DIFFERENTIAL OPERATORS ON VARIETIES
BY
Daniel M. Burns, Jr.
A.B., University of Notre Dame
(1967)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR TIIE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1972
Signature of AuthorD....e
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Department of M hematics,
Certified by0. ...... .
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h' Thesis
Accepted by ..................................
5 1972......
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My 5, 1972
'rvisor•.
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Chairman, Departmental Committee
on Graduate Students
Archive
IUN 29 1972
18,14AOiesss
2.
DIFFERENTIAL OPERATORS ON VARIETIES
By
Daniel M. Burns, Jr.
Submitted to the Department of Mathematics on May 5, 1972
in partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
ABSTRACT
Let X be a (countable, reduced) complex analytic
variety, and let n : X + X be a resolution of
singularities as constructed by Hironaka. The thesis
initiates a study of differential operators on the variety
X by pulling these operators back to operators with
singular coefficients on R. A formal condition is given
on a differential form w on X which guarantees that
it is the pull-back of a differential form on X,
provided X has isolated singularities:
this extends
a result of G. Glaeser, and uses methods due to
H. Whitney and S. Lojasiewicz. Several applications
of this result to differential operators and function
theory are given.
Thesis Supervisor: I. M. Singer
Title:
Norbert Wiener Professor of Mathematics
3.
To Anne, Fritz Lang, and the MBTA
Table of Contents
Page
Introduction
5
51.
Preliminaries
11
§2.
Operators on Varieties Definitions
30
§3.
Blowing Down Forms
49
§4.
DeRham Theory Applications
75
§5.
Miscellaneous Applications; Operators on
Curves
§6.
Two Examples
99
121
Introduction.
This thesis represents an attempt to extend to
singular complex analytic varieties the methods of
partial differential equations.
The results obtained
are the initial steps in one approach at constructing
this generalized elliptic operator theory.
The basic example of geometric significance in the
case of non-singular manifolds is, of course, the Laplacian
of Hodge's theory of harmonic integrals.
For the case of
a singular variety, we again have a DeRham complex of
C
differential forms, the coboundary operator being the
differential operator of exterior differentiation.
The
Poincare lemma fails in the singular case, however, and
the cohomology of the DeRham complex may no longer be
interpreted as the topological cohomology of the variety
in question.
It is a theorem, due to Bloom and Herrera [4],
that this DeRham cohomology contains the topological
cohomology as a direct summand, invariant under the ring
structure on cohomology given by exterior multiplication
of forms.
Thus, the De Rham cohomology contains topological
information about the variety, as well as some analytic
invariants concerning the differentiable nature of the
singularities of the variety.
It is not known at present
what these invariants are; in fact, it is not even known
in general whether the DeRham cohomology for a compact
variety are finite-dimensional.
Trying to demonstrate
the finite dimensionality of the DeRham cohomology by
elliptic equation methods was one of the test questions
in the author's mind when beginning to search for an
elliptic operator theory on varieties.
This modest
desire was unfulfilled; one result contained in this
report is the finite dimensionality of DeRham cohomology
in the case of a compact variety with isolated singular
points, but this is proved by methods other than elliptic
equations.
It is possible to show some simple singularities
for which the Poincare lemma does hold, and we consider
some of the difficulties involved in trying to construct
harmonic integrals even in these simple cases: the
problem, in general, is that we don't know at present how
to construct formal adjoints for differential operators
on varieties.
What we shall consider in this report are the first
steps towards implementing the following philosophy
concerning operators on varieties:
1.
lift all analysis problems from the variety X
to a resolution of singularities
2.
X
of
X.
solve the partial differential equation problems
on the manifold
X
by (hopefully) standard,
non-singular elliptic theory
3.
push the solutions down to
obtaining solutions on
X, when possible,
X.
We have good results on 1., which is quite simple,
although the lifted operators one gets on the manifold X
are, in general, "meromorphic".
This meromorphy already
forces us to very non-standard partial differential
equation situations on
even for
when
X
X
X, which are difficult to handle,
with isolated singularities.
For example,
has dimension, we get operators with very
singular lower order terms.
In the case where
X
has
isolated singularities, we can find simple conditions for
pushing down solutions on
X
to
X.
In fact, the main
positive result of the present work is a criterion for
pushing down, or "blowing down" in more Italianate
phrasing,
a differential form from
case of isolated singularities.
X
to
X
in the
This criterion is
simple enough to let us conclude several corollaries from
it without much difficulty.
is
One of these corollaries
a functional analytic measure of the "number" of
differential operators on functions on a variety of
dimension 1.
In outline, then, this is the content of this
thesis:
Section 1 is preparatory, and mainly presents
a generalization to varieties of a classical result of
Emile Borel.
Section 2 deals with definitions also, and
contains the lifting procedure for a differential operator.
Section 3 contains all of the blowing-down results, which
extend a theorem in the local, non-singular case by Glaeser.
The methods of this section, as well as Glaeser's, go back
to the work of Whitney and Lojasiewicz on differentiable
functions, as detailed in Malgrange's book [20].
Section 4
applies the blowing-down results to the smooth DeRham
cohomology groups of varieties.
The most complete result
here is the finite-dimensionality of these groups for a
compact
X
with isolated singularities.
This case admits
a direct comparison with analogous holomorphic cohomology,
whose finite dimensionality has been proved by function
theoretic arguments in [51.
In section 5 we collect
some more examples of applications of the results of
section 3.
The first result tells which holomorphic
functions on a resolution
singularities
on
X.
X
of an
X
with isolated
blow down to holomorphic functions
The second generalizes to such an
X
a theorem
of Malgrange on ideals of differentiable functions
generated by finitely many real analytic functions.
The last result is the completeness of the space of
smooth functions on a variety of dimension 1 in the
topology given by the semi-norms
any compact set
D
on
X.
K
in
X
sup IDf(x)l,
xeK
for
and any differential operator
This is made possible by putting results of
sections 2 and 3 together to construct a large supply
of differential operators on such an
X.
The method
employed fails in higher dimensions because the
functions from
functions on
X:
X
have too great a codimension in the
for example, they don't contain any
principal ideals of the larger algebra of functions
on
X.
Finally, section 6 uses the method of section 4
to calculate as explicitly as possible the DeRham complex
of the two simplest examples of singular plane curves,
in order to see what might be done about the missing
Hodge theory for the DeRham complex.
10.
The author would like to extend his gratitude and
appreciation to his advisor, Professor I. M. Singer,
as well as to Professor M. Artin, Professor V. Guillemin,
Professor H. Hironaka, and Gerry Schwarz.
11.
Preliminaries.
§1.
Part A of this section recalls basic definitions
concerning complex analytic varieties and fixes notations.
Part B generalizes a classical result of Emile Borel on
formal power series to an analogue for singular varieties.
Finally, part C contains a few necessary comments on
globalizing results of Lojasiewicz and Whitney.
A.
A complex analytic variety is a
Definition:
X
Hausdorff topological space
together with a sheaf
of germs of continuous functions such that every
point x e X has an open neighborhood U with the
0X
following property:
•: U - Y CA ,
Y
there is a homeomorphism
where
is a closed subset of
A
is a polydisc in
A
CN ,
defined by the vanishing
functions.
of a finite number of holomorphic
-I
is isomorphic to
OXIU
4 Oy, where
Furthermore,
Oy = 0 /I
Y
is
the ideal sheaf of germs vanishing on
is
: 0
A, and
the sheaf of germs of holomorphic functions on
Iy
and
Y.
Throughout this thesis it is assumed that complex
analytic varieties are countable at infinity, i.e.,
union of countably many compact subsets.
the
(We have also,
in the definition given, assumed them reduced.)
We also
12.
use the terms complex analytic space, or variety in what
follows.
S(X)
in
X
X, the points
denotes the singular locus of
at which
X
is not locally a complex manifold.
X
is a closed, nowhere dense subvariety of
S(X)
p. 111 and p. 141).
([12],
By means of the local imbeddings
€
: U -
Y
A
used above, one may also consider sheaves
-l(EA/Jy), and
EXIU
AX IU =
-1(AA/K)
: here
EA = sheaf of germs of differentiable functions on
A(AA = sheaf of germs of C-valued real analytic
functions on
on
Y
A)
and
(respectively,
vanishing on
Y).
J
= ideal of Cw germs vanishing
KA = ideal of real analytic germs
These definitions make good sense,
regardless of the local imbeddings chosen by means of
the inverse mapping theorem ([12],
p.
154).
Thus, each complex analytic variety
equipped with three sheaves of rings,
X
comes
OX, AX and E X '
Each of these classes of functions has an associated
notion of differential form.
For general reference on
forms, one may consult [11], 0.20.4,
or [4].
For the
13.
category of holomorphic functions, the definition
adopted here is as follows:
holomorphic 1-forms on
sheaf
S:
U
-(•/Iy
-
Y
the sheaf of germs of
X, denoted
SX, is locally the
00* + O *d(Iy)), using the rotation
A
as before, and further:
A = sheaf of germs holomorphic 1-forms on
d
A
and
is the exterior differentiation operator on
1-forms on
A.
i-forms
= A
Define the sheaf of germs of holomorphic
= OX
by convention.)
Again,
the inverse mapping theorem insures these local
definitions are consistent.
For the real analytic and Co
categories, the
definitions are slightly different:
a local imbedding of
and define
X,
4 :U
Y
-
i = sheaf of germs of
N
see [4].
Consider
A, as before,
Co i-forms which
vanish when restricted to the submanifold Y - S(Y),
which is dense in
Y.
Then define the sheaf of germs
of differentiable i-forms on
where
E
X
by
EXI
U
denotes germs of smooth i-forms on
-1(E/Ny),
A.
may categorically imitate this definition to obtain
One
14.
i
sheaves
AX
of real analytic i-forms on
X.
Again,
the inverse mapping theorem proves independence of the
local imbeddings.
Given a map
f : X
-
Y
of complex varieties,
i.e., the f-pull back of any holomorphic function is
holomorphic, then
Asf: f-
Ey
f
induces pull-backs
0 E X , as well as for the holomorphic and
real analytic categories.
Note, also, that
'q
AP
x
where
is
p Ap
AX
X
defined as
Es =
X
p+q=-s
AX
A =
f
APj q
A
p+q=s A
decomposition
q
$
p+q=s
of forms into
x
sometimes
will sometimes be denoted
wXx
ideal.
Let
If
'
Ex.
Ai
X,x
R
M
EP
xX
A similar comment applies to
Ex(X).
EiX,x' AX
X,x and
B.
X
was, using the
(dz, dF)-type on the polydisc A (similarly for
The stalk EX
will sometimes be denoted
The stalk
EPX'q
be a local ring and
m
its maximal
is a finitely generated R module, denote
A
by
M
the Hausdorff completion of
M
with respect to
)
15.
A
the Krull topology.
a module over
M is
A
A
R, and
M is
A
isomorphic to
lim M/mn.M.
The Krull topology on
M is
is the inverse limit of the discrete topologies on the
M/mnM.
on
If
C, and
R/m
R
is a topological field, such as
is an R/m algebra, then each
is a finite dimensional vector space over
R
M/mn.M
R/m.
For
complete, locally compact fields there is a unique
topology of topological vector space over
the spaces
M/mnM.
Thus,
M
R/m
for
also inherits the inductive
A
limit topology of these topologies.
M
topological vector space over
In all our examples,
R/m
is just
R/m.
is thus a complete
C, and in this case, each of the
M/mnM's
A
above may be normed, and the inverse limit topology on
M
makes
M a Frechet space.
M/mnM
is finite dimensional, it is a nuclear topological
Furthermore, since each
A
vector space.
Hence,
M
is also a nuclear space
([21], p. 103).
Let's look at some examples, first for local rings
of functions at non-singular points of a complex manifold.
If we take
R
to be
OCn,
or simply
00, then
A
00 = CI[z1,".
in
(Zl,...,Zn)
n ]],
,
the algebra of formal power series
where
(l,..,,z
n)
are holomorphic
16.
0
coordinates centered at
(i.e.,
zi(
0
) = 0,
i = 1,...,n).
A
The Krull topology on
A
00
says that
z
if and only if the coefficient of
f
- f
is zero, for all
j
fj
f
= Z
1 ... Z n
n
in
0O
sufficiently large.
in
The
A
Frechet-nuclear topology on
00
is the topology of
simple convergence of coefficients, i.e.,
za
and only if the coefficient of
to
0
in
C
j
as
we get
f
If
R
-
- f
is
f
if
tends
A
or
A
A
E n
C,o
gets large.
in
fj
A
= C[[Zl,...,Zn, z1,...,zn]]
= E
The
same relation as above holds between the Krull and
A
A
Frechet-nuclear topologies on
A
and
Eo ,
and the
isomorphisms above are topological isomorphisms (from
Krull topology to Krull topology, etc., of course).
Considering modules of germs differential forms, one
gets, for example,3
like
fdz
is in
...
C[[zl,...,Zn]]
dz
sn
Cno
,
is composed of sums of terms
1 < i1
,
< .
< n,
where
f
and the topology is the Krull or
Frechet nuclear topology on the
series.
< i
(n)
possible coefficient
17.
Returning to the general situation, given a module
A
M
as before,
there is
always a map from
M -+ M,
which
will be continuous, by definition of the Krull or Frechet
nuclear topology on
or
M.
If
R
is noetherian, e.g.,
A
00, then this natural map is injective, by Krull's
theorem.
The case of
E0o
The natural map,
E
is quite different, however.
Eo
is given by sending a germ of
differentiable function at the origin into its complete
Taylor series expansion.
By the classical theorem of
E. Borel, this map is surjective and its kernel is just
M
m
k=1l
k
where
,
m
is the maximal ideal of
E .
Our first proposition shows that this surjectivity
property is true for all finitely generated
Proposition 1.1:
functions at
in
Eo = germs of differentiable
Let
Rn, let
m
M be a finitely generated
Let
sequence
m
0
=
0
m
k=l
m
m,
*
M-
M
-
E0 modules.
M -+ 0
be its maximal ideal.
Eo
is
module.
The
exact where
and all the arrows are the natural ones.
18.
K
O
module,
Eo
finitely generated
sequence,
n
k=l
We first show that
k=l
mkM)
= ~
)
is the
mk*M = m.*M.
p-l(mk.M)
=
n
r.
Consider
(K + mk.E(r)).
This
k=l
K + m .E(r)
0
which may be seen as
The theorem of E. Borel says that
follows:
o
o
g
given by
theorem^k
theorem,
K'
K'
Consider the submodule
(r) = E(r)/mE(r)
E^(r)
an exact
E (r
where
is a
M
Eo, for some finite
k=l
last term is just
o
there is
0
M
Er)
direct sum of r-copies of
p-1((
Since
(mainly from [20], p. 73)
Proof:
= (K + m *E 0(r)/mE
(r)
(r).
= K', where
) (K' + mk*E r)
of
By Krull's
m = maximal ideal
k=1
of
E^ .
00
Taking inverse images in
(r)
K + m *E 0
(r)
=
k0
(r)
(K + mk .E(r)).
k=l1
it follows that
n
mkM = m .M.
(r) , one gets
Er
Taking
p
of both sides,
Thus, the kernel of the
k=1
natural map
M -+ M
is just
m M,
and
injection whose image is a dense subset.
M/m M
+
M
is an
19.
The following sequence is exact:
E)(r
1 tK'
M/m M
O
-
0,
three terms are
and all
A
finitely generated
Eo
Taking Krull completions
modules.
A
as
E
modules, which preserves exactness ([11, §3, no. 5),
and since
K'
and
are already complete
Eo
M/mmM
§3, no. 1, cor. 1), we see that
([2],
is
A
complete in the Krull topology as
But
module.
E
M/mNM
this is the same as the Krull topology on
as
Eo module, since
^
^k
(Eo/m)
@
k
E
(M/m M) = (E /M ) 0E
(M/m M)
o0
E
= M/mkM.
A
Therefore, the natural inclusion
isomorphism, since
M/m M
+
M
is an
M/mNM is already complete.
Before passing on, one should note that the
proposition applies to the modules
EX
as in
x
1A
above.
Given two local rings
ideals
m
and
R
and
m , an extension
R
with maximal
: R
-
R
of local
20.
a ring homomorphism
rings is
¢
Such a
¢(m)
m
A*
A
¢ : R
induces a map
such that
4
A
+
R ,
continuous in the
Krull topology or the Frechet nuclear topology, if
R
R
and
are
R
Similarly, if
are C-algebras.
