C OMPOSITIO M ATHEMATICA
K LAUS W IRTHMÜLLER
Root systems and Jacobi forms
Compositio Mathematica, tome 82, n
o3 (1992), p. 293-354
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Root systems and Jacobi forms
KLAUS
WIRTHMÜLLER
Fachbereich Mathematik der Universitiit Kaiserslautern, Postfach 3049, W-6750 Kaiserslautern, Germany
Received 8 March 1990; accepted 22 October 1991 (Ç)
Introduction
Let R be a
(classical)
rootsystem,
and letQ
denote its root lattice. For eachelliptic
curve E =C/(Z
+Z03C4)
theWeyl group W
of R acts on the abelianvariety
A =
Q z
E. Thequotient morphism
A ~A/W
of this actionplays
animportant
role in
understanding
deformations ofsimply-elliptic
surfacesingularities [Looijenga 2].
In fact it was for that purpose that the
W-variety A
was first introduced in[Looijenga 1].
On A there is anessentially unique
minimalample
line bundle 2on which W acts too. The
algebra
of W-invariant sections of 2 and its tensor powers has been studied in loc.cit., [Bernshtein
andShvartsman],
and[Saito 1, 2].
Later, Looijenga
reconsidered the same situation in the moregeneral setting
of affine root
systems [Looijenga 3].
In that context Anaturally
appears as afamily AH
of abelianvarieties, parametrized
via iby
thecomplex
upper halfplane
H. It was also notedby Looijenga
that the natural action of the modular groupSL2(Z)
lifts to thatfamily,
as well as to theample
line bundle2H
onAH .
Some
aspects
of that action have been discussed in the work of[van Asch]
and[Kac
andPeterson].
The purpose of this paper is to
study
the invarianttheory
ofAH
and2H
withrespect
to the action of the modular group(or
asubgroup
r of finiteindex)
aswell as of W We form the
quotients by
rand,
as a firststep,
seek to extend thefamily
over the cusps of r. This is achieved in a natural and
quite explicit fashion, using
a toroidal
embedding technique
from[Wirthmüller 2].
The invariants that we then
study
form abi-graded algebra
over thering
ofmodular
forms, graded by weight (referring
to behaviour withregard
tor)
andindex
(referring
to theappropriate
power of2).
We call these invariants Jacobiforms since in the
special
case when R isof type Al they
reduce to(weak)
Jacobiforms in the sense of
[Eichler
andZagier].
We succeed in
determining
thealgebra
of invariants for alltypes
of rootsystems excluding E8.
The invariantalgebra
turns out to be apolynomial algebra
over thering
of modularforms,
withgenerators
that do notdepend
onthe
particular
choice of the group r.The result has an
application
insingularity theory,
to deformations of fatpoints
in theplane
withdefining
idealThis
complements
the paper[Wirthmüller 1]
and will appear elsewhere.The author is
grateful
to thereferee,
whosuggested
variousimprovements
toSection 1.
The paper is
organized
as follows:1. Toroidal
embeddings
andreflexive
sheavesWe describe a class of toroidal
embeddings
associated with certainproperly
discontinuous group actions on open cones. This class is
sufficiently general
forour purposes but on the other hand
explicit enough
to enable us tocompute
thecohomology
of certain reflexive sheaves that arisenaturally
in the same context.2. A
family of
abelian varietiesWe introduce in detail the
family AH ~
H referred to at thebeginning,
as well asits
compactification.
We also describe the line bundleYH
and its extension as areflexive sheaf.
Using
the results of Section 1 we thencompute
thecohomology
of that extension.
3.
Jacobi forms
We define the notion of Jacobi form and formulate our main result
concerning
the structure of the invariant
algebra,
in Theorem(3.6).
4. Construction
of
Jacobiforms
We first discuss some
auxiliary
resultspertaining
to theproblem
ofextending
sections of a sheaf that are
given
on somearrangement
of divisors. These results facilitate an inductive construction of Jacobiforms, by picking
a suitable subroot
system
of R of smallerrank,
andextending
Jacobi forms associated with that sub rootsystem
and itsconjugates
under kK5. The individual root systems
The construction
prepared
in Section 4 is carried out on acase-by-case basis,
and
thereby
theproof
of the main theorem isfinally
achieved.1. Toroidal
embeddings
and reflexive sheaves(1.1)
Let V be a finite dimensional real vector space, and let A c V be a lattice with rank A = dim KDEFINITION. A set E
consisting
of cones in V is called anadmissible fan
if thefollowing
hold:(i)
Each (J E 1: is arelatively
open A-rationalpolyhedral
cone; it does notcontain any affine lines.
