Annals of Mathematics, 152 (2000), 369–382
Projective varieties invariant by
one-dimensional foliations
By Marcio G. Soares
To Jacob Palis on his 60th anniversary
1. Introduction
This work concerns the problem of relating characteristic numbers of onedimensional holomorphic foliations of PnC to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a
one-dimensional holomorphic foliation F of M is a morphism Φ : L −→ TM
where L is a holomorphic line bundle on M . The singular set of F is the
analytic subvariety sing(F) = {p : Φ(p) = 0} and the leaves of F are the leaves
of the nonsingular foliation induced by F on M \ sing(F). If M is PnC then,
since line bundles over PnC are classified by the Chern class c1 (L) ∈ H2 (PnC , Z)
' Z, one-dimensional holomorphic foliations F of PnC are given by morphisms
Φ : O(1 − d) −→ TPnC with d ≥ 0, d ∈ Z, which we call the degree of F.
i
We will use the notation F d for such a foliation. Suppose now V −→ PnC is an
irreducible algebraic variety invariant by F d in such a way that the pull-back
i∗ F d of F d to V has a finite set of points as the singular set. The problem we
R
address is the relation between d and the degree d0 (V ) = V c1 (i∗ O(1))dimV
of V .
This question was first considered by Poincaré in [12], in the case of plane
curves invariant by foliations of P2C with a rational first integral. More recently Cerveau and Lins Neto [3], Carnicer [2] and Campillo and Carnicer
[1] addressed this question when the invariant variety is a curve. In [14] we
considered this problem for smooth hypersurfaces and in this work we treat
the case of smooth algebraic varieties, concentrating on complete intersections,
where effective computations are feasible.
To obtain the result we first calculate the number of singularities of the
foliation in the invariant variety. To this end we impose the condition of
nondegeneracy of the foliation along the variety; i.e., the linear part of a vector
field representing the foliation on the variety has only nonzero eigenvalues at
the singular points. With this at hand we can use Baum-Bott’s theorem [4]
370
MARCIO G. SOARES
and relate the degrees of the foliation and of the variety through a polynomial.
Next we need a positivity argument saying that this polynomial is positive
whenever it represents the number of singularities of a foliation leaving the
variety invariant. We then try to obtain relations between the degrees. The
positivity argument given here follows from the ampleness of the normal bundle
of a smooth projective variety together with the vanishing theorem of [10]
which, in turn, is a consequence of the work of Kamber and Tondeur [8] on
foliated bundles.
2. Statement of results
Let F d be a one-dimensional holomorphic foliation of PnC , n ≥ 2, of degree
d ≥ 2, given by a morphism
O(1 − d) −→ TPnC
Φ
with singular set sing(F d ) = {p : Φ(p) = 0} which we assume to have codimension greater than 1. Such a foliation F d is represented in affine coordinates
(z1 , . . . , zn ) by a vector field of the form
X = gR +
where R =
n
P
i=1
d
X
Xj
j=0
zi ∂z∂ i is the radial vector field, g ∈ C[z1 , . . . , zn ] is homogeneous
of degree d and Xj , 0 ≤ j ≤ d, is a vector field whose components are homogeneous polynomials of degree j. Since codim sing(F d ) ≥ 2 we have either g 6≡ 0
or g ≡ 0 and Xd cannot be written as hR with h ∈ C[z1 , . . . , zn ] homogeneous
of degree d − 1. In this case X has a pole of order d − 1 along the hyperplane
at infinity and it is worth remarking that, in case g 6≡ 0, g = 0 is precisely
the variety of tangencies of F d with the hyperplane at infinity, whereas in case
g ≡ 0 this hyperplane is invariant by the foliation.
If i : V −→ PnC is a smooth algebraic variety invariant by F d , we say
d
F is nondegenerate along V if its pull-back to V , i∗ F d , is nondegenerate;
i.e., sing(F d ) ∩ V is a finite set of points and, at all points p ∈ sing(F d ) ∩ V ,
¡ i¢
det ∂Y
∂zj (p) 6= 0 for any vector field Y , tangent to V at p and representing
i∗ F d . Observe that we necessarily have V 6⊂ sing(F d ) and that a singularity
of F d which lies on V may be degenerate, when considered as a singularity of
F d in PnC .
