Annales de l’institut Fourier (AIF)

Esteves, Eduardo

Vainsencher, Israel

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ANNALES
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L’INSTITUT FOURIER
Eduardo ESTEVES & Israel VAINSENCHER
A note on M. Soares’ bounds
Tome 56, no 1 (2006), p. 269-276.
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Ann. Inst. Fourier, Grenoble
56, 1 (2006) 269-276
A NOTE ON M. SOARES’ BOUNDS
by Eduardo ESTEVES & Israel VAINSENCHER (*)
Abstract. — We give an intersection theoretic proof of M. Soares’ bounds for
the Poincaré-Hopf index of an isolated singularity of a foliation of CPn.
Résumé. — Nous employons des outils de la théorie d’intersection résiduelle
pour donner une démonstration de l’inegalité obtenue par M. Soares pour l’indice
de Poincaré-Hopf d’une singularité isolée d’un feuilletage de CPn.
1. Introduction
Let f : (Cn, 0) → (Cn, 0) be a holomorphic map germ such that f−1(0) =
{0} in some neighborhood of 0. The Poincaré–Hopf index i of f at 0 is the
topological degree of f ; cf. [6, p. 88]. Algebraically,
i = dimC C[|x1, . . . , xn|]/〈f1, . . . , fn〉,
where x1, . . . , xn are local coordinates for Cn at 0; see [3, Ch. 5, §2]. In [5]
M. Soares addresses the question of finding bounds for i. Using Mather’s
theory, he replaces the germ by a suitable polynomial map of an appropriate
degree k. The latter map is reinterpreted in terms of a certain foliation of
degree δ (equal to k or k−1) of CPn. By this one means a global holomorphic
section σ of TCPn(δ− 1) = TCPn⊗O(δ− 1). The foliation has an isolated
singularity at 0 ∈ Cn ⊂ CPn of the same index as the original map germ.
When all the singularities of σ are isolated, the bound is deduced from
Baum–Bott’s formula, which expresses the sum of indices as the degree,∑n
0 δ
i, of the top Chern class cn(TCPn(δ − 1)). Soares then proves the
Keywords: intersection theory, singularities, foliations.
Math. classification: 32S65, 14C17, 37F75.
(*) The first author is partially supported by PRONEX, Convênio CNPq 41/96/0883/00,
and by CNPq, Proc. 300004/95-8.
The second author is partially supported by CNPq.
270 Eduardo ESTEVES & Israel VAINSENCHER
general case by means of a clever analytic deformation argument in order
to get rid of possible wrong dimensional components in the singular locus
of the foliation.
Our purpose here is to replace the deformation argument by a direct
application of Fulton’s intersection theory. Our main result, Theorem 3.1,
gives bounds for the Poincaré–Hopf index of an isolated singularity of a fo-
liation of CPn. There is a general bound, and special bounds for foliations
that admit smooth invariant hypersurfaces not passing through the singu-
larity. The case of nonlinear hypersurfaces was not contemplated in [5].
2. Intersection Theory
We start giving a proof of the following lemma (cf. [7, Ex.(4), p. 16]).
Lemma 2.1. — Let Z be an irreducible component of a scheme Z. Let
Z1, . . . , Zn be subvarieties of Z distinct from Z. Then any relation mZ +∑
miZi = 0 in the Chow group A?(Z) implies m = 0.
Proof. — The relation means that there are subvarieties Vj ⊆ Z and
rational functions rj ∈ R(Vj) such that the corresponding divisors of zeros
and poles [rj ] add up to mZ +
∑
miZi. If m 6= 0 then Z must appear as
zero or pole of some rj . In particular, Z ⊆ Vj ⊆ Z. Since Z is an irreducible
component of Z, we must have Z = Vj . This is absurd since zeros and poles
are of codimension one. 
Let Y be an algebraic scheme over C, and let E → Y be a vector bundle
of rank r generated by its sections. Let sE be the zero section. Let V ⊆ E be
a m-dimensional subvariety. Let Z = sE−1(V ). We recall the construction
and some properties of the Gysin class s?E [V ] for the specific case at hand.
We form the cartesian diagram,
Z = sE−1(V ) ↪→ V
↓ 2 ↓
Y ↪→
sE
E
The bottom embedding is regular with normal bundle E. Take the normal
cone C = CZV to Z in V . This cone is a pure m−dimensional subscheme
of the restricted bundle EZ . The corresponding cycle [C] lives in the Chow
group Am(EZ), which is isomorphic to Am−r(Z) via the pullback asso-
ciated to the structure map π : EZ → Z. By construction, s?E [V ] is the
unique class in Am−r(Z) such that π?(s?E [V ]) = [C] holds.
