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 d^{0}(S) £ d^{0}() + 2, regardless of the singularities of being dicritical or nondicritical.
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
d^{0} tangency divisorChoose affine coordinates (z_{1},..., z_{n}) 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 d^{0}() and Y_{i}(z_{1},..., z_{n}) is a polynomial of degree £ d^{0}(), 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 d^{0}() (observe that g(z_{1},..., z_{n}) = 0 is precisely tang(H,)).
DEFINITION. Consider a pencil of hyperplanes = {H_{t}}t , with axis L^{n  2}. The tangency divisor of with respect to is
LEMMA 2.1.
is a (possibly singular) hypersurface of degree d^{0}() + 1.PROOF. Let p be a point in L^{n  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 T_{p}
L^{n  2} or, T_{p} together with L^{n  2} determine a hyperplane H, and hence we have p tang(H,) , so that L^{n  2}. Now, let p L^{n  2} be a regular point of and choose a generic line , transverse to L^{n  2}, passing through p and such that L^{n  2} and determine a hyperplane H_{b}, distinct from H. This line meets at p and at d^{0}() further points, counting multiplicities, corresponding to the intersections of with tang(H^{b},). Hence has degree d^{0}() + 1.EXAMPLE. If we consider the twodimensional 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 L^{k + j  2} is a linear subspace, then the jth polar locus of is defined byfor 0 £ j £ n  k. If L^{k + j  2} is a generic subspace, the codimension of _{j}() in is precisely j. The jth 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 = {H_{t}}t , with axis L^{n  2} generic, so that codim(_{n  k}(),) = n  k. If q
_{n  k}(), then T_{q} meets L^{n  2} in a subspace W of dimension at least n  k  1. If T_{q}L^{n  2} then any hyperplane H_{t} contains T_{q}, if not, a line T_{q}, L^{n  2}, W consisting of a point determines, together with L^{n  2}, a hyperplane H_{t} such that T_{q}H_{t}. Since is invariant, we have T_{q}T_{q}H_{t}, in case q is not a singular point of , where is the leaf of through q. This implies q tang(H_{t},) , so that _{n  k}() . Also, it follows from the definition of that is not contained in it.THEOREM I. Let be a onedimensional holomorphic foliation on of degree d^{0}() 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. ThenPROOF. Observe that we may assume
_{n  k  j}() _{n  k  j  1}() and henceBézout's Theorem then gives
COROLLARY 1. Let
sing() be a smooth irreducible complete intersection in , which is not a linear subspace, defined by F_{1} = 0,..., F_{k} = 0 where F[z_{0},..., z_{n}] is homogeneous of degree d, 1 £ £ k and invariant, where is as in Theorem I. If _{n  k  j}() but _{n  k  j  1}() thenwhere is the Wronski (or complete symmetric) function of degree in k variables
Observe that if is a smooth irreducible hypersurface, this reads d^{0}() + 2 d^{0}(). In (Soares 1997) we showed d^{0}() + 1 d^{0}(), but assumed to be a nondegenerate foliation on .
Also, in (Soares 2000) the following estimate is obtained, provided n  k is odd and is nondegenerate: if 1 £ k £ n  2 then
We remark that this estimate is sharper than that given in Corollary 1.
4. THE TWODIMENSIONAL CASE
As pointed out in Corollary 1, whenever we have a smooth irreducible invariant plane curve S, the relation d^{0}(S) £ d^{0}() + 2 holds because (S) = d^{0}(S)(d^{0}(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 S_{reg} denote the regular part of S and, for a generic point p in , we consider the subset of S_{reg} consisting of the points q such that p T_{q}S_{reg}. 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, m_{q} denotes the Milnor number of S at q and m_{q} denotes the multiplicity of S at q. Because _{1} is a finite set of regular points in S, revisiting Lemma 3.1 we conclude:
_{1}S.Also, sing(S) Í sing(), so that
sing(S)
Sand hence
_{1}sing(S) S.It follows from Bézout's theorem that
(S) + m_{q} (d^{0}() + 1)d^{0}(S)
Therefore we obtain the
THEOREM II. Let S be an irreducible curve, of degree d^{0}(S) > 1, invariant by a foliation on , of degree d^{0}() 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 m_{q}= 1) then
d^{0}(S) d^{0}() + 2.Theorem II illustrates one obstruction to solving Poincaré's problem in general, since we cannot estimate the sum å_{q}(m_{q} 1) when dicritical singularities are present. However, if S is an irreducible invariant algebraic curve, which is a nondicritical separatrix, then it follows from (Brunella 1997) that
where the sum is over all singular points q of S, B_{1}^{q},..., Br_{q}^{q} are the analytic branches of S at q, and GSV denotes the GomezMont/Seade/Verjovsky index.