i
Let Vn−k −→ PnC , n ≥ 2, be a smooth irreducible algebraic variety of
n−k
n−k
n−k
codimension k. Define V[i]
as follows: V[0]
= Vn−k , V[1]
is a generic
n−k
n−k
hyperplane section of Vn−k and V[i]
is a generic hyperplane section of V[i−1]
,
371
ONE-DIMENSIONAL FOLIATIONS
n−k
i ≥ 2. In what follows it is irrelevant whether we regard V[i]
as a subvariety of
¡
¢
n−i
n−k
PnC or of PC and we let χ V[i] denote the Euler-Poincaré characteristic of
¡ n−k ¢
n−k
is just the degree d0 (Vn−k )
. Of course, by Bézout’s theorem, χ V[n−k]
V[i]
of Vn−k .
We then have:
Theorem I. Let Vn−k −→ PnC be a smooth irreducible algebraic variety invariant by F d , a one-dimensional holomorphic foliation of PnC of degree
d ≥ 2, nondegenerate along Vn−k . Then the number of singularities of F d in
Vn−k , noted N (i∗ F d , Vn−k ), is
i
∗
N (i F , V
d
n−k
Xh ¡
¡ n−k ¢ n−k n−k
¢
¡ n−k
¢i n−k−j
n−k
χ V[n−k−j]
) = χ V[n−k] d
+
.
d
−χ V[n−k−j+1]
j=1
Alternatively,
∗
N (i F , V
d
n−k
)=
n−k
X
"
j
X
j=0
#
i
(−1) %i (V
n−k
) dn−k−j
i=0
where %i (Vn−k ) is the ith class of Vn−k . Moreover, N (i∗ F d , Vn−k ) > 0.
n−k
−→ PnC is a smooth irreducible complete interSuppose now that V(d
1 ,...,dk )
section defined by F1 = 0, . . . , Fk = 0 where F` ∈ C[z0 , . . . , zn ] is homogeneous
of degree d` , 1 ≤ ` ≤ k. Then we have:
i
Corollary.
n−k
is invariant by F d , a one-dimensional
Assume V(d
1 ,...,dk )
holomorphic foliation of
PnC
n−k
of degree d ≥ 2, nondegenerate along V(d
.
1 ,...,dk )
n−k
Then the number of singularities of F d in V(d
is
1 ,...,dk )
¡
∗
N i F
d
n−k
, V(d
1 ,...,dk )
¢
=
n−k
X
"
j=0
= (d1 . . . dk )
" j
n−k
X X
j=0
(k)
where Wδ
variables
j
X
δ
³
(−1) %δ
n−k
V(d
1 ,...,dk )
´
#
dn−k−j
δ=0
#
δ
(−1)
(k)
Wδ (d1
− 1, . . . , dk − 1) dn−k−j
δ=0
is the Wronski (or complete symmetric) function of degree δ in k
(k)
Wδ (X1 , . . . , Xk ) =
X
X1i1 . . . Xkik .
i1 +···+ik =δ
As an application of Theorem I and the corollary we have:
372
MARCIO G. SOARES
Theorem II.
n − k odd. Then
n−k
Let V(d
and F d be as in Theorem I and suppose
1 ,...,dk )
d≥
n−k
%n−k (V(d
)
1 ,...,dk )
n−k
%n−k−1 (V(d
)
1 ,...,dk )
.
In order to avoid trivialities, the invariant varieties considered in Theorem
II are not linear subspaces of PnC . The theorem gives, in case k = n − 1, that
(n−1)
d ≥ W1
(d1 − 1, . . . , dn−1 − 1),
and hence
µ
d
0
1
(V(d
)
1 ,...,dn−1 )
≤
d
1+
n−1
¶n−1
.
³
´
n−1
V
= d1 (d1 − 1)j (see
)
≤
d
+
1,
since
%
Also, if k = 1 we obtain d0 (Vdn−1
j
d1
1
[14]).
Example 1. Let V`2n−1 , ` ≥ 3, be the smooth hypersurface in P2n
C defined
2n−1
`
` + X`
+ X2n
is invariant by the
by X1` + X2` + · · · + X2n−1
2n+1 = 0. Also, V`
foliation F of degree ` − 1 on P2n
C defined by the vector field (affine coordinates
X2n+1 = 1)
∂
∂
+ (z2` + 1)
∂z1
∂z2
·
n
X
`−1
+
)
(z2`−1 z2i−1 − z2i
Z = z2`−1 z1
i=2
∂
∂z2i−1
+
(z2`−1 z2i
+
`−1
z2i−1
)
¸
∂
.
