Inventionesmath. 42, 239- 272 (1977)
/F/ve//t/onr
mathematicae
9 by Springer-Verlag 1977
Hirzebruch's Proportionality Theorem
in the Non-Compact Case
D. Mumford
Harvard University,Department of MathematicsCambridge Massachusetts02138, USA
Dedicated to Friedrich Hirzebruch
In the conference on Algebraic Topology [-7] in 1956, F. Hirzebruch described a
remarkable theorem relating the topology of a compact locally symmetric
variety:
x =o/r,
D = bounded symmetric domain,
F=discrete torsion-free co-compact group of automorphisms of D
with the topology of the extremely simply rational variety /), the "compact
dual" of D. (See w for full definitions.) His main result is that the Chern
numbers of X are proportional to the Chern numbers of/5, the constant of
proportionality being the volume of X (in a natural metric). This is a very useful
tool for analyzing the structure of X. Many of the most interesting locally
symmetric varieties that arise however are not compact: they have "cusps". It
seems a priori very plausible that Hirzebruch's line of reasoning should give
some relation even in the non-compact case between the chern numbers of X
and of/5, with some correction terms for the cusps. The purpose of this paper is
to show that this is indeed the case. We hope that the generalization that we find
will have applications.
The paper is organized as follows. In w1, we make a few general definitions
and observations concerning Hermitian metrics on bundles with poles and
describe an instance where such metrics still enable one to calculate the Chern
classes of the bundle. This section is parallel to work of Cornalba and Griffiths
[6]. In w2, which is the most technical, we prove a series of estimates for a class
of functions on a convex self-adjoint cone. In w3, the results of w1 and w2 are
brought together, and the Proportionality Theorem is proven. One consequence
is that D/F has the property, defined by Iitaka [8], of being of logarithmic
general type. Finally, in w4, we analyze the step from logarithmic general type to
general type and reprove a Theorem of Tai that DIE is of general type if F is
sufficiently small.
240
D. Mumford
w1. Singular Hermitian Metrics on Bundles
In this section we will not be concerned specifically with the locally symmetric
algebraic varieties D/F, but with general smooth quasi-projective algebraic
varieties X. When X is not compact, we want to study the order of poles of
differential forms on X at infinity, and when E is moreover a vector bundle on
X, we want to study Hermitian metrics on E which also "have poles at infinity".
This situation has been studied by Cornalba-Griffiths [6]. The following idea of
bounding various forms by local Poincar6 metrics on punctured polycylinders at
infinite is due to them. More precisely, we choose a smooth projective compactification .~:
XcX
where ) ~ - X is a divisor on X with normal crossings.
Then we look at polycylinders:
[A = u n i t disc]
A~)f
\r=dimX
where Ar n ( X - X ) = ~
!
uni~ of coordinate hyperplanes~
{Z 1 = 0 , Z 2 = 0 , . . . ,
Zk=O
.}
hence:
Ar n X = ( A * ) k x A r-k.
In A* we have the Poincar6 metric:
ds 2 =
Idzl 2
[zl2(log Izl)2
and in A we have the simple metric [dzl 2, giving us a product metric
x d ' - k which we call cotv).
on
(A*) k
Definition. A complex-valued C ~ p-form I/on X is said to have Poincar~ growth
on X"- X if there is a set of polycylinders U, c ,~ covering A"- X such that in
Hirzebruch's Proportionality T h e o r e m in the N o n - C o m p a c t Case
241
each U,, an estimate of the following type holds:
[q(tl, ..., tp)l 2< C, ''(p)t'wv.~ol,tO . . . . . c~(mtv.,p, tp)
tp tangent vectors to X at some point of U~n X).
it is not hard to see that this property is independent of the covering U, of
- X (but unfortunately it does depend on the compactification .~). Moreover, if
~1, q2 both have Poincar6 growth on ) ~ - X , then so does ~/1 A q2. This leads to
the basic property:
(all t 1. . . . ,
Proposition 1.1. A p-form 11 with Poincard growth on X - X has the property that
for every C ~ (r-p)-form ~ on X,
j" I~A~I< +oo
X-X
hence tl defines a p-current [t/] on f(.
Proof Since ( has Poincar6 growth, we are reduced to checking that if t/is an rform with Poincar6 growth, then
In a polycylinder U,, this amounts to the well-known fact that for all relatively
compact V c c U , ,
the Poincar6 metric volume of Vc~(A*kxA "-k) is
finite. QED
Definition. A complex-valued Coo p-form ~/ on X is good on ff if both q and dq
have Poincar6 growth.
The set of all good forms q is differential graded algebra for which we have
the next basic property:
Proposition 1.2. I f q is a good p-form, then
d([~]) = [d,].
Proof. 1 By definition of d([q]), this means that for all C o o ( r - p - 1 ) - f o r m s ( on
2,
J d~^~=-
X-X
S q^d~.
X-X
This comes down to asserting that if U, is a tube of radius e around _~ - X. then
lira ~ (q A r = O.
~000~
If we take, for instance, r = 2 and set up this integral in local coordinates x, y on
-~ near a point where ) ( - X has 2 branches x = 0 and y = 0 , then this comes
i
C o m p a r e Cornalba-Griffiths [6], p. 25
242
D. Mumford
down to the assertion
[dx] 2
]dyl
=0
lim x5>_~Ixla(log Lxl)2 lYI"I(log Lyl)I
which is easy to check. The general case is similar.
QED
Next, let/~ be an analytic rank n vector bundle on )?, let E be the restriction
of/~ to X and let h: E ~ I E be a Hermitian metric on E. For such h we define
" g o o d " as follows:
Definition. A Hermitian metric h on E is good on X if for all x e X - X
and all
bases e~ .... , e, of/~ in a neighborhood A' of x in which J ? - X is given as above
k
by Iq zi=O, if hij=h(ei, ej), then
i=1
i) [hJ,(deth)-l<=C
log]zll
, f o r s o m e C > 0 , n>__l,
i=
ii) the 1-forms (Oh. h-1)ij are good on 2c~ U.
The first point about good Hermitian metrics is that given (E, h), there is at
most one extension/~ of E to 2 for which h is good. This follows from:
Proposition 1.3. I f h is good, then for all polycylinders A~c2 in which X - X
is
k
given by l~ zi=0,
i=I
F(A', ft.) = {s~F(A" c~X, E)] h(s, s) <=C . (~. log Izzl)2n, for some C, n}.
Proof The inclusion " c "
is immediate. As for " ~ " , if s = ~ a i ( z ) e ~ is a
i=1
holomorphic section of E on A r n X , for which h(s, s) is bounded as above, then
it follows that
la~(z)l _< C'(~ log Izil)2m,
Therefore
for suitable C', m.
z~ 9 ai(z ) is bounded on Ar, hence is analytic, hence at(z ) is
meromorphic with simple poles on J ? - X . But as no inequality
1
iz12 ~ C(log Izl)zn
holds, at(z) is in fact analytic.
QED
The main result of this section is the following:
Theorem 1.4. I f ft. is a vector bundle on X and h is a good Hermitian metric on E
=/~lx, then the Chern forms Ck(E, h) are good on $2 and the current [Ck(E, h)]
represents the cohomology class Ck(ff~)~H2R(x, tI~).
Hirzebruch's Proportionality Theorem in the Non-Compact Case
243
Proof. Let h* be a C ~ Hermitian metric on/~. Define
O=dh . h -1,
K=~O,
O* =dh* . h *-1,
K* = 00".
Intrinsically, K and K* are Hom(E,E)-valued (1,1)-forms and 0 - 0 " is a
Horn (E, E)-valued (1, 0)-form. According to results in Bott-Chern [5], for each k
there is a universal polynomial Pk with rational coefficients in the forms K, K*
and 0 - 0 * such that on X:
Ck(E, h) - Ck(E, h*) = d(Tr Pk(K, K*, 0 - 0")).
Now K, K* and 0 - 0* are forms good on 9(, hence the (k, k)-form Tr Pk(K, K*, 0
-0") is good on 3(. It follows that Ck(E, h) is good on )~ and that
[Ck(E , h)] = d [ t r Pk] + [ek(E, h*)].
QED
represents c~(g)
w2. Estimates on Cones
The results of this section are purely preliminary. We have isolated all the
inequalities needed for the general proportionality theorem which involve only
the cone variables (cf. w3, definition of Siegel Domain), and worked these out in
this section.
The object of study then is a real vector space V and
CcV,
C an open, convex, non-degenerate ((7~(pos. dim. subspace of V), or equivalently, 3 1 e V , / > 0 on C) cone. Most of our results relate only to those C which
are homogeneous and self-adjoint; for any C, we let G c G L ( V ) be the group of
linear maps which preserve C, and say C is homogeneous if G acts transitively. If,
moreover, there is a positive-definite inner product ( , ) on V for which
(~= {x~V] (x, y ) > 0 , all y ~ C }
we say C is self-adjoint. The classification of these is well known (see [1], p. 63),
as in the fact that all such arise by considering formally real Jordan algebras V
and setting
C = {x 2 [ x~ V, x invertible}.
