Characteristic Classes of Hypersurfaces and Characteristic Cycles

15 Pages·0.19 MB·English

by  Adam Parusinski

CHARACTERISTIC CLASSES OF HYPERSURFACES
AND CHARACTERISTIC CYCLES
8
9
9
1
Adam Parusin´ski and Piotr Pragacz

n
a
J

1
2

]
G Abstract. We give a new formula for the Chern-Schwartz-MacPhersonclass of a hypersurface,
A generalizingthemainresultof[P-P],whichwasaformulafortheEulercharacteristic. Twodifferent
. approaches are presented. The first is based on the theory of characteristic cycle and the work
h
t of Sabbah [S], Brianc¸on-Maisonobe-Merle [B-M-M], and Lˆe-Mebkhout [L-M]. In particular, this
a
m approach leads to a simple proof of a formula of Aluffi [A] for the above mentioned class. The
[ second approach uses Verdier’s [V] specialization property of the Chern-Schwartz-MacPherson

classes. Some related new formulas are also given.
1
v
2
0
1
1 Introduction and statement of the main result
0
8
9 Let X be a nonsingular compact complex analytic variety of pure dimension n
h/ and let L be a holomorphic line bundle on X. Take f ∈ H0(X,L) a holomorphic
t section of L such that the variety Z of zeros of f is a (nowhere dense) hypersurface
a
m in X. Recall, after [A], that the Fulton class of Z is
:
v
Xi (1) cF(Z) = c(TX|Z −L|Z)∩[Z],
r
a
where TX denotes the tangent bundle of X. Note that if Z is nonsingular then
cF(Z) = c(TZ) ∩ [Z]. By c (Z) we denote the Chern-Schwartz-MacPherson class
∗
of Z, see [McP]. We recall its definition later in Section 1. After [Y] we shall call
(2) M(Z) = (−1)n−1 cF(Z)−c (Z)
∗
(cid:0) (cid:1)
the Milnor class of Z. This class is supported on the singular locus of Z; it is
convenient, however, to treat it as an element of H (Z).
∗
Typeset by AMS-TEX
1
2 CHARACTERISTIC CLASSES OF HYPERSURFACES
Example 0.1. Suppose that the singular set of Z is finite and equals {x ,... ,x }.
1 k
Let µ denote the Milnor number of Z at x (see [M]). Then
x i
i
k
M(Z) = µ [x ] ∈ H (Z)
x i 0
i
i=1
X
- see, for instance Suwa [Su], where this result is generalized to complete intersec-
tions.
Consider the function χ : Z → Z defined for x ∈ Z by χ(x) := χ(F ), where F
x x
denotes the Milnor fibre at x (see [M]) and χ(F ) its Euler characteristic. Define
x
also the function µ : Z → Z by µ = (−1)n−1(χ−11 ).
Z
Fix now any stratification S = {S} of Z such that µ is constant on the strata of
S. For instance, any Whitney stratification of Z satisfies this property, see [B-M-M]
or [Pa]. Actually, it is not difficult to see that the topological type of the Milnor
fibres is constant along the strata of Whitney stratification of Z. Let us denote the
value of µ on the stratum S by µ . Let
S
(3) α(S) = µ − α(S′)
S
S′6=XS,S⊂S′
be the numbers defined inductively on descending dimension of S. (These numbers
appear as the coefficients in the development of µ as a combination of the 11 ’s –
S
see Lemma 4.1.)
The main result of the present paper is
Theorem 0.2. In the above notation,
(4) M(Z) = α(S)c(L| )−1 ∩(i ) c (S),
Z S,Z ∗ ∗
S∈S
X
where i : S → Z denotes the inclusion.
S,Z
When X is projective, (4) was conjectured in [Y]. Under this last assumption,
the equality
(5) M(Z) = α(S) c(L| )−1 ∩c (S)
S ∗
Z S∈S Z
Z X S
was proved in [P-P]; hence the theorem gives, in particular, a generalization of the
main result (5) of [P-P] to compact varieties.