REMARK. Let S be a nondicritical separatrix of , so that d^{0}(S) £ d^{0}() + 2. Assume equality holds in the expression in Theorem II, which amounts to
Hence we conclude d^{0}(S) = d^{0}() + 2 and S has only ordinary double points as singularities.
5. INVARIANT SMOOTH IRREDUCIBLE CURVES
Ssingd_{1} + ^{ ... } + d_{n  1}
d^{0}() + nso that
d^{0}(S)
provided codim sing() 2. In the general case we have:
COROLLARY 3. Let S sing() be an invariant smooth irreducible curve of degree d^{0}(S) > 1, where is a onedimensional holomorphic foliation on of degree d^{0}() 2, with singular set of codimension at least 2. Then the first class (S) of S satisfies
(S) (d^{0}() + 1)d^{0}(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) (d^{0}() + 1)d^{0}(S).
PROOF. Since S is a curve which is not a line, we have to consider only (S) = d^{0}(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) = 2d^{0}(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
Sand Bézout's theorem implies
N(, S) (d^{0}() + 1)d^{0}(S).The first class of a smooth irreducible curve S in was calculated by R. Piene (1976), and is as follows:
(S) = 2(d^{0}(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
2d^{0}(S)  (S)  (d^{0}() + 1)d^{0}(S).
REMARK ON EXTREMAL CURVES. We can obtain an estimate for d^{0}(S) in terms of d^{0}() and n 3, provided S is nondegenerate (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 nondegenerate curve in of degree d^{0}(S) 2n, Castenuovo's bound is (Arbarello et al. 1985):
g(n  1) + m,
where
d^{0}(S)  1 = m(n  1) + .
The inequality
g + 1
together with S extremal give, performing a straightforward manipulation:
d^{0}(S) 2(d^{0}()  1)(n  1) + .ACKNOWLEGMENTS
RESUMO
Palavraschave: folheações holomorfas, variedades invariantes, classes polares, graus.
LINS NETO A. 2000. Some examples for Poincaré and Painlevé problems. Preprint IMPA.
 ARBARELLO E, CORNALBA M, GRIFFITHS PA AND HARRIS J. 1985. Geometry of Algebraic Curves, volume I. Grundlehren der mathematischen Wissenschaften 267, SpringerVerlag.
 BRUNELLA M. 1997. Some remarks on indices of holomorphic vector fields. Publicacions Mathemŕtiques 41: 527544.
 CARNICER MM. 1994. The Poincaré problem in the nondicritical case. An Math 140: 289294.
 CERVEAU D AND LINS NETO A. 1991. Holomorphic Foliations in P^{2}_{C} having an invariant algebraic curve. An Institut Fourier 41(4): 883904.
 FULTON W. 1984. Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge  Band 2, SpringerVerlag.
 LAMOTKE K. 1981. The Topology of Complex Projective Varieties after S. Lefschetz. Topology 20: 1551.
 MILNOR J. 1968. Singular points of complex hypersurfaces. An Math Studies 61.
 PIENE R. 1976. Numerical characters of a curve in projective nspace, Nordic Summer School/NAVF. Symposium in Mathematics, Oslo, August 525.
 PIENE R. 1978. Polar Classes of Singular Varieties. An scient Éc Norm Sup 4^{e} série 11: 247276.
 POINCARÉ H. 1891. Sur l'Intégration Algébrique des Équations Differentielles du Premier Ordre et du Premier Degré. Rendiconti del Circolo Matematico di Palermo 5: 161191.
 SOARES MG. 1997. The Poincaré problem for hypersurfaces invariant by onedimensional foliations. Inventiones mathematicae 128: 495500.
 SOARES MG. 2000. Projective varieties invariant by onedimensional foliations. An Math 152: 369382.
 TEISSIER B. 1973. Cycles évanescents, sections planes et conditions de Whitney. Astérisque 89: 285362.
Publication Dates

Publication in this collection
12 Dec 2001 
Date of issue
Dec 2001
History

Accepted
18 July 2001 
Received
28 June 2001