R
and
r e R, m e M,
: M
M
continuous map
,
then
-
and
x : M
modules respectively, and
is a 4-homomorphism of modules, i.e.
for
M
M
M
4(rm) = ¢(r)*.(m),
induces a natural,
M .
w : Y
For example, if
+
X
is a differentiable map of analytic spaces, then
: EX,w(y
7
s
*'s
E
)
are all
is a local homomorphism and the
R -homomorphisms.
The following lemma is useful on several occasions.
Lemma 1.2.
4 : E
Eo
-
F
F
be Frechet spaces, and let
be a dense subspace of
Proof:
open map.
F
E,
be a surjective, continuous linear map.
is injective,
¢,E 0
Let
4
E.
If
4
restricted to
Eo
is a topological isomorphism.
By the open mapping theorem,
Hence,
Let
giving
E
4
is an
the induced topology from E,
is a homeomorphism onto its image
being given the subspace topology in
4(E
o )
F.
= Fo ,
Thus,
21.
: F
there is a continuous, linear map
*.E 0 = Id
that
on
The map
E o.
continuity to the closure of
since
on
E.
is dense in
E
Fo,
i
-*
such
E
extends by
which is all of
Hence,
0
F,
is injective
E.
One application is to the following theorem.
Theorem 1.3:
variety, and
X
Let
a point of
x
be a complex analytic
X.
Then the natural map
^s
A
X,x
+ E
X,x
Proof:
x
is
an isomorphism.
The map is an isomorphism in the case of
a regular point of
X.
For a general point
since the problem is local about x, consider
closed analytic set in a polydisc
to the origin
0
of
Cn.
X,o
0 -+ As
ES
0 +AX,
EX,
Ao
X,o
6,o
X,o
,
X
x
as a
corresponding
We have the following exact
commutative diagram (see §lA):
Ns
A C Cn
x s X,
22.
Taking completions,
and using the proof of the previous
proposition, we get another diagram (exact, commutative):
s
N
+
X,o
0
S
AmE
,o
+·
A
A
SEs
AA,o
A,o
A
Es
X,o
X,o
4-
4,
0
0
A
A
Hence,
AX 0
SEs
is
X,o
X,o
A
AS
The map
surjective.
AX, °
4
X,o
s
X,o
is clearly continuous in the
Frechet nuclear topology, and
AX,
is a dense subspace
A
of
A3
X,O'
if
A 9
S
By the previous lemma, the theorem is proved
E
S
is
injective,
ie,
S
if
A
00 S
m *E
X,o
S{01
in
EXo*
Take a germ
Let
X =
i=1
Xi
wa
As
,2'
defined on an open set
be a decomposition of
X
locally at
U.
0
23.
n : X -+ X
Let
into irredundant irreducible components.
be a local resolution of singularities as in the result
of Hironaka ([131).
of
in
0
Xi
We may take open neighborhoods
Ui
([121):
so that
(1) U } Xi " Ui
(2)
Let
Ui - U i C S(X)
= closure in
X
note that each
Xi
is a proper map,
of
0 e X.
is a connected complex manifold.
-1
n- (Ui)
of
- (U i - U i n
contains a neighborhood
w e A
Assume that
m *E
X,o
x i C 7-
(0)n Xi,
s ~ (
Xsn *w E A.
X,x i
i = l,...,r.
s
mCO
*E.
X,x
.
since
Hence,
n
i.
in
and let
But this says that the
i
Hence,
w E 0
is an isomorphism from
w = 0
V
Then,
X n w
By uniqueness of analytic continuation,
for each
n
X,O'
Taylor series of the analytic form
Xi,
Since
is a connected manifold.
n( Xi)
S(X));
is
Xs
0
w
at
0
xi.
on
on
V -
X -
n- (S(X)) -+ X - S(X).
AX
Xo' by
by definition
S(X)tf
V,
24.
It would, of course, be pleasing to construct a
proof which doesn't use the resolution of singularities
at such a basic level.
i.e.,
For the case of functions,
s = 0, a more elementary proof may be
for
constructed from theorem 3.4 of Chapter VI of [20].
This
result of Malgrange is still very non-trivial, and it
is not clear how one might extend Malgrange's method to
the case of forms on a variety.
The last proposition in the section relates the
Frechet nuclear topology introduced above with the
natural maps
X sr
E
As
Ey
where
: EX
X,x
Y,y'
X,x
-+ E
Y,y
fr(y)
induced from the maps
= X.
one must consider several points in
Xs
and
(or Xs
k
EX
$ E 5YYi
Xx
i=l
)
For later purposes,
w-1(X)
will denote the map from
(or
Esx
->
k A
$ E
i=lYi
y
whoeth
whose
it h
(or
s
"(yi
)
respectively)
E
factor is the map
EXx
simultaneously,
+ E
= x, for each
Y,
Xsr
: EXx
respectively),
i = 1,...,k.
E
where
25.
Proposition 1.4:
map, where
space.
For
Let
w : Y -+ X
x e X
}
{yl,..,Yk
and
sw
image of the natural map
S
is an analytic
X
is a manifold, and
Y
be a differentiable
C 7-1(x),
S
the
A
: EX
X,x
E
has
i=l
closed image in the Frechet nuclear topology.
Proof:
(after [71, prop. VIII)
The map
A
7s
is continuous in the Frechet nuclear topologies.
By a
5s 7T
basic theorem of Dieudonne and Schwartz [ 61,
has closed range if and only if the dual map has weakly
(EX,x)'.
closed range in
But, if considering
a closed subvariety of a polydisc
S
by proposition 1.1 above,
A Ci CN,
EX
2N
( )
Now
A
^
Xo
)
A,o
(EAo)',
copies of
i : (EsS
(E
and
x = 0,
is surjective
A
closed subspace.
as
AS
E.
and induces an inclusion
and
X
)'
(Es
,o
)'
as a
is the direct sum of
^
(EA)'
has the same
algebraic and topological duals, namely
EA
since
A
(EA,o)'
is just finite linear combinations of the Dirac
function at
0
subspace of
(EeA,o
and its derivatives.
)'
is closed, hence
Thus, any linear
26.
k
i :
As
(EE
A s
(ED)'
i=lY,1
:
(s)'
Intersecting with
(Es
has closed image.
But
)'
has closed range..
,o
it follows that
)'
AX,
(Xs I),
is Frechet nuclear, and
Eo
([21], p. 147), implying that
therefore reflexive
This
"weakly closed" is the same as "closed".
concludes the proof.
In this section we simply note that several
C.
theorems of
stated locally for r-tuples of
[-0]
globally for manifolds and vector
functions are valid
The theorems of [20] we are interested in
bundles.
are Whitney's extension theorem (Chap. I, Theorem 4.1)
and Lojasiewicz's theorem on regular situation of two
closed analytic sets (Chap. IV, cor. 4.4).
First of all, given a section
f
of an
A
EX module
M
on
X,
denote by
by the natural map
induced by
f
in
Mx .
Mx
-
Mx of the germ
Now if
X
(E)
fx
This is to be considered
"taking the Taylor series expansion" of
let
the image in
A
A
Mx
f(x)
is a manifold, and
E
f
at
x.
a vector bundle over
X,
be the sheaf of germs of smooth sections of E.
27.
Choose local coordinates around
trivialize
E
locally.
E,
section of
f(x)
is a smooth
f
is the Taylor series expansion
of the r-tuple of functions given by
to the local basis of
and
with r-tuples of formal power
In this identification, if
series.
X
As noted in section B above,
(E)x
this identifies
in
x
E.
f
with respect
Whitney's extension theorem
gives necessary and sufficient conditions (called
A
regularity conditions) for a collection
r-tuples of formal power series, for
subset of
{vx }x
x e K
K
of
a closed
X, to arise as the collection of Taylor
A
series
{f(x)})x
on
when
X,
of a smooth r-tuple of functions f
K
X
is an open subset of
Rn
.
However,
it is easy to check that Whitney's conditions on the
A
collection
{vx) x
coordinates in
K
are still satisfied if we change
X, and change the basis for the
r-tuples differentiably:
use Whitney's theorem to
prove this invariance as follows.
{vx xc K
Interpreting
as a collection of r-tuples of formal power
series in one local trivialization, suppose these
series satisfy Whitney's regularity conditions, and
28.
A
A
hence, by Whitney's theorem, {vx x
K
for some smooth r-tuple of functions
is
{f(x)}xeK
f.
Changing
coordinates and bases gives a new r-tuple of functions,
f', but still representing the same section of
{vx }x
Hence, when one interprets the
K
E.
as formal
power series again, in the second trivialization, they
A
are already given as
{f'(x)})x K
and hence, using
Whitney's theorem in the other direction, the new
formal power series satisfy the regularity conditions.
It is a simple matter to check that with this sort of
invariance, one can use smooth partitions of unity to
solve the corresponding global problem.
Thus, we may
A
speak of a regular gield
of
E
over
X, for
K
{vx }x
K
of formal sections
closed in a manifold
X, and
A
Whitney's theorem says that
Taylor series
over
{f(x)})xK
{v XEK
is given as the
of a smooth section of
E
X.
Lojasiewicz's theorem, when seen from the same
point of view,
says the following:
Proposition 1.5 (Lojasiewicz):
Let
X
be a real
analytic manifold (countable at infinity), and let
Y
29.
Z
and
{w zz)
are Whitney regular formal sections of
Z
A
A
over
E
the vector bundle
Z,
vt = wt,
{Wz}ze
{Vy }yy
then
X, and
Y
regular formal section over
is also a Whitney
Z.
where
y e Y,
is a smooth section of
fl
w z = f2 (z),
then there is a section
of conditions, i.e.,
vt's
z E Z,
f
f(y)
A
provided the
)
E, and
A
A
similarly,
fl (y
as
vy
In other words, if we can obtain
for
for
A
A
t C Y
{v }
If
be two closed analytic subsets.
and
f2
a smooth section,
which satisfies both sets
= vy
and
f(z) = wz,
A
and
wt
's
agree on
Y
Z.
30.
§2.
Operators on Varieties Definitions
In this section several definitions of a differential
operator on a variety (all equivalent) are reviewed at
first.
Then we examine what real vector fields may be
lifted when we blow-up a complex sub-manifold of a
complex manifold.
Using the observations made, and the
construction of the resolution of singularities given
by Hironaka, it follows that any differential operator
on a singular variety may be pulled back to a meromorphic
operator (in an appropriate sense) on a resolution of
singularities.
Differential operators on varieties allow of several
possible definitions,
probably the most general being
based on Grothendieck's definitions in
([11],
IV.16.8),
which make it possible to talk of a differential
operator from one sheaf of modules to another.
consider on a complex variety
X
We can
many different sheaves
of rings and their corresponding modules, i.e., the
sheaves
OX, AX and
X
considered in section 1.
Each
of these sheaves of rings has its own associated differential
operators, analogous to holomorphic operators, and
operators with real analytic or differentiable coefficients
31.
Cn
for domains in
.
While making a succession of
definitions which are categorical in nature, we will
RX
simply speak of a sheaf of rings
(Inductive definition)
Definition 1.
-
which is C-linear, but not necessarily a
N
D
homomorphism of RX-modules.
operator from
M
is a differential
if it is locally of finite
N
to
order at each point in
given
i.e.,
X;
any sufficiently small neighborhood
that some integer
k}
{fl,*.f
MJU
-
x
denotes the commutator:
If
k
containing
of
x
[f 2 *'"* [fk,
[fl'
then
Rx(Ux),
f E RX(Ux),
Ux
x E X, and
such
k, given any k-tuple
is the O-homomorphism from
for
M, N
Let
a homomorphism of sheaves
D
be RX-modules, and
D : M
A, respectively.
O0, AA or
which is
is a
X
A, we also speak of
closed subvariety of a polydisc
RA
where
X,
If
for our interests.
X
R X = OX, AX or
on
NIU .
a
D] ]
- D(Feo),
a e M(U )).
is the minimal such integer for a
x, then
K-1
]
[ , ]
(Here
[f, DI(a) = f*D(a)
...
Ux
is called the order of
D
32.
near
x.
That
says that
D
D
x
clearly just
is an Rx-module homomorphism near
Definition 2:
above.
is of order 0 near
(Jets definition)
A C-linear sheaf map
Let
D : M - N
x.
M, N
is
be as
a differential
operator if it locally factors through the sheaf
of germs of k-jets of sections of M, for some
jk : M * Jk(M).
means of the natural map
M,
RX-modules
([111],
Jk(M)
is
Jk(M)
k, by
For
defined categorically as in
IV.16.7, where it is denoted
pX(M); Jk(M)
is
closer to analytic and differential geometric conventions).
Here
jk
is the map which sends a germ to its k-jet at
each pt.
If the postulated factoring is given by the
D
commutative diagram
M -- > N
,
then
4 is called
jk
Jk(M)
the coefficient homomorphism of
D.
We are tacitly assuming, by referring to
we may categorically construct a product
sheaf of rings
subvariety.
of
RX'
RX xX ,
Jk(M),
X x X
that
with a
for which the diagonal is a
This is clearly possible for our examples
33.
Definition 3.
Let
(Ambient neighborhood definition)
RX = OX, AX
be RX-modules, where
M, N
Assume further that
M and
generated as RX-modules,
N
i.e.,
or
X9
are locally finitely
near any point
x c X,
we may find surjective sheaf homomorphisms
:
RX + M and
r-copies
:
ED
RX + N.
s-copies
x c X
if a sufficiently small neighborhood of
M
0 + Ker (T)
is
A
embedded as a closed subvariety of a polydisc
one has sheaf exact sequences on
Thus,
CN
A:
RA
@
@
r-copies
+
0
and
0
Ker (W) +
@
RA
It
0
s-copies
M, I
Here
by
0
denote extension of
to be sheaves on all of
homomorphism
D : M ~+
D
of
M
to
N
M, N
A.
respectively
A C-linear sheaf
gives rise to
in the representation above, and
differential operator if locally at any
D
is a
x E X, there
34.
exist
¢
and
9
as above giving a commutative
diagram:
0
-
Ker (T)
RA ÷ M
+
0
O
r-copies
0
-
Ker (W)
S
÷
Rs-copies
s-copies
where
D
is given by a system of
operators on
A, such that
An O -differential operator
D =
Z
a (z)
lal<k a
--
,
RA-differential
D : Ker (W)
D
where
+
Ker ( ).
is one of the form
z = (zl,...,zN)
coordinates on
A, a = (al,...,aN)
and the
are holomorphic functions on
aa(z)
are
a
An A -differential operator
is a multi-index,
D
is given by
,
where the
A.
~a+B
D =
b
I
+B <k
(zs)
azwhe
3a,B
b
's
a,
are C-valued, real analytic functions; for
A-differential operators, the b ,8's may be C.
35.
The equivalence of these three definitions of
differential operators when all three apply, is
essentially given in [111, IV.16.8.8 (cf. also [31).
The final definition we give seems more limited in
scope, but is more suited for our philosophy of
analyzing differential operator problems on resolutions
of singularities.
Let
X
be a complex analytic variety, and let
~7T
X -) X
be a resolution of singularities as constructed
by Hironaka ([13],
[14]).
Y = n-1(S(X)).
Let
define a "real-semimeromorphic function"
poles on
Y
to be a C
function on
small enough neighborhood
on
U
(X - Y)
as
U
of
f
First,
on
X
with
X - Y, which for a
y e Y, may be written
g s Cw(U)
g/4, where
¢
and
is
a real analytic function with zeroes only contained in
This definition is local, and
S~
Y.
will denote the sheaf
X,Y
of rings of terms of such functions on
X.
One may also
speak of real-semimeromorphic forms with poles on
replacing
g
Y,
in the definition above by a smooth form w;
denote the sheaf of i-forms of such a type by
i
S&
X,Y
A differential operator
D : E.,
X
S,
X,Y
(S~
= s
X,Y
is a C-linear sheaf
X,Y
.
36.
homomorphism which is locally representable in a real
analytic coordinate neighborhood
f
E
jaj<k
and where
b
-
f ,
for
as
f E E.(U),
b
X
a ax
x = (Xl,...,X2n)
coordinate functions on
operator
U
D : Ei * Sj
X,Y
X
U.
