with only non-degenerate critical points, such that at the boundary, ?~ is transverse to ~S I. Then ..... we use the Leray spectral sequence of ~. One has by direct.
POLAR
CURVES,
AND THE
RESOLUTION
FILTERED
MIXED
ON THE VANISHING
Joseph
University Postbus
connect
we
relation mixed
and
Johns
The Netherlands.
L~ D.T.,
F. M I C H E L
proofs
between
and
invariants
decomposition
give
Hodge
of M a t h e m a t i c s
Hopkins
Baltimore,
the polar
Waldhausen
Zucker*
Department
Institute
of Leiden
Leiden,
ABSTRACT.
STRUCTURE
Steven
9512
RA
paper
HODGE
COHOMOLOGY
Steenbrink*
Mathematical
2300
OF SINGULARITIES
C. W E B E R
associated
the polar
structure
on
have
curve
obtained
results
filtration
results
singularity
to its M i l n o r
of t h e s e
its
21218
U.S.A.
of a p l a n e
of m o s t
University
MD
with
fibration.
and
In t h i s
investigate
of the M i l n o r
fibre
which the
and
the the
cohomology.
INTRODUCTION
Let X = f-l(0). linear some
f: Let
form.
~ =
A polar
component
of the
c a n be o b t a i n e d filtration
F1 ~
polar
filtration. [6],
(~,f):
of
These
of ~
b y L~
of the
we describe
filtration.
*Supported by the SLOAN foundation.
first
in
have
f , and fibre
been
[4] a n d
history
~
of t h e
exponent
quotients they
of
studied
[5]. W e
germ,
is a g e n e r a l
Puiseux
. The polar of
of t h e M i l n o r
properties
function
where
is the
resolution
~Fg
summarized
÷ (~2,0) f
discriminant
F2 C...
paper,
be a h o l o m o r p h i c
(~2,0)
quotient
for a d i s c u s s i o n
In t h i s polar
and
+(~,0)
from a good
WEBER papers
(~2,0)
give
of
of f
rise
to a
MICHEL
and
f , the b y L~,
refer
to these
subject.
the Hodge-theoretic
properties
of
the
179
The groups
Hi(Fj),
structures;
those on
H i ~)F j', F.j _ I
carry natural m i x e d Hodge
HI(Fj,Fj_I)
are of a p a r t i c u l a r l y simple kind
(see our T h e o r e m 2 in ~3). In order to prove our result, results of
we felt obliged to reprove most of the
[6]. This is done i n ~ 2: we claim no p r i o r i t y there,
though
our proof is s u b s t a n t i a l l y d i f f e r e n t and gives a slightly more general result.
The g e n e r a l i z a t i o n to functions of more than two variables,
p r e d i c t e d by LE,
is treated in ~4.
We thank L~ for s u g g e s t i n g this p r o b l e m to us; we also thank the I n s t i t u t e of A d v a n c e d Study in P r i n c e t o n for its h o s p i t a l i t y d u r i n g J a n u a r y 1986, where most of this work was done.
~i. R E S O L U T I O N OF PLANE CURVE SINGULARITIES.
Let
f:
(~2,0)
+ (~,0)
X = f-l(0). We assume that variety
0 G X
function,
is a singular point of the reduced
X red .
Let
~: Z ÷ ~2
in which curves,
be a germ of an a n a l y t i c
0 E ~2
give the minimal good e m b e d d e d r e s o l u t i o n of
X ,
is replaced by a connected union of smooth rational
such that
for
normal crossings.
~ = f=, ~-i(0)
is a divisor on
Z
with
It is c o n s t r u c t e d by s u c c e s s i v e l y blowing up points,
c r e a t i n g a tower of m o d i f i c a t i o n s
(i)
~j
~J
~
~
3
If we w r i t e produce point.
~j = f~j,
Zj+ 1 Thus,
o n l y if
a point of ~j-l(0)
~j -l(0) C
the p r o c e s s can be d e s c r i b e d by saying that in
Zj. In case
units, of Z
is b l o w n up to
fails to have normal crossings at that
there is one a d d i t i o n a l e x c e p t i o n a l curve in
Z3
f
of course),
has only one distinct Zj+ 1
Zj"
b e y o n d what one has
irreducible
factor
(up to
is always o b t a i n e d by b l o w i n g up a p o i n t
Ej. The process terminates being the final
Ej+ 1
Zj+I,
after a finite number of steps,
with
180
The graph for any
of the r e s o l u t i o n
sequence
exceptional
corresponding
set,
El.AS
w i t h order
~ a
~*
defined
if and only
the case w h e r e
f
down as a "mobile"
The g r a p h
V
by "distance"
if the chain has
698),
w i t h one
first e x c e p t i o n a l
forms
a partially
to the d i s t i n g u i s h e d
from
~ to a*
one i r r e d u c i b l e
([i] p.