ANNALES DE L’INSTITUT FOURIER
M. SOARES’ BOUNDS 271
If Z ⊆ Z is an irreducible component of dimension m − r, then, as
explained in [2, p. 95 and §7.1, p. 120], we may write
s?E [V ] = i[Z] +
∑
iZ′ [Z ′] in Am−r(Z),
where i is positive and the Z ′ are subvarieties of dimension m − r of Z
distinct from Z. Furthermore, i is at most the length of OZ,Z , with equality
if V is smooth. (In fact, equality holds if OZ,Z is Cohen–Macaulay.)
Proposition 2.2. — Let Y be an algebraic scheme over C, and let
E → Y be a vector bundle of rank r generated by its sections. Let sE be the
zero section. Let V ⊆ E be a m-dimensional subvariety. Let Z = sE−1(V ).
Then s?E [V ] can be represented by a nonnegative cycle in Z. Furthermore,
suppose Z ⊆ Z is an irreducible component of dimension m − r and V is
smooth. Then
s?E [V ] = i[Z] +
∑
iZ′ [Z ′] in Am−r(Z),
where the Z ′ are subvarieties of dimension m− r of Z distinct from Z, the
iZ′ are nonnegative and i is the length of OZ,Z .
Proof. — We show first that s?E [V ] can be represented by a nonnegative
cycle in Z.
Let F → Y be a trivial bundle endowed with a surjection of bundles
ρ : F → E over Y . The existence is assured by the hypothesis of global
generation. We have sE = ρ ◦ sF . Hence, we may write
Z = sE−1(V ) = sF−1ρ−1(V ).
In view of general properties of the intersection class [2, Prop. 6.5, p. 110],
we have
s?E [V ] = s
?
F ρ
?[V ] = s?F [ρ
−1(V )] in Am−r(Z).
Thus we may assume E is trivial. Now proceed by induction on the rank
r of E. If r = 1 then s?E [V ] is the same as the intersection class of V by the
principal Cartier divisor Y = Y×{0} of E = Y×C [2, prop. 6.1(c), p. 94].
Now, if V 6⊂ sE(Y ) then s?E(V ) is represented by the effective Cartier
divisor Y · V ∈ Am−1(Z), i.e., we have s?E(V ) ∈ A
>
(Z) as asserted. If
V ⊆ sE(Y ) then Y · V = 0 in Am−1(Z) [2, Def. 2.3, p. 33], because the
normal bundle to Y×{0} in Y×C is trivial. Next, for the inductive step,
we write E = G ⊕ O, with G trivial of rank r − 1 and O = Y×C. Let j
denote the inclusion of G in E, so j(g) = (g, 0). Then sE = j ◦ sG. Now, on
one hand s?E [V ] ∈ Am−r(Z), and on the other hand s?E [V ] = s?Gj?[V ] by
[2, Thm. 6.5, p. 108]. The bundle E is a rank-1 bundle over G, under the
TOME 56 (2006), FASCICULE 1
272 Eduardo ESTEVES & Israel VAINSENCHER
natural surjection, and j is the zero section. Hence the case r = 1 allows
us to write
j?[V ] =
∑
mi[Wi] ∈ A
>
m−1(j
−1(V )),
with mi > 0 and the Wi denoting subvarieties of dimension m−1 of j−1(V ).
By induction, each s?G[Wi] lies in A
>
m−1−(r−1)(sG
−1(Wi)), hence maps to
A
>
m−r(sE
−1(V )). We are done with the first statement by linearity.
As for the last statement, the class s?E [V ] can be represented by a cycle
of Z where [Z] appears with coefficient i equal to the length of OZ,Z . By
Lemma 2.1, [Z] has the same coefficient in any representation of s?E [V ]
by a cycle of Z. In particular, by the first statement just proved, i is
the coefficient of [Z] in a representation of s?E [V ] by a nonnegative linear
combination of prime cycles of Z. 
Remark 2.3. — In order to prove the first statement of Proposition 2.2,
we could have invoked Fulton’s treatment of positivity, to the effect that
s?E [V ] has a representation as a nonnegative linear combination of prime
cycles [2, Thm. 12.1(a), p. 212]. Though the result only asserts that s?E [V ]
is in A
>
m−r(Y ), the proof actually gives that s?E [V ] lies in A
>
m−r(Z); this
refinement is necessary for our proof of the second statement. Though we
could have quoted the more general [2, Thm. 12.2, p. 218], we reproduced
above a small variation of the proof given to [2, Thm. 12.1(a), p. 212] for
the reader’s convenience.
3. Foliations of CPn
Theorem 3.1. — Let P ∈ CPn be an isolated singularity with Poincaré–
Hopf index i of a one-dimensional foliation of degree δ of CPn. Then
i 6
n∑
j=0
δj .