(U),
S..
a
X,Y
are the real analytic
Similarly, a differential
is a C-linear sheaf homomorphism
which is locally given by a matrix of differential
operators from
E,
S.
-
X
free over
E
X
all
and
.
X,Y
Note that
Ei
is locally
X
S.1
is locally free over
X,Y
S.
, for
X,Y
i.)
(Resolution definition)
Definition 4.
sheaf homomorphism
A C-linear
- E
is a differential
operator
aif
differential
there
exists
operator
D : EX
operator if there exists a differential operator
D : Ei
X
Si
XIY
such that the following diagram commutes,
for all open sets
U
37.
D
-- > E (U)
EX(U)
E(n-l(U))---->
X
Thus,
for
Sj
(n-I(u)).
X,Y
* ( w)) ) =
w c Ei(U), ,D(xi
this case, we say that
J7*(D(w)).
lifts to
D
D
on
In
X, and that
Since
7
is
almost everywhere, it follows that
D
is unique for a
D
blows-down to
given
D
on
D
on
X.
an isomorphism
X.
This last definition of differential operator is
also categorical in nature:
an analogous construction
may be performed on real analytic forms, and on
holomorphic forms as well, though for holomorphic forms,
you allow the operators
D
to have meromorphic
coefficients, and not real-semimeromorphic
coefficients.
We wish to show that the last definition is
equivalent with the preceding ones, when the definitions
overlap.
Once this is done, we will have shown that
problems concerning differential operators on a variety,
at least on its sheaves of forms, may be lifted to
38.
"meromorphic" problems on the manifold
X, where
more or less conventional analysis methods may be
applied.
It remains, of course, even in these
preliminary considerations, to show that relatively
simple criteria exist for "blowing-down" the results
of analysis on the manifold
X:
this problem is taken
up in the next section.
Proposition 2.1:
equivalent.
All four definitions above are
In particular, any differential operator
on smooth forms (in any of the senses given by the
first three definitions) lifts to a real-semimeromorphic
operator on smooth forms on a resolution of singularities
X
of
X.
Proof:
We already know that the first three
definitions are equivalent.
that a differential operator
Thus, it suffices to show
D
in the sense of
definition 4, is also one in the sense of definition 1;
and conversely, a differential operator in the sense
of definition 3 is also one in the sense of definition 4.
First consider a C-linear sheaf homomorphism
D
D:Ei
X
EJ
X
which fits
into a commutative
diagram:
39.
-D> E (U)
Ex(U)
Ji r*f
+X 7T
(*)
Ei(X
-D > SY
(U))
X,Y
for every open set
U
X, for some fixed res olution
as earlier, and
r : X -I X
operator on
X
( -l(U
D
with poles on
relatively compact,
U
i.e.,
w
a real-semimer omorphic
Now, if
Y.
U
is
is compact in
X, then
is a proper map .
On
r-1(U)
is compact, since
7w-l~ ),
the system of differential operators represented
by
have, therefore, a finite order,
D
say
< k.
But,
one checks easily by an inductive calculation in local
coordinates that for such a
W
f
-
f*'D
- D(fw),
E (w-(17(U)),
X
< k - 1.
functions,
any
(i.e.,
D,
the operator
[f, D]), for a fix ed
w E Ei(-1 (U)),
Hence, given any
X
is of order
(k+l)-tuple of smooth
{fl,..,' k+l}, the operator
[fl',2',.,[fk+1,D]...]
is of order
< - 1,
i.e., it
40.
}
{gl, ""gk+l
.
let
EX(U).
I
X 7r ([gl,[g 2,.*
in
S5
Egl,[g 2 '*""
(-1 (U)),
EX(U), and
in
be a (k+l)-tuple of functions from
From diagram (*) it
=
w
Consider any
is identically zero.
is
easy to see that
[gk+l,D]... ]()w))
[gk+lD]". .1(Xin *w) = 0
where
=
for
gi,
= 1 ,
. .
,k+l.
X,Y
Since
XJ
we see that
EJ(U).
is
injective from E (U)
[gl'
Since
w
g2***'
'"] (m)
[gk+l'D]
was arbitrary,
Sj
(-1(U)),
X,Y
to
= 0
in
the conditions of
definition 1 above are verified.
D : E i -+ E
EX
Conversely, assume
x
is a differential
x
operator in the sense of definition 3 above (in
of an ambient space).
We want to show that, given a
resolution of singularities
Hironaka ([13]
terms
and [14]),
real semi-meromorphic on
i : X
+
X
in the sense of
then we can "lift"
X with poles in
D
Y =
to a
D
-1(S(X)).
41.
D
Recall "lift" means that
for every open set
In order to do this, we
X.
U
fits into a diagram (*)
examine a simple situation for lifting which will,
in fact, suffice for our purposes when we've examined
the process Hironaka uses to resolve singularities.
Consider the process of blowing up a point (say
the origin) in
(zl,...
In
Cn.
Cn x pn-1
n) coordinates on
Cn,
with
Pn-,
homogeneous coordinates on
(wl,...,wn)
and
we take the
subvariety defined by the equations:
for all
Cn
i,
j
between
Cn x pn-l,
projection onto
1
.
wiz j = ziwj,
Call this locus
be induced by
w : Cn + Cn
and let
Cn
n.
and
the
It is a basic fact that the
construction above is independent of the choice of
coordinates in
automorphism
Cn
,
and thus given a holomorphic
¢ : Cn
.
Cn ,
holomorphic automorphism
we may cover it by a
~n
~n
C .
4 : C
0 E Cn , and we
that the construction is local about
may blow-up
0
in
U
and get
any open neighborhood of
in
C .
Now if
fis
0
in
U
U,
-
Cn
Note, also,
,
and
where
U =
U
is
-1 (U)
a holomorphic vector field
42.
defined on a neighborhood
U
of
0 E Cn,
for
U
relatively compact and small enough, there exists a
: U + Cn
one parameter family of maps
C, i.e.,
defines
and
f E C (U)
If
C(0) = 0,
0
is defined.
C(f)(p) =
5
(here
then
which
f((p))j(, 0 , for p E U,
lies in an E-disc about 0 in C).
(0) =
= 0
for all
ý
for which
Putting the two observations above
together, we see that there is a family of local
5
automorphisms
(
Let
: U - Cn
covering the maps
45.
be the holomorphic vector field which they
define:
(f)(p) =
Since the
's
i
f(E (p))I= 0,
cover the
O&'s, we have
5
Thinking of the vector fields
operators, note that
((w f) = W (C(f)),
df*(E) = ý
of the differential operator
5
(
and
dw*,()
= 5.
as differential
says that
f E C (U).
for
, f EC (U).
for
to
Thus
(
is a'lift"
U.
Next, note that by complex conjugation, one may
lift any anti-holomorphic vector field on
vanishes at
0.
D
on
which
Note further, that if one can lift a
differential operator
a
U
D
acting on functions on
U
U, then one can clearly lift the operator
to
f-D,
43.
It follows, therefore, that any smooth
f s C'(U).
where
vector field
5
on
U
vector field
5
on
U, with poles only along
In fact,
lifts to a real-semimeromorphic
= E ai (z,z)
write
i
=
a(
azj
z
++
i,j
Since
i
-
z
bi
's
are in
on
U.
Set
i-
only along
z zjezj.,
zi
lift, and the
C0(U), the operator
1
= Tr (¢);
i
(0),
('/i
and
meromorphic lifting of
write
b
biz
*
E
i,j
EJ
j
and
i
+ E bi(Z)
i
i
=
If
ai, b i E C (U).
where
(
- (0):
ýC
aiz 's
and
lifts to a
E'
is real analytic and vanishes
is
to
the required semi-
U.
More generally, we may blow-up along a linear
subspace
Ck x 0
just
Ck x Cn
.
Introduce coordinates
and get, for the blow-up along
Ck,,for
(yj*.*k)
Ck
in
Ck
Cn C Ck+n x pnl,
given, as before, by
ziwj = wizj, and no conditions on the
yj's.
Again,
this is invariant with respect to coordinate changes
which take
Ck x 0
into itself.
earlier shows that
for
j =
,...,k.
and
Yj
Finally, if
5
to a
Ck.
on
in
on
we may lift
C k x C , for any vector field
E's defined only on open neighborhoods of
Ck+n
< k
of order
Ck x Cn
order
O*D
=
n
E
C zi
i=l
D =
acting on functions,
U
of
0
in
to a real-semimeromorphic operator
D
Ck x Cn.
in
< 1.
< 1,
D
on the opne neighborhood
lifts
n -(U)
write
is a holomorphic
Ck x {0},
Any differential operator
on
f
The process is again a local one, and
applies to
0
E'
lift to the blow up,
aYj
function which vanishes along
(f*T)*E
The same argument as
This is so for operators of
In fact, for an operator
D
of order
lifts to an operator without poles, where
.
In general,
N
E
Dk+Do,
•
if
D
where
is
of order
< k,
E2,'s are vector fields
R=l
the
D 's
of order
(k-
are operators of order
0.
#k*D = Z 0k5 .D
2
k-1
.kD ()-D*D
+
o.
< k -
+).(kk-1
+kDo
= E(D
By induction,
1,
and
is
Do
D
)
(RE) and
-
45.
k-I
( k-1 D)
¢kD,
lift,
for every
k,
k
as does
kD , and hence,
as a sum of products of liftable terms,
liftable, to a smooth operator
D'.
is
also
Again,
~
D'/
k = D
is
a real-semimeromorphic
lift
of
D.
Now consider a differential operator (in the sense
of definition 3),
In
order to lift
D : EX + E X
D
to a
for
D : E,
X
a complex variety.
S..
X
we may do so
X,Y
locally, and patch them together (since the lift to
1(U)
I(U)
C
C X
X of
of a
a
D
D
mentioned earlier).
defined
defined on
on
U
U
X
is
unique,
Let us consider a
U
open in
as
X
such that:
(i)
U
may be considered a closed subvariety of a
polydisc
(ii)
in
CN
A, the closed subvariety given by
S(X) C U
functions
(iii)
A
is defined by finitely many holomorphic
fl'",
fr"
we have the following commutative diagram
46.
T l(U)
u
U(S)
÷ A(s) =-
(1)
U(l)
IT
U
-
i
where each
i = 1)
for
a(i)
is the blowing-up of
S(i)
.
(or
Here
A( i - l)
(or A,
along a closed submanifold contained in
S(U (i - 1 ) ) , and where
U(i -l)
A
U, for
7
(i)
U (i
)
i = 1)
is the proper transform of
under the transformation
is the map induced by
a(i)
.
This
diagram results from the method used by Hironaka to
resolve the singularities of
(iv)
there is a differential operator
which induces the operator
order
< k
on
D : EU
-
EU,
L : EA
and
L
-
s'k.*L
r
= E f.I..
i
i=l
We will
lifts to a smooth operator
L'
EA
is of
A.
Given all of this, let
prove that
X, [13].
on
47.
A, where
s
is the number of blow-ups used to resolve U.
In fact, we have already shown this if
f.
each
1
Thus
on
Set
A(1)
o(1)*(f
k *L
=
()
lifts to a smooth operator
()()
(since it is contained in
S(U(1))
o(1)-(S(U))).
on
s,
we see that
L'
on
A.
S(U(1)),
(s-l)k*L(1)
function which vanishes along
U:
A - a 1 (S(U))
lifts to the desired
g E C (A)
U, then
a - S(U),
defines an operator on
A
type.
vanishing along
Thus
L'
a
U,
also
is an isomorphism
1
a•(S(U))
and that
is
*Sk*L
X
and thus takes a function
U
into a function of the same
defines an operator
lifts the operator
is any
L'(g)
U - U
an everywhere dense open subset of
on
therefore, to
this is because
to
A(2)
and proceeding by induction
It is clear that if
vanishes along
and vanishes
and
ýsk.L lifts,
(s-1) kL(l),
an operator
A(1),
blown up to get
A(l)
L(l)
= Z(1)*(f i)((1)*() i)
is holomorphic on
i )
on the submanifold of
of
since
vanishes along the submanifold which is
blown-up.
Each
s = 1,
sk D,
and
D'
D'/ýsk
on
is
U, which
a
4 8.
real-semimeromorphic operator which lifts the operator D,
= n (4).
W
where
Since
singular locus of U,
7- l (s(U))
D
¢
vanishes only along the
has poles only in
= Y CU.
The proof of the proposition for forms of higher
degree is similar, but messier.
We will not use this
part of the proposition later, so we will omit its proof
here.
490
§3.
Blowing Down Forms
In this section, a few basic results are proved on
"blowing-down" smooth functions and forms from a
resolution of singularities to the resolved variety.
Although the problem is a natural one, there is very
little in the literature about it, except for a theorem
of G. Glaeser [71, which will be our starting point:
Theorem:
Let
U, V
respectively, and let
Rn , Rk
be open sets in
4 : U -+ V
be a real analytic
map such that:
(1)
(2)
is a closed subset of
W(U)
if
K
is a compact subset of
then there is a compact set
that
(3)
C(U),
K' C U
such
ý(K') D K.
de, is surjective on an open dense subset
of
If
V
U.
f e E(U)
respect to
(In particular,
n > k).
satisfies the "2-point condition" with
4, then there is a
g s E(V)
such that
0*(g) = f.
A function satisfies the "n-point condition" with
respect to
0
if, for any
x E O(U), and
xl , . . . , x n C U
50.
¢(x
with
) = ...
1
gx c Ex
= 4(x ) = x,
such that
(gCx)
=
there exists a
f
s Ex
i
i = 1,...,n.
,
i
The "finite point condition" for a function
a map
¢
¢,
for
means that
for every
f
n.
f
and
satisfies the n-point condition
We can also speak of the n-point
condition for i-forms as well, replacing rings of formal
power series with modules of formal i-forms, and
i^A
with
c
.
Xi
We shall first make two very simple extensions
of Glaeser's result.
The first generalizes this result
to include forms.
Proposition 3.1:
w e ES(U)
let
on
U
Rn
1
< ...
T c ES(V)
y',...,yn
Let
< i
-
V
be as above, and
such that
XSC*(T)
P
the same for
s
runs over all such
= W.
RP .
V
is a multi-index of degree
< p
ý.
be the standard coordinates
x' ... ,
and
I = (il,...,i s )
1<
$ : U
satisfy the 2-point condition for
Then there exists
Proof:
Let
set
dx
i
= dx 1 ...
S-multi-indices
the
s,
i
dx s
dx
's
If
with
If
I
form a
51.
basis for
on
U,
E s ( V)
when
over
dy 's
Similarly for
E(V).
J = (Jl,...,Js),
1 < jl
With respect to these bases, write
< .*.
< js < n.
xs( *)
as a matrix,
,
X (* )dxI = Ea dy
J
where the
I's
a 's
The
sequences.
are restricted to increasing
J's
and
are (real-valued) real analytic
A
functions on
U.
Thus, a formal s-form
T Z= I
written
gI
,
dx
EX ,
x
A^^
may be
and
AT
4 (
=
Xs
A
T s Es
I)aj dy
in
E , 0(y) = x.
I,J
w e ES(U),
Given
2-point condition for
in particular, that
for every
E fjdy
=
y e U.
,
w = E fjdy
write
w
f
a E(U).
with respect to
"y =
(T
XS
),
The
implies,
(y)
E
y)
Expanding this, we see
(gI)a JI dy , for some
gi's
^*
^
J
Equating coefficients of
E E(y).
dy 's, we see
f
= E
^
I
in
E,
Let
for each
N = (
y e U.
), S M = (n),
s
so that the
)A
(gI )
a 's define
52.
A :
a linear map
$
E(U)
-1
N copies
matrix multiplication.
is
if and only if, for every point
l , . . . , a M)
L
A
gI
E4 (y)
Th. 1).
^*
^
Ey,
i =
,...,M
Since every power series
gI
on
0(y)
of a
V, the equations
^I
fj = E
I
(gi)aj
(fl,*..f
M)
say that the M-tuple of functions
satisfy the conditions of Whitney's
(fl',*,f
and hence
in the image of A
in
is the power series at
differentiable function
^
by
y E U, there exists
ai = bi
such that
Exp. 25,
([18],
A
E(U)
By a theorem of Malgrange, an
(bl,...,bM)
M-tuple of functions
(a
$
M copies
M) E L.