∂z2i
Note that the bound d0 (V) = d + 1 is attained. Observe that in P2n
C a
smooth hypersurface defines a hamiltonian vector field which can be used to
foliate it.
Example 2. The elliptic quartic curve E4 can be realized as the complete
intersection in P3C defined by the quadrics Q1 = {X12 + X22 + X32 + X42 = 0}
and Q2 = {X1 X3 + X2 X4 = 0}. The foliation F of degree 2, on P3C , defined
by the vector field (affine coordinates X4 = 1)
∂
∂
∂
+ (−z1 z22 + 2z2 z3 − z1 )
+ (−z1 z2 z3 − z22 + z32 + 1)
∂z1
∂z2
∂z3
¶n−1
µ
d
has E4 as invariant curve. Note that the bound d0 (V) = 1 +
is
n−1
attained.
Z = (−z12 z2 + z1 z3 )
ONE-DIMENSIONAL FOLIATIONS
373
3. Proof of Theorem I
Choose a hyperplane H∞ transverse to Vn−k and such that H∞ ∩ Vn−k ∩
sing(i∗ F d ) = ∅. Since a vector field X representing F d has a pole of order d−1
along H∞ , the same holds for the representative X|Vn−k so that it defines a
section of TVn−k ⊗ i∗ O(d − 1). This section has isolated nondegenerate zeros
by hypothesis, and so, according to Baum-Bott’s theorem [4] applied to the
top Chern class:
Z
∗ d
n−k
N (i F , V
)=
cn−k (TVn−k ⊗ i∗ O(d − 1))
Vn−k
where integration is over the fundamental class of Vn−k . Since
cn−k (TVn−k ⊗ i∗ O(d − 1))
= cn−k (TVn−k ⊗ i∗ O(−1) ⊗ i∗ O(d))
=
n−k
X
¡
¢
cj TVn−k ⊗ i∗ O(−1) c1 (i∗ O(d))n−k−j
j=0
=
n−k
X
¢
¡
cj TVn−k ⊗ i∗ O(−1) c1 (i∗ O(1))n−k−j dn−k−j ,
j=0
and
¡
cj TV
n−k
µ
¶
j
¢ X
j−i n − k − i
ci (Vn−k )c1 (i∗ O(1))j−i ,
⊗ i O(−1) =
(−1)
j−i
∗
i=0
we get
(1)
cn−k (TVn−k ⊗ i∗ O(d − 1))
µ
¶
¸
j
n−k
X·X
j−i
j−i n − k − i
n−k
∗
ci (V
c1 (i∗ O(1))n−k−j dn−k−j
=
(−1)
)c1 (i O(1))
j−i
j=0
=
i=0
j
n−k
X·X
j=0
i=0
µ
j−i
(−1)
¶
¸
n−k−i
n−k−i n−k−j
n−k
∗
ci (V
d
)c1 (i O(1))
.
j−i
Following Fulton [5, 14.4.15], recall that the cycle associated to the j th polar
locus of Vn−k is given, for a general linear subspace Lk+j−2 of PnC , by
µ
¶
j
X
i n−k+1−i
n−k
k+j−2
[V
ci (Vn−k )c1 (i∗ O(1))j−i
(L
)] =
(−1)
j−i
i=0
374
MARCIO G. SOARES
and that the j th class %j of Vn−k is defined to be the degree of [Vn−k (Lk+j−2 )],
so that
µ
¶
Z X
j
i n−k+1−i
(−1)
%j =
ci (Vn−k )c1 (i∗ O(1))n−k−i
j−i
Vn−k
i=0
n−k−j
∗
since the degree is computed
through
multiplication
. Now,
¡n−k−i+1
¢ ¡n−k−i
¢ ¡n−k−i¢by c1 (i O(1))
using Stifel’s relation
=
+
we
get
`−i
`−i
`−i−1
j
X
Z
i
(−1) %i =
i=0
Vn−k
j
X
µ
(−1)
j−i
i=0
It follows from (1) that
N (i∗ F d , Vn−k ) =
Z
¶
n−k−i
ci (Vn−k )c1 (i∗ O(1))n−k−i .