(ii)
If 6 E E then each face of abelongs
to E.(iii)
If 03C3 E E and T e S then à n 1-r is a union of faces of 6.(iv)
The interior ofis an open convex cone 7 c
V,
and c7
for each J E E.(v)
If C c V is any closed A-rationalpolyhedral
cone with C cII:I
then Cmeets but a finite number of a E E.
Note that if E is an admissible fan then
by (iv),
each J E E is either contained in Ior else is
disjoint
from it.By property (v)
then{03C3 ~ I | 03C3 ~ 03A3}
is alocally
finitecovering
of I.For any a E X we use the
symbol St(Q)
to denote the star of a(with respect
toE),
i.e. the union of all z E E with a c i. WhileSt(03C3)
need not be open inII:I
theintersection
St(Q)
n I isalways
open in 1. We also record thefollowing property
of the star.LEMMA. For each creX and any
y~I
one hasProof.
We choose a closed A-rationalpolyhedral
cone C’ c 1 u{0}
contain-ing y
in itsinterior,
andpick
somepoint
z~03C3~. Then the convex conespanned by
C’ and{z}
is a closed A-rationalpolyhedral
cone contained inJY-1.
By (v)
it meetsonly
a finite number of 03C4~E. Since theparametrized
linesegment
maps into C the
point (1- t)y
+ tz is contained in one and the same i E E for allt E
(0, 1) sufficiently
close to 1. Wenecessarily
have r ESt(u),
and thereforefor those t.
D
Let us agree to call a subset (D c E of an admissible fan closed if i ~ 03A6
implies
that all faces of 03C4
belong
to 03A6. Weput
and write
by
abuse oflanguage.
Writing Vc = C R V
we let for each such 4Ydenote the torus
embedding
determinedby 03A6 (i.e. by
the collection(à 1 u
E03A6},
inthe sense of
[Kempf
etal.] p. 24).
Weput
and let
denote the open subset
comprising
allpoints
that can berepresented by
vectorsfrom V + iI c
Vc.
(1.2)
Let 03A3 be an admissiblefan,
and letagain
I denote the interior ofIMI.
LEMMA. Let r c
GL(V)
be asubgroup
such that A and E are r’-stable. Then r has theSiegel property:
For any two cones u E 03A3 n I and i E E one has 03C3 n
yi
=Qf for
all but afinite
number
of
y Er.In
particular
r acts on E n 1 withfinite isotropy
groups039303C3
c r.Proof.
Let03C3 ~ 03A3 ~ I;
thenSt(03C3)
is the union of a finite number of 03C4 ~ 03A3 ~ I.Thus the set of all indivisible lattice
points
that span a one-dimensional face inSt(u)
is a finite subset of A which contains a basis of V. Since theisotropy
group039303C3
preserves this setr (f
is finite.Let now i ~ 03A3 be
arbitrary.
Wheneveryi
meets 03C3 then 03C3 cyr
and therefore yi cSt(u).
There areonly finitely
many cones with thisproperty,
and all are contained in 7. TheSiegel property
follows since039303C4
is finite. D(1.3)
PROPOSITION. Assume that 03A3 isadmissible,
and that A and 1 are F- stable. Then the(discrete)
group F actsproperly
onI(A, 03A3).
Inparticular
thetopological quotient I(A, 03A3)/0393
is alocally
compactHausdorff
space.Proof.
Let v, v’ EI(A, E);
in order to prove properness we constructneighbour-
hoods U of v and U’ of v’ such that U n
y U’
=0
for all but a finite number of y E r. Thepoint
vbelongs
to the stratumV/(
+03C3)
cT(A, 1)
for some Q ~03A3,
and isrepresented by z = x + iy ~ V
+iI,
say. In view of Lemma(1.1)
we maychoose y in
St(03C3)
n I. Let 03C4 ~ 03A3 n I be the conecontaining
y; then the setrepresents
an openneighbourhood
U of v inI(A, E). Similarly,
aneighbourhood
U’ of v’ is defined. The
Siegel property
thenimplies
Uny U’
=0 for
all butfinitely
many y e r asrequired.
Thé
last clause of theproposition
holds sinceI(A, 1:)
islocally compact
Hausdorff,
and the action of r is proper. DCOROLLARY. Under the
assumptions of
theproposition I(A, 1)/F
inherits aquotient analytic structure from I(A, E),
and in this way becomes a normal Cohen-Macaulay analytic variety.
Proof. By [Kempf
etal.]
p.52,
toroidalsingularities
are normal and Cohen-Macaulay.
Bothproperties
descend to thequotient. D (1.4).