Thus,
=
f
h a ,
theorem,
with
I
h I e E(U).
y E U, one gets:
Passing to formal power series at
^
^
fj =
hla j =
^I
^*
^
Hence,
that
^*
1(hI
I
de,
-
^I
(g )a
^
(g
^A
, with
^AI
))a
g
A
E
0(y).
A
= 0
in
E .
The assumption
y
is surjective on a dense subset of
dually, that
XS(
same dense set.
*
)
U
means,
is injective on the C-image of that
Thus, the matrix
(a )
has to have a
53.
N x N minor when evaluated at points
non-zero
arbitrarily close to any given
arbitrarily close to
Hence, there
minor which is non-zero at points
N x N
is at least one
y s U.
d, is a
This minor, call it
y.
real analytic function which is non-zero arbitrarily
A
A
d c E
y, and hence
close to
is non-zero (uniqueness
of power series expansion), and is the corresponding
(a )
minor of the matrix
A
However,
A
h
C(h
I
-
A*
A
)A
) )a
¢ (g
with entries from
= 0,
implies
^
A*
4 (g l )) = 0, for every
d(hI -
E
I, and hence
A
A
E
(gi) = 0, since
hI -
is an integral domain.
Now the 2-point condition for
x = 0(y)
= ¢(y'),
A
A
y (gI)
A
one set of
says that if
then we may simultaneously solve the
A*
A
hI -
equations
w
gl's
= 0
and
hI
A
-
Ex .
Thus, each
Glaeser's theorem quoted above,
where the
I,
T = E gdx
I
, gives
with
A
in
the 2-point condition with respect to
each
4y ,(g)
g 's
w =
are in
hi
satisfies
4, so that by
h ! = ¢ (g!),
E(V).
for
Setting
4, (T).
I
Q.E.D.
54.
- theorem globalizes
The next extension of Glaeser's
the domain and range of the map in question.
Proposition 3.2:
manifolds,
X, Y
Let
countable at infinity.
be real analytic
c : X -+ Y
Let
be a
real analytic map such that
(1)
O(X)
(2)
for every compact
is closed in
compact
(3)
such that
0, then
w =
* (T),
O(K') D K.
where
T e ES(y).
It is easy to see, using a partition of
unity, that the question is local on
Y
Y, i.e., we
RP . By a theorem
is an open set in
of Grauert ([8]),
we may also assume
X
properly
imbedded as a real analytic submanifold of
Let
of
U CRk
a
satisfies the 2-point condition with
Proof:
may assume
there is
X.
w s ES(X)
respect to
K C O(X),
is surjective on an open dense subset
do4
of
If
K' CI X
Y
R ,$
large.
be a real analytic tubular neighborhood
X, with real analytic projection
we consider the composed map
i
=
n : U
+
X.
If
o n : U
+
Y,
then
P, U, Y satisfy the conditions of Glaeser's theorem.
55.
By functoriality, the s-form
s *((w)
2-point condition for the map
J. By the previous
Xs
proposition,
(W) = XSi
T e ES(Y).
some
Since
XAs
w =
injective,
(T)
Xs
X sr
=
: Es(X)
satisfies the
XNs* (T),
-
for
is
Es(U)
(T).
The next extension of these methods is to the case
w : X
of a map
-
which is
X
a resolution of singularities.
It would be reasonable to conjecture that a smooth form
SES(X)
T
was the pull-back of a smooth form
E ES(X), if
w
satisfied some simple formal condition
D,
on its Taylor series at points of
e.g., if it
We show here
satisfied the "finite-point condition".
that this is, in fact, so if all possible Taylor series
obstructions to blowing-down vanish.
Theorem 3.3:
singularities
in
as above,
ES, for every
Proof:
assume
X
Let
w : X
+
and let
y e D, then
X
be a resolution of
w e ES(X).
w =
The question is local on
sw(T),
X,
is a sub-variety of a polydise
Assume first that
X
If
w = 0
T
E Es(X).
i.e., we may
A C CN
is a resolution of singularities
of the type constructed by Hironaka ([131).
That is,
56.
X
is constructed by a succession of blowing-ups as
follows:
A(n)
(n
X = X
=
A
. (n)
7(n)
C1
X(1) (
(l)
+7T )
x = x ( o)
Each
a(i)
(0)
A(i1)
: A()
closed submanifold of
and
X( i
)
respect to
is
the blowing-up of a
A(i -l ), contained in
is the proper transform of
a(i) a
neighborhood of
Let
X
U c A
imbedded in
function on
in
a neighborhood of
A, supported in
smooth s-form on
A
E,S
for every
X.
Then
X(i-1)
A, and let
X.
p : U-
Let
p
X
be a
U, and identically 1
u =
A, and by construction,
y e D.
with
be a smooth tubular
be the retraction of the tube onto
C
S(X(i-1),
(WSp is a
(t)
u = 0
y
in
Now, throw away the form
u
A
and keep only the collection of formal forms
uy,
y c X.
57.
Since it is the field of Taylor series of a smooth form,
it
is
a regular Whitney field of formal forms,
we'll simply call
Set
of
A,
and
u.
-1 (S(X));
Y =
Y
X = D.
Y
which
is an analytic subvariety
Consider the regular Whitney
field of formal S-forms on
Y
which is identically
A
zero.
Since
u = 0
fields, call it
v,
D, the union of these two
along
is well-defined.
theorem of Lojasiewicz (cf. [20],
v
is a regular Whitney field on
a smooth form
v
and
Y
or prop. 1.5 above)
By Whitney's extension theorem ([20],
there is
X
A, the regular-situation
are analytic subvarieties of
implies that
Since
on
field of Taylor series along
A
Y
Y U X.
or §lc above),
such that
X.
v
is its
In particular,
A
v(y) = 0,
for every
y E Y.
Thus
2-point condition for the map
v
a : A
satisfies the
*
A,
which is
proper, holomorphic and biholomorphic on an open dense
subset.
Hence,
v = Xs *(a),
diagram:
proposition 3.2 applies,
for some
a~
ES(A).
and
Consider the
58.
Then
X --- >
A
X ->
A
XSW*(T)
s
.s
(a)
i *
(a) =
si*(v) = m,
concluding the proof.
Now if
w : X -+ X
is a resolution of singularities
not of the type constructed by Hironaka, construct the
following situation:
Z r-> Z
> X'
+p
+7'
X
Here
Z
X'
T-> X
is a resolution of
is the fibre product
of Hironaka's type,
X xX X',
global Hironaka resolution of
[14].
X
Z,
and
Z -r->
is
a
as constructed in
The maps in the diagram are all proper, holomorphic
and even biholomorphic except at points over
w
Z
as in the hypotheses of the theorem,
has vanishing Taylor series along
S(X).
Xs(p-r)
Given
(W)
(w-.pr)-1(S(X)),
and
59.
p' n : Z
-
is an isomorphism outside this nowhere
X'
dense subvariety.
E ES(X'),
c
a unique
S 5(p.r)*(
Thus,
) = XS(p'*r)
by proposition 3.2 above.
M(),
Now
for
i
has vanishing Taylor series along
7- 1 (S(X)) = p'*r((r p r)- 1 (S(X))):
we proved this in
the course of proving proposition 3.1.
Xs
argument there shows
0 = p'
setting
r.
(
q
) :
)
-E
In fact, the
Es
r' : X' -+ X
Since
is
is injective,
now a
Hironaka-resolution, conclude, by the first half of
the proof, that
ý = Xs(T, *(T),
through the diagram above, gives
T E Es
(
Xs W* (
) = W.
X).
Chasing
The above theorem has several consequences:
Proposition
3 .4:
Let
X
be
a complex analytic
variety with isolated singularities, and let
* : X
-
X
be a resolution of singularities.
If
wE E S(X)
satisfies the finite-point condition for w,
then
s
w =
We will first
Lemma 3.5:
and
x c X
E (X).
T
T*(T),
prove a lemma.
Let
X
be a complex analytic space,
a point such that
X
is irreducible at
x.
60,
Let
r : X
Xsi*
:
Xr
:7E
x
X
-
+
be a resolution of singularities.
S-1
E
Proof:
y c 7(
x).
The proof consists of a comparison with
the real analytic category.
If
x, there is nothing to prove.
x c S(X).
T
is injective, for every
y
Then
Since
X
X
is non-singular at
Thus, we may assume that
is irreducible at
x, we may assume
X - S(X)
is a connected complex manifold ([12]), shrinking
X
x, if necessary - this is clearly permissable,
about
since the proposition is a local one at
X =
r 1(X
- S(X))
an element
x e X
T
is a connected manifold.
s
E Ax
Let
U
is connected.
If
Xs,
(T) = 0
in
as a form on some neighborhood of
-l (U -
(U n S(X)))
-1(U),
-1(U)
T
= 0
AS
y.
U;
U - (U 0 S(X))
then
Xs*()
=
Since
is connected and dense in
is connected.
analytic continuation,
Thus,
Consider
is defined on
T
We can assume that
T.
Thus
be an open neighborhood of
such that the germ of form
we still call it
x c X.
By the uniqueness of
As T(T) = 0
as a form on all of
by definition is therefore
on all of
w-l(U).
U - U r S(X),
and
0 E AS
Now consider the commutative diagram
0,
61.
c_ A
AS Cx
r
7
Xsw1
A>
AS 2Z> E
x
x
s^ *
J
Trr
As c
Es
y
y
where the horizontal arrows are natural inclusions.
proposition 1.4,
X r (Es)
x
is closed in
given their natural Frechet topologies.
*
mapping, where
: Ex
Xsw
mapping theorem,
Es, both spaces
y
By the open
(E x
is
x)
is given the subspace topology
A
A
of
an open
½
(Es)
x
X
s
X
By
Es.
y"
As
Now
is
x
a dense subspace
of
Es
and
x'
restricted to that subspace is injective.
lemma 1.2,
this means
Asw
By
is injective on all of
S
E .
X·
Returning to the proof of the theorem, let
N
an isolated singularity of
X.
Write
X =
U
x
(i)
X(i)
i=l
where each
X(i)
irreducible at
and
x(i)f
X
is a complex analytic space,
x
)
and non-singular away from
= {x,
for
i X jB
([12])
x,
- again,
be
62.
we have to shrink
X
Let
singularity.
about
X(i) =
X
is a manifold and
Let
w e ES(X)
with respect to
for each
exists a
X
have
yi E X(i)n w-1({x})
X
^
c
(i)
7 and then
i.
AS
i.
E 1. for all
al
denotes the natural inclusion, set
If
y
s(w i))
for some
i.
S
o
X n (T)
) =
(T
1
),
= my,
-(x),
A
Xw7
such that
x
Es
Y
we
The finite-point condition
sA*
Es(X)
Xs(
(iA
is any other point in
By the previous lemma,
: w-1(x).
be chosen
Xsw Y(T)
=
yri
Ex
y E X( i)
and hence
^
A*
such that
By naturality of
y
Let
AS
e Ex
T
A
.
i P J.
for
-,
The finite-point condition says there
says there is a
E
=
i.
induced by
for each
X
n : each
A
^(
^)
- hi(w
)
i
X(
by
X,
satisfy the finite-point condition
7.
h i : X(i)
If
{x}), closure in
-l(x(i)
be the proper transforms of the
is the only
x
to ensure
x
^ ^
^(i)
hi(a) = T
we have
i.e.,
By proposition 1.1,
A
(a) = m
in
y
^
Es(X
Xs())*())
W
=
X
* (T)
=
y
in
( )).
A
for any
we may choose a
63.
A
ES(X)
TE
- Asw (T).
' =
y
:
so that
w
W' =
((y),
A
T = T
A
x
Hence,
in
W
y
x
ES(X).
= 0,
Let
for every
and, by the previous theorem,
for some
si*(a),
ES(X).
oa
Subtracting gives
W = A r ( ca +t).
Remark 1:
The finite-point condition can be
replaced by the
max (2,N)-point condition,
where
N = number of locally irreducible components of
at
X
x.
Remark 2:
In the case of a general singular X,
the finite-point condition implies that for every
there is a unique
A
A
T
ES(X)
x
ES(X),
y
for every
y c w- (x).
unique formal solution for
T
^
(T)
A
=
W
y
Thus, there is a
at every point of
We make the following conjecture:
If
w
satisfies the finite-point condition for
W =
S*
XSr
-1
A
in
such that
x e X,
s w*(T),
for some
T
E Es ( X).
X.
E S(X)
w, then
The conjecture,
therefore, amounts to showing that the formal solution
above is a differentiable solution, i.e., the problem
solved in the non-singular case by Whitney's extension
64.
theorem.
As a final extension of the technique given above,
we prove a Glaeser theroem for holomorphic maps to a
variety
X
with isolated singularities.
Theorem 3.6:
Let
f : X'
analytic spaces, where
and
X'
X
be a map of complex
has isolated singularities,
Assume
is arbitrary.
X
-
f
satisfies the following
conditions:
(1)
f(X')
(2)
for every
is closed in
compact in
K
compact
X'
(3)
X' - f-1 (S(X))
(4)
df*
w E ES(X').
T
f(K') D K.
is dense in
X'
If
w
X' -
f-1 (S(X)).
satisfies the finite point
f, then
w = As
(T),
e ES(X).
Proof:
K'
is surjective on a dense set in the
condition with respect to
some
f(X'), there is
such that
regular points of
Let
X
Construct a fiber product square
for
65.
It 1
X
X'
xX
- --
> X'
+7f'
+f
> X
X
where
is a resolution of singularities, and
i7
are the natural induced maps.
proper and surjective.
and let
Let
Z = the closure of
X x
subvariety of
. p
Z
Z
+
X'
Note also that
U = X x X X' U
in
i', f'
w'
(f.7')-
X xX X'
Z
is
1
(S(X)),
is a
and we get a diagram
q
X'
S•+f'
+1f
X+X
where
p
is a resolution of singularities, and the maps
in the square are inherited from the previous square.
The map
q
is proper and by condition (3) above, surjective.
Pull back the form
functoriality,
for the map
a
down to
f
a
w
to
y = Xs(q-p) w
on
Z,
and by
satisfies the finite point condition
of complex manifolds.
In order to blow
X, it is first necessary to check
66.
(1)-
conditions
f(Z) = f'(Z), since
First,
{(xi,xi)}
i,
Let
of proposition 3.2
(3)
be a sequence in
converge to
x C f'(Z).
Since
ki
in
f(X'),
f(xi) 's
converge to
x -
is in
{Tr(xi)} U {wI(x)}
since the
X
7(x), and
assumption (2),
there is
such that
yj c K'
such that
X.
We want
Z, and a point
w(y) £ S(X),
if necessary,
7-l(s(x)).
i.
for every
Hence,
The set
is compact, and contained in
are in
f(X')
f(X'), the
a compact set
f(y~) =
w(xi)'s
is closed.
{7(xi )} J {7r(x)} C f(K').
that
Z
{(xi, xi)},
r(x
i ) SS(X),
we assume that
is surjective.
if and only if
U
we replace the sequence
one for which each
p
in
is dense in
is in
Z
in
(y,y')
U
x
above.
K' C X'
such
Thus, there are
r(xi),
for every
Passing to a subsequence if necessary, we can assume
Y!
+'
y'
(xi, y )
(x,
y')
E
K',
since
is in
is in
U,
Z,
K'
is compact.
(Xi, yj)
and
-
Now each
(x, y')
f'((x, y')) = x.
implies
67.
Next, let
K C f(Z)
find a compact
is proper,
a
such that
K' C Z,
D
be an open subset of
f'(Z)
KE
K (0 D
w(KE) C f(X).
) =
f(K')
K C f'(K')
is dense in
will
f'(Z).
is dense in
KE
Let
K C KE ,
such that
7r(E).
= K", compact in
{xi I
such that
p
K .
and
Consider
By assumption (2), there is a compact
such that
and let
f(y
KE -
is compact.
q -1 (K')
Since
7-1 (S(X)) C X.)
still stands for
K' C X'
f(K) D K.
compact, with
-l (f(X')) - D n 7-1 (f(X'))
(Here
We will
1 (f(X') - f(X') n
-l
S(X)) =
f'(U) =
do,
K C Z
be a compact set.
xi
Let
x
be a point of
be a sequence of points in
x.
+
w(xi )'
Z.
Consider
yi s K'
Choose
for every
i.