All convex non-degenerate cones C carry several canonical metrics on them.
First of all, there is a canonical Finsler metric on C, which is analogous to the
Caratheodory metrics on complex manifolds ([10], p. 49):
V x ~ C , t~Tx, c~- V, let:
]/(t)]
p~(t)=sup
.
l~C l(x)
244
D. Mumford
(Another canonical Finsler metric, analogous to Kobayashi's metric in the
complex case, can also be defined9 First introduce on the positive quadrant IR+
9 dx 2 dy 2
x IR+ the metric ~ + ~ - ;
then we have on any cone the definition:
,
fcanonical length in
)
Px(t)=~the cone C ~ ( R x + R t ) ~
=
-~ a~
if a 1 > 0 and a2<O are determined by x + a 1 t, x + a 2 teC, q~C. We won't need
this second metric however.)
The advantage of the Finsler metric is that (as in the complex case) it
behaves in a monotone way when you replace C by a smaller (or bigger) cone:
Proposition 2.1. i) I f C is" an open convex non-degenerate cone in V, and a t C,
x6C, t6V, then
px+,(t)c <=px(t)c.
ii) I f CI c C z are 2 open convex non-degeherate cones in V, then for all x6C~,
t~V,
px(t)c~ > px(t)c2.
(The proofs are easy.)
Now suppose C is homogeneous and self-adjoint. Then one can introduce a
Riemannian metric on C as follows. Chose a base point e~V which we take as
the identity for the Jordan algebra and let ( , ) be an inner product on V in
terms of which G=tG. Then ([1], p. 62), C is self-adjoint with respect to ( , ) .
Moreover, K =Stab(e) is a maximal compact subgroup of G and if g =1 @ p is a
Cartan decomposition with respect tO K and * : G ~ G the Cartan involution,
then:
1) (gx, g * y ) = ( x , y ) , hence K=exp(~) acts by orthogonal maps while P
= exp (p) acts by self-adjoint maps,
2) (gx)- 1 = g*(x- 1) (here x - 1 is the Jordan algebra inverse).
Now identifying Te,c with V, we use ( , ) to define a Riemannian metric on
C at e: since it is K-invariant~ it globalizes to a unique G-invariant Riemannian
metric on C, which we write ds~. For later use, we need the following formula:
Lemma. Take tl, t2 ~ Tx, c, and let f ( x ) =
inverse9 Then
ds2(tl, t 2 ) =
( tz,X-1)
where x-1 is the Jordan algebra
-D,~f
where Dr, is the derivative of functions on V in the direction t 1.
Proof Let g~exp(p) carry e to x. Then by G-invariance:
ds 2, x(tl ' t2 ) = ds 2, ~(g- 1tl ' g- 1t2 ) = ( g - i tl ' g - I t2 )
Hirzebruch's Proportionality Theorem in the Non-Compact Case
245
and
D , f ( x ) = Dg i t~(fo g)(e).
But
f o g(x) = (t2, (gx) -1 )
= (t2, g - 1 . (X- 1))
= ( g - l t 2 , X-1 ).
If 6xe V is small, then (e + 6 x ) - ~ = e - 6 x +(terms of lower order), hence
Dg-ltl(fo g ) ( e ) = - ( g - ~ t 2, g-1 t~).
QED
On a homogeneous cone C, any 2G-invariant metrics are necessarily comparable, so we can deduce, from the monotone behavior of the Finsler metric p,
a weaker monotonicity for ds~:
Proposition 2.2. i) I f C is a self-adjoint homogeneous cone in V and a t (_., then
there is a constant K > 0 such that
ds~,x+,(t,t)<=K.dS~,x(t,t )
all teV, x e C .
ii) I f C 1 c C2 are 2 self-adjoint homogeneous cones in V, then there is a
constant K > 0 such that
dsZc2,x(t,t)<K.ds~,,x(t,t),
all teV, x e C 1.
(The proofs are easy.)
The main estimates of this section deal with the following situation:
C a self-adjoint homogeneous cone,
G = Aut~
C) (o means connected component),
C, = cone of positive definite n x n Hermitian matrices,
p: G --* G L(n, C) a representation,
H: C ~ C, an equivariant, symmetric map with respect to p.
Here (p, H) equivariant means:
H(gx)=p(g)H(x)tp(g),
all x e C , geG,
(1)
while (p, H) symmetric means:
p ( g * ) = H ( e ) . ' p ( g ) - l . H ( e ) -1,
all geG.
(2)
Note that A~-.H(e).LA-1~ H(e)-1 is just the Cartan involution of GL(n, ~) with
respect to the maximal compact subgroup given by the unitary group of H(e):
calling this ~, we can rewrite the symmetry condition (2):
p(g*) = p(g)".
(2')
246
D. M u m f o r d
Condition (2) is actually independent of the choice of e: if e'eC is any other
point, and g~--~g*' is the corresponding Cartan involution then one can check
that (1)+ (2) imply:
p(g*') = H(e')tp(g)- ' H(e')- '.
(2")
We will also need for applications the slightly more general situation where p is
fixed but H depends on some extra parameters t ~ T with T compact. In this
case, we ask that (1) and (2) hold for each H t. Note that we can change
coordinates in Cn, to get a new pair:
p'(g) =ap(g)a- 1,
n~(g)=aHt(g)'5,
satisfying the same identities. In this way, we can, for instance, normalize the
situation so that
H,o(e)=l.
hence:
p(K) ~ U(n),
,
, (self-adjoint matrices, commuting~
ptexp 0) ~ I.~with Hi(e), all t
;
This normalization will not affect our estimates. The first of these is:
Proposition 2.3. For all 2 > O, there is a constant K > 0 and an integer N such that
II~(x)ll and IdetH~(x)l-l<K.(x,x) ~,
all x ~ ( C + 2 . e ) .
Proof. Let A ~ e x p ( p ) be a maximal R-split torus. Then C = K . A . e and C + 2 e
= K . (A e + 2 e). Write x = k(a(e)). Then
Ilnt(x)ll = IIn,(ae)ll
= lip(a)" Hi(e). p(a)ll
IIp(a)ll 2. Ilnt(e)ll.
We may change coordinates in Cn by a unitary matrix so that all the matrices
p(a) are diagonalized. Write then p(a)=zi(a)-6ij where Zi: A ~ R* are characters. Now we may coordinatize A by
A
~" , A . e ~ - R ' + c V ,
e~--~(1,...,1).
Let a~,..., a r be coordinates in R'+. Then
zi(a)= I~I aj',,
jffil
sijER.
Hirzebruch's Proportionality Theorem in the Non-Compact Case
247
Note that if a e ~ C + 2 e , then ae~F,+ +(2 ..... 2), i.e., ai>2, all i. Hence
Up(a)[] 2 < max (zi(a)) z
l <i<n
r
<=K 1 9K 2 ((ae, ae)) ~. . . . ,j)r
hence
IlHt(kae)ll < KI "Kz" Ka ((kae, kae)) t. . . . . j)r.
The same proof works for [det[-1.
QED
Next, let ~s V, and let D~ be the derivative on V in the direction ~. We wish
to estimate the matrix-valued functions
(DeH,). H/~: C ~ Mn(~).
To do this, we prove first:
Proposition 2.4. For all 1 <_<_a,fl <=n, let (Dr t. H;-1)~p be the (~,fl)th entry in this
matrix. There is a linear map
C~t~,t: V---~V
depending continuously on t such that
(Dell,. H;- 1),~(x)= ( C,e,,(~), x - 1>.
Moreover C,g,, has the property:
For some reason, I can't prove this by a direct calculation, but must resort to
the following trick:
Lemma. Let C c V be a convex open cone, let e~C, and let f: C--~r be a
differentiable function. Suppose that for all W = V, dim W = 2, and e~ W, f is linear
on C c~ W. Then f is linear.
Proof The hypothesis means that
f(ax+be)=af(x)+bf(e),
all x~C, a,b~IR+.
So we may extend f to all of V by the formula
f(x)=f(x+ae)-af(e),
provided x + a e ~ C .
248
D. Mumford
Note that
f ( t x ) = f ( t x + t a e ) - at f (e)
= t ( f ( x + a e ) - af(e))
= tf(x).
Thus for all x~V, n=>l:
f ( x ) :__f(x/n) -f(O)
n
.'.f(x)=Dxf(O)
and the right hand side is linear in x.
QED
We prove (DcHt. H;-1),o(x -1) is bilinear in ~,x. Since it is linear in ~, it
suffices to find a basis {~k} of V such that (DckH t 9Hi- l),~(x- 1) is linear in x. In
fact, we will show that for every ~eC, (DcHt. Hi-1)~(x -'1) is linear in x. To do
this, by the lemma, it suffices to show that for every W c V with dim W = 2, ~ e W,
(D~H t 9H7 1),o(x-1) is linear on C n W. We will do this for all a, fl at once, so in
verifying this we can change coordinates in C".
