A formula for Euler characteristic of line singularities on singular spaces

8 Pages·0.16 MB·

by  Guangfeng Jiang

0
0
0 A FORMULA FOR EULER CHARACTERISTIC OF LINE
2
SINGULARITIES ON SINGULAR SPACES

n
a
J
GUANGFENG JIANG

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1
Abstract. We prove an algebraic formula for the Euler characteristic of the

] Milnor fibres of functions with criticallocus a smoothcurveona space whichis a
G
weighted homogeneous complete intersection with isolated singularity.
A
.
h
t
a
m
1. Introduction
[
For an analytic function germ f : (X,0) −→ (C,0) with critical locus Σ ⊂ X,
1
v there is a local Milnor fibration induced by f. We are interested in the topology of
2 the Milnor fibre F of f in the case when dimΣ = 1. It is well known that in this
7
case the homotopy type of F is not necessarily a bouquet of spheres in the middle
0
1 dimension. The calculation of the Euler characteristic χ(F) of F is of importance.
0
There is a so called Iomdin-Lˆe formula [6] which expresses the Euler characteristic
0
0 of the Milnor fibre of f by that of the series of f with isolated singularities.
/ The question we are interested in is that if there is a way to express χ(F) by some
h
t “computable” invariants determined only by (f,Σ,X). When X is Cm, the singular
a
m locus Σ of f is a one dimensional complete intersection with isolated singularity, and
: the transversal singularity type of f along Σ is Morse, Pellikaan [9] answered this
v
i question positively. Pellikaan’s formula expresses the Euler characteristic in terms
X
of the Jacobian number j(f), δ and the Milnor number µ(Σ) of Σ. These numbers
r
a can be computed directly by counting the dimensions of certain finite dimensional
vector spaces. The development of computer algebra makes this kind of algebraic
formulae more and more important and popular. In this article we answer the
question by proving a similar formula for function germs with line singularities on a
weighted homogeneous space X with isolated complete intersection singularity (see
Proposition 7). Remark that for a function germ f with isolated singularity on a
weighted homogeneous complete intersection with isolated singularity, Bruce and
Robert [1] have proved an algebraic formula for the Milnor number of f.
1991 Mathematics Subject Classification. 32S05,32S55, 58C27.
Key words and phrases. line singularities, Milnor fibre, Euler characteristic.
This article comes from a piece of the author’s thesis which was finished at Utrecht University
advised by Prof. D. Siersma, to whom this author is indebted very much for many discussions,
questions and remarks. This work was partially supported by JSPS, NNSFC, STCLN.
1
2 GUANGFENG JIANG
2. Non-isolated singularities on singular spaces
Let OCm be the structure sheaf of Cm. The stalk OCm,0 of OCm at 0 is a local
ring, consisting of germs at 0 of analytic functions on Cm. The ring OCm,0 is often
denoted by O , or simply by O when no confusion can be caused. The unique
m
maximal ideal of O is denoted by m or m.
m m
Let (X,0) ⊂ (Cm,0) be the germ of a reduced analytic subspace X of Cm defined
by an ideal h of O, generated by h ,... ,h ∈ O. Let g be the ideal generated by
1 p
g ,... ,g ∈ O. The germ of the analytic space defined by g at 0 is denoted by
1 n
(Σ,0). Write O := O/h and O := O/g.
X Σ
Let Der(O) denote the O-module of germs of analytic vector fields on Cm at 0.
Then Der(O) is a free O-module with ∂ ,... , ∂ as basis, where z ,... ,z are
∂z1 ∂zm 1 m
the local coordinates of (Cm,0). Der(O) is a Lie algebra with the bracket defined
by [ξ,η] := ξη−ηξ for all ξ,η ∈ Der.
Define Der (O) := {ξ ∈ Der(O) | ξ(h) ⊂ h}, which is the O-module of loga-
h
rithmic vector fields along (X,0) and a Lie subalgebra of Der(O) [10, 1]. When h
is a radical ideal defining the analytic space X, Der (O) is often denoted by D .
h X
Geometrically, D consists of all the germs of vector fields that are tangent to the
X
smooth part of X. When X is a weighted homogeneous complete intersection with
isolated singularity, one can write down precisely all the generators of D (cf. [13]).
