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553
Documenta Math.
A Modular Compactification
of the General Linear Group
Ivan Kausz
Received: June 2, 2000
Communicated by Peter Schneider
Abstract. We define a certain compactifiction of the general linear
group and give a modular description for its points with values in arbitrary schemes. This is a first step in the construction of a higher rank
generalization of Gieseker’s degeneration of moduli spaces of vector
bundles over a curve. We show that our compactification has similar properties as the “wonderful compactification” of algebraic groups
of adjoint type as studied by de Concini and Procesi. As a byproduct we obtain a modular description of the points of the wonderful
compactification of PGln .
1991 Mathematics Subject Classification: 14H60 14M15 20G
Keywords and Phrases: moduli of vector bundles on curves, modular
compactification, general linear group
1. Introduction
In this paper we give a modular description of a certain compactification KGln
of the general linear group Gln . The variety KGln is constructed as follows:
First one embeds Gln in the obvious way in the projective space which contains
the affine space of n × n matrices as a standard open set. Then one successively blows up the closed subschemes defined by the vanishing of the r × r
subminors (1 ≤ r ≤ n), along with the intersection of these subschemes with
the hyperplane at infinity.
We were led to the problem of finding a modular description of KGln in the
course of our research on the degeneration of moduli spaces of vector bundles.
Let me explain in some detail the relevance of compactifications of Gln in this
context.
Let B be a regular integral one-dimensional base scheme and b0 ∈ B a closed
point. Let C → B be a proper flat familly of curves over B which is smooth
outside b0 and whose fibre C0 at b0 is irreducible with one ordinary double
point p0 ∈ C0 . Let C˜0 → C0 be the normalization of C0 and let p1 , p2 ∈ C˜0
Documenta Mathematica 5 (2000) 553–594
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Ivan Kausz
the two points lying above the singular point p0 . Thus the situation may be
depicted as follows:
(C˜0 , p1 , p2 )
kkk
kkkk
k
k
k
u kk
k
C˜0
SSSS
SSSS
SSSS
S)
C0
/C
b0
/B
where the left arrow means “forgetting the points p1 , p2 ”. There is a corresponding diagram of moduli-functors of vector bundles (v.b.) of rank n:
v.b. E on C˜0
together with an
∼
isomorphism E[p1 ] → E[p2 ]
v.b. on
C˜0
RRR ∼
R=
RRR
f2
(
l
lll
vlll f1
v.b. on
C0
/
b0
v.b. on
C/B
/B
where E[pi ] denotes the fibre of E at the point pi (cf. section 3 below). The
morphism f1 is “forgetting the isomorphism between the fibres” and f2 is “glueing together the fibres at p1 and p2 along the given isomorphism”. The square
on the right is the inclusion of the special fibre. It is clear that f is a locally trivial fibration with fibre Gln . Consequently, f1 is not proper and thus
{v.b. on C/B} is not proper over B. It is desirable to have a diagram (∗):
generalized
v.b.-data on
˜
(C0 , p1 , p2 )
v.b. on
C˜0
uu
uuuf
u
zuu 1
KK
KK
KK
%
f
2
generalized
v.b. on
C0
/
generalized
v.b. on
C/B
b0
/B
where the functors of “generalized” objects contain the original ones as open
subfunctors and where {generalized v.b. on C/B} is proper over B or at least
satisfies the existence part of the valuative criterion for properness. The motivation is that such a diagram may help to calculate cohomological invariants
of {v.b. on Y } (Y a smooth projective curve) by induction on the genus of Y
Documenta Mathematica 5 (2000) 553–594
Compactification of the General Linear Group
555
(notice that the genus of C˜0 is one less than the genus of the generic fibre of
C/B).
In the current literature there exist two different approaches for the construction of diagram (∗). In the first approach the “generalized v.b.” on C0 are
torsion-free sheaves (cf. [S1], [F], [NR], [Sun]). The second approach is by
Gieseker who considered only the rank-two case (cf. [G]). Here the “generalized v.b.” on C0 are certain vector bundles on C0 , C1 or C2 , where Ci is built
from C0 by inserting a chain of i copies of the projective line at p0 . (Cf. also
[Tei] for a discussion of the two approaches). Of course, this is only a very
rough picture of what is going on in these papers since I do not mention concepts of stability for the various objects nor the representability of the functors
by varieties or by algebraic stacks.
In both approaches the morphism f2 is the normalization morphism (at least on
the complement of a set of small dimension) and f1 is a locally trivial fibration
with fibre a compactification of Gln . In the torsion-free sheaves approach this
compactification is Gr(2n, n), the grassmanian of n-dimensional subspaces of a
2n-dimensional vector space. In Gieseker’s construction the relevant compactification of Gl2 is KGl2 . An advantage of Gieseker’s construction is that in contrast to the torsion-free sheaves approach, the space {generalized v.b. on C/B}
is regular and its special fibre over b0 is a divisor with normal crossings.