We shall construct certain reflexiveanalytic
sheaves of rank one on thequotients I(A,1:)/r. They
arise from thefollowing type
of data.DEFINITION. Let E be an admissible fan. A characteristic
triple (03C0, b, 03BB)
for 03A3consists of an
epimorphism
of lattices with kernel rank one, a
preferred generator b
E ker 1t,and, finally,
acharacteristic section
The latter is
by
definition a section of 03C0 :~
Vsatisfying
the two axioms:(i)
For each Q e X themap  |03C3
is the restriction of a linear map from Rato el
and
(ii) 03BB(03C3
nA)
cÃ
for each 03C3~03A3 with dim 0" = 1.With each characteristic
triple (03C0, ô, A)
we associate a sheafL(03C0, ô, A)
onT(A, 03A3)
as follows. Let I c v denote the interior of
JY-1
andput I = 03C0-1(I) c In V,
consider the set of cones
where
(R+ = (0, oo)).
ThisË
is an admissible fan withrespect
toÃ,
andI is
the interior of||.
Inparticular
we have torusembeddings
and the
epimorphism x
inducesmorphisms
of varietiesalso denoted x
by
abuse oflanguage.
NOTATION. Let X be an admissible fan. Then for each r c- 1 we
put
and for each
d c- N,
clearly
these are closed subsets of 1:.LEMMA. Let
(n, b, Â)
be a characteristictriple for
the admissiblefan 03A3,
and let03A6 c 03A3 be an
arbitrary
closed subset.Then the inverse
image of T(A, (03A6)
underT(A, 03A30) T(, 03A3)
isT(Â,,bo)
whereIf
the characteristic section A has the propertythen the restriction
is an
algebraic C*-principal
bundle.Proof.
The first statement is obvious. To prove thesecond,
we choose for each 03C3 ~ 03A6 a linear extensionAa of  1 a-
that sends A intoÃ. Writing
C* =T(Z)
wehave an
algebraic isomorphism
which is a trivialization of
T(A, ilio) h T(A, (D)
over the affine open subsetT(A, 03A303C3). If
J’ e O is another cone thecorresponding
trivializations differ overby
anautomorphism
ofT(A, 1:;’(1
nwhich
is a translation on each fibre.n The lemma shows in
particular
thatalways
is aC*-bundle;
we let 2’ be the sheaf of local sections of the associated line bundle andput
where j: T(, 03A31) T(, 03A3)
is the inclusion.DEFINITION. We refer to
L(03C0, ô, A)
as thereflexive sheaf
associated to the characteristictriple (n, ô, Â).
(1.5).
We collect a fewproperties
of the sheaves2(n, À, ô)
which are immediateconsequences of the way
they
are constructed.(i)
Let(03C0, ô, Â)
be a characteristictriple. Any splitting
of the exact sequencewill
identify
the characteristic section A with a certain real-valuedfunction y
on111.
If i:T(A) 4 T(, 03A3)
is the inclusion then2(n, ô, A)
will appear as an(!JT(A,1:f
submodule sheaf JIf of
i*T().
The space of sections of that sheaf overT(A, 03A303C3)
is
where
A = Hom(, Z)
denotes the lattice of characters ofT(A).
In theterminology
of[Kempf
etal.],
-4Y is thecomplete
coherent sheaf ofT(A)-
invariant fractional ideals characterized
by
the function ord JI which is theconvex
interpolation of IL
on|03A31|,
see loc. cit. p. 29. This fact may serve as a characterization of the class of sheaves that can be obtained from characteristictriples.
It also proves that2(n, ô, A)
is a reflexive sheaf of rank oneindeed,
assuggested by
ourterminology.
(ii) 2(n, ô, Â)
restricts to an invertible sheaf overT(A, 03A6) (03A6
c 03A3 a closedsubset)
if andonly
if03BB(|03A6|
nA)
cÂ
holds.(iii)
Let us call two characteristictriples (03C0, ô, À)
and(03C0’, b’, Â) isomorphic
ifthere exists a linear
isomorphism À’
such that n’ 0 ç = n,~(03B4)
=03B4’,
and~°03BB
= A’. Thenisomorphic triples give
rise toisomorphic
sheaves onT(A,1:).
(iv)
If m is a non-zerointeger
thenis
isomorphic
to the reflexive hull of the mth tensor power2(n, ô, À)m (m
>0) respectively
of(2(n, b, A) v)lml (m 0).
(v)
The action ofT(Â)
on itselfby
translation induces on action ofT(Ã)
onthe bundle
as a group of bundle
equivalences,
the action on the basebeing through
thehomomorphism T(A) ~ T(A). Therefore,
for each 6 e E the torusT(A)
acts onthe space of sections of
L(03C0, 03B4, A)
overT(A, EJ. Writing S = L(03C0, ô, 03BB)
we havea
decomposition
into
T(Â)-weight
spaces, indexedby
the character latticeA v
=Hom(A, Z).