Our proof of the theorem is based on a formula due to Sabbah [S], which allows
one to calculate the Chern-Schwartz-MacPherson class of a subvariety in terms of
the associated characteristic cycle. In the case of hypersurface Z, this characteristic
ADAM PARUSIN´SKI AND PIOTR PRAGACZ 3
cycle was calculated in [B-M-M] and [L-M] in terms of the blow-up of the Jacobian
ideal of a local equation of Z in X. So the proof of Theorem 0.2 is obtained
by putting this local description and the global data together, and expressing the
characteristic cycle of Z in terms of the global blow-up of the singular subscheme
of Z. Here by the singular subscheme of Z we mean the one defined locally by the
ideal f, ∂f ,... , ∂f , where (z ,... ,z ) are local coordinates on X.
∂z1 ∂zn 1 n
Th(cid:16)e approach used(cid:17) leads to a very simple proof of a formula for the Chern-
Schwartz-MacPherson class of hypersurface in terms of some divisors associated
with the above blow-up. This formula was originally obtained by Aluffi [A] by
different methods. Some new formulas for the Chern-Schwartz-MacPherson classes
of the constructible functions χ and µ are also given.
Finally, we show, using Verdier’s specialization property of the Chern-Schwartz-
MacPherson classes (see [V], and also [S] and [K2]) how to prove another conjecture
of Yokura, which, combined with a result from [Y], gives an alternative proof of
Theorem 0.2 in the case of projective X. We find that this specialization argument
somewhat better explains the essence of the theorem.
1. Chern-Mather classes and Chern-Schwartz-MacPherson classes
We start by recalling some results of Sabbah [S]. Let for X as in the introduction,
T∨X denote the cotangent bundle of X. Let V be an (irreducible) subvariety of
X. Denote by c (V) (resp. c∨ (V)) the Chern-Mather class of V (resp. the dual
M M
Chern-Mather class). Let us recall briefly their definitions. Let ν : NB(V) → V
be the Nash blow-up of V. By definition on NB(V) there exists the “Nash tangent
bundle” T which extends ν∗TV0, where V0 is the regular part of V. Define the
V
following elements of H (V)
∗
c (V) = ν c(T )∩[NB(V)]
M ∗ V
(6)
c∨M(V) = ν∗(cid:0)c(TV∨)∩[NB(V)](cid:1),
where T∨ is the dual bundle of T . It(cid:0)is easy to see tha(cid:1)t
V V
(7) c∨ (V) = (−1)dim Vc (V)∨,
M M
where for a homology class a = a +a +a +... , where a ∈ H (V), we denote
0 1 2 i 2i
a∨ = a −a +a −... .
0 1 2
By T∨X ⊂ T∨X we denote the conormal space to V :
V
TV∨X = Closure (x,ξ) ∈ T∨X | x ∈ V0, ξ|TxV0 ≡ 0 ,
and by C(V) ⊂ PT∨X its proj(cid:8)ectivization. Let π : C(V) → V be t(cid:9)he restriction of
the projection PT∨X → X to C(V). Then by [S], we have
c∨ (V) = c(T∨X| )∩π c(O(−1))−1 ∩[C(V)]
M V ∗
(8)
(cid:16) (cid:17)
c (V) = (−1)n−1−dim Vc(TX| )∩π c(O(1))−1 ∩[C(V)] ,
M V ∗
(cid:16) (cid:17)
4 CHARACTERISTIC CLASSES OF HYPERSURFACES
where O(−1) is the tautological line bundle on PT∨X restricted to C(V).
Let now ϕ be a constructible function on X,
ϕ = a 11 ,
j Y
j
X
where Y are (closed) subvarieties of X and a ∈ Z. By the characteristic cycle of
j j
ϕ we mean the Lagrangian conical cycle in T∨X defined by
⊕a
(9) Ch(ϕ) = Ch i C j ,
 Yj,X ∗ Yj 
j
M(cid:0) (cid:1)
 
where C is the constant sheaf on Y and i : Y → X denotes the inclusion.