Each
KC - KS
r
K,
D
so that
(xi , y1)
is in
K",
and, passing to a subsequence if necessary, we assume
yj + y'
in
K'.
f'((x, y')) = x,
So
(xi, YI)
Hence
+
want a point
z'
is surjective at
So, let
z
df,
and
is surjective
be a point of
arbitrarily close to
z'.
s K',
K C f'(K").
Finally, it remains to show
almost everywhere.
(x, y')
z
Z, and we
such that
df*
The following are closed, nowhere
68.
Z : f-1 (D), (q*p)-1(S(X')),
dense subvarieties of
and
p-1(S(Z)).
(f-1(D)
of the theorem.)
is nowhere dense by condition (3)
Consequently, we may assume
the complement of their union.
close to
x'
such that
is surjective at
y'.
+
F(O)
(x, y')
E X - S(X),
0
of
y'
and
df,
such that
f(O) EX - S(X).
E X x X'
X x X X'
> X
x =
If
is contained in
the definition of fibre-product, near
map
arbitrarily
is the projection of a holomorphic
fibre bundle over
then
By
By the implicit function theorem,
there is a neighborhood
flo : 0
y' E X'
x = f(y')
is in
x' = q.p(z).
Let
assumption (4) above, there is a
z
K- (x)
X xX X'.
(x, y')
e X,
By
the
is just the projection of the
pullback of the bundle
flo : 0
-
f(O)
over
-1
7(f(O)), and hence
neighborhood.
dft
is surjective on that
Such points
(x, y')
are in
U, and
may be chosen arbitrarily close to the original
z,
verifying condition (3) of proposition 3.2.
Thus, we may blow
a
down to a form
the lemma below says that
components of
X.
Thus,
f(Z)
T
T
on
X.
is a union of connected
is uniquely determined on
69.
these components, and arbitrary elsewhere on
fix
T
on those components not in
follows.
Let
T1 E ES(X)
for every
x e S(X),
for every
x' e f-1(x).
X.
To
f(Z), proceed as
be a smooth s-form such that,
S
=
Xf (,x)
A
As
x '
in
E x ,(X'),
This is possible formally, by
the finite-point condition, and the existence of
T,
locally follows from proposition 1.I , and globally by
a partition of unity.
T
Let
s-form blown down from
a
be equal to the unique
on the components in
and on the other components equal to
Sn*(T 1 ).
an easy formal calculation to check that this
blow down to
f(Z),
It is
T
will
X.
Lemma 3.7:
Let
f
: Z -+ X
be a holomorphic map
of complex manifolds such that:
(1)
f(Z)
is closed in
(2)
for every compact
K' C Z
(3)
Then
f(Z)
df*
X
K
compact with
in
f(Z),
there is a
f(K') D K,
is surjective on a dense set in
is open in
X,
i.e.,
of all connected components of
f(Z)
X
Z,
is the union
which intersect it.
70.
Remarks:
The condition
as is shown by the map
(x, y)
f(Z)
be closed is necessary,
(x, xy)
-
from
C2
to
This is just an "affine piece" of the blow-up of the
C2
origin in
The map
f
need not be an open map:
X = C , Z = C2 = C2
with the origin blown-up, and
f = the projection of
all of
f-
1
C2 , but
f
again, let
Z
onto
X.
The image of
f
is
is not an open map at any point of
(0).
If
f
is
assumed proper,
subvariety of
f(Z)
is an analytic
X, by the proper mapping theorem.
Condition (3) then says that at a point
dimension of
f(Z) = dimension of
x E f(Z), the
X, hence
f(Z) = X
Condition (2) is probably not necessary, and
nearby.
could probably be removed by use of a dimension theory
argument,
Proof:
Suppose, first of all, that at
the dimension of
f-l(x),
the map
X
f
were
1.
x e f(Z),
Then, at any point of
would be open.
be the origin of a small disc, and
For, taking
z E f-1(x)
x
the
origin of a small ball mapping into the disc by
condition (3), there is a 1-complex dimensional disc
to
C2
71.
through
z
non-trivial.
disc to
C
f
such that
restricted to this disc is
But a non-trivial holomorphic map of the
has open image.
For higher dimensions, we
cut things down to a one dimensional situation.
By condition (3),
f
such that
there is an open dense set
restricted to
are no boundary points of
and let
x E f(Z)
a small open ball
both
Let
f(U)
and
xl E B(x) 0
U
is open.
f(Z).
is closed; let
B(x)
x, it must intersect
around
f(U),
2
-x
l
x2 E B(x) 0 (X - f(Z)).
)j0 < t
{xl + t(x
)
Take a new ball
2(xl, x 2 )
2 -xl)10
K
< t < to}
which contains
centered at
and call it the origin
0
of our ball in
of the real line
set of
Z
£(x
such that
l
,
2
) o4 f(Z)
be the
xl.
x
X
0
B(x)
Then
£(xl,x 2 ) r (X - f(Z)).
in
L = complex line through
in
K = 9(xl,x
< 11.
is a limit point of
B
Taking
in non-empty open sets.
and
connected component of
x I + to(x 2 -xl
Assume the contrary,
x e X - f(TZ).
X - f(Z)
{xl + t(x
We show there
be such that
Consider the real line segment
given by
UC Z
+ to(x2-xl)2
X.
Let
which is the complexification
x 2 ) C B.
Let
K'
f(K') D K.
Let
{t
i }
be a compact
be a
72.
K A B
sequence of distinct points in
t
i
+
0 E B, and let
for each
i.
z i E K'
such that
be such that
ti
f(zi )
Passing to a subsequence, if necessary,
we may assume the
zi's
zO E K',
converge to a
and
f(z ) = 0.
Consider the closed subvariety f-1(L) of f-1(B),
-l
we have z o and z i : f
(L), for every i. Let
be representatives of the local irreducible
c1
z 0 . Since
at
f- (L)
components of
U
Yj forms an
Y1,***,Ys
j=l1
f-
in
zo
open neighborhood of
zi's; say
contains infinitely many of the
f
: Y1
-
is a map such that
L
non-constant near
of
If
f(z o ) = 0
f(Y1)
Y1 .
and
Now
is
f
contains a neighborhood
0 e L, we are done, since this would contradict the
0
fact that
is a limit of points in
By shrinking
consider
zo
n
around
Y1
zo
L
(X - f(Z)).
if ne cessary, we may
the center of local coordinates
n = dimension of
(l,.,n)
zo .
Y
(L), at least one
Z
at
(n
zo, such that the projection
( 1"**,d),
d = dimension of
Y1
at
zo,
73.
restricted to
Y1, represents
of a neighborhood
p : Y1 -
Let
B'
B'
of
Y1
0
in
as a branched cover
(l'''***'k)-space
be the projection.
Let
([12]).
D = the
closed, nowhere dense analytic subvariety such that
p : Y
Y,
- P
1
(D)
-p(D)
-
B' - D
Y 1.
is dense in
Even after shrinking,
zi 's converging to
Y1
z0 .
analytic subvariety of
> 1, and hence
p(f-l(0))
is a topological covering;
Y1-
Note that
contains infinitely many
Y1
through
f -1(0)
> 1.
neighborhood
in
0
Therefore,
in
Y1
zo
is dense in
which is of codimension
through
f-1 (0)
Hence,
is a closed subvariety of
B"
in
p
of
B"
-1
(L')
such that
0
p- (0) = z o .
such that
B'
is a closed
of codimension
Y 1.
Now
through
Hence, there is a
B',
and a complex line
at
C
zo
-
.
C
L'
L' n (B" n (D u p(f-l(0)))) = {0}.
is a curve in a neighborhood of
f- 1 (0)r) p-1(L') = z o .
Let
a a local irreducible component of this curve at
let
0,
z
C =
zo, and
be a resolution of the possible singular point
Then
nC),
and
f(C) = f.r(C),
and
o
f
frw(i(z ))
o
= 0
in
L.
74.
Hence, by the 1-dimensional case mentioned initially,
f(C)
contains a neighborhood of
0 e L; as already
mentioned, this leads to a contradiction, which proves
the lemma.
As a final remark, the above lemma extends to the
case where
f
Z
and
X
are complex analytic spaces, and
satisfies the conditions of the last theorem.
conclusion is then that
irreducible components of
f(Z)
X.
The
is a union of global
75.
§4.
DeRham Theory Applications.
In this section we study the smooth deRham
cohomology
HDR(X)
especially for an
X
X
such that
X,
with isolated singularities.
msE X
Definition:
on
of the analytic variety
sheaf of germs of forms
0,
x
for every
is clearly a sub-complex of
EX.
x e S(X).
w
mSE
There is a natural
exact sequence
0 -+ ms Ex
E-
+
E aS -+ 0
A
where
EX,
along
S(X).
then
is to be thought of as formal forms in
S
EX S
For example, if
is supported at
X
has one singular point x,
x, and there its stalk is
Ex(X), the formal Poincare complex of
in general,
if
Y C X
EX, S
is supported on
is any subset of
of germs of functions
for every
y E Y.
Then
f
X
X, let
such that
my EX
is
X
at
S(X).)
my
f
x.
(Clearly,
More generally,
be the EX-ideal
= 0
in
Ey
a subcomplex of
EX,
76.
and the quotient complex is supported on
closed.
and
Y
For example, if
X
Y, if
Y
is
is a non-singular manifold
a closed submanifold, one can take global sections
and get a short exact sequence of complexes:
0 + myE(X)
+
EX(X)
-
EX,y(X)
-
0.
The arguments
given below will apply to the calculation of the long
exact cohomology sequence associated to these complexes,
and it is the topological cohomology exact sequence of
the pair
(X, Y)
with C-coefficients.
Generally speaking, the approach to controlling
HDR(X)
the groups
given here is to use the blowing-down
results of the previous section to control
H (msEX(X)).
By a comparison with the holomorphic category, some
measure of control on
H (EX,S(X))
is obtained, for
with isolated singularities.
consider the complexes
First,
7 : X
-
X
-I
1
Let
be a resolution of singularities, as
constructed by Hironaka.
w
mS*EX(X).
(S(X))
is a divisor with normal crossings, i.e.,
given by local equation
(zl,...,zn)
We may thus assume that
z 1 ...zj = 0, where
is some appropriately chosen local
X
77.
coordinate system on
X.
By the blowing-down theorem
of the previous section,
: mSE(X)
Asr
+ mDE"(X)
X
is an isomorphism, where
D = w-1(S(X)).
H (mSE(X))
ý))
= H*(mDE(X)),
Thus,
s
since
is a map of
X
complexes.
Now the complex
mDEZ(X)
X
is
the complex of global
mDE'.
sections of the complex of sheaves
--
exact sequence:
E.
X
Consider the
_A
0 + mDE
-
X
The complex
X
EX
+
X
-
EX
0.
X,D
is, of course, exact at the stalk
level On degrees > 0) - this is just the Poincare lemma -
and
on
is a fine resolution of the constant sheaf
E.
X
X.
The complex
mD*E,
C
is clearly also a resolution
X
of
C
at points not in
and let
such that
(z1 ,...zn)
D
D.
Let
y
be a point of
D,
be local coordinates centered at
is defined by
z ... zk = 0, locally, and
y
78.
z j = x j + iy,
where each
being real coordinates at
near
may be written
t
xj ' s
the
y.
and
yJ'
s
Any smooth s-form
w =
E
a
,
dxI dx
w
where
I
I,J
and
J
are increasing multi-indices, each ranging over
1,...,n,
mD-E"
X
the
of
III + IJI = S.
and
near
a
y,
If
w
is a section of
and we write out
w
as above,
each of
's has vanishing Taylor series at every point
D.
It is,therefore, easy to see that
r
aI j
=
the
n 's
j
H ((x
S
j=l
is
IJ
where we may choose
a uniquely determined smooth function.
r
kN
Write
2)
to be arbitrary non-negative integers, and
IJ
then
yj2)nj
n2
) 2 + (y
N
+ (yj ) 2 N, and set
a
=
be the ball of radius
c
about
2
R= ((x
N
N*a
J=1
Now, let
in
xj
.
B
yJ-coordinates,
E'(B )
the complex
y
The Poincare lemma says that
is exact except in dimension 0,
X
where its
cohomology is
exactly the constants,
C.
More explicitly, the lemma is proved by constructing
operators
ks
: Es (B
X
)
Es
-
X
(B E)
as follows:
In a
79.
=
III + IJI = s, define ks(a ,Jdx Idy)
1
(I
1 j(tz)t
a
0
z
dt)(
k+x k
(-)
x
k
dx
J m+k+lk
dy +
(-1)
k=1
I
I-i
where
k
dx
=
dx
i
... dx
k.
k-1
dxik+l
k+l
dx
... dx
,
and
Extend this definition linearly
and check that on
Es(B )
X
Es(B ):
X
Id , for s > 1
dksS + ks+d
s+1
where
6y
Co
X
aI J =
=
ks
ra
Id-6y,
for
s = 0,
is the Dirac function at
w E mDEs(BE)
N
,
J,
o
I,J
I
for every
E k (a jdx IdyJ),
s Js I,J
d,J
J
dxI dy ,
a
N.
each
y.
If
then each
Considering
)
=
k (a
s
jdx dy
TI 3
is a smooth
1
(f 1 a
(tz)tS- 1 dt)ti j)
s-I form.
Finally,
x
k=1
1I
dy J-J
similarly for
to all of
J = (J
al,Jdxldy , I = (il,..., i)
basic s-form
where
I
-k
dx dy
)
80.
0
a I j(tz)tS-ldt =
,
=
0
1 4N(tz)a• ,j(tz)ts-ldt
1 t 2Nr N(z)a N
(tz)ts- dt
1
=
Since
kS
N
is
arbitrary,
CO q
maps
m
t 2 Nr+sa NJ(tz)dt.
(z)
summing over
00
.
E
(B
)
into
m
Sco0
mDE(B ),
X
is exact in all dimensions.
e
+
0,
(B
).
Since
6
X
identically zero on
as
shows that
-E
X
I,J
rDEL(B )
the complex
X
is
Taking direct limits over
c,
of the complexes considered, gives that the
complex of sheaves
oo
mDE
is exact at the stalk level
X
indimensions > 0,and the sheaf itresolves in
dimension 0 is
= the constant sheaf
C
X-D
extended by
to all of
0
C
on
X - D
X.
Consider again the exact sequence:
0
CO
mDE:
M
E
E: X X
-
E
B
XD
- 0.
(")
81.
Hence, for y e D, 0
E"
mDE
+
X,y
E"
X,y
+
0
is exact in dimensions greater than
in dimension 0.
Thus,
E_
an
X,D,y
The sequence of cohomology shows that
sequence.
is
E0
X,D,y
0, and has kernel C
is a fine resolution of
X,D
= constant sheaf
of
X.
C
on
D
extended by
0
to all
Taking global sections and passing to cohomology
on (*),
gives
...
Hi(mDE(X))
X
Hi(E:
(X))
X,D
Hi(E(X))
R
X
...
By the general result on fine resolutions of sheaves,
Hi(E'(X))
X
- Hi(X;C)
1i(E" (X)) = Hi(D;C), and
X,D
and
therefore, from the long exact sequence of topological
cohomology for the pair
Hi(mD(E'(X))
Hi (X,D;C)
In particular, since
Hi(mSEX(X))
SX
(X,D), it follows that
1=
Hi(mDE:(X))
Hi(X-S(X);C)
c
X
H(X-S(X);C).
-
Hi(msE
X)),
X(
and carries only topological
82.
information about the regular points of
Proposition 4.1:
In particular, for
*
X
(mSE(X))
X.
Thus,
H• (X-S(X);C).
H (mSEX(X))
compact,
is
finite-dimensional.
H (EX(X)) = HDR(X),
In order to control
remains to find something out about
This we are only able to do for
it
H (E XS(X)).
X
with isolated
X
with one singular
singularities.
Consider the local case of
point
x .
Again,
singularities, with
crossings.
x
: 0
r :
X -+ X
D =
denotes a resolution of
)
-l(x
a divisor with normal
Consider the exact sequence of stalks at
) mx *E•,
E;,X
E~,Xo
with a result of T. Bloom ([51,
0.
In analogy with
prop. 3.1),
we show the
following:
Proposition 4.2 :
H (E
an isolated singular point of
X
H (E'
X.
)
for
83.
Proof:
mx Ex
0
,E ) =0.
X x0
We'll show
H (m o
x0
= lim mx E(U),
where
U runs over a basis
Since
lim is
0
4.