Weclaim that
while
H0(T(, 03A303C3); 2)’
is trivial for all other 1 EÃ v.
Indeed the kernel of thehomomorphism T(Ã) ~ T(A)
acts withweight
1 on all sections ofT(Ã) ~ T(A),
and for
given
1 EÃ v
with03B4, l
= 1 theunique
section of ’
A withimage
ker 1represents
agenerator
s, EHO(T(A); 2)’.
This section s, isregular
onT(A, 03A303C3)
ifand
only
ifl°03BB
0 holds on J. Theresulting decomposition
is just (1.6)
in an invariantguise.
For
later
use we record thefollowing
facts.(1.8)
PROPOSITION. Let(03C0, ô, A)
be a characteristictriple.
(a)
For each 03C3 e E there exists aninteger
m > 0 such that thereflexive
hullof L(03C0, ô, A)m
is invertible onT(A, Ea).
(b) L(03C0, ô, A)
is aCohen-Macaulay sheaf.
Proof. (a)
follows from(ii)
and(iv) by choosing
msufficiently
divisible so that03BB(03C3
nA)
is contained inÀ
+(1/m)Z03B4.
Likewise,
to prove the localproperty (b),
we may work overT(A, 03A303C3).
If wedetermine m as in
(a)
thenThus the sheaf
is invertible on
T(mA, 03A303C3),
inparticular
it is aCohen-Macaulay
sheaf there.If q
denotes the
quotient morphism
corresponding
to the exact sequencethen we have
on
T(, 03A303C3).
The latter sheaf is a direct summand of aCohen-Macaulay sheaf,
and the assertion follows. D
(1.9)
In this subsection we fix an admissible fan 03A3in V,
and let I denote theinterior of
|03A3|
as usual. For any closed subset (D c 03A3 we defineAs the latter
description
showsI(A, 03A6).
is analgebraic
subscheme ofT(A, 03A6)
rather than a mere
analytic
space. Note thatI(A, 03A6).
need not be of finitetype though
italways
is solocally.
Let now
(À h A, ô, A)
be a characteristictriple
for E.Following
theprocedure
described in
[Kempf et al.]
p.42,
the(algebraic) cohomology
groupsHi(I(A, 03A6).;
2(n, Ô, 03BB)
QI(,03A6).)
can becomputed combinatorially.
To this end weput,
for each characterl~v
with03B4, l
= 1(1.10)
PROPOSITION. For each 1EÃ v
with(b, 1)
= 1 there is a natural additiveisomorphism
Proof.
Almost verbatim that of loc. cit. p. 42.0 (l.ll)
We make the sameassumptions
as in theprevious
subsection.DEFINITION. A
subgroup
r cGL(V)
is admissible(with respect
to03A3)
if(i)
A and 1: areI-’-stable,
and(ii)
the orbit set(X
nI)/r
is finite.Our aim is to prove an
analogue
of(1.10)
for thequotient
ofI(, 03A3). by
anadmissible group 0393. We first note:
(1.12)
PROPOSITION. Theanalytic subspace I(A, 1)*/F of I(A, 1)/F
is compact.Proof.
Let J e 1 n I. SinceSt(03C3)
is aneighbourhood
of 0’ in I the closure of thestratum
Vc /(A
+03C3)
inI(A, E)
is acompact analytic
space. As r is assumed to actadmissibly
there are but a finite number of r-orbits in £ nI,
cf.(iii)
of thedefinition in
(1.2).
ThereforeI(A, 1)’/F
is coveredby
a finite number ofcompact
spaces, and the
proposition
follows.0
DEFINITION. A group r c
GL(V)
is said to actadmissibly
on the character- istictriple (03C0, 03B4, A)
if ô is a fixedpoint
ofr,
and if the action of 0393on P
is admissible withrespect to Î
and descends to an admissible action on V thatmakes JY-1 4. V equivariant
too.If r acts
admissibly
on(03C0, ô, 03BB),
and ifis the induced
analytic quotient
then2(n, ô, A)
descends to a reflexiveanalytic
sheaf
on
I(A, 03A3)/0393. Since q
islocally
a finitemapping
this sheaf still is a Cohen-Macaulay
sheaf as in(1.8).
Recall that for each closed (D c E and each
1 E Ã v
with(03B4, l
= 1 we havedefined a
topological pair
If 03A6 is r-stable then r also acts on the
disjoint
sumvia
This action is
properly
discontinuous with finiteisotropy
groups, and has a finite Hausdorffquotient
(1.13)
THEOREM. Let 03A3 be an admissiblefan in V,
and let(n, b, Â)
be acharacteristic
triple for 1,
on which r cGL()
actsadmissibly.