Very recently, Nagaraj and Seshadri have generalized Gieseker’s construction
of the right part of diagram (∗), i.e. the diagram
generalized
generalized
/
v.b. on
v.b. on
C0
C/B
b0
/B
to arbitrary rank n (cf. [NS], [S2]). Nagaraj’s and Seshadri’s “generalized
vector bundles” on C0 are certain equivalence classes of vector bundles on one
of the curves C0 , . . . , Cn , whose push-forward to C0 are stable torsion free
sheaves.
Without worrying about stability I have recently (and independently from Nagaraj and Seshadri) constructed the full diagram (∗) at least at the level of
functors (details will appear in a forthcoming paper) and I have reasons to
believe that the fibres of the corresponding morphism f1 should be represented
by KGln . The present paper is the first step in the proof of this fact.
The compactification KGln of Gln has properties similar to those of the “wonderful compactification” of algebraic groups of adjoint type as studied by De
Concini and Procesi (cf. [CP]). Namely:
1. The group Gln × Gln acts on KGln , extending the operation of Gln × Gln
on Gln induced by right and left multiplication (cf. 5.6).
2. The complement of Gln in KGln is a divisor with normal crossings with
irreducible components Yi , Zj (i, j ∈ {0, . . . , n − 1}) (cf. 4.2).
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Ivan Kausz
3. The orbit closures of the operation of Gln × Gln on KGln are precisely
the intersections YI ∩ ZJ , where I, J are subsets of {0, . . . , n − 1} with
min(I) + min(J) ≥ n and where YI := ∩i∈I Yi , ZJ := ∩j∈J Zj (cf. 9.4).
4. For each I, J as above there exists a natural mapping from YI ∩ ZJ to the
product of two flag varieties. This mapping is a locally trivial fibration
with standard fibre a product of copies of PGlnk (the wonderful compactification of PGlnk ) for some nk ≥ 1 and of one copy of KGlm for some
m ≥ 0 (cf. 9.3).
Our main theorem 5.5 says that KGln parametrizes what we call “generalized isomorphisms” from the trivial bundle of rank n to itself. A generalized
isomorphism between vector bundles E and F is by definition a diagram
⊗
E = E0
o
E1
⊗
o
E2 . . . En−1
⊗
o
En
∼
/ Fn
⊗
/ Fn−1
. . . F2
⊗
/ F1
⊗
/ F0
= F
with certain properties, where the Ei and Fj are vector bundles of the same
⊗ / indicates a morphims of the
rank as E and F and where the arrow
source into the target tensored with a line bundle to be specified. Cf. 5.2 for a
precise definition.
The wonderful compactification PGln of PGln is contained as an orbit closure
in KGln , in fact Y0 ∼
= PGln . Therefore theorem 5.5 implies a modular description of PGln . One of the reasons why I decided to publish the present
paper separately from my investigations on the degeneration of moduli spaces
of vector bundles on curves is the fact that PGln has been quite extensively
studied in the past (cf. [Lak1] for a historical overview and also the recent paper [Tha2]). Although some efford has been made to find a modular description
for it, up to now only partial results in this direction have been obtained (cf.
[V], [Lak2], [TK]). In section 8 we explain the connection of these results with
ours. Recently Lafforgue has used PGln to compactify the stack of Drinfeld’s
shtukas (cf. [Laf1], [Laf2]).
Sections 4 and 5 contain the main definitions: In section 4 we give the construction of KGln and in section 5 we define the notion of generalized isomorphisms.
At the end of section 5 we state our main theorem 5.5. Its proof is given in
sections 6 and 7. In section 8 we define complete collineations and compare our
notion with the one given by previous authors, in section 9 we study the orbit
closures of the operation of Gln × Gln on KGln and in section 10 we define an
equivariant morphism of KGln onto the Grassmannian compactification of Gln
and compute its fibres.
My interest in degeneration of moduli spaces of bundles on curves has been
greatly stimulated by a workshop on conformal blocks and the Verlinde formula, organized in March 1997 by the physicists J¨
urgen Fuchs and Christoph
Schweigert at the Mathematisches Forschungsinstitut in Oberwolfach. Part of
this work has been prepared during a stay at the Mathematical Institute of
the University of Oxford. Its hospitality is gratefully acknowledged. Thanks
are due to Daniel Huybrechts for mentioning to me the work of Thaddeus, to
M. Thaddeus himself for sending me a copy of part of his thesis and to M.
Documenta Mathematica 5 (2000) 553–594
557
Compactification of the General Linear Group
Rapoport for drawing my attention to the work of Laksov and Lafforgue. I
would also like to thank Uwe Jannsen for his constant encouragement.
2. An elementary example
This section is not strictly necessary for the comprehension of what follows.
But since the rest of the paper is a bit technical, I felt that a simple example
might facilitate its understanding.
Let A be a discrete valuation ring, K its field of fractions, m its maximal ideal,
t ∈ m a local parameter and k := A/m the residue class field of A. Let E and
∼
F be two free A-modules of rank n and let ϕK : EK → FK be an isomorphism
between the generic fibers EK := E ⊗A K and FK := F ⊗A K of E and F .