Furthermore, if the foliation admits a smooth invariant hypersurface of
degree d not passing through P then
i 6
n∑
j=0
δj(1− d)n−j .
Proof. — Let σ be the section of the bundle TCPn(δ− 1) corresponding
to the foliation. For the first bound, we apply Proposition 2.2 taking
(1) Y = CPn,
ANNALES DE L’INSTITUT FOURIER
M. SOARES’ BOUNDS 273
(2) E = TCPn(δ − 1),
(3) and V as the image of CPn under the section σ.
The bundle E is globally generated. Indeed, tensoring the Euler exact se-
quence,
0 → O → O(1)n+1 → TPn → 0
by O(δ − 1), we obtain the following exact sequence:
(1) 0 → O(δ − 1) → O(δ)n+1 → E → 0.
Since O(δ) is globally generated, so is E.
Furthermore, Z = sE−1(V ) is precisely the singular set of the foliation.
Since P is isolated, Z = {P} is a component of Z of the dimension stipu-
lated by the proposition. So we may write
s?E [V ] = iP [P ] +
∑
iP ′ [P ′] in A0(Z)
for suitable points P ′ ∈ Z, distinct from P , and nonnegative iP ′ . The
coefficient iP is the length of OP,Z . This length is the Poincaré–Hopf index
i of P . So i is at most the degree of s?E [V ]. Now,∫
Pn
s?E [V ] =
∫
Pn
cn(E) =
n∑
j=0
δj ,
the last equality following from the exactness of (1), through the Whitney
formula,
c(O(δ))n+1 = c(E)c(O(δ − 1)).
Assume now that there exists a nonsingular invariant hypersurface H ⊂
CPn of degree d such that P 6∈ H. Let TCPn(log H) be the elemen-
tary transformation of TCPn at H by the normal bundle. In terms of
sheaves, TCPn(log H) is simply the kernel of the natural homomorphism
h : TCPn → NHCPn, where NHCPn is the normal bundle to H in CPn.
(Also, TCPn(log H) is the dual of the bundle of logarithmic forms on CPn
with poles along H.)
Since H is an invariant hyperspace, σ factors through a section σ′ of
TCPn(log H)(δ− 1). To prove the second bound, we apply Proposition 2.2
taking
(1) Y = CPn,
(2) E = TCPn(log H)(δ − 1),
(3) and V as the image of CPn under the section σ′.
The sections σ and σ′ coincide off H. So Z = sE−1(V ) coincides off H
with the singular set of the foliation. Since P 6∈ H, the set Z = {P} is
a component of Z of the dimension stipulated by the proposition, and of
TOME 56 (2006), FASCICULE 1
274 Eduardo ESTEVES & Israel VAINSENCHER
multiplicity i. As with the first bound, to be able to apply Proposition 2.2,
and derive the second bound for i, we need only show that E is globally
generated and
(2)
∫
Pn
cn(E) =
n∑
j=0
δj(1− d)n−j .
For this purpose, consider the homomorphism g : O(1)n+1 → O(d) ob-
tained from the partial derivatives of the equation defining H. We claim
that TPn(log H) is also the kernel of g. Indeed, the following diagram com-
mutes,
0 −−−−→ O −−−−→ O(1)n+1 −−−−→ TPn −−−−→ 0∥∥∥ gy hy
0 −−−−→ O −−−−→ O(d) −−−−→ NHPn −−−−→ 0,
where at the top is the Euler sequence and at the bottom is the natu-
ral sequence associated to the identification NHPn = O(d)|H. Since both
sequences are exact, the snake lemma yields the claim.
Now, since H is smooth, g is surjective, and the following piece of the
Koszul complex is exact,
O(2− d)(
n+1
2 ) −−−−→ O(1)n+1 g−−−−→ O(d) −−−−→ 0.
Thus E is a quotient of a sum of copies of O(δ + 1 − d), and hence is
globally generated if d 6 δ + 1. This inequality holds, as proved by Soares
in [4]. (The inequality d 6 δ + 1 was basically known to Zariski; see [1,
Thms. 7 and 8].) In addition, E sits in an exact sequence,
0 → E → O(δ)n+1 → O(δ − 1 + d) → 0.
The Whitney formula,
c(O(δ))n+1 = c(E)c(O(δ − 1 + d)),
yields (2). 
4. Soares’ bounds
We describe here the reasoning of Soares in [5] and reobtain his bounds.
Let f = (f1, . . . , fn) : (Cn, 0) → (Cn, 0) be a holomorphic map germ. Let
i = dimC C[|x1, . . . , xn|]/〈f1, . . . , fn〉.