j−i
cn−k (TVn−k ⊗ i∗ O(d − 1))
Vn−k
=
j
n−k
X·X
j=0
¸
(−1) %i dn−k−j .
i
i=0
Let us now recall a consequence of Lefschetz’ theorem on hyperplane sections [9]. If X is a smooth irreducible algebraic variety and Ht∈P1C is a pencil of
hyperplanes with axis Ln−2 then the Euler-Poincaré characteristics are related
by
χ(X) = 2χ(X ∩ H) − χ(X ∩ Ln−2 ) + (−1)dim X %dim X (X)
where H is a generic element of the pencil. Applying this to Vn−k we get:
¢
¡ n−k ¢
¡ n−k ¢
¡ n−k ¢
¡
= χ V[1]
− χ V[2]
+ (−1)n−k %n−k (Vn−k ).
χ Vn−k − χ V[1]
By repeating this argument, using the Piene-Severi comparison theorem [11]
(which says that the class of a hyperplane section of X is %dim X−1 (X)) we
obtain
j
X
¡
¢
¡ n−k ¢
¡ n−k
¢
(−1)i %i Vn−k = χ V[n−k−j]
− χ V[n−k−j+1]
i=0
so that
Xh ¡
¡ n−k ¢ n−k n−k
¢
¡ n−k
¢i n−k−j
n−k
N (i∗ F d , Vn−k ) = χ V[n−k]
+
.
d
−χ V[n−k−j+1]
d
χ V[n−k−j]
j=1
It remains to show N (i∗ F d , Vn−k ) > 0. To this end we invoke the vanishing theorem of [10, théorème 2] which states that, if Vn−k is foliated by
i∗ F d without singularities, then any polynomial on the Chern classes of the
normal bundle NVn−k of Vn−k in PnC must vanish in dimension greater than
375
ONE-DIMENSIONAL FOLIATIONS
2s, where s is ¡the complex
codimension of i∗ F d . Now, codim i∗ F d = n − k − 1
¢
and, since det NVn−k is ample [6], it follows from the hard Lefschetz theorem
¡
¡
¢¢n−k
¡
¢
[9] that the rational class c1 det NVn−k
is a basis of H2n−2k Vn−k , Q ,
¡
¢n−k
is nonzero. This finishes the proof of Theorem I.
and therefore c1 NVn−k
4. Proof of the corollary to Theorem I
n−k
Let us calculate cj (V(d
). Set c1 (i∗ O(1)) = h. It is well known [7]
1 ,...,dk )
n−k
that the total Chern class of V(d
is given by
1 ,...,dk )
n−k
)=
c(V(d
1 ,...,dk )
(2)
(1 + h)n+1
.
k
Q
(1 + d` h)
`=1
Recalling that the Wronski functions are defined by
∞
X
1
(k)
=
(−1)δ Wδ (d1 , . . . , dk ) tδ
k
Q
(1 + d` t) δ=0
`=1
we have that (2) becomes
n−k
c(V(d
)
1 ,...,dk )
n−k
X
=
j=0
j=0
n−k
)
ci (V(d
1 ,...,dk )
"
=
i
X
#
X µ n + 1¶
(k)
δ
(−1) Wδ hj
i
i+δ=j
" j
n−k
X X
=
so that
"
µ
δ
(−1)
δ=0
µ
δ
(−1)
δ=0
n+1
i−δ
n+1
j−δ
#
¶
(k)
Wδ
hj
#
¶
(k)
Wδ (d1 , . . . , dk )
0 ≤ i ≤ n − k.
hi ,
Hence,
(3)
¡ n−k
χ V(d
)=
1 ,...,dk )
Z
¡ n−k
¢
cn−k V(d
1 ,...,dk )
n−k
V(d
,...,d
1
=
· n−k
X
δ=0
k)
µ
δ
(−1)
n+1
n−k−δ
¶
¸
(k)
Wδ (d1 , . . . , dk )
Z
hn−k
n−k
V(d
,...,d
1
k)
µ
¶
· n−k
¸
X
n+1
(k)
δ
Wδ (d1 , . . . , dk ) .