Yj j Yj,X j
For a general definition of the characteristic cycle of a sheaf, we refer the reader
to [B]. The characteristic cycle of a constructible function admits the following
interpretation. Let F(X) and L(X) denote the groups of constructible functions
on X and conical Lagrangian cycles in T∨X respectively. It is known that the
assignment
(10) T∨X 7→ (−1)dim VEu ,
V V
where Eu stands for the Euler obstruction (see [McP] and also [S], [K1]), defines a
V
naturaltransformationofthefunctorsofLagrangianconicalcyclesandconstructible
functions, that is an isomorphism. In particular, we have an isomorphism between
L(X) and F(X). The operation of taking the characteristic cycle is the inverse of
this isomorphism; that is, it is given by
(11) Ch(Eu ) = (−1)dim VT∨X.
V V
Since every constructible function is a combination of the Eu ’s (see [McP]), this
V
allows “in theory” to compute Ch(ϕ) for a constructible function ϕ. However, even
for ϕ = 11 , this would involve not only the Euler obstruction of V itself but also
V
of some subvarieties of V.
Now we associate with a constructible function ϕ on X its Chern-Schwartz-
MacPherson class (abreviation: CSM-class). Let π : SuppPCh(ϕ) → Suppϕ be
the restriction of the projection PT∨X → X. Set
(12) c (ϕ) = (−1)n−1c(TX| )∩π c(O(1))−1 ∩[PChϕ]
∗ Suppϕ ∗
(cid:16) (cid:17)
– an element in H (Suppϕ). We note that, in particular, by (8), (11) and (12) one
∗
has
(13) c (Eu ) = c (V).
∗ V M
ADAM PARUSIN´SKI AND PIOTR PRAGACZ 5
If V ⊂ X is a (closed) subvariety, we will write c (V) = c (11 ) as is customary.
∗ ∗ V
Note that (12) is in agreement with [McP] because for 11 = b Eu , where
V i i Yi
b ∈ Z and Y ⊂ X are (closed) subvarieties, we have
i i
P
c (11 ) = b c (Eu ) = b c (Y ) = c (V).
∗ V i ∗ Y i M i ∗
i
i i
X X
Thus, denoting by π : SuppCh(11 ) → V the restriction of the projection PT∨X →
V
X, we have
(14) c (V) = (−1)n−1c(TX| )∩π c(O(1))−1 ∩[PCh(11 )] .
∗ V ∗ V
(cid:16) (cid:17)
2. Characteristic cycle of a hypersurface (local case)
n
Suppose that U ⊂ C is an open subset and Z ⊂ U is a hypersurface of zeros of a
holomorphic function f : U → C. Let J denote the Jacobian ideal ∂f ,... , ∂f
f ∂z1 ∂zn
n
of f, where (z ,... ,z ) are the standard coordinates of C . Consid(cid:16)er the blow-u(cid:17)p
1 n
π : Bl U → U of J . Recall that we may interpret it as follows
J f
f
∨n−1 ∂f ∂f
Bl U = Closure (x,η) ∈ U ×P |x ∈/ SingZ,η = (x) : ... : (x) ,
J
f ∂z ∂z
1 n
(cid:26) (cid:20) (cid:21)(cid:27)
where SingZ denotes the singular subscheme of Z.
Remark 2.1. Bl U can be also interpreted as the projectivization of the relative
J
f
conormal space T∨ ⊂ T∨U (see [B-M-M, §2], where we put Ω = X = U). Then
f
by the Lagrangian specialization all fibres of the restriction of f˜: T∨U → U −→f C
to T∨ are conical Lagrangian subvarieties of T∨U. In particular, every irreducible
f
component of f˜−1(0)∩T∨ is conormal to its projection on U. For details we refer
f
to [B-M-M, §2] and to references therein.
Let Z be the total transform π−1(Z) of Z in Bl U and Z = D be the
J i
f
i
decomposition of Z into irreducible components. Set Ci = π(Di) and dSenote by IC
i
the ideal defining C . Then define
i
n = multiplicity of I along D
i C i
i
m = multiplicity of f along D
i i
p = multiplicity of J along D
i f i
Let us now record the following result.
6 CHARACTERISTIC CLASSES OF HYPERSURFACES
Proposition 2.2. : m = n +p .
i i i
Proof. Observe that by Remark 2.1 we have D = PT∨ U. Let x be a generic point
i C
i
of C and choose a system of coordinates (z ,... ,z ) at x such that C = {z =
i 1 n i 1
... = z = 0} in a neighborhood of x. Then, over a neighborhood of x,
k
∨k−1
(15) D = C ×P ,
i i
where
∨k−1 ∨n−1
P = {[η : ... : η ] ∈ P |η = ... = η = 0}.