X
For f ∈ O, the ideal J (f) := {ξ(f) | ξ ∈ D } is called the (relative) Jacobian
X X
ideal of f. Obviously, when X is the whole space Cm, namely, h = {0}, then
J (f) = J(f), the Jacobian ideal of f.
X
Let S = {S } be an analytic stratification of X, f : (X,0) −→ (C,0) an analytic
α
function germ. The critical locus ΣS of f relative to the stratification S is the union
f
of the critical loci of f restricted to each of the strataS , namely, ΣS = Σ .
α f Sα∈S f|Sα
We denote ΣS by Σ when S is clear from the context. If the dimensionSof ΣS is not
f f f
positive, we say that f defines (or has) isolated singularities on X. If the dimension
of ΣS is positive, we say that f defines (or has) non-isolated singularities on X. If
f
ΣS is one dimensional smooth complex manifold, we say that f defines (or has) a
f
line singularity on X.
For an analytic space X embedded in a neighborhood U of 0 ∈ Cm, there is a
logarithmic stratification S := {S } of U (see [10, 1]). In general, S is not
log α log
locally finite. If S is locally finite, then X is said to be holonomic.
log
Let X be of pure dimension. The collection S′ = {X ∩ S | S ∈ S } is a
log α α log
stratification of X which will be called the logarithmic stratification of X in this
article. Especially, when X has isolated singularity in 0, then {0} and the connected
components of X \ {0} form a holonomic logarithmic stratification of X. So 0 is
always a critical point of any germ f : (X,0) −→ (C,0)relative to this stratification.
Hence for f ∈ m, Σ = {p ∈ X | ξ(f)(p) = 0 for all ξ ∈ D } is the critical locus of
f X
f relative to the logarithmic stratification. Obviously Σ = X ∩V(J (f)).
f X
EULER CHARACTERISTIC OF LINE SINGULARITIES 3
Definition 1. Let h = (h1,... ,hp) ⊂ g = (g1,... ,gn) be ideals of OCm,0. Define a
subset of O, called the primitive ideal of g relative to h:
g := {f ∈ g | ξ(f) ⊂ g for all ξ ∈ Der (O)}.
h
Z
h
In the following we always assume that h and g are radical, X = V(h) and
Σ = V(g). In this case, g is denoted by g or g when no confusion can be
h X
caused by this. R R R
Remarks 2. (1) When X is smooth this definition was given by Pellikaan [7, 8]. It
is straightaway to verify that g is an ideal of O, and g2+h ⊂ g ⊂ g always
h h
holds. And for g ⊃ h (i = 1,R2), we have g ∩ g = (g ∩Rg );
i h 1 h 2 h 1 2
(2) Geometrically, the relative primitive idealRcollectRs all theRfunctions whose zero
level surfaces pass through Σ and are tangent to the regular part X of X
reg
along Σ∩X .
reg
(3) The singular locus (relative to S′ ) of f is Σ := V(J (f)) ∩ X, and Σ ⊂
log f X f
f−1(0) if f(0) = 0. If f ∈ g, then Σ ⊂ X ∩ V(J (f)) = Σ . Conversely,
X f
for f ∈ m, we have f ∈ gR when Σ ⊂ Σ and g is radical. The reason is:
f
J (f) ⊂ g, and since f takRes finite values on Σ and 0 ∈ Σ, f| = 0, so fk ∈ g
X f Σ
for some k ∈ N. Hence f ∈ g since g is radical.
(4) The relative primitive ideals have been generalized to higher relative primitive
ideals in [5]. Under the assumption that h is pure dimension, g is radical, and
the Jacobian ideal of h is not contained in any associated prime of g, it was
proved that the primitive ideal g is the inverse image in O of the second
h
symbolic power of the quotient idReal g¯ := g/h of O . Remark that the results
X
in [5] generalized the results of [11, 7, 8].
3. Transversal singularities
Let (X,0) ⊂ (Cn+1,0) be the germ of a reduced analytic space with isolated
singularity in 0. Let Σ be a reduced curve germ on (X,0) defined by g and have
isolated singularity in 0. A germ f ∈ g is called a transversal A singularity
1
along Σ on X if its singular locus Σf =R Σ, and, for P ∈ Σ \ 0,f has only A1
singularity transversal to the branch of Σ containing P. It was proved in [3], that
f is a transversal A singularity along Σ on X if and only if the Jacobian number
1
j(f) := dim(g/(h+J (f)) < ∞.