= (d1 · · · dk )
(−1)
n−k−δ
δ=0
376
MARCIO G. SOARES
n−k−(q)
To calculate the Euler-Poincaré characteristic of the variety V(d1 ,...,dk ,1q ) , obn−k
tained by cutting V(d
successively by q generic hyperplanes, we either
1 ,...,dk )
n−k−(q)
add q extra equations of degree 1, or regard V(d1 ,...,dk ,1q ) as a complete inter-
section in Pn−q
, given by k equations of degrees d1 , . . . , dk . Doing it this last
C
way we have, from (3):
" n−k−q
#
µ
¶
X
¡ n−k−(q) ¢
n
−
q
+
1
(k)
Wδ (d1 , . . . , dk )
χ V(d1 ,...,dk ,1q ) = (d1 · · · dk )
(−1)δ
n−q−k−δ
δ=0
and
¡ n−k−(q+1) ¢
χ V(d1 ,...,d ,1q+1 )
k
" n−k−q−1
µ
X
= (d1 · · · dk )
(−1)δ
δ=0
n−q
n−q−k−1−δ
#
¶
(k)
Wδ (d1 , . . . , dk )
.
dq
in the formula of Theorem I is (by Stifel’s relation):
The coefficient of
¡ n−k−(q) ¢
¡ n−k−(q+1) ¢
χ V(d1 ,...,dk ,1q ) − χ V(d1 ,...,d ,1q+1 )
k
µ
¶
· n−k−q
¸
X
n−q
(k)
δ
= (d1 · · · dk )
Wδ (d1 , . . . , dk ) .
(−1)
n−q−k−δ
δ=0
Setting q = n − k − j we have that Theorem I reads
¢
¡
n−k
N i∗ F d , V(d
1 ,...,dk )
" j
#
µ
¶
n−k
X X
(k)
δ k+j
= (d1 · · · dk )
Wδ (d1 , . . . , dk ) dn−k−j .
(−1)
j−δ
j=0
δ=0
Now, Lemma 2 of Todd ([15, p. 200] tells us that
µ
¶
p
X
(k)
p−i k + p − 1
(k)
Wi (d1 , . . . , dk ).
Wp (d1 − 1, . . . , dk − 1) =
(−1)
p−i
i=0
Taking the alternate sum and using Stifel’s relation, we arrive at:
µ
¶
j
j
X
X
k+j
(k)
(k)
(−1)δ Wδ (d1 − 1, . . . , dk − 1) =
(−1)δ
Wδ (d1 , . . . , dk ).
j−δ
δ=0
δ=0
Thus we recover the classical formulas of Severi [13] and Todd [15], for the
classes of a smooth complete intersection:
¡ n−k
¢
(k)
%j V(d
= (d1 · · · dk )Wj (d1 − 1, . . . , dk − 1) , 0 ≤ j ≤ n − k.
,...,d
)
1
k
This finishes the proof of the corollary.
377
ONE-DIMENSIONAL FOLIATIONS
5. Proof of Theorem II
Let x1 , . . . , xk be nonnegative integers. Then
Lemma 1.
(k)
(k)
(k)
W1 (x1 , . . . , xk ) Wδ−1 (x1 , . . . , xk ) − Wδ (x1 , . . . , xk )
(k)
(k)
(k)
≥ W1 (x1 , . . . , xk ) Wδ−2 (x1 , . . . , xk ) − Wδ−1 (x1 , . . . , xk ).
(k)
Proof. Just observe that every monomial appearing in Wi (x1 , . . . , xk )
(k)
(k)
also appears in W1 (x1 , . . . , xk ) Wi−1 (x1 , . . . , xk ) with coefficient at least 1.
In particular both sides of the inequality are nonnegative. Now, the left side
is a sum of monomials of degree δ and the right side is a sum of a smaller or
equal number of monomials of degree δ − 1. Since x1 , . . . , xk are nonnegative
integers the result follows.
Note that Lemma 1 gives
(k)
W1 (d1 − 1, . . . , dk − 1)
≥
min
(k)
(k)
(k)
Wδ−1 (d1 − 1, . . . , dk − 1) − Wδ−2 (d1 − 1, . . . , dk − 1)
Lemma 2.
d<
(k)
Wδ (d1 − 1, . . . , dk − 1) − Wδ−1 (d1 − 1, . . . , dk − 1)
2≤δ≤n−k
.