0
U
of neighborhoods of
H (m E
O
xo .
exact, we have
) = lim H (m0 E(U)) = lim H (U,x ;C),
+
o
+
o
U
U
the last equality by the previous proposition.
H (U,x ;C) = 0
are two ways to show
basis of neighborhoods of
xo .
may be triangulated ([17]),
call this triangulation
T.
neighborhood of
Tn,
x0
triangulation of
subdivision of
at
in
X
T.
H (m.
X0
"E
X
X,
w-l(U)
U
in a
One way, you note that
with
Let
xo
Un = open star
where
Tn
is the
U 's
The
are a basis of neighborhoods
x .
) = 0.
Hence,
xo, they
H (Un,xo;C)
= 0,
A second way to see this is to
0
U
X
as a vertex;
xo, and since they are star-shaped about
note that as
in
for all
by the n-th iterated barycentric
are contractible onto
and
There
runs over a neighborhood base of
x
runs over a neighborhood basis of
D
84.
in
X.
H (U,x ;C)
Fit
= H (7( (U),D;C)
into the long
exact sequence
- Hi
...
Taking
lim
r-1(U),D;C)
Hi
(U); )
-
...
Hi(D;C)
preserves exactness:
U
...
lirm H (Ux
v
;C)
) lim H (-i
(
1(U);C)
-
H1i(D;C)
-
By continuity properties of sheaf cohomology, however,
the limit map
lim Hi(-1(U);C)
+
Hi(D;C)
is an
U
isomorphism, and exactness says
It is
H (Ex
lim Hi(U,x 0 ;C) = 0.
) which can be controlled for
an isolated singular point.
To do so,
let
xo
2,x
denote the stalk-complex of germs of Grauert-Grothendieck
holomorphic differential forms ([4]).
We recall here two
theorems concerning this complex:
Theorem BH ([4]1):
)
H (Qx
X,xO
is finite dimensional,
.
85.
for
x
E X
an isolated singularity.
Theorem B ([51):
X0
o
X
H (
H ('
~
for
an isolated singularity.
An intermediate complex is needed to compare
H (,x
)
H (E" ).
with
Let
K" C Q* denote the
subcomplex of sheaves defined by
KX (U) = {W
i (U)0p(w)
X
for for every
U
restriction map.
sends
Ki
= 0
open in
2iX(U
- Ug 1 S(X))},
X
in
X, and
p
The differential
d
denotes the
on
2X
clearly
into itself.
X'
forms on
X,
is the quotient complex
First note that each
Since
i
X
coherent.
is a coherent O -module:
X
is coherent, it suffices to show
KX
is
To see this, consider a resolution of
singularities
sheaf
W'i
X
R'/K'.
*(Q.i )
X
f : X
on
X.
+
X,
and consider the direct image
Since
7
is a proper holomorphic
86.
0,
map, and
X
a coherent 0O-module, a theorem of
X
Grauert ([ 9 ]1)
r,(Q )
implies
For an open set
U,
is coherent on
x
Q i(T-I(u)), and
7,(Qi)(u)
X
A)T
Xi 7
X
i)
induces a Qhomomorphism
. Now
l(Kof OX-modules
i(U)) = 0
X
. (T-l(U)),
- 1 (U), and
r-l(U - U n S(X))
since
is
Qi
,
in
is dense in
the sheaf of germs of holomorohic
sections of a vector bundle on a manifold.
isomorphism, if
then
Conversely,
t : T-1 (U - U n"S(X)) -+ U - U r S(X)
since
w = 0
Thus, K
If
it
(w) = 0
U - U n
on
i > dim X.
singularity
is an
- Urn S(X)),
w e Ki(U).
and hence is coherent.
is a regular point, it follows directly
S(X).
1
Ki
= 0, and
X,x
Note also that
Again, suppose
xo .
-1(U
),
T
from the definition that
supported on
on
S(X), which says
= ker (
x e X
X.
Since
Ki
X
i
K
X
= 0
is this
for
has one isolated
is coherent and supported
87.
at
xo, we have, by the analytic Nullstellensatz, that
K
0
is a finite dimensional vector space over
Xx'0
for each
i,
(Note that K•
Xx
dimesion o
imbeddig
Xax
i > imbedding dimension of
X
for
X,x
at
C,
x .)
Consider the short exact sequence of complexes:
0 + K'Xx
'o
-+*Xx
'
+ Xx
(#)
+ 0
Taking cohomology, the finiteness of
KX
X,
xo
and
Theorem BH above together imply:
Theorem BH':
for
x0 e X
H (0•"
Xx
)
is finite dimensional,
an isolated singularity.
One may also pass to completions as
in sequence (#),
0 Xx
-modules
and get induced differentials in the
limit such that the following diagram commutes:
0
-
K
Xx
-+ 0
,x
II
0
-+
Ký
'0
-
'"
X,x
-+ 0
(I ')
4.x
X,X
'0
XIx
'O
+ 0
88.
Here we've used the exactness of the completion functor
([11
or [221), and the fact that some power of the
maximal ideal of
OX
K
annihilates
Xx0
X• Xo.
the inverse system for the completion of
so that
K~ ,
consists
XxO
of isomorphisms beyond a finite stage.
Taking cohomology,
using Theorem B above and the five lemma,
H (5
Theorem B':
Xx
) = H (I "
Xx 0
),
yields:
for
x
e X
an isolated singular point.
Xx C
Clearly,
'X
0
on
X
at
Also,
Aso•
xo
xo
where
E
wher
E ,
'X
= germs of forms
of type (i,0), as defined in section 1.
-Ii
where the bar denotes complex
0X
is a module over
~X,Xx
S
germs at
X'
X ,x Cxo
C EXo,i
Ex 0 '
Xxx °0
x0
conjugation.
Eio
= ring of
x
0X
o
of anti-holomorphic functions on
:
is a natural map
1
X,x 0
Xx
+E^j
Xx
'0
'
X.
There
where the
tensor product is just the algebraic tensor product of
two C-vector spaces.
2
EX,
The map
4
takes
wl
and then projects them into
"2
to
Xo
89.
where
W
whereXx
2
Note that the
X,x
Qi
's
Xx
are all modules of real analytic forms as in section 1.
We wish to show that if
the map
n : X
let
¢
X
-
is locally irreducible at
just defined is injective.
xo$
To do this, let
be a resolution of the singularity at
nw-1 (x).
x
le0
X
xo, and
Consider the following diagram:
÷EAi ,9J
Xx
xi
Xx
0 xJn *
W*
•i•
xi+j *
+
®
X,x o
XIX
X,Xo
As in the proof of Theorem 1.3, each vertical map is
injective, and the top horizontal map (defined as
was above)
X
is
injective,
is non-singular.
as is
immediate to see,
Consequently,
an analogous
The space
^ij
EXx
: Q 1 xo 0
since
4
is injective.
'J
X
o
A
There is
0
^iJ
X, x
may be naturally considered a
Frechet nuclear topological vector space (cf., section 1
above).
For the same reasons as there, the spaces
90.
i 3
X,x o0
and
sort.
also have natural topologies of this
X,X0
0'
X,x
If we endow
r-topology or
completion,
with either the
Xx
e-topology, we therefore get the same
^
0, 1
denoted
which is then also
x
Xo'
ox
a Frechet nuclear topological vector space ([21]).
an example, as noted in section 1, for
at
xo, with local coordinates
at
Xo
= E , °
Exn
x
o'
= c[[zl",..,z
= oX,x
non-singular
(zl,...,zn)
= C[[zl,...,Zn;
A
A
X
1,
As
centered
Zl,...,Zn]],
x
X,xo
= c[[Z1 ,...,
11],
each with the topology of pointwise convergence of
It's
coefficients.
A
AA
Sgo
O
easy to check here that
Xx O0
, and generally from this fact it
= EX
follows that
O
Q
Xx
0
0
o
fX
O
X~xo
that this is also true for
least if
S:
oi
X,x o
X
is
=^E ,j
Xx
x
irreducible at
Xx
X,x0
X,x
•X
O
We wish to show
0o
singular in
x .
X, at
The map
is continuous, and consequently,
91.
A
we get a map
:
X,X
X,x
'
9x0
O
Proposition 4.3:
xo, then
:
X
If
^
at
^iXEx$,x
j
A
is locally irreducible
^
Xx '0
X,x o
also continuous.
o
+
^iXJ
i9
Xx
is a topological
isomorphism.
Proof:
Embed
X
locally at
A
subvariety of a polydisc
x
= 0.
x
in some
as a closed
CN , and assume
Consider the following commutative diagram
A,o
%
A,o
+restriction
A^Ao
AO•
4-
Q
AIs
00
•0j
Al
+restriction +restriction
Ai,j
X,o
X,o
X,o
X,o
X,o
All three vertical arrows are onto: the left-hand side
was shown to be so in section 1, the right-hand side
is elementary, and the middle one follows from exactness
properties of tensoring topologically with a Frechet
nuclear topological vector space [10].
that
Ai
QAo
Al,o
^X,
Xio
(Note first
is surjective, by exactness of
92.
0A,o modules [22].)
completion for finitely generated
A
By commutativity,$
Amapping
4
mapping theorem,
'j
X,o
,i ®
Xo
above.
is
theorem,
open.
But
4
and
Xo
X,o0
,i
X,o
-,j
X,o
restricted to
4), which was shown to be injective
is just
4
By lemma 1.2,
is
an isomorphism.
A I
Consider the double complex
At
QXO
X,o
the differential from
A
is
d 8 1,
and from
Sit i
Xo
X,o
A^
AI
0'i
0
X,o
X,o
(X'
X,o
where
'
' i+l
oX,o
j
+
ti ^
0
A
X,o
to
Xj
0
to
X,o
i
(-1)
By the open
A
,i
X,o
dense in
must be surjective.
X,o
^
A
j+1
X,o
A
@ d.
4
The isomorphism
A
of complexes
:
A
above is
an isomorphism
A
"
@0-'
X,o
X,o
E
considering the
X,o
double complex as a single complex in the standard way.
C
Corollary 4.q:
0 E X
A
H (Ek
)
H(^"
X,o )
an isolated singular point, and
irreducible at
Proof:
X
H (
" ),
X,o
for
locally
0.
First of all, it should be noted that all
the differentials for the complexes above are continuous
93.
in the Frechet nuclear topologies.
and B', each of
Theorems BIH'
Secondly, by
H ($"')
X,o
and
H (A' )
X,o
(which are isomorphic, of course) are finite dimensional.
^
Consequently,
the differentials of the
A
0 X,o
'
and
X,o
0'
complexes are homomorphisms of Frechet nuclear
topological vector spaces (i.e., they each have closed
But then the corollary follows by Grothendieck's
images).
KtInneth formula for such complexes of topological vector
spaces [101].
As noted in the proof above, we have the
Corollary 4.5:
0
H (E
o )
an isolated singularity of
irreducible at
is finite dimensional, for
X, and
X
locally
0.
Next, a simple induction procedure will show that
the last corollary is true, for any isolated singular
point.
First, we prove a formal analogue of the
Nullstellensatz.
X
Let
sets in a neighborhood
such that
X /) Y = {0}.
of germs of C
which vanish on
and
A
Y
be complex analytic
of the origin
Let
JX C EA
0
in
CN
denote the ideal
functions on a neighborhood of the origin
X;
Jy
is similarly defined for
Y.
94,
00oo
As above,
is the ideal of germs at
moEA,o
0
functions
)
is
with all derivatives at the origin vanishing.
Proposition 4.6:
+ Jy + ME ,o
/(JX
E
finite dimensional.
Proof:
0
A,o
First, let
defined
by
X
IX, Iy
denote the ideals in
The analytic
Y, respectively.
and
0A
Nullstellensatz says that
o/(Ix + Iy)
dimensional, and hence, so is
7A o/(TX +
is finite
Again,
y).
denotes complex comjugation.
E
Next,
Ao
/(JA r + J
J
00o
Y
+ mEE
0 A,0
A
)
E
A
A,
/(Jr + J)
A
Y
--
where J X =
cp eX i,o
X
= J
A
X
X
and likewise for
EAo/(IX + Iy +
Let
n
o
A
,o
/mEA
o
X
is the
+Eo
Ao
X
6o
Y, it suffices to show
X
+
)
E A,
is finite dimensional.
be chosen large enough so that
m(OAo )nc IX + Iy
coordinates for
IX + Iy.
+ mEA
N
C ,
If
(zl,...,zN)
therefore
z1
are the standard
1
- N
...zN
is in
Consequently, any monomial in the zi's and
95.
zj's
of degree > 2n
+ -Y)'EA,o.
+
X + I
Since such monomials generate
2n
m(E ,
, we have
Ato eo
the ideal
Theorem 4.7:
then
will be contained in
H (E ,o
Proof:
)
If
0 c X
m(E
)2n C
is an isolated singular point,
is finite dimensional.
The proof proceeds by induction on the
number of local irreducible components of
X
(I + Iy + IX + Iv)*E
X
Y
XA(o
X
at
is irreducible, we are done by the corollary
Suppose
S
X = V
i=l
,
and irreducible at
s > 1, where the
X.
X i ( Xj = {0},
0, and
0.
If
4 .5-above.
are distinct
for ifj.
s-1
Let
Y =
U
Xi .
Since
0
is an isolated singularity
i=l
of
X, we may assume
Y t X
s
= {0).
Consider the short exact sequence of complexes:
0 -+ M'
(,)
- E
-+ 0
consists
o
ofwhich
Yoforms
germs vanish when restricted
E,,
consists of germs of forms which vanish when restricted
to
Y.
Hence, there is a short exact sequence
96.
o - M'
+
-+ Q' -+ 0
E*
(*)s
S,O
Now a germ
w
in
is in
EX
M
if there is a
S,O
0
in
in
CN
neighborhood of
imbedded at
to
XsS
0
)
(we still consider
such that
w
m(EX
sufficiently large.
m(EAo )n
Y.
n
for
C Mi ,
We
S,O
Consider the case
i = 0.
is onto, for an
f E m(EX
)n
m(Ex
+
)n.Ei
so
X
is the restriction
of a smooth form which vanishes on
wish to see that
,Oso
CN
Since
)n
s,O
we may extend it to
preceeding proposition,
F = F 1 + F 2 + F3,
extension of
if
n
f
is
As in the
large enough,
F 1 E JX,
where
F 1 = 0, i.e.,
We may assume
)n.
F e m(E A
F2 E Jy,
F2 + F 3
F 3 Em (E ,).
also is an
to a neighborhood of
0,
The Whitney
A
field of Taylor series
Whitney field, and is
F 3 (x),
for
x e Xs,
in
EA,o,
i.e.,
0
is
a regular
F3(0) = 0.
By the same regular situation argument as in section 3,
Theorem 3.3, find a smooth function
neighborhood of
for
x e Xs,
for •
s
,
0
and
nd
in
F '
3
CN
F
'
on a
N^
such that
F 3 (x) = F 3 (x),
is identically zero on
Y.
97.
F3 - F '
Consequently,
is an extension of
F2 + F '
and
is zero on
0
to a neighborhood of
f
vanishes on
Y.
F2 + F 3
X, and so
Thus,
Cn
in
)ntC Mo
m(EX
S,O
for
n
large enough.
i,
For higher
)n*E
m(E X
S,O
form
may be written as a sum of germs of the
S,O
f*w,
f E m(E X
any extension of
w
simply note that every germ in
w
restricts to
w
)n
and
w
S,O
E
Let
W
be
S,O
to a neighborhood of
in
EX
and take
0 e CN , i.e.,
F
an
S,O
extension of
which vanishes on
f
an extension of
f*w
Y.
which vanishes on
Then
F*w
is
Y.
The preceeding argument implies that the complex
in sequence (*)S above,
is
vector spaces,
H (Q')
Since
H (M')
Hence,
)
H (E'
s,o
Q"
a complex of finite dimensional
is finite dimensional.
is finite dimensional, by corollary 4.4,
is finite dimensional,
Returning to
(*)y,
is finite dimensional by induction hypothesis.
H (E' o
YHence,
H (Eo)
is also finite dimensionalo
Hence, H (E'
X~o)
isalso finite dimensional,
98.
Let
Corollary 4.8:
X
be a compact analytic space
with isolated singular points.
cohomology of
X
is
Then the C -deRham
finite dimensional.