1 n k+1 n
Let ζ : E → U denote the blow-up of the product of J and I . So
f C
i
∂f ∂f
E = Closure x,[z (x) : ... : z (x)], (x) : ... : (x) |x ∈/ SingZ
1 k
∂z ∂z
1 n
(cid:26) (cid:27)
(cid:16) h i(cid:17)
k−1 ∨n−1
in U ×P ×P . Then ζ factors through π
E −→ Bl U
J
f
(cid:31) |
ζ (cid:31) | π
ց ↓
U
and there exists at least one irreducible component, say B , of the exceptional
ij
divisor of ζ which projects surjectively onto D . Let γ(t) = z(t),v(t),η(t) be an
i
analytic curve in E such that z(0),v(0),η(0) is a generic point of B , z (t) ≡
(cid:0) ij k+(cid:1)1
... ≡ z (t) ≡ 0 and f z(t) 6= 0 for t 6= 0. Then we have for t 6= 0
n (cid:0) (cid:1)
(cid:0) (cid:1)
k−1
v(t) = [z (t) : ... : z (t)] ∈ P
1 k
∂f ∂f ∨n−1
η(t) = z(t) : ... : z(t) ∈ P
∂z ∂z
1 n
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
and η(0) = η (0) : ... : η (0) : 0 : ... : 0 by (15).
1 k
Since z((cid:2)0),η(0) is a generic point of D(cid:3) the following equality would imply the
i
proposition
(cid:0) (cid:1)
ord (f ◦ζ) γ(t) = ord f z(t)
0 0
(16) ∂f ∂f
(cid:0) (cid:1) (cid:0) (cid:1)
= ord z (t),... ,z (t) +ord z(t) ,... , z(t) .
0 1 k 0
∂z ∂z
1 n
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
We show (16). First we note that we may suppose that (z ◦ ζ,... ,z ◦ ζ) is
1 k
generated by z ◦ ζ at γ(0) and ζ−1J is generated by ∂f ◦ ζ at γ(0), where
i0 f ∂z
j0
ADAM PARUSIN´SKI AND PIOTR PRAGACZ 7
j ∈ {1,... ,k}) by (15). We have
0
k
d ∂f (cid:5)
f z(t) = z(t) z (t)
i
dt ∂z
i
i=1
(17) (cid:0) (cid:1) X (cid:0) (cid:1)
∂f (cid:5) k ∂∂zf z(t) z(cid:5)i(t)
= z(t) ·z (t) i · ,
∂zj0(cid:0) (cid:1) i0 Xi=1 ∂∂zfj0(cid:0)z(t)(cid:1) z(cid:5)i0(t)
 (cid:0) (cid:1) 
(cid:5)
where z stands for dzi. Note that the quotients make sense since z ◦ζ generates
i dt i0
ζ−1(z ,... ,z ) and ∂f/∂z ◦ζ generates ζ−1J .
1 k j0 f
We may suppose that η = 1 and v = 1, which corresponds to choosing affine
j0 i0
k−1 ∨n−1
coordinates on P ×P . Since
(cid:5) (cid:5)
lim [z (t) : ... : z (t)] = lim [z (t) : ... : z (t)]
1 k 1 k
t→0 t→0
we get
k ∂f z(t) z(cid:5) (t) k η (t) v (t) k
∂z i i i
lim i · = lim · = η (0)v (0).
t→0Xi=1 ∂∂zfj0(cid:0)z(t)(cid:1) z(cid:5)i0(t) t→0 Xi=1 ηj0(t) vi0(t)! Xi=1 i i
 (cid:0) (cid:1) 
This last sum is nonzero by the transversality of relative polar varieties, see, for
instance, [H-M, 8.7, Lemme de transversalit´e]. Consequently, (17) implies
∂f
ord f z(t) −1 = ord z(t) + ord z (t)−1
0 0 ∂z 0 i0
j0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
which gives (16), as required. (cid:3)
In the following theorem, the equality (i) and the second equality in (ii) were
established in [B-M-M] (see also [L-M]).