What we do is this: we let ~ be a new base point of C and we change
coordinates so that Hto(~) = I .. This reduces us to verifying (Dell t 9H;-1),~(x-1) is
linear on C ~ W when dim W = 2 , eeW. Now any such W is part of a subspace of
V of the form A. e, where A c exp (p) is a maximal IR-split torus (of course, this p
corresponds to the new choice of e). Moreover, as above, we can diagonalize p
on A:
p(a 1.... , a,) = (zi(a) fig,i),
Zi(a) = ~I a~~j"
j=l
Then
Ht(a e) = p(a) z Ht(e),
p(a)H t (e) = Ht(e ) p(a).
Since e=(1 ..... 1), it follows:
o,(n,(ae)) = i~1 ff~ai H'(ae)"
Using this, one calculates:
r
(Dell t 9H i- a),a(ae)= ( ~ 2s~---2/ 6~
~j= l aj l
and since on Ae, x -1 is given by (a 1..... a,)~---~(a~ 1.... , a ; 1), this proves that
(D~Ht.Hi-1)~,tj is linear in x -1 on every W = A e .
Now say ~, q~C, (r q ) = 0 . For a suitable coice of maximal torus A, we have
~, r/cA. e. In the coordinates (a 1..... a,) on A . e , let
~=(~1 . . . . . ~,),
,7 =(,71 .... ,,7~).
Hirzebruch's ProportionalityTheorem in the Non-Compact Case
249
Since ( , ) on Ae makes A self-adjoint, it is a quadratic form of the type (ae, b e)
= ~ 2 i a / b i, so (r t / ) = 0 means that for every i, r
or r/i=0. A calculation like
that just made shows:
(D,H,. HZ1)~#(oe)= ( ~ 2s~j~ t .~#
\j= 1 aj 1
i.e.,
(D~" H," Ht-1)~I3(x-1)'A.e= (j~12s~j~ jaj" 6~,).
This is clearly zero if x = ~.
QED
For the next result, suppose 6 is any vector field in the manifold of values of
t. Then
Proposition 2.5. For all vector fields 6 on T,
( 6 H I 9 H t 1)(X)
is independent of x, i.e., depends on t alone.
Proof. By equivariance:
3 Ht(g x ) = p(g). 6 Ht(x) . 'p(g)
hence
(6Ht "Ht 1)(g. e) = p(g). (6H t 9H t- 1)(e). p(g)- 1.
By symmetry:
p(g*). 6H,(e) = 6Ht(e ).'p(g)- I
hence
p(g*). (3Hr. H;- ')(e). p(g,)- i = (6 H,. n i- ')(e).
Together, these imply the Proposition.
QED
Proposition 2.4 gives us estimates on iIDcHt 9H;-lll. To work these out, we fix
a maximal flag of boundary components of C. In the notation of [1], p. 109,
choosing this flag and the base point eeC is equivalent to choosing in the
Jordan algebra V, a maximal set of orthogonal idempotents:
e = / 3 1 + ... +/3r.
Let
C~ = boundary component containing ~+ 1 + " " + ~,.
Then
250
D. Mumford
is the flag. Also let
~=CwC~wC2u...wC,
w(O)
and let
A = i ~'%"
i=1
Let P be the parabolic group which stabilizes the flag { C~}. Our estimates are
based on:
Proposition 2.6. (1) Let ~ e C and let ~'leV satisfy
( ~ , ~ ) = O~
_
~ <G,@=o.
t/~C
Then for every compact set o~cP, there is a K > 0 such that:
[(~'l,x-1)l<Kl/ds2,~(~l,~l),
all xeo~.a.e.
(2) Let ~leC, ~'leV be as above. L e t ~ 2 E C . Then for every compact set a~cP,
there is a K > 0 such that:
[ds2,x(~'~, ~2)] ~ KF/ds~,x(~, ~1)" ]/dscZ,~(r ~2)Proof. We will use the Peirce decomposition of V defined by the idempotents %:
v=| %
i_<j
where
a i + aj
XE Vij ~ E aie," x = - ~ -
x.
This decomposition is orthogonal with respect to ( , )
in the subspace @ Vw If ~IECk, then note that
and
Ck
is an open cone
k<i~_j
~1• |
%
i<__j<_k
and
@ V~j~(the boundary component of C corresponding to e 1 +... + ek).
i<-_j<=k
This boundary component is open in
i<j<k
or
~.)
(~ Vii. Thus
i<j<k
Hirzebruch's ProportionalityTheorem in the Non-Compact Case
251
To prove (1), let x = g a e , g~co, a e = ~ a i e i. Then
<~,1,x - 1) = <g-1 ~'l,(ae)-
=
-<(g
i=l
1)
~0.,~i>
ai
where (g- 1 ~'l)ii is the component of (g- 1 ~'1) in V/i. But co preserves the flag, so
g- 1~6(~) Vij
i<--j
k<j
tOO. Thus
<~'I'X--I>=
i = k~+ l
l((g-l~])"'ei>"
As g varies in co, ((g-l~'l)ii, ei) is bounded, hence
i=k+l ai
Let e ~k~= ek+ ~ +... + er be the base point in C k and note that, again by compactness of co, there is a K 2 > 0 such that
g--l~l~Ck+KEe~k),
all g~co,
hence
ds2,x(KEetk),K2e~k))<ds~,x(g-l~l,g-l~O,
all x e C .
Thus
-1 ~ y,, g-1 ~1)
ds2,gae(~l, 41)=ds2,ae(g
> g 2 d s 2 a e (e(k), e(k))
= K2ds2,e(a- I e(k), a- i etk))
= K 2 i ~k+l~<ei'ei>
The same procedure proves (2). If ~ e C k, ~2 e Cz, the argument goes like this:
2
r
dsc,g.e(~l.~2)=<
a - 1 (g - l~i),a-1(g-l~2)>
1
= ~ --<(g-x~'l)ij,(g-1~2),j>
[i~_j\aiaj
it<i|
~k < j /
<- g ~
~ ~"
)_2 - ~
i= 1 ai ~ j = k + l aj
<K~K2I,/ds2~(r162
9 1,/ds2~(~2,~2).
QED
252
D. M u m f o r d
We can put everything we have said together as follows: Suppose N = dim V
and 41 ..... ~N~Cspan V. Define a simplicial cone a c C by
N
i=1
Let l~: V ~ I R be a dual basis: li(~j)=6~j. Then for all p,H t as above, we have the
estimates:
Proposition 2.7. For all vector fields 6,6' to the T-space, and a ~ C there is a
constant K > 0 such that for all x~Int(a + a):
K
= li(x ) - li(a ) '
II6H,. n ~ 1(x)II <=K,
IIDr
Ht- l(x)l[ <
[[O~,(o~jn,. H? 1)(X)I [
IIOr
K
<=(li(x ) -li(a)). (lj(x) - lj(a)) '
9n,- 1)(x)ll = 0,
K
[16(oe,n,. n 21)(x)ll < l,(x)- l~(a)'
116(6'H~. a,- 1)(x)l[ __<g.
Proof Combine (2.4), (2.5) and (2.6) and the formula ds2(tl, tz)= - D t l ( ( t 2, x - 1 ) )
to get estimates in terms of ds2(r ~i) on sets co. A. e. Then apply (2.2)(i) and (ii)
to the inclusion Int(a+a)--~ C, plus Ash's theorem that I n t ( a + a ) c c o . A .e if co
is large enough ([1], Ch. II, w plus the formula
ds z = ~ dl2i
i=a lZi
for the canonical metric on the homOgeneous convex cone a.
QED
There is one final estimate that we need. For this, we first make a definition:
Definition. A linear map T: C " ~ C" is called p-upper triangular if the following
holds: For all maximal R-split tori A c G , let X ( A ) = H o m ( A , ffJm) be the
character group of A. As is well known, there is a basis 71 ..... 7, of X(A) such
that the weights of A acting on V are contained in 7i+Yj, i ~ j (and contain 2~,
1 < i < r). Partially order the' characters by defining
EnlYi2>Emi~i
if
ni>mi,
all i.
Diagonalize p(A):
r
@ v~.
2EX(A)
Then T is p-upper triangular if for all A, all 2o~X(A),
T( |
,~-->3,o
V3=- |
A~)-o
V~.
Hirzebruch's Proportionality Theorem in the Non-Compact Case
253
The estimate we need is:
Proposition 2.8.
If T
is p-upper triangular, then for all a t C(F), there is a constant
K > 0 such that
Ilnt(x).'T.n,(x)-lll < K ,
all x E C ( F ) + a .