X
There exist admissible linear forms l (see [6]) such that {l = 0}∩Σ = {0} and
{l = 0} intersects both X and Σ transversally, and {l = 0}∩X∩f−1(0) has isolated
singularity at the origin. We assume {l = 0} is the first coordinate hyperplane
{z = 0} of Cn+1.
0
Let D be generated by ξ0,ξ1,...ξs. Denote by D0 the submodule of D gener-
X X X
ated by those ξi such that if we write ξi = n ξi ∂ , then ξi ∈/ g, and by D1 the
j=0 j∂zj 0 X
submodule of D generated by those ξi sucPh that if we write ξi = n ξi ∂ , then
X j=0 j∂zj
P
4 GUANGFENG JIANG
ξ ∈ g, thus D = D0 +D1 . Denote
0 X X X
J0(f) := D0 (f) = {ξ(f) | ξ ∈ D0 }, J1(f) := D1 (f) = {ξ(f) | ξ ∈ D1 }.
X X X X
If f is clear from the context we just write J0 and J1.
Lemma 3. Let f ∈ g, and z ,z ,... ,z be the coordinates of Cn+1 such that
0 1 n
z0 = 0 is admissible. TRhe transversal singularity type of f along every branch of Σ
is constant at all the points of Σ\{0} if and only if
O
dimC < ∞.
(cid:18)g+((J1 +h) : J0)(cid:19)
Proof. The inequality means that Σ ∩ V((J1 + h) : J0) = {0}. For ǫ > 0 small
enough, let P ∈ Σ ∩ {z = t}, 0 < |t| < ǫ. P ∈/ V((J1 + h) : J0) if and only
0
if (J0) ⊂ (J1) , where (Ji) is the localization of the ideal at P. Since z = 0
P P P 0
intersects both Σ and X transversally and X is smooth at P, we can choose the
local coordinates such that locally the branch of Σ containing P is the first axis.
Furthermore, we can arrange the coordinate transformation such that under this
transformation D0 and D1 are preserved. By this we mean that any derivation
X X
of X with the first component non-zero at P will remain non-zero at P and any
derivation of X with first component zero at P will remain zero at P. We let
x = z ,y ,... ,y be the new local coordinates in a neighborhood of P in X, then
0 1 n−p
at P, (J0) = ∂f O and (J1) = ∂f ,... , ∂f O . By [9], the inequality
is equivalePnt to(cid:0)∂txh(cid:1)e coXn,Pstancy of tPhe tr(cid:16)a∂nys1versal ∂syinn−gpu(cid:17)lariXty,Ptype of f. (cid:3)
Let X, Σ and f ∈ g be the same as before. There is an integer k such that for
0
all k ≥ k , f = f +R1 xk+1 defines an isolated singularity at O. Let µ(f ) be the
0 k k+1 k
Milnor number of f . If X is a weighted homogeneous complete intersection with
k
isolated singularity, by [1] 7.7, we have
O
µ(f ) = dim .
k
(cid:18)h+J (f )(cid:19)
X k
By the exact sequence
g+J (f ) O O
X k
0 −→ −→ −→ −→ 0,
h+J (f ) h+J (f ) g+J (f )
X k X k X k
we know that
g+J (f ) O
X k
µ(f ) = dim +dim (3.1)
k
(cid:18)h+J (f )(cid:19) (cid:18)g+J (f )(cid:19)
X k X k
Define
O O O
Σ Σ
e := dim , σ(ΣX,0) := dim , multi (Σ) := dim .
k x
g+J (f ) (cid:18)J (x)(cid:19) (cid:18)(x)(cid:19)
X k X
EULER CHARACTERISTIC OF LINE SINGULARITIES 5
Then obviously e = σ(ΣX,0)+kmulti (Σ), and
k x
g+J (f ) g
X k
= .
h+J (f ) (h+J (f ))∩g
X k X k
Hence
g
µ(f ) = σ(ΣX,0)+kmulti (Σ)+dim (3.2)
k x
(cid:18)(h+J (f ))∩g(cid:19)
X k
4. Line singularities
In this section, we assume that Σ is a line in Cn+1 defined by the ideal g, and X
is a space with isolated complete intersection singularity defined by h ⊂ g.