Then the(analytic) cohomology of 2(n, b, A)/r
overI(A, 03A3)/0393
can becomputed
via acanonical
isomorphism
H*(I(A, 03A3)/0393; 2(n, b, A)/r (D I(,03A3)/0393)
~H*(Y(L;
n,b, Â), Y’(L;
n,b, Â); C).
Proof.
We first prove the existence of a normalsubgroup
F’ c r of finiteindex,
with theproperty:
(1.14)
Whenevery E r’
and 03C4 ~ 03A3 ~ I are such that1-r n y-T n 1 :0 Qf
then y = 1.Let F c A be the set of indivisible
generators
of one-dimensional cones inL,
and
put
Since r is admissible it acts on
F2
with a finite number oforbits,
and we find apositive integer m
such thatThen
is a normal
subgroup
of finite index in r which satisfies(1.14). Indeed,
letyen
and T E E n 7 be such that t n
yî
n I isnon-empty.
This intersection containssome
03C3 ~ 03A31 n I, spanned by v E F,
say. We haveand
by
definition of r’ thisimplies
yv = v. Thus y mapsSt( eT)
intoitself,
and therefore must be theidentity.
We now choose r’ so that
(1.14)
holds. For each 03C4~03A3 nI the unionthen is a
disjoint
one; thereforeI(A, 03A3)/0393’ is
coveredby
the affine varietieswith i
running through
asystem
ofrepresentatives
of 03A3 n 1 mod r’. ThusI(A, 03A3)/0393’
itself has the structure of acomplex algebraic variety,
which iscomplete by (1.12).
In view of the GAGAprinciple
we maycompute
thecohomology
groups inquestion
asalgebraic
rather thananalytic cohomology (for
F’ inplace
ofr).
Likewise
(1.14) implies
that for each 03C4~03A3 n 7 the unionis
disjoint,
and that therefore thepair
defined
by
r’ is coveredby
thepairs
(L
E(E
nI)/r’).
The assertion of the theorem for the group r’ inplace
of r’ nowfollows upon
comparing
theCech complex
of the affinecovering
ofI(A, 03A3)/0393’
(with
coefficients in(03C0, b, A)/r’
(DI(,03A3)/0393’)
to that of thecovering
of thepair ( Y(E;
7r,b, Â), Y’(03A3;
n,b, 03BB)) (with
constantcomplex coefficients).
The
isomorphism
thus obtained isequivariant
withrespect
to the finite groupr/ r’.
The theorem therefore follows ingeneral
uponpassing
to ther/ r’-fixed
parts
of thecohomology
groups. n(1.15)
Wekeep using
the notation from theprevious
subsections. There is acriterion of
ampleness
for the sheaves2(n, b, A)/r,
as follows.(1.16)
THEOREM. Let 1 be an admissiblefan
inY,
and let(7r, b, Â)
be acharacteristic
triple for
E.Further,
let r cGL(V)
actadmissibly
on thetriple (7r, Ô, Â).
Then the
sheaf 2(n, b, A)/r
(DI(,03A3)./0393
onI(A, 03A3)/0393
isample if
andonly if
thecharacteristic section A is
strictly
convex on E n 1 in the sense thatfor
each03C3 ~ 03A3 n I there exists an 1 E
 ’
such that03B4, l> = 1
10
Â
0everywhere on |03A3|,
and10 A = 0
exactly
on 03C3.Proof.
Since there areonly finitely
many r-orbits in 03A3 n IProposition (1.8)
implies
that for suitable m > 0 the reflexive hull of!R(n, ô, 03BB)m
is invertible.Replacing (Ã , ô, A) by (Ã + (1/m)Z03B4 A, (1/m)ô, 03BB)
we thus may assume that Y:=L(03C0, ô, A)
itself is invertible.First suppose that
2 Ir Q OI(,03A3)/0393
isample,
and fix any 03C3 e X n 7. The closure S of the stratumVc/(A +Cu)
in7(A, X)’
is acompact analytic variety,
and thequotient mapping
is a finite
morphism.
At the costof raising
fil to a suitable tensor power we mayassume that for each s E S the stalk
Ls O (9,,,
isisomorphic
to(9s,,,
as ars-
module. It then follows that
which shows that S
Q9 (!Js
is anample
sheaf on S.The
variety S
is the torusembedding
ofT(A/(A
nR03C3))
definedby projecting
all members of E that are contained in
St(Q). Applying
theampleness
criterion[Kempf et al.]
p.48,
Theorem 13 we conclude that  isstrictly
convex at a. Thisbeing
true for all J E E n I iteasily
follows that 03BB isstrictly
convexglobally, using
the fact that each A-rational
compact segment
inJY-1
meetsonly finitely
many 03C3 ~ 03A3.Assume now that  is
strictly
convex on E n 7. We choose a finite closedsubset 03A6 c X
sufficiently large
so thatI(A, 03A6).
maps ontoI(A, E)’/r. Arguing
asin
[Demazure 1] p. 568,
Théorème 2 we find apositive integer
m and a finitenumber of characters
1 E Ã v
with03B4,l
= m such that thecorresponding
sectionsin
HO(I(, E)’;
Lm QOI(,03A3).)
define anembedding
ofI(A, 03A6).
in aprojective
space.