Proof. Take a as a base point of C(F) and pick any maximal torus A so that A e
-IR+," a = ( 1 ..... 1), C ( F ) = K . a . e . Therefore
C(F)+ a = { k a e l k ~ K , a = ( a
1 ...,an), ai> 1, all i}.
Change coordinates in C n so that
p ( a ) = |\/21(a)0"..
0 )
2,(a)
Now
nt(x).t ~. Ht(x )- 1 = p(k)nt(e ) p(a) 2. 'p(k) .t ~. tp(k)- 1 p(a)- 2 H t (e)- ' p(k)- t
so it suffices to bound [Ip(a)2.t(p(k) - 1. T.p(k)).p(a)-2[[. This means we wish to
bound:
12i(a)2 2j(a)- 2(p(k )- 1. T. p(k))~i [
when k ranges over K, and a = ( a 1..... a,) satisfies ai> 1, all i. This is equivalent
to:
(p(k)- 1. T. p(k))ji ~ 0 ~ 2j ~ 2 i
(,)
(weights of A being partially ordered as in the definition). But for all i define
W,=p(k)( |
r
~j > hi
(where eieC" is the ith unit vector). N o t e that k A k - 1 m a p s W~ into itself and that
W~ is one of the sums of weight spaces referred to in the definition. Therefore
T ( ~ ) ___~ , hence
( p ( k ) - ' rp(k))e,~ (~ ff?,.ej
Zj > 2i
which is precisely (,).
QED
w3. The Proportionality Principle
Let D be an r-dimensional bounded symmetric domain and let F be a neat 2
arithmetic group acting on D. Then X = D / F is a smooth quasi-projective
2 Recall that following a definition of Borel, a "neat" arithmetic subgroup F of an algebraic
group ~ c GL(n,r is one such that for every x~F, x~e, the group generated by the eigenvalues of
x is torsion-free. Every arithmetic group F' has neat arithmetic subgroups of finite index
254
D. M u m f o r d
variety, called a locally symmetric variety, or an arithmetic variety. In [1], Ash,
Rapoport, Tai and I have introduced a family of smooth compactifications ) f of
X such that X - X
has normal crossings. We must first recall how .g is
described locally. At the same time, we will need various details from the whole
cumbersome appartus used to manipulate D so we will rapidly sketch these too.
All results stated without proof can be found in [1].
By definition, D ~ K \ G, where G is a semi-simple adjoint group and K is a
maximal compact subgroup. Inside the complexification Gc of G, there is a
parabolic subgroup of the form P§ 9K e (P§ its unipotent radical which is, in fact,
abelian and Kr the complexification of K) such that K=Gc~(P+ .K~) and
G. (P§ Kc) open in Gc. This induces an open G-equivariant immersion
O'
,/5
K\G
P+.Kr162
Here I~ is a rational projective variety known as a flag space and G C is an
algebraic group acting algebraically on/}. L e t / ) be the closure of I) in D. The
maximal analytic submanifolds F c / 5 - D are called the boundary components
of D. For each F, we set
N(F) = {geG I g F =F},
W(F) = unipotent radical of N(F),
U ( F ) = c e n t e r of W(F), a real vector space of dimension k, say,
V(F)= W(F)/U(F): known to be abelian, centralizing U(F). Via "exp", we
get a section and write W(F) set-theoretically as V(F). U(F). Also
dim V is e v e n - let it be 2 l.
Next splitting N(F) into a semi-direct product of a reductive part and its
unipotent radical, we decompose N(F)further:
N(F)=!Gh(F ) 9GI(F) 9M(F)). V(F). U(F),
direct product mod finite
central group
where
a) G 19M . V- U acts trivially on F, Gh mod a finite center being Aut~
b) Gh 9M . V. U commutes with U(F), G~ mod a finite central group acts
faithfully on U(F) by inner automorphisms
c) M is compact.
Here F is said to be rational if F n N(F) is an arithmetic subgroup of N(F). Mod F
there are only finitely many such F, and if F1,..., Fk are representatives:
k
Xw
U (Fi/F c~N(Fi)),
i=1
with suitable analytic structure is Satake-Baily-Borel's compactification of X.
Hirzebruch's ProportionalityTheorem in the Non-CompactCase
255
Next, for each F, we define an open subset D r c / ) by
DF=
U
geU(F)c
g-D.
The embedding of D in D r is Pjatetski-Shapiro's realization of D as "Siegel
Domain of 3rd kind". In fact, there is an isomorphism:
Dr~ U(F)r
xF
such that not only N(F) but even the bigger group G h 9( M " Gl) C 9 V•" Ur acts by
"semi-linear transformations":
(x, y, t)F--~(Ax +a(y, t), Bty+b(t), g(t))
(A, B t matrices, a, b vectors) and
O = {(x, y, t) I Im x + l,(y, y)e C(F)}
where C ( F ) c U(F) is a self-adjoint convex cone homogeneous under the G,action on U(F) and lt: ~ l x ~ 1 ~ . U(F) is a symmetric IR-bilinear form.
Moreover
U(F)~-group of automorphisms of D: (x, y, t)~--~(x+a, y, t), aeU(F),
U (F)c = group of automorphisms of D (F): (x, y, t ) ~ ( x + a, y, t), ae U(F)r
W(F)~-group of automorphisms of D: (x, y, t)~-~(x +a(u, t), y+b(t), t) and
the group V(F) acts, for each t, simply transitively on the space ~ of
possible y-values.
There is a technical lemma which we will need about this action:
Lemma. Let toeF, eoeC(F ), and let uoeU(F)r be the map (x, y, O~--*(x+ieo, y, t).
Let e=(ieo, O, to) be a base point of D, so that Stabs(e)= K, a maximal compact of
G and Stab6r ) = K~. 1~ Moreover, Staba,(eo)= K z is a maximal compact in G I.
Since G t ~ Stab (0, 0, to), u o(G t) Uo i ~ Stab (e) and we may look at
0~: G 1 -c~176
, Stab~(e) ------~m~
Kr
I f * is the Cartan involution of G l with respect to Kl, then:
ct(g*) = ct(g).
Proof. This is a straightforward calculation, for instance, using the fundamental
decomposition of g = L i e G via sl(2)rcg (cf. [1], p. 182) and the description of
L i e ( M . G~) in this decomposition for the standard boundary components Fs
given in [1], p. 226.
We now describe local coordinates on X. Recall that )f is not unique but
depends on the choice of certain auxiliary simplicial decompositions. We need
not recall these in detail. The chief thing is that each X is covered by a finite set
of coordinate charts constructed as follows:
1) take a rational boundary component F of D,
2) take {~1, .-., ~k} a basis of Fc~ U(F) such that ~ i ~ C ( F ) c U(F) and in fact,
~i~C(F)L)Clk.)C2u...uCI=C,
where C ( F ) ~ _ C I ~ C 2 ~ . . . ~ _ C t is a flag of
256
D. Mumford
boundary components (cf. w
~ , ..., ~m~C(F),
and at least one ~ is in C(F): say
~ra+ 1,-.., ~keC(F) - C(F),
3) let li: U ( F ) e ~ r be dual to {~i}, i.e., li(~j)=6i i,
4) consider the exponential:
D
~(U(F)r
)
(e 2~'~'1~ .....
e2~'~%y, t)
D/F c~ U (F)c(IE *k x •' x F)
X
5) Define (D/Fc~U(F)) ~ to be the set of P e C k x C ~ x F
neighborhood U such that
which have a
U c~(r k X tEz x F) c (D/F ~ U (F)).
Note that
m
(o/rc~ U(F)) ~ ~i~=1 {(z, y, t) l z = ( z , , ..., Zk), Zi=0} =S(F, {~i}).
6) The basic property of X is that for suitable F, {~}, the covering map p
extends to a local homeomorphism
~: (D/F c~ U (F))~-o s
and that every point of X is equal to/~((z, y, t)), where z~-- O, some 1 _ i_< m, for
some such F, {~i}.
We now come to the main results of this paper. Let Eo be a G-equivariant
analytic vector bundle of rank n on D. E o is defined by the representation
a: K--~GL(n, C)
of the stabilizer K of the base point e+ ~D in the fibre ~2" of E o over e+. We
complexify a and extend it to P+ 9 K c by l~ting it kill P§ Then a defines a G~equivariant analytic vector bundle E o on D also. In the other direction, we can
divide E o by F obtaining a vector bundle E on X. Since K is compact, E o carries
a G-invariant Hermitian metric ho, which induces a Hermitian metric h on E.
We claim:
Main Theorem 3.1. E admits a unique extension E to X such that h is a singular
Hermitian metric good on .Y.
Hirzebruch's
Proportionality
Theorem
in the Non-Compact
Case
257
These various bundles are all linked as in this diagram:
Eo
-
r estfi_cti_on - ~ E 0 _ _ d e s c e n t
1
D ,
_ +
E - - -~x!"-~si-~-
1
D
~
E
1
*X
,)(.