Lemma 4. Let X be a space with isolated complete intersection singularity, con-
taining a line Σ, which is chosen to be the first axis of a local coordinate system of
Cn+1. If X is weighted homogeneous with respect to this coordinate system, then
D = Oξ +D1 and
X E X
J (f ) = (ξ (f ))O +J1(f )
X k E k k
⊂ (ξ (f ))O+D1 (f)+D1 (xk+1),
E k X X
where J1(f ) = {ξ(f ) | ξ ∈ D1 } ⊂ g.
k k X
Proof. Let x,y ,... ,y be the local coordinates of (Cn+1,0) in the statement of the
1 n
n
lemma. We know by [13] that the Euler derivation ξ = w x ∂ + w y ∂ ∈ D ,
E 0 ∂x k k∂y X
k
k=1
P
where w ,w ,... ,w are the weights of the coordinates x,y ,... ,y respectively.
0 1 n 1 n
n
Let ξ = ξ0 ∂ + ξk ∂ ∈ D . Since O is a principal ideal domain and X has
∂x ∂y X Σ
k
k=1
P
isolated singularity at O, ξ¯0 ∈ (x¯). Let ξ¯0 = x¯ξ¯0. Then ξ′ = ξ − 1 ξ0ξ has the
1 w0 1 E
coefficient of ∂ in g. (cid:3)
∂x
Lemma 5. Let Σ and X be the same as in Lemma 4. Let f ∈ g have transversal
A1 singularity along Σ. For k > 0 sufficiently large, we have R
J˜:= ξ (f)g+D1 (f)+h = (h+J (f ))∩g
E X X k
Proof. By Lemma 4, we have
(h+J (f ))∩g = (ξ (f ))∩g+D1 (f )+h.
X k E k X k
For a ∈ (ξ (f )) ∩ g, a = a xk+1 + a ξ (f) ∈ g. Since ξ (f) ∈ g, a ∈ g. Hence
E k 0 0 E E 0
(ξ (f ))∩g = (ξ (f ))g and
E k E k
(h+J (f ))∩g = (xk+1 +ξ (f))g+D1 (f )+h
X k E X k (4.1)
⊂ (xk+1 +ξ (f))g+xkD1 (x)+D1 (f)+h
E X X
Since j(f) < ∞, there is a k such that when k > k , xkg ⊂ h + J (f) =
1 1 X
(ξ (f))+D1 (f)+h.
E X
6 GUANGFENG JIANG
By Lemma 3, there is a k such that xk2 ∈ g + ((D1 (f) + h) : (ξ (f))), and
2 X E
xk2ξ (f) ∈ (ξ (f))g+D1 (f)+h. Hence when k > k +k ,
E E X 1 2
a (xk+1 +ξ (f))+b xk ∈ J˜ and (h+J (f ))∩g ⊂ J˜.
0 E 0 X k
On the other hand, there is an integer n >> 0 such that when k ≥ n (see (3.2))
1 1
xkg ⊂ (h+J (f ))∩g ⊂ J˜⊂ g.
X k
Then for k > n , by the equality in (4.1)
1
xk+1
J˜⊂ ξ f + g+xk+1g+D1 (f)+h
E(cid:18) k +1(cid:19) X
= (ξ (f)+xk+1)g+D1 (f)+h+xk+1g
E X
= (h+J (f ))∩g+xk+1g
X k
⊂ (h+J (f ))∩g+xkg
X k
⊂ (h+J (f ))∩g+xk−n1J˜
X k
⊂ (h+J (f ))∩g+mJ˜.
X k
By Nakayama’s lemma, (h+J (f ))∩g = J˜. (cid:3)
X k
Lemma 6. Under the assumption of Lemma 5, we have
h+J (f) (ξ (f))
X ∼ E
1) L := = ;
J˜ (ξ (f))∩J˜
E
2) Ann(L) = g+((D1 (f)+h) : (ξ (f)));
X E
O O
∼
3) L = = .