Enlarging m
we can arrange the choice of the characters 1 so that all these sections havesupport
inI(A, 03A6).
Let now z and z’ be
points
inI(A, 03A6)~
with Fz ~ 0393z’.By
thepreceding,
we findan even
larger
m and a sectionthat vanishes on all but one
point
of 0393z u0393z’,
and hassupport
inI(A, 03A6)~.
As thenumber of
points
any r-orbit can have inI(A, 03A6)~
isbounded,
m can be chosenuniformly
for all choices of z and z’ inI(A, 03A6)~.
Since the action of r on(n, 03B4, A)
isadmissible the sum
is
locally
finite and thus defines an invariantanalytic
section inBy
construction this sectionseparates
the orbits rz and rz’.Likewise one can construct invariant sections that
separate tangent
vectors.We omit the details. D
2. A
family
of abelian varieties(2.1)
Let R be a reduced irreducible finite rootsystem,
and letbe the dual of the affine root
system
obtainedby completing
R v. A choice of abasis
of R also determines bases of
R, R^
and hence ofR;
we letao E R
denote theextra base root. Note that R is
canonically
embedded inR.
We think of
R
as realized in a real vectorspace P
of(minimal)
dimension r + 2.Following [Looijenga 3]
we introduce the Tits cone+
cF
ofR;
its interiorI c V
determines the tube domainLet
Q
= 7 Rrespectively Q
= ZR denote the root lattices of R andR,
and defineb E Q
as the smallestpositive
linear combination of the base roots that isorthogonal
to()v = R^.
Then as the coefficient of ao in ô is 1 theprojection
will send the sublattice
Q ~ e bijectively
toQ/Z03B4.
In this way we will oftenidentify
these twolattices;
inparticular
this allows us to think of R as a rootsystem
in E If weput
then
(2.2)
The rootsystems
R andR generate Weyl
groupsthe first
of which, W,
preserves thesubspace RQ ~ .
We also have an extendedWeyl
groupof R,
which is a group of affineautomorphisms of K
These groups also act on the tube domains Q andS2,
as is clear from the definitions.The structure of
17V
and W^ can be understood as follows. Let T cW
be the kernel of theprojection homomorphism
(which
is defined since ô is a fixedpoint
ofW).
The obvious grouphomomorph-
isms fit into exact sequences
In order to
give
anexplicit description
of these extensions we introduce the W- invariant scalarproduct (? ?)
onQ,
normalizedby
therequirement
(note
that b - ao is a short root ofR).
We shall also think of(?|?)
as defined onthe vector space
RQ,
and sometimes evenon Q
orRQ (with
a radicalspanned by ô).
In order to fix coordinates
on P
we choose avector p e P
which isorthogonal
on R v and is normalized
by
We then
assign
to each coordinatetriple (u, z, 03C4) ~ R
xRQ
x R thepoint
Each element of T
c W
can now be writtenfor a
uniquely
determined t EQ,
and this sets up anisomorphism
between T andthe additive group
Q,
cf.[Kac] p. 69
or[Looijenga 3] p. 28.
This formulalikewise describes the extensions
(2.3);
indeed the action of W on Tby conjugation corresponds
to the natural action of W on the latticeQ,
ande
x Tbecomes a
Heisenberg
group with centre 7Lb.Note that the actions of
W on Î,
and of W onfi
areproperly
discontinuous with finiteisotropy
groups, andyield
Hausdorfftopological respectively analytic quotients.
This followsdirectly
from theexplicit description
of the actionsthough
it also ispart
of thegeneral theory
of rootsystems [Looijenga 3].