to the
sense
if the
is a tree,
of v e r t i c e s
makes
for each
two v e r t i c e s
corresponding
the set
description
of a v e r t e x
between
intersect.
vertex such,
following
-consists
and an edge
curves
distinguished curve,
of b l o w - u p s
curve,
-the
passes
factor,
ordered
vertex:
through
the graph
~.
In
can be layed
e.g.
O
T°I
(2)
~. o~
~,
0 We have
started
presence
above
employing
of a c o m p o n e n t
The p i c t u r e is the p r o c e s s by b l o w i n g tangent
0 .........
(2) arises
0
the n o t a t i o n
up p o i n t s
until
to an e x c e p t i o n a l
component°
through
the i n t e r s e c t i o n
blow-up
This p r o c e s s a crossing
then
point.
until
an arrow
of
directions. X
of two e x c e p t i o n a l
1. In the m i n i m a l
At the next
resolution
---G
X
O
have
transform The
between
no longer
one moves
one cannot
becomes
components.
transform
stage,
up a p o i n t
in d e c r e a s i n g
in the c h a i n a c o m p o n e n t the p r o p e r
the
One b e g i n s
the p r o p e r
mobile.
REMARK
of
This produces, stage,
denotes
X.
that b l o w i n g
transform
At the next
inserts
repeats
Recall
the t a n g e n t
the p r o p e r
passes
that
transform
as follows.
in the graph.
through
0
that d i s t i n g u i s h e s
a chain
two.
0--
of the p r o p e r
order,
succeeding
0
Om
upward
these
passes in the
181
at the would
top
of
have
been
In the watching chain
(2),
for t h e n
case,
all of the
component
last
step
in the
resolution
process
superfluous.
reducible
several
the
must
irreducible
arrows
in the
one
perform
components.
in the p i c t u r e ,
same
the b l o w i n g s
or d i f f e r e n t
some
This
gives
perhaps
points,
up w h i l e
in the
meeting
initial
the
same
e.g.
© One
proceeds
as before,
intersection
REMARK
2.
with
to at l e a s t
intersects described
THE
the p r o p e r
In the m i n i m a l
connected
~2.
going
E R.
"up"
from
of the p o i n t s
of
transform.
resolution,
3 other
This
each
for any p o i n t
points,
is e a s i l y
their
deduced
exists from
the
u G V
which
is
~ 4 u
such
that
construction
above.
RELATIVE
POLAR
CURVES
OF
f.
Let ~)
(3)
H f3 j
be
the
]
factorization
f. N o t e
of
that
any
directional
derivative
of
1 f For
is d i v i s i b l e generic
by
f. 3
directions
by the v a n i s h i n g
for e a c h u,
the
j, a n d b y n o h i g h e r
topological
type
of the
power
curve
of
of
vj-i (4)
D u f / ~ fj 3
is c o n s t a n t : Let
(5)
it is c a l l e d
~
be a l i n e a r
~:
(~2,0)
a relative
f o r m on
. (~2,0)
~2.
polar One
curve
defines
associated the m a p p i n g
fj.
defined
to
f.
182
by
~ =
(%,f).
components polar
of the
curve
PROPOSITION where
the
~oint.
critical
F. We m a k e
puts
each
(1.2)).
component
f
f-l(0)
Ai
it has
of
of
%
but
the p o l a r
to the =
, the
~ , other
elementary,
f-l(0),
A = ~(F).
As s u c h ,
locus
of
F ~
choice
the
l: O u t s i d e restriction
Moreover,
One then
For g e n e r i c
level
union
than
useful,
of the
those
of
X
, is a
observation:
curve
is the
curve
of ~ has
set of p o i n t s
the
coordinates
a critical
{(0,0)}.
If
(z,w)
of
A
denote
is t a n g e n t
a Puiseux
to the
z-axis
mapping
onto
on
~2,
([11],
Prop.
series
r,
Z = a.w l with
ri
the
formula
at
L~ also
general
~
must So
0,
a
and
for t h e
linear
f:
Thus
the
=
{Y}'
if a n d
has
(M,0)
near
the 0
gI"
non-
o I = O.
gI = gy = m point.
only
components points,
r(~*)
holds,
of
X
has m
different
from from
E ,.
function
÷ (6,0)
have
if where
it f o l l o w s
(and t h e o r e m )
local
if in t h e of the
i.e.