Proof We saw in w1 that/~, if it existed, has as its sections the sections of E with
growth O
log Izil
\\i=
1
along X - X .
/
To see that the set of these sections
t
defines an analytic vector bundle on )(, it suffices to check this locally, e.g., on
(D/Fc~ U(F)) ~. But now the bundle /~0 restricts to a bundle E r on D e with
N(F). U(F)r acting equivariantly. N o w note that the subgroup U(F)~ acts
simply transitively and holomorphically on the first factor U(F) of D v in its
Siegel D o m a i n presentation. Since 117~x F is contractible and Stein (F is another
bounded symmetric domain), it follows that E v has a set of n holomorphic
sections e 1, ..., e, such that
i) e i is U(F)e-invariant,
ii) el(x), ..., e,(x) are a basis of Ev(x ), all xeD~.
Dividing by Fc~ U(F), E F descends to a vector bundle E~ on tE *k x C l x F, which
is also globally trivial via the same basic sections e l, ..., e,. We can then extend
E} to ~2Rx C tX F so as to be trivial with these basic sections. We must show
that along S(F, {~}) the sheaf of sections of this extension is exactly the sheaf of
log IzJ
sections of E ) on (D/Fn U(F)) with growth O
k
on the coordinate
i
hyperplanes U (zl =0). Equivalently, this means that h(ei, Q) and (det h(e i, ej))-1
i=1
have this growth. To do this, it is convenient to use a 2nd basis of E), which is
C ~ but not analytic. Note that V(F). U(F)e acts simply transitively on the 1st 2
factors of D r. So we can find e'l . . . . . e',eF(De, Ee) such that
i') e i is V(F). U(F)r
ii') e'i(x) ..... e',(x) are a basis of Er(x), all xeDv.
iii') On (0, 0) x F, e'i = % hence are holomorphic sections.
{e~} and {el} are related by an invertible U(F)e-invariant matrix S; so that
and Idet SI-1 are uniformly bounded on subsets
([~*)kx
ISul
compact subset)
k
ofll]ZxF
c ( C ) xlI]~xF.
Therefore it is enough to check that h(e~, ej), (det h(e~, e~))-1 have the required
growth.
N o w if geG~(F), note that because g normalizes V. Ue, ge'i is another V. Ueinvariant section of E}. Therefore
g" (el)= i rig(t)" ej
j=l
(here t is the coordinate on F).
258
D. M u m f o r d
Since g(U(F)r x 112t x {t})= U(F)e x i~ l x {t}, it follows that for each t,
g~-~p,(g)=(rij(t))
is an n-dimensional representation of Gt(F ). In fact, as g(0, 0, t) =(0, 0, t), this is
just the representation of the stabilizer of (0, 0, t) restricted to G~(F). This shows
that Pt is a h o l o m o r p h i c family of algebraic representations of G, (this is not
trivial because G, has a positive dimensional center). Since Gt is reductive, we
may change our basis {el} so that Pt is in fact independent of t.
N o w consider the functions hit = h(e'i, e~) on D. Since h, el and e~ are U . Vinvariant, so is hlj , hence in Siegel domain notation, it is a function of u = Im x
+ lt(y, y) and t, i.e., is a function on C(F)• F. F o r each fixed t~F, and variable
u~C(F),
U,(u) = (hlj(u , t))
is a map
(cone of pos. def. n x n].
!
C(F)-* C, = \Hermitian matrices
I claim that (p, Ht) satisfy the hypothesis of w
In fact,
Ht(gu), J = h(ge I, gej)
= Y'. r,k(g) h(e'k, el) rjt(g )
k,l
= ( p ( g ) . Hi(u). 'p(g))ij.
Let e = ( i e 0, 0, to) be a base point of D. Since h is a K-invariant metric on E0,
h(ke~, kej)(e) = h(e'i, ej)(e),
all k~K.
Complexifying, we get too:
h(ke'i, kej)(e) = h(e'i, ej)(e),
all k6Kq:.
Let uo~U c be given by (x, y, t)~--~(x + ie o, y, t). Then for all g~Gl(F), UogUol(e)
=e, so U o g U o l = k . p , where k ~ K c , peP+. By the lemma above, u0g*Uo 1
= k . p' for some other element p'EP+. Therefore:
h(e'i, ej)(e)=h(pe'i, p' ej)(e)
(since P+ acts trivially on Eo(e))
= h(kpe'i, kp' ej)(e)
= h(u o gUo l(e'i), u o g* u o 1(ej))(e)
= ~ rik(g) h(e'k, el)(e) ~ l(g*)
(since u o e'i = e'i, all i)
or
Hto(eo ) = p(g) H,o(eo ) tp(g.).
In fact, the same holds if t o is replaced by any t~F as follows easily using the
Gh(F)-invariance of h and the fact that Gz(F ) and U(F)r c o m m u t e with Gh(F ).
Hirzebruch's Proportionality Theorem in the Non-Compact Case
259
Thus we have the full situation of w In particular, we have available all the
bounds of w2.
As above, to describe local coordinates near boundary points of )(, choose a
simplicial cone:
k
~isC(F).,~l <_i<_m
~ = ~ IR+-~i C C(F);
i=1
and let Ii be dual linear functionals on U(F)r Then if (x, y, t) are Siegel domain
coordinates on D(F), and zi=e 2'~tax), then (z, y, t) at points where at least one z i
is 0(1 <i<m) are local coordinates on )f. Moreover each point PsS(F, {~i})c)f
has an open neighborhood whose intersection with X is contained in the image
of
{(x, y, t) ] u = Im x + It(y, y)~a + a, y s Y', t6F'}
for some a as above, as C(F), Y' (resp. F') a relatively compact subset of Y (resp.
F). Note that l o g ] z i [ = - 2 n l i ( I m x ). At all points P~S(F,{(i}), we have to
estimate h(s'i, sj) -+1 in terms of
choose C large enough
SO
that ~
<
1 in
a
neighborhood of P ) . This is the
same as estimating h(sl, sj) -+~ in terms of
(y, l,(u) + c) 2"
(choose C large enough so that l~(u)+ C > 0 in a neighborhood of P). But if l~(u)
+ C > 0 , (~li(u)+ C) 2u is comparable with (~l,(u)2) u, hence with (u, u>u. This is
exactly the estimate that Proposition 2.3 gives us. Next we have to estimate the
connection and the curvature. Now in terms of a holomorphic trivialization of
E o, the connection is given by 0h. h -1. What we have is good control of
61h.h -1 and 6 2 ( 6 t h . h -1) for all vector fields 61, 6 2 in terms of the real
analytic trivialization given by {el}. Write
e i = ~ aij ej,
j=l
A = matrix (aifl,
H ~ = matrix h(e i, ej).
Then
Ha"=A. H.'A.
From this you calculate:
Connexion in~
trivialization ~ = OH"". (H a")- 1
{ei}
/
t~A.A-I+A.OH.H-1.
A - I + A 9H . (OtA 9t A - 1 ) H - 1 A - 1 ,
260
D. Mumford
connexion in\
trivialization ~ = d(O H a" . ( H a") - 1)
{e~}
/=d(Oa
. A - X ) + d ( a 9 O H . H - x . A - x)
+ d ( A . H . (O'A. t ~ - a ) . H - 1 . A - l )
=d(OA. A-a)+dA
9( O H . H - X ) 9A - x
+ A 9 d ( O H . H -x)- A - x + A . O H . H - x 9 A - x 9 d A 9 A - x
+dA.
[ H . ( O * A . ' A - 1 ) H - X ] . A-X
+A(dH.
+A.
H - x ) [ U 9(991. tA-x). H - X ] . A -x
[ H . d(O ~ . t A - l ) . H - x ] 9 A -1
+ A . EH . (O'A . tTt- X) . H - X] . ( d H . H - X) . A - x
+ A . [ H . (OtA 9 ~Zl-x) 9 H - x ] 9 A - x . d A . A -1.
Therefore, since A is a C a metric on ){, to show that the connexion and its
differential have Poincar6 growth on )( it suffices to prove that the 4 forms:
0 H . H -1,
d ( O n . H - 1),
H.(0~I.'d-1).
H. d(0~.'4-1).
H-l, '
H-X
have Poincar6 growth on X. To check this for the first two, note that H is a
function on C ( F ) x F , hence it suffices to bound 6 1 H . H -1, 6 2 ( 6 x H . H -x) for
all vector fields fix, 6z on C ( F ) x F. But the Poincar6 metric is given in Siegel
coordinates by:
ds z = ~
[dzilZ z.
z ~ - E [ d y ' I E + E Idt't2
Izil= ( l o g - C' )
tdl~(x)[ 2
=Y" (/,(Im x ) + C') 2
~_~ Idyll 2 +~" Idt~l2
(choose C large enough so that l i ( I m x ) + C > O near the boundary point in
question). Therefore the bounds of Proposition 2.7 imply that OH. H -x and
d ( O H . H - 1 ) have Poincar6 growth.