(2.4)
There is a natural action of the modular groupSL2(Z) on Pc
andS2,
expressed by
the formulaIn fact this defines an action of a semi-direct
product
WSL2(Z),
the latterbeing
determined as follows:conjugation by
the matrixacts
trivially
on 7Lb x W cW ^ ,
and sendszt~QT ~ 6x6
toSince
SL2(Z)
actsproperly discontinuously,
with finiteisotropy
groups, and witha
Hausdorff analytic quotient
on the W^-invariant coordinate ivarying
in theupper half
plane H
the sameproperties
hold for the group W^SL2(Z)
withrespect
to its action on the tube domain Q.We now fix a
subgroup
r cSL2(Z)
of finiteindex,
andstudy
the action of ron the
C*-principal
bundleNote that
fi
cVc
and 03A9 cVc
can be described as the inverseimages
of theupper half
plane H
under thecomplex
coordinate ion Vc
resp.Vc,
cf.[Kac], [Looijenga 3]. Adjoining
the fibre C viawe pass to a line bundle on
Q/Q -
T. Letfi
denote thecorresponding
invertiblesheaf.
Dividing
nowby
the action of r we obtain ananalytic quotient
together
with the coherentanalytic
sheafon A’. For any i E H the fibre
ArT of p’
over the orbit ri E C’ is thequotient
of theabelian
variety 03A903C4/QT by
theisotropy
groupFB
of i, with a non-reduced structure if this group contains elements other than+1,
i.e. if 03C4 is anelliptic
fixedpoint
of r. Note that in case r contains -1 thegeneral
fibreA’
is not anabelian
variety
but its(singular)
Kummerquotient.
The sheaf S’ on A’
always is, by construction,
a reflexiveCohen-Macaulay
sheaf of rank one.
(2.6)
Let C be thecompact
curve obtained from C’by adding
the cusps of r. We wish to extend A’ and Y’ over C.Quite general compactifications
of families of abelian varieties have been constructedby Namikawa,
cf.[Namikawa].
Thepresent
situation isparticularly simple
as the abelian varieties inquestion
aremere
products
of severalcopies
of anelliptic
curve, andonly
the modulus of thatcurve is allowed to vary. In fact
quite
anexplicit compactification
of A’ isreadily
at hand if a construction from
[Wirthmüller 2]
is used. LikeNamikawa’s
it is oftoroidal
type and,
based on the discussion of reflexive sheaves in Section1,
italso
provides
a natural extension of £F’.We
proceed
to describe thiscompactification.
AsSL2(Z)
actstransitively
onthe set of cusps of r it suffices to deal with the standard cusp at i~. A
typical punctured neighbourhood
of this cusp isrepresented by
the affine upper halfplane
and as is well
known,
for c 1 each orbit of theisotropy
group0393i~
in ic + H isthe intersection of ic + H with a full r-orbit. If the cusp at i oo is a
regular
one wehave
for some
positive integer
s.Similarly,
in the case of anirregular
cusp we haveNote that the matrix
acts on
V
as the translationby
the vectors03B2.
(2.9)
The rootbasis {03B10,..., 03B1r} together
with thevector - sfl
constitutes a mixed root basis in the sense of[Wirthmüller 2]
Section 9. ItsDynkin diagram -9 comprises
theDynkin diagram
ofR
as the fullsubdiagram -9b,.,,k
whileDwhite
consists
of just
one vertex realizedby
thevector - s03B2.
Theremaining edges
of -9are determined
by
the valuesEXAMPLE.
If
R is oftype B3
then the
Dynkin diagram
ofR
isand -q is the mixed
diagram
Following
loc. cit. the mixed root basisgives
rise to a toroidalembedding X(-9)
of
Q/Â
whereA
denotes the lattice of rank r + 2Let K
c v
be the convex conespanned by
the orbitWp;
then K c+ since p
lies in the closed fundamental chamber
C
ofR.
Theanalytic
spaceX(-9)
constructed in loc. cit.
is, by definition,
the union of a copy of the upper halfplane H,
and the open subset of the affine torusembedding
determinedby A
andK, comprising
allpoints representable by
vectors inQ.
(2.10)
Rather thanworking
withX(-9) directly,
we use the cone K in order toconstruct a
(non-affine)
torusembedding
ofT(A),
whereTo this end we need an admissible fan
EK
forA,
which we take to be the set ofprojections
of all proper faces of K under the linear map K Thefollowing proposition
willshow,
inparticular,
thatEK
is admissible indeed.We let
1:K
denote the set of proper faces of the closed coneand
put
K. is the
topological boundary
of Kin P.
Thusby
definition.(2.11)
PROPOSITION.(a) 03C0(K) = I+,
andfor
each(z,-r)E1
the setis a closed ray in
R, bounded from
below.(b)
The restrictionof
n:V
- Vis a
homeomorphism.
(c) KBR_03B4
is theimage of
a sectionwhich is characteristic with respect to
EK
andà ~
A.Proof. Clearly 03C0(K)
is contained inI+.
On the other hand{(z, T)
E7r(K) ! r
=1}
is a
non-empty
convex set which is invariant under the lattice of translations T ~Q,
whence this set is the full affine space{(z, 03C4)~V|03C4= 1}.