Theorem 2.3. (i) Ch(11Z) = (−1)n−1 niTC∨ U
i
i
X
(ii) Ch(χ) = Ch(RΨ C ) = (−1)n−1 m T∨ U
f U i Ci
i
X
(iii) Ch(µ) = (−1)n−1Ch RΦ C = p T∨ U
f U i Ci
i
(cid:0) (cid:1) X
For a definition of the complexes of nearby cycles RΨ and vanishing cycles RΦ ,
f f
we refer the reader to [D-K]. The first equalities in (ii) and (iii) are well-known
and follow from the local index theorem, see for instance [B-D-K] and [S, (1.3) and
(4.4)].
8 CHARACTERISTIC CLASSES OF HYPERSURFACES
Proof of (iii). By the definition of µ we have
Ch(µ) = (−1)n−1 Ch(χ)−Ch(11 ) .
Z
Hence, using Proposition 2.2, the asserti(cid:0)on follows. (cid:3) (cid:1)
Let Y denotes the exceptional divisor in Bl U . Since D = PT∨ U, we can
Jf i Ci
rewrite the assertions of the theorem as the following equalities.
(i) [PCh(11 )] = (−1)n−1([Z]−[Y])
Corollary 2.4. Z
(ii) [PCh(χ)] = (−1)n−1[Z]
(iii) [PCh(µ)] = [Y]
Observe that these equalities already take place on the level of cycles.
Remark 2.5. Since f belongs to the integral closure of J (see [LJ-T]) the nor-
f
malizations of the blow-ups of J and f, ∂f ,... , ∂f are equal. Hence Corollary
f ∂z1 ∂zn
2.4 holds true if we replace the blow-u(cid:16)p of the former(cid:17)ideal by the blow-up of the
latter one.
3. Characteristic cycle of a hypersurface (global case)
Let X, L, Z, f be as in the introduction. Let B = Bl X → X be the blow-up
Y
of X along the singular subscheme Y of Z. Let Z and Y denote the total transform
of Z and the exceptional divisor in B, respectively. The following description of the
CSM-class of Z was established by Aluffi [A] by different methods.
Theorem 3.1. ([A]) Let π : Z → Z be the restriction of the blow-up to Z. Then
[Z]−[Y]
c (Z) = c(TX| )∩π ,
∗ Z ∗
1+Z −Y
(cid:18) (cid:19)
where on the RHS, Z and Y mean the first Chern classes of the line bundles associ-
ated with Z and Y i.e. those of π∗(L| ) and O (−1), the latter being the canonical
Z B
line bundle on B.
Proof. To get a convenient description of B, we use (after [A]) the bundle P1 L
X
of principal parts of L over X (see e.g. [At]). Consider the section X → P1 L
X
determined by f ∈ H0(X,L). Recall that P1 L fits in an exact sequence
X
0 → T∨X ⊗L → P1 L → L → 0
X
and the section in question is written locally as (df,f) = ∂f ,... , ∂f ,f , where
∂z1 ∂zn
(z1,... ,zn) are local coordinates on X. It follows that th(cid:16)e closure of the(cid:17)image of
ADAM PARUSIN´SKI AND PIOTR PRAGACZ 9
the meromorphic map X > PP1 L induced by (df,f) is the blow-up B → X.
X
Thus we may treat B as a subvariety of PP1 L. Clearly, the total transform Z of
X
Z equals B ∩ P(T∨X ⊗ L). The canonical line bundle O (−1) = O(Y) on B is
B
the restriction of the tautological line bundle O(−1) on PP1 L. Observe that the
X
bundle O(−1) restricted to Z is contained in (T∨X ⊗ L)| (because f ≡ 0 over
Z
Z). Hence O (−1)| is the restriction of the tautological line bundle O (−1) on
B Z P
P = P(T∨X ⊗L). Using the natural identification P(T∨X ⊗L) ∼= P(T∨X) the line
bundle OP(−1) corresponds to the line bundle OP(−1)⊗L on P = P(T∨Xe ). Thus
OeP(1) on P corresponds to OP(1) ⊗ L on P. Hence, using the characteristic cycle
e
formula (14), we get
e e
c (Z) = (−1)n−1c(TX| )∩π c O (1)⊗π∗L| −1 ∩ PCh(11 )
∗ Z ∗ B Z Z
[Z]−(cid:16)[Y(cid:0)] (cid:1) (cid:2) (cid:3)(cid:17)
= c(TX| )∩π
Z ∗
1+Z −Y
(cid:18) (cid:19)
because by (theglobalanalogueof)Corollary2.4, wehave theequality[PCh(11 )] =
Z
(−1)n−1 [Z]−[Y] . (cid:3)
(cid:0) (cid:1)
By Corollary 2.4, we have [PCh(χ)] = (−1)n−1[Z] and [PCh(µ)] = [Y]. There-
fore, using similar arguments, we get the following result.