Suppose n − k odd and 1 ≤ k < n − 1. If
( (k)
)
(k)
Wδ (d1 − 1, . . . , dk − 1) − Wδ−1 (d1 − 1, . . . , dk − 1)
(k)
(k)
Wδ−1 (d1 − 1, . . . , dk − 1) − Wδ−2 (d1 − 1, . . . , dk − 1)
¢
¡
n−k
≤ 0.
then N i∗ F d , V(d
1 ,...,dk )
(k)
Proof. To avoid cumbersome notation write Wδ (d1 − 1, . . . , dk − 1) as
(k)
Wδ . By the corollary
i
h
¢
¡
(k) n−k−1
n−k
n−k
d
=
d
(d1 · · · dk )−1 N i∗ F d , V(d
+
1
−
W
1
1 ,...,dk )
i
i
h
h
(k)
(k)
(k)
(k)
(k)
+ 1 − W1 + W2 dn−k−2 + 1 − W1 + W2 − W3 dn−k−3 + · · ·
i
h
(k)
(k)
+ 1 − W1 + · · · + (−1)j Wj dn−k−j
i
h
(k)
(k)
+ 1 − W1 + · · · + (−1)j+1 Wj+1 dn−k−j−1
i
h
i
h
(k)
(k)
(k)
(k)
+ · · · + 1 − W1 + · · · + Wn−k−1 d + 1 − W1 + · · · − Wn−k .
378
MARCIO G. SOARES
Grouping the terms pairwise, always assuming the term of highest degree in d
to be odd we get:
i
¡
¢ h
(k) n−k−1
n−k
(d1 · · · dk )−1 N i∗ F d , V(d
d
=
d
+
1
−
W
1
1 ,...,dk )
i
h ¡
¡
(k)
(k) ¢
(k)
(k)
(k) ¢ n−k−3
d
+ d 1 − W1 + W2 + 1 − W1 + W2 − W3
+ ···
h ¡
i
¡
(k)
(k) ¢
(k)
(k)
(k) ¢
+ d 1 − W1 + · · · + Wj
+ 1 − W1 + · · · + Wj − Wj+1 dn−k−j−1
i
h ¡
¢ ¡
(k)
(k)
(k)
(k)
(k) ¢
+ · · · + d 1 − W1 + · · · +Wn−k−1 + 1 − W1 + · · · +Wn−k−1 − Wn−k .
The term preceeding dn−k−j−1 can be regrouped as
¡
¡
(k) ¢
(k)
(k)
(k)
(k) ¢
d + 1 − W1 + − W1 d + W2 + W2 d − W3
¡
(k)
(k)
(k)
(k) ¢
+ · · · + − Wj−1 d + Wj + Wj d − Wj+1 .
Now, Lemma 1 and the hypothesis imply
(k)
d + 1 − W1
≤0
and
(k)
(k)
(k)
(k)
−Wj−1 d + Wj + Wj d − Wj+1 < 0,
¡ ∗ d n−k
¢
so that N i F , V(d1 ,...,dk ) ≤ 0. This proves the lemma.
Lemma 3. Let x1 , . . . , xk be nonnegative integers. Then, for j ≥ 1
´
³
(k) 2
(k)
(k)
(x1 , . . . , xk ) ≥ Wj−1 (x1 , . . . , xk )Wj+1 (x1 , . . . , xk ).
Wj
Also,
(
min
(k)
Wj (x1 , . . . , xk )
)
(k)
Wj−1 (x1 , . . . , xk )
1≤j≤n−k
(k)
=
(m)
Wn−k (x1 , . . . , xk )
(k)
Wn−k−1 (x1 , . . . , xk )
.
(m)
Proof. Let us write Wj (x1 , . . . , xm ) as Wj . The proof is by induction
on the number of variables. If k = 1 then
³
´
(1) 2
(1)
(1)
j−1 j+1
(x1 ) = x2j
= Wj−1 (x1 )Wj+1 (x1 ).
Wj
1 ≥ x1 x1
³
´
(k−1) 2
(k−1)
(k−1)
Assume it holds for k − 1, so that Wj
≥ Wj−1 Wj+1 . Observe that
this inequality implies
(k−1)
(∗)
W1
(k−1)
W0
(k−1)
(k−1)
≥
W2
(k−1)
W1
≥ ··· ≥
Wj
(k−1)
Wj−1
(k−1)
≥
Wj+1
(k−1)
Wj
(k−1)
≥ ··· ≥
Wn−k
(k−1)
Wn−k−1
.