= l/m,
a tangent
case
proof
a
of
is a t y p i c a l
intersects
for w h i c h forms
has
is c l e a r l y
is a r u p t u r e
latter
curve
component
Our
~
~
that
If the
case
of
be m i n i m a l ,
point
E ,
(Note
considers
singularity
Such
1 of ~ l .
the polar
a polar
invariants
always a multiple
at t w o or m o r e
0.)
a
when ~ E ~ .
in 6 2 ,
X
J.
numerical
is
is a r u p t u r e
of
of
0. T h e n
g~
Then
singularities.
modifications
isolated
~ # ~*.
tangents. f
CI >
= 2, C I = 0.
, ~*
transform
for t h e
g~ = g I
i (c~-l)
two distinct
X
V,
resolution):
and
by R e m a r k
Here
factorization
those
# B
of the o r i g i n
the multiplicity the
of
from the
subchain
as e a c h
By Lemma C,
blow-up
least
= 2
~ i > O,
~ ~ ~*,
01 = C I - m
4.
~ B
= i, C I > 0,
and
(first)
the p o s s i b i l i t i e s f r o m the m i n i m a l
with
{a},
~ B
element
by induction
is a 3 - e l e m e n t
> 2, or
point.
3.
a minimal
.
can be deduced
We now discuss V
contains
= m + + m
~_
...
> rg.
and
0
is a r a t i o n a l
can g e n e r a l i z e
OF THE MILNOR
the n o t a t i o n
set of all
(M,0)
In the m o r e
one
complicated,
FILTRATION
We keep
that
is true
fundamental
much
POLAR
be the
this
polar
i,
Bi 6
and
HODGE
sections.
exponents
Choose
MIXED
~
of
f
STRUCTURE.
Let
at
{rl,...,rg }
0,
sufficiently
and
large
assume (see
[5],
let r,
F i-- {z~ ¢211zI< Then
Fg
is d i f f e o m o r p h i c
filtration
FIC
diffeomorphism invariant An
class
of the
the m i n i m a l increasing
The
. . . C Fg
chain
V(i) V(i)
g-tuple
f
at
description
of
of
f
subsets
0 of
V(i)
) ri}.
connected
subgraphs
and
fibre
a polar
(loc. the
Fi the
f(z)
of
f
= 6}.
and
f i l t r a t i o n ' of
(F 1 .... ,Fg)
, using
= {=eVir(=)
are
~ Bi6 ~
to the M i l n o r
of the
resolution
P(z)l
is c a l l e d
g e r m of
alternative
(19)
~,
cit. can
the f. The
is an a n a l y t i c Th~or~me
42.7)).
be g i v e n
results
of ~ 2 .
of
V,
i = l,...,g,
of
V.
To these,
in t e r m s We
define
of an
by
there
correspond
curves
(20)
E(i)
=
U
E
~£V(i) with
(21)
neighbourhoods
U(i)
=
~J
U
=~V(i) in Z
(cf.(12)).
PROPOSITION the
8. W i t h
filtration
notation
as above,
for
6 > 0
s u f f i c i e n t l ~ small,
191
U-~
f-I(6)C...cU--T~
~-I(6)
is diffeomor~hic
to a polar filtration of
PROOF.
let
For
=eV
~ = U~
~
U
Let
f. h : Z \ f-l(0) ÷ ]R be given
-r.
by
hi= l~I" Ifl
i.
If
~eV
with
r(~)
B i r(a) ) r i ~ ki(6)< B i. It follows, Fi= {z~f-l(6)lhi(z)¢B i}
and that the boundary of
is contained
in 0 { U ~ U ~ f - l ( 6 ) I r ( ~ ) < r i ~ r ( ~ )
We will construct our diffeomorphism
}.
in such a way that it will differ
from the identity only on these open sets. In local coordinates (s,t)
near E ~
E8
on
Z, we will have
u=: I~I< l, u~: itl< 1 sdtd~
m
~(s,t)=
with
u
, ~(s,t)=
m
u s ~t
a unit. Then hi(s,t)=
lul° Is]-altl b with
a > 0, b ; 0.
We claim that there exists a diffeomorphism which maps this,
U~ ~
f-l(6)
we may t a k e
to
~
slightly
above still holds,
~
bigger
and work on
U' ~
following
lemma,
LEMMA D. Let
e.
0