To check the result for the last two, we need to know what sort of a function
A is. Firstly, since e i and e'i are both Ue-invariant, A is Ue-invariant, i.e., is a
function of y and t alone. Therefore it has derivatives only in the y and t
directions, so to say these have Poincar6 growth is just to say they are bounded
along X'. Next, for all t o , F , the action of Von vector space of points (y, to) puts
a complex structure on V. Thus it defines a splitting Vr Vto
+ @ Vto, where V.to+
are complex subspaces and V~o acts trivially on the points (y, to), while V~o
+ acts
simply transitively. If we fix one t o, then Vt+ still acts simply transitively and
holomorphically on the vector spaces of points (y, t) for t near t o. Thus it is
natural to choose our holomorphic basis e i to be in fact V~+ . Ue-invariant.
Hirzebruch's Proportionality Theorem in the Non-Compact Case
261
Moreover, it is easy to see that G~. V normalizes Vto+ 9 Uc for all t o, hence that
the action of Gt " V in terms of the holomorphic basis is given by
g(el)= ~ ~ij(t).ej
j=l
where
g ~ - ~ ( g ) = (matrix ~ij(t))
is a representation of G~. E Comparing p and ~, we get:
P(g) = g* A - l " Pr(g)" A.
Since ei=e'i on (0, 0 ) x F , P(g)=Pt(g) for all g~G, and A(0, t)=I,. Thus if v(0, t)
= (y, t),
I. = p(v)= A(y, t)- 1. ~t(v) . I,,
or
A(y, t)=pAv).
Now we use the simple:
Lemma. Let a be an algebraic representation of G I 9 V, and let a o be the restriction
of a to G I. Then for all veV, a(v) is ao-upper triangular.
Proof. Let A ~ GI be a maximal IR-split torus. As in w there is a basis 71 . . . . . 7,
of the character group of A such that A acts on the vector space U(F) containing
the cone C(F) through the weights 7i +7i- Then its action on V(F) is through the
weights 7i (cf. [1], p. 224). Now if V/(F)~ V(F) is the root space corresponding to
7i, and if we diagonalize a(A):
r
|
w~,
~eX(A)
then
V,(F)(W~)= W~+~,
hence V(F) acts in a ao-upper triangular fashion.
QED
Thus A(y, t) is p-upper triangular for all y, t, hence so are
A-I.t~A
and
d(A-I.OA).
Applying Proposition 2.8, it follows that
H.t(A-I.SA).H-X
and
H.'(d(A-X.~A)).H-X
are bounded in a neighborhood of every point of )~, as required. This completes
the proof of the Main Theorem.
A natural question is whether these vector bundles/~ are in fact pull-backs of
vector bundles on less blown up compactifications of D/F. Thus Baily and Borel
262
D. Mumford
defined in [2] a "minimal" but usually highly singular compactification (D/F)*
of D/F. Unfortunately /~ is only rarely a vector bundle on (D/F)* (we will see
below one case where it is however). However, in [1], Ash, Rapoport, Tai and I
defined not only smooth compactifications of D/F but also a bigger class of
compactifications with toroidal singularities (cf. [9]). These are important
because when you try to resolve (D/F)*, often there is a D/F with relatively
simple structure on the boundary but still with some toroidal singularities. It is
easy to see that the construction above of/~ goes through equally well on all of
these compactifications: it gives vector bundles on all of them such that
whenever compactification a dominates compactification b, then extension a is
the pull-back of extension b.
The Main Theorem, plus Hirzebruch's original proof of his proportionality
theorem for compact locally symmetric varieties X, gives us easily the proportionality theorem in the general case:
Proportionality Theorem
3.2. As above, fix:
X = an arithmetic varieiy D/F, D = K \ G,
.~=a smooth compactification as in [1],
= compact dual of D.
Then there is a constant K, which in terms of a natural choice of metric on D is the
volume of X, such that for all:
Eo = Gr
analytic rank n vector bundle on
defined by a representation of Stabo~(e ) trivial on P+,
ff~= corresponding vector bundle on X,
the following formula holds:
C~(/~)=(__ 1)dimX .
K . c~(ff~o),
all ,=(ct I . . . . . ~,), ~, ~zi=dim X.
Proof As above, choose a G-invariant Hermitian metric h o on E o. By the Main
Theorem, h o defined a " g o o d " Hermitian metric h on E, hence its Chern forms
Ck(E, h) represent the Chern classes of /~. But on D, Ck(E, h) are G-invariant
forms, so:
c~(ff~)= ~ c~(E, h)
X
=
S
c~(Eo ' ho)
F' al Domain
FeD
= vol (F). c~(Eo, ho)(e).
Now if Gc is a compact form of G:
Lie G = t 0 0 ,
Lie Gc = f ~ ip
Hirzebruch's Proportionality Theorem in the Non-Compact Case
263
then /~o has a unique GC-invariant Hermitian metric/~ equal to h 0 at e. So
/3
= v o l (/)). c'(/~ o,/~)(e).
T h e n - a n d this is the essence of Hirzebruch's remarkable p r o o f - a simple local
calculation shows (cf. [7]):
ek(E, h)(e)=(- 1)k Ck(Eo, h~).
This proves the result.
To apply this result, it is important to describe the bundles E as closely as
possible. Firstly, we can characterize their sections, precisely as a special case of
the general definition of automorphic forms given by Borel [3]. Let
p : K-* GL(n, ~) be a representation of K, and let
Eo=GxK~"
be the associated G-equivariant vector bundle over D = K ' - . G (i.e., E o--set of
pairs (g,a), mod(g,a)~(kg, p(k)a)). E o has a complex structure as follows:
complexify p and extend it to K c.P+ to be trivial on P+. Then E o is the
restriction to D to the bundle
/~o = Gr x tKr247
on /), and in the definition of/~o, everything is analytic. Borel introduces a
measure of size on G by:
Hglla = tr(Ad g*- 1. g)
* - C a r t a n involution on G w.r.t. K,
and defines holomorphic p-automorphic form f to be a function
f: G--,~"
such that
(1) f(kgT)=p(k)f(g),
all keK, 7~F,
(2) f induces a holomorphic section of Eo,
(3) [f(g)l<C'llgt]~,
some n > l , C > 0 .
Then one can show:
Proposition 3.3. In the above notation:
F(X, E) ~- {vector space of holomorphic p-automorphic forms}.
Sketch of Proof The problem is to check that the bound (3) is equivalent to
requiring that the corresponding section of g over X has growth O((~ log [zil)2")
264
D. Mumford
along Jr. But
Vx~D:
IlgllG
defines a measure of size on D and on X by:
Ilxllo=llgllG
i f x corresponds to the coset K . g ,
V ~ e X , image of x: II~lLx=min IIy(x)llo=min
~'EF
~er
IIgTLIG.
Then holomorphic p-automorphic forms are clearly holomorphic sections s of/~
over X, such that
h(s,s)(x)<Clllxll~x,
some n > l , C1>0.
But if d o is a G-invariant distance function on D, then it is easy to see (using G
= K . A. K) that do(x, e) and log Ilxll are bounded with respect to each other. In
another paper [4], Borel has proven that if x is restricted to a Siegel set
= co. A t 9e c D, then
min do(x T, e) ,~ do(x, e),~ d a( a(x), e)
~EF
(here x=co(x)a(x).e and ~ means the differences are bounded). Applying this
to a subset of a Siegel D o m a i n of 3rd kind of the type {(x,y,t)lyeV',teF',
Re xeU', Im x-l~(y, y)ea + a} where U ' c U, V' c V,, F' c F are compact subsets
and a c C(F) is a simplicial cone and aeC(F), we see that
min do((x, y, t) 7, e)
"t~F
can be bounded above and below by expressions
C210g((Imx-l,(y,y),e)),
hence
IlXEIx can
eeC(F), C 2 > 0
be bounded above and below by expressions
( I m x - l,(y, y), e)",
e~ C(F), n ~ 1.
Describing a as li> 0 as above (ll linear functionals on U), this is of the same
size as
( Z li(Im x) + C3)"
and as z i = e 2~ Z,(x), this is equal to
(-~log(bzil/C4))".
QED ~
Next, there are 2 particular equivariant bundles where we can describe /~
more completely:
Proposition 3.4. a) I f E o = 0~, the cotangent bundle, with canonical G-action, then
E = O ~ (log), the bundle on X whose sections in a polycylinder A " c X such that
A nn ( ~ ? - X ) =
~) (coordinate hyperplanes)
i= 1 \ z i = 0
Hirzebruch's Proportionality Theorem in the Non-Compact Case
265
are given by
f
i=1
dz i
zi
a~(z)--+ f
bi(z) dzv
i=k+l
b) I f E o is the canonical line bundle ~2"n, then E is the pull-back of an ample
line bundle (_9(1)on the Baily-Borel compactification X* of X. The sections of (9(n)
are the modular forms with respect to the nth power of the canonical automorphy
factor given by the jacobian, hence (9(n), n ~ O, is the very ample bundle used by
Baily and Borel to embed in X* in IPN.