Then i is finite if and only if f−1(0) = 0 in some neighborhood of 0. If
i < ∞ then i is the Poincaré–Hopf index of f at 0.
ANNALES DE L’INSTITUT FOURIER
M. SOARES’ BOUNDS 275
Assume from now on that i is finite. Then Mk ⊆ 〈f1, . . . , fn〉 for some
integer k, where M = 〈x1, . . . , xn〉 denotes the maximal ideal. For each
i = 1, . . . , n let f (k)i be a polynomial of degree at most k such that f
(k)
i −fi ∈
Mk+1. Then
〈f1, . . . , fn〉 = 〈f (k)1 , . . . , f (k)n 〉,
and hence the Poincaré–Hopf index of f is the same as that of the polyno-
mial map f (k) = (f (k)1 , . . . , f
(k)
n ). So, in order to bound i, we may and will
assume from now on that f is a polynomial map of degree k.
(The map germs f and f (k) are said to be K-equivalent, in the sense of
Mather; see [5]. It might be that f is K-equivalent to a polynomial map of
degree smaller than the minimum k such that Mk ⊆ 〈f1, . . . , fn〉. If two
map germs are K-equivalent, they have the same Poincaré–Hopf index. The
converse might not be true, though.)
If k = 1, then i = 1. Suppose k > 1. Consider on Cn the vector field
X = f1
∂
∂x1
+ · · ·+ fn
∂
∂xn
.
Since i is finite, 0 is an isolated singularity of X. Extend X to a foliation of
CPn. The homogeneization process is as follows. Write X = Y0 + · · ·+ Yk,
where Yj is a homogeneous vector field of degree j. Then form
X ′ = Yk + x0Yk−1 + · · ·+ xk0Y0.
The homogeneous vector field X ′ on Cn+1 induces a nonzero section σ′ of
TCPn(k− 1), i.e., a foliation F ′ of CPn of degree k. The index i is also the
Poincaré–Hopf index of F ′ at P = (1 : 0 : · · · : 0).
Note that the hyperplane H at infinity, given by x0 = 0, is left invariant
by F ′, as X ′(x0) = 0. However, H might be in the singular locus of F ′.
This is the case when Yk is a multiple of the radial vector field,
R = x1
∂
∂x1
+ · · ·+ xn
∂
∂xn
.
In fact, since the Euler vector field,
E = x0
∂
∂x0
+ · · ·+ xn
∂
∂xn
,
induces the zero section of TCPn, if Yk = GR then also
X ′′ = −Gx0∂/∂x0 + x0Yk−1 + · · ·+ xk0Y0
induces σ′. In this case, let Xh = X ′′/x0; otherwise let Xh = X ′.
Let F be the foliation of CPn induced by Xh. Note that F coincides with
F ′ off H. Thus P is an isolated singularity of F with Poincaré–Hopf index
TOME 56 (2006), FASCICULE 1
276 Eduardo ESTEVES & Israel VAINSENCHER
i. So we may apply Theorem 3.1. If Yk is a multiple of the radial vector
field, then F has degree k − 1, and hence
i 6
n∑
j=0
(k − 1)j .
If Yk is not a multiple of the radial vector field, then F has degree k and
leaves invariant the hyperplane at infinity. Thus
i 6 kn.
These were the bounds obtained by Soares in [5].
BIBLIOGRAPHY
[1] E. Esteves, “The Castelnuovo-Mumford regularity of an integral variety of a vector
field on projective space”, Math. Res. Lett. 9 (2002), p. 1-15.
[2] W. Fulton, Intersection theory, Springer, New York, 1985.
[3] P. Griffiths & J. Harris, Principles of algebraic geometry, John Wiley & Sons,
New York, 1978.
[4] M. Soares, “The Poincaré problem for hypersurfaces invariant by one-dimensional
foliations”, Invent. math. 128 (1997), p. 495-500.
[5] M. Soares, “Bounding Poincaré-Hopf indices and Milnor numbers”, Math.
Nachrichten 278 (2005), no. 6, p. 703-711.
[6] T. Suwa, Indices of vector fields and residues of singular holomorphic foliations,
Hermann, Paris, 1998.
[7] I. Vainsencher, Classes características em geometria algébrica, IMPA, Rio de
Janeiro, 1985.
Manuscrit reçu le 15 mars 2004,
accepté le 10 octobre 2004.
Eduardo ESTEVES
IMPA
Estrada D. Castorina 110
Jardim Botânico
22460-320 Rio de Janeiro RJ (Brasil)
esteves@impa.br
Israel VAINSENCHER
UFMG
Departamento de Matemática
Cidade Universitária
30123-970 Belo Horizonte MG (Brasil)
israel@mat.ufmg.br
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