This proves03C0(K)=I+.
Let
C c V
denote the closed fundamental chamber ofR
as before. Let y EC
nÎ
bearbitrary. By [Looijenga 3] (2.4)
the convex hull of the orbitWy
meets
C exactly along
the setwhere
A - = (-~, 0].
As the chambers induce alocally
finitecovering
of theopen Tits cone
Î
thisimplies
that K nÎ
is closed in(put
y =03B2).
Fory~
C n I n
K it alsoimplies
and since
C
is a fundamental domain for the action ofW
on+
we concludeGiven
(z, r) ~I,
the set{u
ER|(u, z, i)
EK}
thereforeequals [t, oo)
for some t ER,
or t = - oo . In fact this last
possibility
is ruled out at once sinceWp,
and henceK,
iscompletely
contained in the half space{(u, z, i) ~|
u0}.
Thiscompletes
the
proof
of(a).
We now know that for any y E I the set
03C0-1(y)
~K is a closed ray inK
Thisray intersects K. in its end
point,
which we define to beÂK(y).
Wecomplete
thedefinition of
03BBK: I + ~ by putting AK(O)
= 0. ThenAK 1 a-
is linear for each o- EEK
because n is linear.
The fundamental chamber
C
meets but a finite number of cones fromf.
n.
Since
C has
the formthis means that
03C0C
meets but a finite number of cones fromI:K
nI,
which in turnimplies
thatEK
n 7 is alocally
finitecovering
of I. The section IK
n1, being piecewise linear,
thus is seen to be a continuous inverse to K. n~
I.In
particular
we now haveproved (b),
and in order tocomplete
theproof of (c)
it remains to show the
integrality property
for each 03C3 E
EK
with dim Q = 1. Theimage 03BBK(03C3)
EÎK
is a one-dimensional face ofK,
and in view of the classification of faces of Kgiven
in[Wirthmüller 2]
p.230,
Theorem 9.5 that face is
W-conjugate
to the rayR+03B2~.
We therefore mayassume
03BBK(03C3)=R+03B2,
and the assertion follows sinces·03C003B2~V
is aprimitive
vector in A. D
REMARK. It is not
generally
true that03BBK(I+ n A) ce À.
To what extent this failsmay be read off from the classification of faces of K. In
fact,
up to-conjugacy
these faces are classified
by
thesubdiagrams
of -9 in the sense of loc. cit. It isreadily
verifiedby inspection
that theonly
03C3 EEK
with03BBK(03C3
nA) Â
are thosewith
03BBK(03C3) corresponding
to one of thesubdiagrams
(R
oftype E7),
or(R
oftype E8).
The numbersmarking
the vertices that are not included in thesubdiagram
are theirmultiplicities
in thegreatest
root b - ao ofR;
in all casesthe
g.c.d.
of these numbers is the smallestpositive integer m
with03BBK(03C3
nA)
c(1/m).
(2.12)
We letlk and Â,
retain themeaning
from theprevious
subsection.LEMMA. The group
T acts . freely
onEK
n I.Proof.
Let t~T and03B1 ~ 03A3 k ~ I,
and assumet(03C3) =
J. As the coordinatei : V - R is T-invariant t must fix the
relatively compact
setThe
barycentre
of this set will then be a fixedpoint
of t. But t acts as a translationon the affine space
{(z, 03C4)~ Vlr = 1},
and thus t = 1 follows. 0The lemma shows in
particular
that the action of T on V is admissible withrespect
to1,,
and a fortiori T actsadmissibly
on the characteristictriple (n, Ô, 03BBK).
We therefore have an associated reflexiveanalytic
sheafL(03C0, b, 03BBK)/T
on
I(A, 03A3K)/T.
If the cusp at foc isregular,
and if- 1~0393
then the open subsetnaturally
extends the restriction ofA’ i
C’ over thepunctured neighbourhood
of the cusp at ioo.
By
construction the sheaf!l’en, b, 03BBK)/T
extends L’ in the same sense. In aslightly
different way, this is also true if-1 E r,
or if the cusp isirregular.
In the former case the extension isprovided by
thequotients
ofI(A, 1,)IT
andL(03C0, 03B4, 03BBK)/T by
the involutionLikewise,
if the cusp isirregular
one wouldbegin
the construction with 2s inplace
of s, and then form thequotients by
Using
the action ofSL2(Z)
in order to handle all other cusps of r we obtain aproper
analytic morphism
extending p’,
as well as a reflexive rank one sheaf Y on A with.P A’ -
Y’.(2.13)
Wecompute
thecohomology
of Ef and its powersalong
the fibresA,
=p-1(c),
for all c~C.At first we look at the fibres of the
projection
which is a smooth