379
ONE-DIMENSIONAL FOLIATIONS
Now note that, since
(k)
(k−1) j−1
xk
Wj−1 = W0
(∗∗)
(k−1) j−2
xk
+ W1
(k−1)
+ · · · + Wj−1
we have
(k)
Wj
(k−1) j
xk
= W0
(k−1) j−1
xk
+ W1
(k)
(k−1)
(k−1)
+ · · · + Wj−1 xk + Wj
(k−1)
= Wj−1 xk + Wj
and
(k)
(k−1) j+1
xk
Wj+1 = W0
(k)
(k−1) j
xk
+ W1
(k−1)
= Wj−1 x2k + Wj
(k−1)
(k−1)
+ · · · + Wj−1 x2k + Wj
(k−1)
xk + Wj+1
(k−1)
xk + Wj+1 .
With this at hand we get
´
´
´
³
³
³
(k) 2
(k) 2
(k)
(k−1)
(k−1) 2
Wj
= Wj−1 x2k + 2Wj−1 Wj
xk + Wj
and
´
³
(k)
(k)
(k) 2
(k)
(k−1)
(k)
(k−1)
Wj−1 Wj+1 = Wj−1 x2k + Wj−1 Wj
xk + Wj−1 Wj+1 .
Hence,
(∗ ∗ ∗)
³
´
(k) 2
Wj
³
´
(k)
(k)
(k−1) 2
(k)
(k−1)
(k)
(k−1)
− Wj−1 Wj+1 = Wj
+ Wj−1 Wj
xk − Wj−1 Wj+1 .
Let us consider the term
(k)
(k−1)
Wj−1 Wj
(k)
(k−1)
xk − Wj−1 Wj+1
³
´
(k)
(k−1)
(k−1)
.
= Wj−1 Wj
xk − Wj+1
Using (∗∗) we obtain
³
´
(k)
(k−1)
(k−1)
Wj−1 Wj
xk − Wj+1
´
³
(k−1) j
(k−1)
(k−1)
(k−1)
(k−1)
= Wj
xj−1
xk + W1
Wj
− W0
Wj+1
k
³
´
(k−1)
(k−1)
(k−1)
(k−1)
+ W2
Wj
− W1
Wj+1
+ ···
xj−2
k
´
³
(k−1)
(k−1)
(k−1)
(k−1)
(k−1)
(k−1)
xk − Wj−1 Wj+1 .
− Wj−2 Wj+1
+ Wj−1 Wj
By (∗), all the coefficients of x`k are nonnegative, for ` ≥ 1. Taking this into
(∗ ∗ ∗) we conclude that
³
´
´
³
(k) 2
(k)
(k)
(k−1) 2
(k−1)
(k−1)
Wj
− Wj−1 Wj+1 ≥ Wj
− Wj−1 Wj+1 ≥ 0
380
MARCIO G. SOARES
by inductive hypothesis. From this it follows that
( (k) )
(k)
Wj
Wn−k
min
.
= (k)
(k)
1≤j≤n−k
Wj−1
Wn−k−1
Lemma 3 is proved.
(
Let α =
Lemma 4.
min
)
(k)
Wj
and
(k)
Wj−1
1≤j≤n−k
(
(
(k)
β = min W1 ,
min
2≤j≤n−k
(k)
− Wj−1
(k)
(k)
Wj
(k)
))
.
Wj−1 − Wj−2
Then α ≥ β > α − 1.
(k)
Wj
Proof. Write aj =
bj =
(k)
− Wj−1
(k)
(k)
Wj
≥ 0 for 1 ≤ j ≤ n − k and b1 = W1
≥ 0,
(k)
Wj−1 − Wj−2
Lemma 3
≥ 0 for 2 ≤ j ≤ n − k. Now, a1 = b1 and aj ≥ bj since by
(k)
(k)
Wj−1 Wj
Therefore,
(k)
(k)
Wj−1
min {aj } ≥
1≤j≤n−k
α=
(k)
(k)
(k)
(k)
− Wj Wj−2 ≥ Wj−1 Wj
´
³
(k) 2
− Wj−1 .
min {bj }. Write
1≤j≤n−k
min {aj }
1≤j≤n−k
and
β=
min {bj }.