Proof. Using Siegel Domain coordinates (x, y, t) on D(F), a U(F)e-invariant basis
of t2~tF) is given by {dx i, dyi, dtk}. Therefore these span the corresponding bundle
on X near the boundary F. But here
{Z i = e2ni li(x), Yi' tk}
are coordinates and these differentials are ~dzi, dy2, dtk~. This proves (a).
~ zi
)
To prove (b), recall that X* is set-theoretically the union of X and of
F / F n N ( F ) for all rational boundary components F. Moreover, if
P e F / F n N ( F ) c X * , then there exists a neighborhood U ~ X * and an open set
V c D such that V maps to Uc~X and V is a (Gt(F). V(F). U(F))nF-bundle
over U n X. Now say {sl} is the U(F)r
holomorphic basis of ~o on
D(F) used to extend E over the F-boundary points of .(. If we verify that each sl
is (G~(F). V(F). U(F))nF-invariant it follows that E is trivial on U n X and
moreover, if n: X ~ X * is the canonical birational map, then {s~} are a basis of
~z,/~ over U. Thus ~./~ is a vector bundle which pulls-back to E on X. Now in
the case in question, E is a line bundle, sl can be identified with the differential
form
(Ai dx,) A (Aj dyj) A (A
dtk)
k
on D(F), and G~. V. U acts on it by multiplication by the Jacobian determinant
in the Siegel Domain coordinates. But Baily-Borel ([2], Prop. 3.14) showed that
the Jacobian on (Or" V. U)nF was a root of unity. Since F is neat, it is one and
(Gt" V. U ) n F indeed fixes sl. The last assertion is just a restatement of Proposition 3.3 for this special case. Q E D
The following consequence of the proportionality principle seems to be more
or less well known to experts, but does not seem to be contained in any
published articles:
Corollary 3.5. Let L = (~2~6)- 1 be the ample line bundle on D, and let
P(1) = z( L|
be the Hilbert polynomial of D. Let 7r: R - * X* map a smooth compactification of
X onto Baily-Borers compactification. Let n 1 = d i m ( X * - X ) . Then there exists a
266
D. Mumford
polynomial Pl(l) of degree at most n 1 such that for all l > 2 :
dim
] ~ vol(X)* P(l-
[cusp forms on D
I_w.r.t. F of weight lJ
1) +PI(I).
Proof. The R i e m a n n - R o c h t h e o r e m gives us a "universal p o l y n o m i a l " Q such
that if L is any line bundle on a s m o o t h projective variety W,, then
)~(L) = O(c, (L) ; c1(f21) ..... c,,(f2~v)).
Therefore if n = d i m D,
( - 1)" vol(X) 9P( - l) = ( - 1)n vol(X) 9Z((O~-)| ')
=(-
1)"vol(X).
Q(lc,(O~); c,(f2~) .... ,c.(O~))
= Q(lcl(Y2 ~ (log));
C 1 (~c~l(log)) .....
c,(f2~-(log)))
by Proportionality T h e o r e m 3.2.
Consider a typical term
[lc x (f2~(log))] k. c~(f2~-(log)),
]c~]+ k = n.
N o w by Proposition 3.4b:
c 1(f2~-(log)) = re* H,
H an a m p l e divisor on X*. Let n 1 = d i m ( X * - X ) . If k > n 1, the cycle class H k on
X* is represented by a cycle s u p p o r t e d on X alone, hence so is n*H k. Thus if
k>nl
(1. n* H) k. c~(f2~-(log))=(l 9x ' H ) k. c~(f2~-).
Therefore
Q(lct(O~(log)); c, (f2~(log)), ..., c,(O~- (log)))
= Q(lcl (f2~(log)); c I (12~-)..... c,(f2~)) + (polyn. of degree < n:)
= X(f2~(log) |
+ (polyn. of degree < nl)
= ( _ 1), )~(f2~(log)|174
f2~) + (polyn. of degree < nl)
by Serre duality.
Thus for suitable P~ of degree at most nl:
vol(X) 9P(t - 1) = Z(Q~ (log) |
- 1) @ f2~) - P~(l).
But since (f2~(log)) | is generated by its sections and m a p s X to X * of the same
dimension for N~>0, K o d a i r a Vanishing (cf. [13]) applies if 1>2 and we have
h~
( l o g ) ' - ~ | f2]) = vol(X). P ( I - 1) + P~(l).
Hirzebruch's Proportionality Theorem in the N o n - C o m p a c t Case
267
The left-hand side is exactly the space of sections of ~2](log) t which vanish on
the boundary. By Proposition 3.3, these are exactly the cusp forms of weight
I. QED
w4. Applications: General Type and Log General Type
The purpose of this section is to consider the application of the preceding theory
to the question of when D/F is of general type, and to reprove as a consequence
of our theory the following theorem of Y.-S. Tai ([1], Ch. IV, w1).
Tai's Theorem 4.1. I f F is any arithmetic variety acting on a bounded symmetric
domain D, then there is a subgroup Fo c F of finite index such that for all F1 ~ Fo of
finite index, the variety D/~ is of general type.
We recall that if X is any variety of dimension n, we say that X is of general
type, if for one (and hence all) smooth complete varieties )( birational to X, the
transcendence degree of the ring
(~ r(,Y,(n~)|
N=O
is (n+ 1). More generally, the transcendence degree of this ring minus one is
called the Kodaira dimension of X.
Recall that Iitaka [-8] has recently introduced a complementary theory of
"logarithmic Kodaira dimension" for arbitrary varieties Y In fact, he first
chooses a smooth blow-up Y' of Y and then a smooth compactification I7 of Y'
such that I 7 - y ' has normal crossings and defines t21(log) as the complex of 1forms
dz i
i=1
ai(z)---}zi
~
i=k+l
ai(z)dzi
k
if, locally, 17-Y' is given by 1-[ zi=O. By definition Q~(log)=Ak(O~(log)). He
i=l
then looks at the "logarithmic canonical ring":
R = (~) F(F,,t2~(log|
N=0
He shows that this ring, as well as all other vector spaces of global forms with
logarithmic poles (obtained from decomposing (2~(log)| ... | t2~(log) under
the symmetric group and taking global sections) are independent of the choice
of Y' and I7. He then defines the logarithmic Kodaira dimension of Y to be the
transcendence degree of R minus 1. We may restate Proposition3.4(b) in this
language as follows:
Proposition 4:2. I f F is a neat 3 arithmetic group, then D/F is a variety of
logarithmic general type, i.e., its logarithmic Kodaira dimension equals its
dimension.
3 Some hypothesis on elements of finite order is needed because
log general type!
H/SL2(Z)---A l
which is
not of
268
D. Mumford
Proof In fact, by Proposition 3.4, R is just the homogeneous coordinate ring of
the Baily-Borel compactification of D/F.
Note that D/F of logarithmic general type is weaker than saying D/F is
general type.
I would like to add one comment to his theory which, in some cases, makes
it easier to apply: one does not need smooth compactifications, but merely a
toroidal compactification 17 of Y' (cf. [9], p. 54). This means that locally Y' c I7 is
isomorphic to ( ~ * ) " c X , , where X~ is an affine torus embedding (i.e., (r
is
Zariski-open in X,, X, is normal affine and translations by (E*" extend to an
action of ((E*)" on X,). On X~, define f2~,(log) to be the sheaf generated by the
((E*)"-invariant 1-forms. Carrying these over, we define I2-~(log) to be the
coherent sheaf of 1-forms on 17, regular on Y', isomorphic locally to f2Jc~(log). If
17'---~17 is an "allowable" modification of toroidal embedding of Y ([9], p. 87),
then p*(f2~(log_))~12~,(log). In particular, there is always a smooth allowable
modification Y' ([9], p. 94). So Iitaka's spaces of forms with log poles can be
calculated equally well on a smooth 17 or a toroidal 17.
This extension is helpful in checking the analog of the ~tbove Proposition for
the moduli space of curves:
Proposition 4.3. Let OYil(")be the moduli space of smooth curves of genus g with level
n structure. I f n > 3, then OJl(') is of log general type.
--
- - g
Sketch of Proof The proof follows the ideas of [12], w5 very closely. Let Hg be
the Hilbert scheme of e-canonically embedded smooth curves of genus g. Let
H(g")---~H be the covering defined by the set of level n structures on these curves.
Let H_, 9J/_s_be the compactified spaces allowing stable singular curves as well.