Ann(L) g+((D1 (f)+h) : (ξ (f)))
X E
Proof. It is an easy exercise in commutative algebra, we omit it. (cid:3)
From the exact sequence
h+J (f) g g
X
0 −→ −→ −→ −→ 0,
(h+J (f ))∩g (h+J (f ))∩g h+J (f)
X k X k X
we have
h+J (f)
X
µ(f ) = j(f)+e +dim .
k k
(cid:18)(h+J (f ))∩g(cid:19)
X k
Notice that σ(ΣX,0) = multi (Σ) = 1, e = k +1. By Lemma 6, we have
x k
O
µ(f ) = k +1+j(f)+dim
k (cid:18)g+((D1 (f)+h) : (ξ (f)))(cid:19)
X E
Let F and F are the Milnor fibre of f and f respectively. Iomdin-Lˆe’s formula
k k
[6, 12], says that
χ(F) = χ(F )+(−1)dimX(k +1).
k
But χ(F ) = 1+(−1)dimX−1µ(f ). Since in our case σ(ΣX,0) = 1, we have proved
k k
EULER CHARACTERISTIC OF LINE SINGULARITIES 7
Proposition 7. Let X be a space with isolated complete intersection singularity
containing a line Σ, which is chosen to be the first axis of a local coordinate system
of Cn+1. Assume that X is weighted homogeneous with respect to the coordinate
system. For an analytic function f ∈ g with j(f) < ∞, the Euler characteristic
X
of the Milnor fibre F of f is R
χ(F) = 1+(−1)dimX−1(j(f)+ν) (4.2)
where
O
ν = dim . (cid:3)
(cid:18)g+((D1 (f)+h) : (ξ (f)))(cid:19)
X E
Remark 8. Formula (4.2)allows us touse acomputer programtocompute theEuler
characteristic effectively. In fact, we have a small Singular [2] program to calculate
χ(F) for a function with critical locus a line on a hypersurface X. We use it to
check the examples in Example 10.
Let Σ be a line in Cn+1 defined by the ideal g = (y ,... ,y ), X a space with
1 n
isolated complete intersection singularity defined by h = (h ,... ,h ) ⊂ g. By
1 p
changing the generators of g, we can write
p
h ≡ b y mod g2
i ik k
Xk=1
such that the determinant b of the matrix B = (b ) is a non-zero divisor in O .
ik Σ
Note that y ,...y are projected to zero or the generators of the torsion part T(M)
1 p
of the conormal module M = g/(g2 +h) of g/h, and y ,...y form a basis of the
p+1 n
free module N = M/T(M). We call λ(ΣX) := dimCT(M) = dimO/(g+(b)) the
torsion number of the space pair (Σ,X). For f = h y y ∈ g2, define
kl k l
P O
∆ = det(h ) , δ := dim( ).
kl p+1≤k,l≤n f g+∆
Question 9. For f ∈ g2, does the equality ν = 2λ(ΣX) + δ − 1 always hold? Or
f
under what conditions does it hold?
The equality in Question 9 and (4.2) show the geometric meaning of χ(F). By
using Singular [2] we have checked that it holds for all the examples we know.
Example 10. Let X be an A singularity defined by h = xly+xsz2+yz. This is
k,l k,l
an A singularity with k = 2l+s−1 (l ≥ 1,s ≥ 0). Since we take the line Σ with
k
torsion number l as x-axis, we have the definition equation (see [4]). By resolution
of singularities, one can prove that for any function f on X , if it has isolated line
k,l
singularity and A type transversal singularity, then the Milnor fibre F of f is a
1
bouquet of circles (see [11]). Then µ(f) = j(f)+ν.
(1) We consider a function g : X −→ C. For generic (a,b,c) ∈ C3, let g =
k,l
ay2+byz +cz2, for example, take g = y2−yz+ 1z2, a calculation shows that
2
µ(g) = 6l−3.
8 GUANGFENG JIANG
(2) Let f : X −→ C be defined by f = y + 1z2. A calculation shows that
k,l 2
l−1 if s = 0,
µ(f) =  l+3s−2 if 1 ≤ s ≤ l−1,
 4l−2 if s ≥ l.

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Present (till March 31, 2000): Department of Mathematics, Faculty of Science,
Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, 1920397
JAPAN
E-mail address: [email protected]
Permanent DepartmentofMathematics, JinzhouNormalUniversity,JinzhouCity,
Liaoning 121000, People’s Republic of China
E-mail address: [email protected]