1≤j≤n−k
(k)
We know, by Lemma 3, that α =
Wn−k
(k)
Wn−k−1
(k)
β=
. Let us say
(k)
Wm − Wm−1
(k)
(k)
Wm−1 − Wm−2
for some 2 ≤ m ≤ n − k. Then,

(k)
1≥
(k)
(k)
(k)
(k)
Wn−k−1 Wm
W
Wm − Wm−1
β
·
=
= n−k−1
(k)
(k)
(k)
(k)
(k)
α
Wn−k
Wm−1 − Wm−2
Wn−k Wm−1

(k)
W
 1 − m−1 

(k) 

Wm 

.
(k) 

W
 1 − m−2 
(k)
Wm−1
381
ONE-DIMENSIONAL FOLIATIONS
(k)
By (∗) of Lemma 3,
(k)
Wn−k−1 Wm
(k)
1−
1−
≥ 1 and hence, by using (∗) again
(k)
(k)
β
≥
α
(k)
Wn−k Wm−1
Wm−1
(k)
Wm
≥
(k)
Wm−2
1−
(k)
Wm−1
Wn−k−1
1−
(k)
Wn−k
1
(k)
W1
(k)
>1−
Wn−k−1
(k)
Wn−k
>1−
1
.
α
Therefore α ≥ β > α − 1 and the lemma is proved.
¡
¢
n−k
Theorem II follows, since by Theorem I, N i∗ F d , V(d
> 0, which
,...,d
)
1
k
happens, by Lemma 2, for d ≥ β; and as, by Lemma 4, β > α − 1 we obtain
d > α − 1. Now, d is a positive integer and this gives d ≥ α. By the corollary
¡ n−k
¢
(k)
to Theorem I, %j V(d
= (d1 · · · dk )Wj (d1 − 1, . . . , dk − 1) so that
1 ,...,dk )
¡ n−k
¢
%n−k V(d
1 ,...,dk )
α=
¡ n−k
¢.
%n−k−1 V(d
1 ,...,dk )
Remark 1. It is clear from the
of Theorem
II that all that is needed
¡ proof
¢
n−k
%n−k V(d
1 ,...,dk )
to obtain the bound d ≥
¡ n−k
¢ are the following relations in%n−k−1 V(d1 ,...,dk )
volving polar classes: %1 %j−1 − %j ≥ %1 %j−2 − %j−1 and %2j ≥ %j−1 %j+1 for
1 ≤ j ≤ n − k. It would be interesting to know if such relations hold for
i
the polar classes of a variety Vn−k −→ PnC which is not necessarily a complete
intersection.
¢
¡
n−k
is automatically positive
Remark 2. If n − k is even, N i∗ F d , V(d
1 ,...,dk )
so, assuming the foliation is nondegenerate just along the variety, we cannot
use the same arguments as given in Theorem II to relate d to the polar classes
n−k
of V(d
. In [14] we considered the codimension 1 case, regardless of n − 1
1 ,...,dk )
been even or odd, but assumed the foliation was nondegenerate in the whole
of PnC . This allowed us to bound from above the number of singularities of the
foliation in Vdn−1
by dn + dn−1 + · · · + d + 1, the total number of singularities
1
of F d in PnC , whenever n − 1 is even. However, if n − k is even and we make
n−k
n−k+1
n−k
the stronger hypothesis that both V(d
and V(d
⊃ V(d
are
1 ,...,dk )
1 ,...,dk−1 )
1 ,...,dk )
n−k+1
invariant by F d , which is also nondegenerate along V(d
then, using the
1 ,...,dk−1 )
relation
(k)
(k+1)
Wδ (x1 , . . . , xk ) = Wδ
(k+1)
(x1 , . . . , xk+1 ) − xk+1 Wδ−1 (x1 , . . . , xk+1 ),
382
MARCIO G. SOARES
the same argument
in ¢the proof of Theorem II works, only this time
¡ ∗ d given
n−k
we bound N i F , V(d1 ,...,dk ) from above by the corresponding number of
n−k+1
. In this case we obtain precisely the same
singularities of F d in V(d
1 ,...,dk−1 )
relations as in the odd dimensional case.
Acknowlegements. I am grateful to A. Lins Neto, I. Vainsencher and to
R. S. Mol for pointing out mistakes in an earlier version of this work, to
PRONEX - Dynamical Systems (Brasil), to Univ. de Rennes I (France) and to
Univ. de Valladolid (Spain) for support and hospitality.
Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte,
Brazil
E-mail address: msoares@mat.ufmg.br
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(Received April 29, 1997)
(Revised March 31, 2000)