As earlier in this section, consider the
Proof:
short exact sequence
0
Since
Since
+
+
E
ES
S(X) E
E S (X) =
XS
xeS(X) X~x
2
S(X)
dimensional.
isomorphic to
dimensional.
of
msE X
0.
H (E~ S (X))
XPS
H (E
=
xcS(X)
,).
X
is a finite set, this last space is finite
As noted earlier
H (mSEX(X)) is naturally
H (X,S(X);C), which is also finite
Hence,
HI (EX(X)), the C
X, is finite dimensional.
deRham cohomology
99.
§5.
Miscellaneous Applications; Operators on Curves
This section deals with miscellaneous applications
of the blowing down results of section 3.
The first
half contains an extension of Malgrange's theorem on
ideals of differentiable functions to varieties with
It also contains a characterization
isolated singularities.
of holomorphic functions on a resolution of singularities
of a variety with isolated singularities come from holomorphic
functions on the variety:
this is a direct consequence
of §3 and a theorem of Malgrange [191.
These results
would follow for varieties with arbitrary singularities,
if the blowing down conjecture in §3 were true.
The second half deals with results which are more
exceptional in nature.
It is shown that
E(X), for
X
a curve, is complete in the topology generated by
differential operators on
X.
In fact, the method
works for a variety with isolated singularities and a
finite resolution (which will then not be the Hironaka
resoltuion, in general).
in dimensions > 1.
Finite resolutions are rare
In the considerations here, finiteness
of the resolution map is used to construct a large family
of differential operators on the variety in question.
100.
A.
The blowing-down technique of section 3,
together with a result of Malgrange, shows which
holomorphic functions on a resolution of singularities
blow-down to holomorphic functions on the resolved
variety, if the original variety has isolated
singularities.
Proposition 5.1.
Let
X
be a complex analytic space
n : X -+ X
with isolated singularities, and let
resolution of singularities.
function on
X, then
function on
X, if and only if
If
f
condition with respect to the map
Remark:
is a holomorphic
f = 7 (g), with
f
be a
g
a holomorphic
satisfies the finite-point
7.
This proposition clearly has interest
only in the case where the isolated singular points
of
X
are not normal points on the variety, since for
such points, any holomorphic function on
X
blows down
to a weakly holomorphic continuous function, and normality
would say that such a function is, in fact, holomorphic.
Proof:
a
g
i
:: XX--
on
X
One simply applies proposition 3.4 to find
such that
-l(S(X))
(3(X)) + XX -~
f =
(g), and
S(X)
S(X)
is
is aa holomorphic
holomorphic isomorphism,
isomorphism,
g e E(X).
Since
101.
g
is holomorphic on the regular points of
a result of Malgrange ([19])
X.
However,
says that such a function
g
is actually holomorphic.
Next, we show that Malgrange's theorem concerning
ideals generated by finitely many real analytic functions
is also true in the ring
E(X), for
X
with isolated
singularities.
Proposition 5.2.
Let
be real-valued,
gl'*'*''s
real analytic functions on
X,
isolated singular points.
a complex variety with
f e E(X)
A function
may be
A
written
A
f = hlg 1 + ... + hsgs,
in
Ex(X),
with
A
hi C E (X).
f
(i.e.,
is in the ideal generated by
if it is so formally at every point.)
gl,**.gs
A
Furthermore, if
that
hi(x) = 0,
Proof:
is local on
=
=
,...,s,
in
h i's
so
Ex(X).
A partition of unity shows that the question
X, so we may assume
point, call it
f()
f(x) = 0, we may choose the
0.
A
f(0) = h 1 g(0) + ...
X
has just one singular
By assumption,
A
A(0),
+ hgs (0),
for
some
for some
hi,
i
hi,
i = 1...,s$
102.
in
Eo(X).
Let
hlo ,...,h
A
A
and so that
f - h
1
0
be chosen to be real-valued
h (0) = hi, i = 1,...,s.
_
s 0 gs = f'
Then
still
satisfies the
A
assumptions of the proposition,
Let
7 : X -> X
and consider
The
gi's
A
f'(0) = 0 e E (X).
and
be a resolution of the singularity at
f' = T (f'),
gi =
are real analytic, and
to the ideal in
E(X)
i = 1,,...,s,
(gi),
i
f'
generated by the
r .
Furthermore,
w- 1 (0).
Taylore series along
gi's
f'
E(X),
= hlgl
f'
such that
furthermore, we may assume the
Taylor series along
hi's
-1 (0).
may all be written
hi's in
E(X), and
f = hhg1 + ...
h (0)
+ hsgs,
+
hi
'
s
has vanishing
hl',...,h s
+"'
+
hs'gs,
and
hi's have vanishing
By Theorem 3.3 in section 3,
hi = 7 (hi),
= 0.
with
at every
By a theorem of Malgrange
([201 Chap. VI, Th. 1.2'), we may find
in
on X.
belongs formally
point, as is seen simply by pulling-back the
above by means of
0,
i = l,,.,,s,
Finally,
h 2i = h
+ hi,
i
= 1,...,s.
with
103.
The proposition above is also true for complex
valued, real analytic functions, and for submodules of
E(V) = C
sections of C
vector bundle
V
over
X, since
theorems similar to the Malgrange theorem quoted above
are true for these extended situations.
The point
of the method is made by the example given.
Clearly such results go through for more general
provided an adequate blowing-down theorem for
7
X,
: X
-+
X
is known.
The following result will prove useful later in
this section, and is a natural complement to theorem 3.6
above (cf.
[71).
Proposition 5.3:
theorem 3.
above.
f
Let
: X'
sf *(E
Then
s
+ X
(X))
be as in
is closed in
Es(X').
Proof;
is
The subspace
Xsf *(Es(X)),
by proposition 3.4,
determined as the space of all forms satisfying the
finite point condition for
points of
X'
¢ : E s (X')
4
such that
t
( Es , (X')
i=l xi
f.
Let
x1l ,...,xt
t
be
x = f(x 1 ) = ... = f(xt).
be the natural projection,
Let
104.
which is continuous and onto.
A f (Ex(X))
Si= C
it
E.
call it
Then
t
^.
O
Esx ,(X')
i
~-1(E)
By proposition 1.4,
is a closed subspace, call
is a closed subspace of
V(x 1 ,...,xt).
ES(X'),
w e ES(x')
To say that
satisfies the finite point condition is to say that
w{E
IV(x 1 ,...,xt)If(x 1 ) = ...
is equal to
Since this
Xsf*(Es(X)), and is also closed, the
proposition is
B.
= f(xt)}.
proved.
We'd like to look now at a simple observation
concerning differential operators which, in light of the
blowing-down result above, will provide us with "many"
differential operators for some special spaces
7
: X
-
X
be a resolution of singularities,
X.
X
Let
with
isolated singularities.
Proposition 5.4:
Let
D' : E(X)
differential operator on
X.
D'
differential operator
on
X
D
-
be a
blows down to a
if
D'
finite point condition with respect to
For a differential operator such as
point condition means that, given
E(X)
X1 , ...
satisfies the
v.
D', the finite
,xt
in
X
105.
operator
= w(x t ) = x, the induced formal
= ...
wT(x)
such that
A
t A
D' :
$ E. (X)
~
i=1x
t ^
of $ E
A* ^
S(E (X))
i=1
-LI~I~
+
imposed on
i=1
i
xi
into itself.
(X)
y
imply, by proposition 3.4, that
D'
D' : r (E(X))
-
By proposition 2.1, there
*(E(X)).
is a differential operator
D = D'
A
E, (X) takes the subspace
One simply notes that the formal conditions
Proof:
*1
T
t
$
on
iT
D
i.e.,
E(X),
on
D'
X
such that
blows down to
X.
An analogous result holds, of course, for real
meromorphic operators as in section 2 above, with poles
along
sums of
w-1(S(X)).
D 's,
D' : E~(X)
-
The formal conditions concern finite
where, for such an operator
Q(E,(X)), where
Q(E,(X))
of quotients of the integral domain
Theorem 5.5:
and let
w : X
a function
+
Let
X
f e E(X)
X
denotes the field
E,(X).
x
be of complex dimension 1,
be its normalization.
There exists
such that for any differential
106.
operator
D'
on
Proof:
feD'
X,
blows down to
X.
The theorem is clearly local on
we may look at just one singular point in
O,
Let
X
on
the analogous sheaf for
X.
is a short exact sequence of coherent
0
Since
7
so
OX
+
> 7~(0)
X
+
The following
OX
modules:
Q + 0
1
is an isomorphism when restricted to
7
is an isomorphism outside of
at
0
alone.
annihilates
0,
Hence, there is an
Q,
i.e.,
n
and
Q
X - w- (0)
is supported
such that
S(0,
m(OX,o)n.*(0,)o
Choose finitely many generators
fl,...,fs
of
m(OX,o)n
o)
Ve may have to shrink
This implies that
f = fl.l + ..
7 (f) = f
0
X
about
0
.
m(O X $ )n
such that their common zeros are exactly the point
that
0.
X, call it
be the sheaf of germs of holomorphic functions
OX
X;
X,
0.
to do this.)
is the only point where
+ fs' x
vanishes.
We want to show
has the property that
f*E(X)
T
(E(X)).
107.
Let
{Xl*,...,m} =.-
suffices to show, for every
^
^
h
Eo(X)
E
^A*
By proposition 3.4, it
1(0).
g e E(X), that there is an
^
m
A
E. (X)
7 (h) cE
such that
i=1
is equal to
x.
1
f.g(xi ).
9
By the choice of the
above, we
fi's
i=1
m
know that
fi.( s
0..
i=1
f. =
SXO
A
fi'*(O0)o C
(OXo), where
X
Hence, passing to closures,
fi ( @
O..
i=l
~ ) C~
(E (X)) C
A*
mA
fi*(
C
(fi).
)
X,x i
$ E. (X).
i=l Xi
A
Similarly,
e
•, ~ ) C7 (Eo(X)).
i=l X,x i
Hence,
)
X,x i
mA
fC( 0O ~
i=l X,x i
^
U
)
X,xi
7 (E (X)), and passing to closures, yields
A
m
fe(
$
A
A*
E~ (X)) C
i=1 x0
r (E (X)), and the desired property of
is demonstrated.
The theorem follows simply by noting
that
E(X)
foD'
maps
into
* (E(X)), or using the
previous proposition.
The previous theorem says, in effect, that there
are "many" differential operators on a complex curve.
f
108.
One way of measuring how many differential operators
there are on an analytic space is to consider the topology
on
which they generate, and compare it with the
E(X)
Explicitly, the differential
E(X).
usual topology on
operator topology is given by a family of semi-norms
p = p(D,K), where
and
K
D
is a differential operator on
is a compact set of
X,
X, and where
p(f,g) = sup IDf - Dgl ,
for
f
and
g e E(X).
xeK
Denote the topological vector space
topology by
E(X)
with this
It is easy to see that the identity
D.
E(X)
map is continuous from
this is
E(X)
an isomorphism.
-
Since
E(X)D,
and we ask whether
E(X)
is
a Frechet space,
isomorphism would imply that a countable family of the
p's above would generate the differential operator topology.
One could also define the differential operator topology by
a countable family of semi-norms if one knew that the
differential operators are countably generated as
E(X)-module, i.e., if there exists a family of differential
operators
{Di}icZ
,
differential operator
and
D
D = Z fiDi,
on
E(X),
for
for every
fi's E E(X),
109.
all but finitely many identically zero.
For an
X
with
isolated singularities, such a countable family exists.
We won't use this fact in general, so only sketch a
proof:
first, note that the underlying real analytic
space of a complex analytic space with isolated
singularities is coherent (cf.
[191).
For such a space,
the sheaves of real analytic differential operators are
coherent, too, and hence finitely generated locally.
Finally, a result of J. M. Kantor ([16])
says that the
real analytic differential operators generate all
C0 differential operators over
for
X
coherent
(This last is a consequence of Malgrange's
real-analytic.
result that
E(X),
E (Rn)
is flat over
Ao(Rn): [20],
Ch. VII,
Cor. 1.12).
If the topology of
E(X)D
were generated by a countable
would be a Frechet
family of semi-norms, then
E(X)D
space if it were complete.
Finally, if
E(X)D
is a
Frechet space, the open mapping theorem would say that
E(X)
-
E(X)D
is a topological isomorphism, and hence
the
E(X)D topology would inherit several useful
properties from
E(X),
e.g., reflexivity, nuclearity, etc.
110.
This chain of hypothetical reasoning proves valid
for
X
of dimension 1.
In higher dimensions, Kantor
seems to have a counter-example to the "completeness"
For dimension 1, the
stage of the above argument.
completeness follows from the following simple lemma
about C
functions on the disc.
Lemma 5.6:
be a sequence oi
{fi=1,
Let
about
C" functions on the unit disc
A = A(0,1)
C1 .
the coordinate for
If
and if,
sup
xnK
K
on
z = x + iy
for all
re
m,n
A,
E(A), then the
Remark:
in
C,
sufficiently large,
2k
2k
r2kDfn - r2kDf m
compact in
is
0
<
and
{fi )
for arbitrary
D
E > 0,
a differential operator
are a Cauchy sequence in
Note here that
k
E(A).
is a fixed, positive
integer.
Proof:
is where
Clearly, the only place there is a problem
r = 0,
i.e., at the origin.
For convenience,
by radial expansion, we may assume all our hypotheses
are satisfied on
A' = disc of radius 2 about 0 in C'.
111.
This done, we only have to show that the hypotheses
imply
-
sup ID(f
provided
n,m
that
course,
integers.
2k
the r
fm)
< c,
E > 0,
for arbitrary
n
XcT
sufficiently large.
s+t
a
D =
'
for
We may assume, of
and
s
t
non-negative
The proof simply consists of "integrating out"
factor from the estimates of the hypothesis.
In what follows,
stands for a positive
0(n,m)
function, arbitrarily small for
n,m
sufficiently large;
it needn't stand for the same function from line to line,
however.
sup
E.g., our hypothesis is that
Ir2 kD(fn
= 0(n,m),
f
for any
and we wish
D,
XCA
to conclude that
sup ID(fn - f
)
= 0(n,m).
To prove
the desired estimate, note first that, if one can estimate
a
(f
x (fn
fm)
and
(fn -
replacing the sequence
{Yy-
,
fm )
{fi
}
by
0(n,m)
on
T., then
of the hypothesis by
~-i
{
DRr
one can proceed to estimate higher derivatives
by induction.
Secondly, note that if
A1, A2
are bounded
or
112.
functions on
T -
or
T
(0),
and if
then
n
-
f )
are estimated as
X1 Dl(f
)
f
-
+ X2 D2 (f
D2
and
Dl(fn - f m )
differential operators, then if
D2 (f
D1
0(n,m)
-
and
T
on
are
~ -
or
(0),
can also be estimated
f)
Thus, it suffices to show we can estimate
similarly.
S(f - fm )
for then m
1
(f n - fm)
m
rhave
and
0(n,m)
as
--
(0),
(f
SCfn
fm
on
for then we have estimated
cos
and
as
e
r
(f
sin 6
(f
0(n,m)
on
n
-
f
)
n - fm )
a-
sin 6
~(fn
r
m
-
Cos
r
8
T
n
- fm
n
)
m
-_
f
m
(fn
(0); hence, by continuity, on
f)
m
T,
proving the lemma.
So, starting, from the estimates of the hypothesis,
rfn
we want to show how to estimate
1
F 7TnC(fn-
fm)
as
O(n,m)
on
T -
(0).
-
f)
Thus,
and
113.
sup
XE3
and
Ir
sup
xe•
2k
x( f
-
r 2 k 3 (f
T fn
= O(n,m)
m)
- O(n,m).
- fm )
By the reasoning used above, this implies
sup
Ir2k
(f - fm )I
xcl-(O)
sup
r2k cos e a(f
-
fm) + r 2 ksin e
(fn -
f
)
O(n,m)
and
sup
Ir 2 k (
)-
(f
- f )I
xce-(O)
sup
Ir2kCos 8 af
-
fm )
r2ksin 0
(f
- f )I
O(n,m).
The same reasoning shows, by induction, that we can
estimate (for
N > 0,
integer):
=
114.