Let //~), 9J/~"' be the normalization of H g, gJ/j in the coverings H~"),~
("'_._g. The
group G = P G L ( v ) ( v = ( 2 e - 1 ) ( g - 1 ) ) acts on Hg and on H~"), freely on the latter,
so that -9J/g
- "-~
= Hg/G,
~(")
._.g -~
___g
iq(")/C,
,_. We have the diagram:
_
g -
r)(n)
where D and c~ are the universal curves. Recall from [12] the notation:
whenever p: C ~ S is a flat family of stable curves,
2 = Agp,(O~c/s),
O9c/s= relative dualizing sheaf
and if p is smooth over all points of S of depth zero, then A c S is the divisor of
singular curves and
6 = ~Os(Zt ).
Hirzebruch's Proportionality Theorem in the Non-Compact Case
269
Now on all 3 families above, we wish to show 2 1 3 | - 1 and the sheaf of top
logarithmic forms t2"(log) are isomorphic line bundles. Firstly, for p : / ) g ~ Hg,
we proceed like this: a) a simple modification of the proof of Theorem 5.10 [12]
shows:
213 |
6 - 2 _~ A
3g- 3 (p, ((2~/a | o9~/a)).
b) Since //g represents the functor of e-canonical stable curves, Ta,,tcl is
canonically isomorphic to the vector space of deformations of C. This has a
subspace consisting of deformations of the e-canonical embedding where C
doesn't change, and a quotient space of the deformations of C alone:
0 ~ Lie G --~ Tn~, to--+ Ext ~(f21, (tic) --* 0
or dually:
0 -~ H~
| COc)-~
f2~. |
IK(C)--,(Lie G)'---~O.
Therefore globally, we get
0--~ p,(f2b/a | COO/a)--+Q]~ --~ (Lie G)' |
(9.%~.0
hence if m = dim//g,
Hg
.'. Q ~ ( 1 o g ) ~ 213 @ 6 - 1
Secondly, for p: B~n)~I4~ n), 213 @6-1 pulls back to the analogous sheaf on //tgn).
Moreover,_because /4~n)~Hg is ramified only along A which has normal
crossings, H~n) has toroidal singularities and O~(log) pulls back to f2~,)(log).
Finally both bundles descend to ~tg,) and by Proposition l.4 [11], are still
isomorphic. Finally, it is proven in 1-12] (Th. 5.18 and 5.20: cf. diagram in w
that 213@6--1 is sample on ~ " ) . Q E D
In certain cases, there is a way of deducing that coverings of a variety of log
general type are actually of general type. To explain this, suppose we are given a
smooth quasi-projective variety Y,, and a tower of connected 6tale Galois
coverings:
n~: Y~-* Y,,
group F~.
We assume that any 2 covers n,,na are dominated by a third one nr:
y /
"~
270
D. M u m f o r d
Let 17 be a smooth compactification of Y with normal crossings at infinity.
Extend the covering Y~ to a finite covering
7~t I ]7 ___~1,7
be defining 17, to be the normalizations of 17 in the function field of Y~. We now
make the definition:
Definition. The tower {~,} is locally universally ramified over 17- Y if Jot all x~ 17
- Y, we take a nice neighborhood of x:
A n t Y,
A" c~(17-Y)= (union of coordinate hyperplanes)
\ Z 1 = 0 , ...,Zk-~-O
then for all m, there is an ~ and a commutative diagram:
~
~ 11(A")\
(A")\
U s it~
. ~
( m
An / ~
(Z 1 . . . .
.,
zl' ...~Zk~Zk+l~ ..
.,z.).
,Zn~
In other words, lr~-~(A"n(17-Y)) is cofinal in the set of all unramified coverings
of A" c~(17- r).
Then we assert:
Proposition 4.4.
Let Y c Y be as above and let n,: Y,--~ 17 be a tower of coverings
unramified over Y and locally universally ramified over 17-Y. Then if Y is
logarithmically of general type, there is an % such that for all cq such that the
covering n~ dominates n,o, Y~ is of general type.
Proof Let A = 17- Y, n = d i m Y and let O)=f2~z(log). Then we know that
a) for some N, there are differentials r/o.... ,r/,~F(s *N) such
ql/rlo ..... rl,/qo are a transcendence base of the function field C(Y),
b) h~
n+l i f N > N o.
F r o m (b) it follows that
r(~, co|
that
r(A, o~| @~ ~)
has a non-zero kernel for some N: let ~ be in the kernel. Replacing ~/i by ql | ~,
we may assume that all qi are zero on A. Now let's examine locally what
happens to qi when lifted to a covering of I7: let A " c 17 be a polycylinder such
that A " n ( 1 7 - Y ) = V I/In1 z~ . Write out qi:
\i=
1
k
/
rIz, (dz
-az"tN,
i= 1
Zk
\
Z1
!
Hirzebruch's Proportionality Theorem in the Non-Compact Case
271
ai holomorphic. Let w~= z ~TM , l<_i<k,
_ _
w~=z~, k+l<_i<_n and let
TOm: An-~ A"
be the covering of the z-polycylinder by the W-polycylinder. Then
dw i
1 dz i
-
wi
,
l<_i<k,
m zi
dwi=dzl,
k + l <_i<_n
hence
~*(rh)=mkUai'i~lw~" \
-w--1
wk
/"
So if m > N , rc*(th) is a h o l o m o r p h i c differential form on A'. N o w for each xe17
Y, fix a n e i g h b o r h o o d U~ c 17 of this type and a covering
-
which, over Ux, d o m i n a t e s rcN. 17- Y is covered by a finite n u m b e r { U J of U~'s,
so we can find one cover rc~o which d o m i n a t e s all the covers rc,~,). I claim that if
re,1 d o m i n a t e s re,o, hence rc,(x.~, then rc*~(qi) has no poles on a desingularization 17"1
over 17,,. This is clear because it has an open covering by open sets V~ sitting in a
diagram
U~
so g*(rh) regular
rc, l(t/i ) regular on Vi. But now rc, l (qi/tlo) are a transcendence
base of the function field of 17~1so I~, is of general type. Q E D
Let's consider the case Y = D / F again. If F is an arithmetic group, then for
every positive integer n, we have its level n s u b g r o u p F(n), i.e., if
F = ff(TZ.),
~ algebraic g r o u p over Spec Z
then
r(n) = K e r [if(Z) ~ Cg(Z/n Z)].
It's easy to see that
~.: n/r(n)-~ o / r
272
D. Mumford
is locally universally ramified over D / F - D / F . In fact, let F be a rational
boundary component. Then near boundary points associated to F, the pair
D/F c D/F is isomorphic to
D(F)/%(F) c~F c (D(F)/U(F) n 7Z)~
II
C*" x C " • F
~ " x ~ " • F.
Thus if U c D/F is a small neighborhood of a point corresponding in the above
chart to (O,y,t), ~I(Un(D/F)) is isomorphic to U(F)nF. Thus we must check
that for all n, there is an m such that
U(F) n F(m) c n. U(F) c~F.
But if F is rational, U(F) is an algebraic subgroup of the full group (~ which is
defined over ~, so this is clear. So now Proposition 4.2, Proposition 4.4 and this
remark altogether imply Tai's theorem.
It is now known that this same method will show that at least some highnon-abelian levels of 9J~g are varieties of general type too. It is not simple
however to check that the Teichmuller tower is locally universally ramified at
infinite. This has recently been proven by a ingenious use of dihedral level by
T.-L. Brylinski.
References
1. Ash, A., Mumford, D., Rapoport, M., Tai, Y.: Smooth Compactification of locally symmetric
varieties. Math. Sci. Press (53 Jordan Rd., Brookline, Mass. 02146), 1975
2. Baily, W.L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains.
Annals of Math. 84, 744 (1966)
3. Borel, A.: Introduction to Automorphic Forms. In: Symp. in Pure Math., Vol. IX (AMS, 1966),
p. 199
4. Borel, A.: Some metric properties of arithmetic quotients of symmetric spaces and an extension
theorem. J. Diff. Geometry 6 (1972)
5. Bott, R., Chern, S.: Hermitian vector bundles and the equidistribution of the zeroes of their
holomorphic sections. Acta math. 114, 71 (1965)
6. Cornalba, M., Griffiths, P.: Analytic cycles and vector bundles on non-compact algebraic
varieties. Inventiones math. 28, 1 (1975)
7. Hirzebruch, F.: Automorphe Formen und der Satz von Riemann-Roch. In: Symposium Internacional de Topologia Algebraica, Unesco, 1958
8. Iitaka, S.: On logarithmic Kodaira dimension of algebraic varieties. (To appear)
9. Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. Springer Lecture
Notes 339 (1973)
I0. Kobayashi, S.: Hyperbolic manifolds and holomorphic mappings. M. Dekker Inc., 1970
11. Mumford, D.: Geometric Invariant Theory. Berlin-Heidelberg-New York: Springer 1965
12. Mumford, D.: Stability of projective varieties. L'Enseignement Mathematique 1977
13. Mumford, D.: Pathogies III. Amer. J. Math. 89 (1967)
Received May 8, 1977