[Z]
Theorem 3.2.
(i) c (χ) = c(TX| )∩π ,
∗ Z ∗
1+Z −Y
(cid:18) (cid:19)
[Y]
(ii) c (µ) = (−1)n−1c(TX| )∩π .
∗ Z ∗
1+Z −Y
(cid:18) (cid:19)
(The constructible function µ is supported on Y but for later use we consider its
CSM-class in H (Z).)
∗
Remark 3.3. One can add to the above formulas also
[Z′]
c (Z) = c (Eu ) = c(TX| )∩π ,
M ∗ Z Z ∗
1+Z −Y
(cid:18) (cid:19)
where Z′ is the proper transform of Z. This equality for the Chern-Mather class was
establishedoriginallybyAluffi[A]bydifferentmethods. Usingthetechniqueofchar-
acteristic cycles, it is a consequence of the equality P Ch(Eu ) = (−1)n−1[Z′]
Z
(see (11)).
(cid:2) (cid:0) (cid:1)(cid:3)
4. Proof of Theorem 0.2
We start this section with the following fact about the constructible functions µ
and α defined in the introduction.
10 CHARACTERISTIC CLASSES OF HYPERSURFACES
Lemma 4.1. : µ = α(S)11 .
S
S∈S
P
Proof. Fix an arbitrary stratum S and a point x ∈ S . We have
0 0
α(S)11 (x) = α(S)+α(S )
S 0
(cid:18)XS (cid:19) S6=SX0,S⊃S0
= α(S)+ µ − α(S) = µ(x). (cid:3)
 S0 
S6=SX0,S⊃S0 S6=SX0,S⊃S0
 
Now we pass to the proof of Theorem 0.2. Let π : Z → Z be the restriction of the
blow-up B = Bl X → X. We have, rewriting (1) as in [A] and using the projection
Y
formula,
[Z]
cF(Z) = c(TX| )∩π .
Z ∗
1+Z
(cid:18) (cid:19)
Invoking (2) and using Theorem 3.1, we get
M(Z) = (−1)n−1 cF(Z)−c (Z)
∗
[Z] [Z]−[Y]
= (−1)n−1(cid:0)c(TX| )∩π (cid:1) −
(18) Z ∗ 1+Z 1+Z −Y
(cid:18) (cid:19)
[Y]
= (−1)n−1c(TX| )∩π
Z ∗
(1+Z)(1+Z −Y)
(cid:18) (cid:19)
because Y ∩ [Z] = Z ∩ [Y]. If we pass to the characteristic cycle approach, the
equality (18) is rewritten, by Corollary 2.4, in the form
[PCh(µ)]
(19) M(Z) = (−1)n−1c(TX| )∩π .
Z ∗
(1+Z)(1+Z −Y)
(cid:18) (cid:19)
Since µ = α(S)11 by Lemma 4.1, we have
S∈S S
P
Ch(µ) = α(S)Ch(11 )
S
S∈S
X
and hence
[PCh(µ)]
=
(1+Z)(1+Z −Y)
(20)
= α(S)c(L| )−1 ∩π c π∗L| ⊗O (1) −1 ∩[PCh(11 )] .
Z ∗ Z B S
SX∈S (cid:16) (cid:0) (cid:1) (cid:17)
By (14) and the proof of Theorem 3.1, we get
(21) i c (S) = (−1)n−1c TX| ∩π c π∗L| ⊗O (1) −1 ∩[PCh(11 )]
S,Z ∗ ∗ Z ∗ Z B S
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)