2
sk
sup
(fn
Ir2 k aN
sup
xc~-(O)
ar
since
r2
fm )l
-
1 a
m ) I = O(n,m),
n
2k
k aN+l
cos e r
-N+
a
a
- + sin e r
-JT-
2k aN
2kaN+ (1 a )
rN+1
r 2k
=
a
r 2k aN
ax + cos 6 r 2k
ar
rN
2 aN
-sin e r 2k
sr
ar
N
ar
ar
ar
and
= O(n,m)
ar
xE2-(O)
and
N
Ir2 k
In the last step, one should note, from the induction
hypotheses, that if
r2kk
sup
•n
m)
-
=
(n,m),
0r
XCE-(0)
then similarly,
sup
Ir2k aN
3r
XCs-(O)
ax the corresponding statement for
and
is also true.
p2k
Now integrate
For a point in
T -
r
m)' = 0(n,m),
x(fn
-
in place of
out of these estimates:
(0), with polar coordinates (r,e),
115.
•1
-
-(rfn(r,e)
ar
r
fm(r,e))
ai
i+
- fm(p,6))dp +
1 ap +
- fm(1,6))M.
ar
By hypothesis,
i
1
sup
T (fn - fm) I=
0(n,m),
ar
r=1
and thus,
i
S--(fn
ar
-
fm
< I
r
< /
r
0(n m)dp
P
2k
fm)dpl
n -
i+
+
ap
-
+ O(n,m)
O(n,m)(1 + -2ki
r
where we've used
starting from
+ O(n,m)
r2ki 3'2ri+l (f n
ar
-
)
f )
< O(n,m).
i = 2k + 1, we then have shown
Now
116.
a2k+l
a 2k+l
ar
- fm) I
(fn
1
< O(n,m)(l +
r
2k-r
Hence, repeating the process, gives that
S2k
r
Sar 2(f -f
<J
-1
2k+1
2k+l
f n-f m) I dp
fn-fm)(1,O)
n mn
ap
r
1
< f O(n,m)(l + -- 1•)dp + 0(n,m)
P
1
= O(n,m)(l +
r
1
r2k-
The process clearly may be repeated inductively, and
after 2k repetitions, provides the estimate:
2
S--rar
(
(f
-
fm)
O(n,m)(log r- 1 ) + 0(n,m),
Hence, integrating once more,
Tr- n - fmm
I a (f
)
< O(n,m)Ir
= o(n,m),
log r -1
- ri
+ O(n,m)
117.
since
- r log r
0
has limit
as
r
The same
0.
-
procedure obviously may be used to show that
2k 2 k+l
Ir2k
using that,
- fm)J < 0(n,m),
I!.(fn
1
1
2
"
Hence, the lemma
( < 0(n,m).
-
is proved.
As a consequence of the lemma, it is easy to
deduce the following theorem.
Theorem 5.7:
analytic space,
X
Let
W : X
-
be a 1-dimensional complex
X
Then
the normalization.
there is a countable family of differential operators
{Di)
on
X
which blow down to
the usual topology of
Proof:
a function
X,
E(X).
Near a singular point
f = f
theorem 5.5 above.
and which generate
x0
of
X, choose
as constructed in the proof of
Let
F
be a function in
E(X)
which
has the following properties:
(1)
(2)
for every
x0 E S(X),
a neighborhood of
x .
the only zeroes of
F
F
agrees with
are the
xo 's
fx
in
S(X).
118.
Proposition 5.4 says that for any
F = 7 (F).
Let
D
differential operator
{(FD
proven if we show that
of
the topology of
except in
i)
generates the topology
which generate
{Di)
for a family of operators
E(X),
F-D
the operator
Hence, the claim of the theorem is
X.
blows down to
X,
on
There is clearly no problem
E(X).
a neighborhood of a point
Near such a point, write
x
7-1(S(X)).
in
2k
F = r -g, where
ie
z = re
= x + iy
g(x) X 0:
and where
form of
is a local coordinate centered at
F
x,
this follows from the explicit
given in a neighborhood of
w(x)
in the
proof of theorem 5.5.
Hence, we only want to show that
operators of the form
r 2kD
on C
generate the usual topology
functions in a neighborhood of
x.
But this is,
of course, precisely what the previous lemma states,
and the proof is complete.
Corollary 5.8:
E(X) D
is complete, for
X
of
dimension 1, and hence, the topology generated by
differential operators is the same as the ordinary
topology on
E(X).
119.
Proof:
{Di}
and let
X
Let
: X
i
on
Let
X,
{Di }
i.e.,
and generate the topology of
X
be the corresponding family of operators
7 (D h) = Di(7 h),
i > 0.
and every
be the normalization again,
be a family (countable) of operators on
which blow down to
E(X).
X
-
{fj.
If
for every
h e E(X),
E(X)
is a sequence in
such that
= 0(n,m),
sup IDi(fn - fm)l
XEK
K
for any
X, then it follows that
compact
sup IDi(f
0(n,m),
- f)
xcK
for any
K
that the sequence
it has a limit
f e
T
(E(X)).
{fj}
If
E(X)
in
f e E(X).
j, and since
where
f
Di's, the last
By definition of the
every
X,
compact in
Since
n (E(X))
f =
T
(f),
j
=
f,
for every J.
3'
estimate says
Hence,
is Cauchy.
f. E
for
(nE(X)),
is closed in
E(X),
f e E(X),
+
E(X)D, which proves completeness.
f.
f
in
Since we only used
120.
a countable family of
mapping theorem says
isomorphism.
Di's, as noted earlier, the open
E(X) -÷ E(X)D
is a topological
121.
§6.
Two Examples
This section is devoted to two simple examples of
we calculate the DeRham
some of the foregoing theory:
cohomology of the two simples singular plane curve,
and examine the possibilities for Hodge operators on these
curves.
The two curves we shall look at are the projective
plane curves
X1 , X2
X1 : y
2
given by the affine equations:
p2
2
= x (x - 1),
2 = x3 .
2
Each has the origin
0 E C
in the above affine
representation as its only singularity.
simple double point at the origin, and
Pictorially, we have:
X1
X2
has a
a cusp.
122.
lemma at the origin:
satisfies the Poincare
X1
It is easy to check that
in fact, locally at
0,
X1
looks like two straight lines intersecting transversally,
L1
call them
given by any pair of
f2
fl
fl(0) = f2 (0).
conditions on a pair of smooth forms
W2
together" to a smooth form
Ei
= Ei
L1,
fl
for
L1
E
EL 1 •Oo
and
w
for
,
E
L2,o ,1
is a closed 1-form, then
for
EX
1
o
L1
and
There are no
wl
on
LI
and
L2 .
X 1 . Thus,
on
i > 0.
wl = dfl
with
If
and
w = wl
w 22
m2 = df2'
lemma
f2 e EL 22 9o, by the Poincare
If we choose
f 1 ( 0 ) = 0 = f 2 (0)
(we may change them by constants),
f
on
of degree greater than zero to "piece
L2
on
is
X1
function on
functions
CO
such that
L2
on
A Cc
L 2.
and
fl L
= fl
is, therefore, equal to
and
fl
then there is an
= f2'
and
df
m.
Since the Poincare lemma holds at all points of
Xl,
the smooth DeRham cohomology of
X1
is just the
123.
topological cohomology,
dimensions
although
H (X1;C),
0, 1, 2, and 0
HDR(X)
which is
C
in
for higher dimensions.
Thus,
in general is "bigger" than the
topological cohomology of
X, it is not "big enough"
to regain for singular varieties either Poincare Duality
or the Hodge (p,q)-decomposition for the cohomology of a
projective manifold:
that
DR(X1)
for
HDR (X)
we may not write
HI,o(X1) = Ho,1(X2
H•R(X)
dim HDR(X1)
HDR(X
=
is one implies
is zero and clearly
1 ,0(X
1)
@ Hol(X1 ) with
.
In the case of non-singular manifolds, Poincare
duality is closely related to Hodge's *-operator,
which is, in effect, based on local or infinitesimal
Poincare duality in the exterior algebra of a vector
space.
The *-operator, too, is very closely related to
the standard adjoint of the exterior differentiation
operator
d.
Perhaps, in general, one should not expect
to be able to create a theory of harmonic integrals
(which is basically using an adjoint operator to construct
a Hodge decomposition) in a space where Poincare duality
124.
In our example
is not satisfied.
p1,
is
X1
it looks very unlikely that a "good" or
X1
actually an immersion of
2
P ,
into
to construct a formal adjoint for
d
2
is
we might hope
on
by means
XI
This will work on
of the induced metric.
X1 .
X1--X1 C P
Since
natural adjoint will exist.
on
X 1 , whose resolution
X1 , but not
In fact, the picture is the following
E(X 1) d :,E1(X 1 ->
E2 ( 1)
t 7T
E(Xl) --> E1(X1)
where
n (E(X 1 ))
d-> E (X1 )
has codimension 1 in
the metric mentioned above, we construct
and Laplaceans
~i
EIi
on each
gives an operator on
E i (X1 ).
X1 ,
For
which for
in fact, give harmonic representatives for
for trivial reasons.
For
i = 1,
Zi
since the harmonic 1-forms represent
Using
E(X 1 ).
d
s
on
X1'
i = 1 and 2,
i = 2
does,
H2DR (Xl1)
has no kernel,
H1 (X1 ;C)
= 0.
125.
H (X1;C) = HDR(X)
But
is one-dimensional.
The
missing kernel which is supposed to represent
H (X1;C)
comes from the fact that
d
:
(X)
may be considered a differential operator on
E(X 1)
X1
only
if we restrict it to acting on a codimension 1 subspace
of
EI(X1) , namely
d-l(n*(E(X 1 ))).
A one dimensional
complement does not admit the Hodge decomposition on
as a decomposition on
X1
Xl, and this complement represents
H 1 (X1 ;C).
It is hard to imagine a more natural attempt
at a
d
and it is harder still to see how to repair
this
d
,
so that it alters the Hodge decomposition by
a 1-dimensional subspace.
The example
something about.
X2
is
a little
bit harder to say
One can check directly that it, too,
satisfies the Poincare lemma at its singular point, or
one can do it in the following way.
X2
is irreducible at
0,
Note first that
so that the methods of
section 4 apply to show that
H (E 2
2,o0
,(
H (
2X )
X2,o
6 H ( 'i X2,o ) = H (~ X•
2,o
) H ( , .X ),
2,o
in
126.
the notation of that section.
H (Q
)
So, we'd like
to be zero in dimensions > 0,
i.e., in
2,o
dimension 1.
It is a general calculation of Brieskorn
and Mumford ([51, p. 132) which says that
dim H1 (
2 , o
) = 0,
of the quotient
in
0 2
C
,
conclude
0
+
where
Since
K'
H1
Q2,
x 2 .0
2 ,
(f, -f,
f
,o
0.
at
) = dimension over C
dim H2 (j
W
)/(-,
and
X2,o ) = 0,
X2,o
-
X2
To
recall the sequence
0, and it
is
easy to see that K'
QX2,o.
It is easy to
K0 = 0, K 1 = {a(2xdy-3ydx)+bx(2xdy-3ydx)Ia,bEC},
K2 = {a dxdy + bx dxdyla,bEC}.
d(2xdy -
of the two ideals
dim H 2 (nX,
2 ) = 0.
2,o
the complex of torsion forms in
check that
-)
is the defining equation of
f = x3 -S, y2 ,
X2 ,o
o
3ydx) = 5dxdy
it follows that
H (K')
sequence of cohomology,
and
= 0.
H1 P
Since
d(x(2xdy - 3ydx)) = 7xdxdy,
Hence, by the long exact
2,o ) = 0.
is
127.
*
X2
Thus, for
X2
Since
to
is irreducible at
X2 = P'.
Thus
X2
a
- H (X2 ;C).
HDR(X2)
we also have
0,
it is homeomorphic
does satisfy Poincare duality.
It is not possible to analyze quite so directly the
relation of the DeRham complex of
X1
as for
and
X1
and that of
X2
Is it possible that there
earlier.
of
d
is a special formal adjoint
X2
X2
on
d
which
induces a Hodge decomposition on the subspaces of
forms coming from
X2 ?
There are other reasons to believe there might be
"elliptic" operators on
X 2.
X2
has a good "symbol
space" in the sense of Bloom ([23])
if
t
-
(t2 ,t3 )
at
is a uniformization at
meromorphic operator
D =--
d
2
-
2d
2 d-
0.
For example,
0,
then the
is an operator
dt
on
X2
near
0
(i.e., it blows down to
X 2 ), and
its leading term has a non-zero coefficient.
D*D
O=
has a leading term
but
the author is unable at present
a2
,
which is elliptic,
to handle the singular
but the author is unable at present to handle the singular
128.
lower order perturbations of this leading term.
This
operator has the trivial, but encouraging property that
it is formally onto at
A
A
: EX2
0,
i.e., the induced operator
A
+ EX2
is surjective:
in the non-singular
case this is an immediate consequence for any elliptic
operator of the local solvability properties of such
operators.
More examples of such operators are given
in [15], which has some of the best constructive
examples of operators on varieties available at present.
129.
Biographical Note
Dan Burns, Jr. was born in Brooklyn, New York
in 1946.
He graduated from Regis High School in
New York in 1963 and graduated from the University
of Notre Dame in 1967.
Since 1967 he has been a
graduate student at M.I.T., except for the 1969-70
academic year, during which he taught mathematics at
the Boston Public Latin School.
He was supported by
a National Science Foundation Fellowship from 1967
to 1969.
in 1969.
He married Anne Cronin of Milton, Mass.
130.
Bibliography
[11
N. Bourbaki, Topologie ge6nrale, Chap. II, Hermann,
Paris, 1961.
[2]
, Algebre commutative, Chap. III, Hermann,
Paris, 1961.
[31
T. Bloom, Differential operators on complex spaces,
preprint.
[4]
T. Bloom and M. Herrera, DeRham cohomology of an
analytic space, Invent. Math. 7 (1969), pp. 275-296.
[5]
E. Brieskorn, Die Monodromie der isolierten
Singularit.ten von Hyperfl.chen, Manuscripta Math.
2 (1970), pp. 103-161.
[6]
J. Dieudonn6 and L. Schwartz, La dualite dans les
espaces 5
et dZ,9, Ann. Inst. Four. I (1949),
pp. 61-101.
[7]
G. Glaeser, Fonctions composees differentiables,
Ann. of Math. 77 (1963), pp. 193-209.
[8]
H. Grauert, On Levi's problem and the imbedding of
real analytic manifolds, Ann. of Math. 68 (1958),
pp. 460-472.
[9]
,
Ein Theorem der analytischen Garbentheorie,
Publ. Math. I.H.E.S., no. 5, 1960.
131.
[101
A. Grothendieck, Seminaire Schwartz, 1953/54,
especially expos' 24.
11
,
_
El6ments de geometrie algebrique,
Publ. Math. I.H.E.S., nos. 4, 8, ... ,
[12]
1960 -
.
R. Gunning and H. Rossi, Analytic Functions of
Several Complex Variables, Prentice-Hall,
Englewood Cliffs (1965).
[131
H. Hironaka, Resolution of singularities of an
algebraic variety over a field of characteristic
zero, I-II, Ann. of Math. 79 (1962), pp. 109-326.
[14]
, Bimeromorphic
smoothing of a
complex-analytic space, mimeographed lectures,
Harvard Univ., 1971.
[15]
M. Jaffe, Thesis, Brandeis Univ., 1972.
[16]
J.-M. Kantor, Operateurs differentiels a
coefficients derivables sur les espaces analytiques
reels,
[17]
preprint.
S. Lojasiewicz, Triangulation of semi-analytic sets,
Ann. Sc. Norm. Sup. Pisa, ser. III, v. XVIII (1964),
pp. 449-474.
[18]
B. Malgrange, Seminaire Schwartz, 1959/60,
exposes 21-25.
132.
[19]
B. Malgrange, Sur les fonctions differentiables
et les ensembles analytiques, Bull. Soc. Math.
France 91 (1963), pp. 113-127.
,
[20]
Ideals of Differentiable Functions,
Oxford Univ. Press, Bombay, 1966.
[21]
H. Schaeffer, Topological Vector Spaces,
MacMillan, New York, 1966.
[22]
J. P. Serre, Geom6trie alg6brique et geom6trie
analytique, Ann. Inst. Fourier 6 (1956),
Spp.
1-42.
[23]
I. M. Singer, Future extensions of index theory
and elliptic operators, in Prospects in Mathematics,
Princeton Univ. Press, Princeton, 1971.