We can choose A-bases of E and F such that ϕK has the matrix presentation
diag(tm1 , . . . , tmn ) with respect to these bases, where mi ∈ Z and m1 ≤ · · · ≤
mn . Now let a0 := 0 =: b0 and for 1 ≤ i ≤ n set ai := − min(0, mn+1−i ) and
bi := max(0, mi ). Note that we have
0 = a0 = · · · = an−l ≤ an−l+1 ≤ · · · ≤ an
and
0 = b0 = · · · = bl ≤ bl+1 ≤ · · · ≤ bn
for some l ∈ {0, . . . , n}. Let
En ⊆ · · · ⊆ E1 ⊆ E0 := E
be the A-submodules defined by
tai+1 −ai In−i 0
Ei
Ei+1 :=
Ii
0
and Fn ⊆ · · · ⊆ F1 ⊆ F0 := F
Fi+1 :=
,
Ii
0 t
0
bi+1 −bi
In−i
Fi
,
where Ii denotes the i × i unit matrix. Then ϕK induces an isomorphism
∼
ϕ : En → Fn and we have the natural injections
Ei ֒→ mai −ai+1 Ei+1
Fi+1 ֒→ Fi
,
m
,
bi −bi+1
Ei ←֓ Ei+1
Fi+1 ←֓ Fi
.
Observe that the compositions Ei+1 ֒→ Ei ֒→ mai −ai+1 Ei+1 and Ei ֒→
mai −ai+1 Ei+1 ֒→ mai −ai+1 Ei are both the injections induced by the inclusion A ֒→ mai −ai+1 . Furthermore, if ai − ai+1 < 0 then the morphism of
k-vectorspaces Ei+1 ⊗ k → Ei ⊗ k is of rank i and the sequence
Ei+1 ⊗ k → Ei ⊗ k → (mai −ai+1 Ei+1 ) ⊗ k → (mai −ai+1 Ei ) ⊗ k
is exact. This shows that the tupel
(mai −ai+1 , 1 ∈ mai −ai+1 , Ei+1 ֒→ Ei , mai −ai+1 Ei+1 ←֓ Ei , i)
Documenta Mathematica 5 (2000) 553–594
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Ivan Kausz
is what we call a “bf-morphism” of rank i (cf. definition 5.1). Observe now
that if ai − ai+1 < 0 and (f, g) is one of the following two pairs of morphisms:
g
f
E ⊗ k −→ (m−ai Ei ) ⊗ k −→ (m−ai+1 Ei+1 ) ⊗ k
g
f
Ei ⊗ k ←− Ei+1 ⊗ k ←− En ⊗ k
,
,
then im(g ◦ f ) = im(g). The above statements hold true also if we replace the
Ei -s by the Fi -s and the ai -s by the bi -s. Observe finally that in the diagram
0
ker(En ⊗ k → E0 ⊗ k)
WWWWW
WWWW+
/
∼
/
/
k)
im(Fn ⊗ k → F0 ⊗ k) / 0
En ⊗ k = Fn ⊗ k
0 ker(Fn ⊗ k → F0 ⊗
WWWWW
WWWW+
im(En ⊗ k → E0 ⊗ k)
0
the oblique arrows are injections.
All these properties are summed up in the statement that the tupel
Φ :=
((mbi −bi+1 , 1), (mai −ai+1 , 1), Ei ֒→ mai −ai+1 Ei+1 , Ei ←֓ Ei+1 ,
∼
Fi+1 ֒→ Fi , mbi −bi+1 Fi+1 ←֓ Fi (0 ≤ i ≤ n − 1), ϕ : En → Fn )
is a generalized isomorphism from E to F in the sense of definition 5.2, where
for a ≤ 0 we consider ma as an invertible A-module with global section 1 ∈ ma .
Observe that Φ does not depend on our choice of the bases for E and F . Indeed,
it is well-known that the sequence (m1 , . . . , mn ) is independent of such a choice
and it is easy to see that En = ϕ−1
K (F ) ∩ E, Fn = ϕK (En ) and
Ei = En + mai E
,
Fi = Fn + mbi F
for 1 ≤ i ≤ n−1, where the +-sign means generation in EK and FK respectively.
Observe furthermore that by pull-back the generalized isomorphism Φ induces
a generalized isomorphism f ∗ Φ on a scheme S for every morphism f : S →
Spec (A). Of course the morphisms f ∗ Ei+1 → f ∗ Ei etc. will be in general no
longer injective, but this is not required in the definition.
3. Notations
We collect some less common notations, which we will use freely in this paper:
• For two integers a ≤ b we sometimes denote by [a, b] the set {c ∈ Z | a ≤
c ≤ b}.
• For a n × n-matrix with entries aij in some ring, and for two subsets A
and B of cardinality r of {1, . . . , n}, we will denote by detAB (aij ) the
determinant of the r × r-matrix (aij )i∈A,j∈B .
• For a scheme X we will denote by KX the sheaf of total quotient rings of
OX .
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Compactification of the General Linear Group
• For a scheme X, a coherent sheaf E on X and a point x ∈ X, we denote
by E[x] the fibre E ⊗OX κ(x) of E at x.
• For n ∈ N, the symbol Sn denotes the symmetric group of permutations
of the set {1, . . . , n}.
4. Construction of the compactification
Let X
(0)
:= Proj Z[x00 , xij (1 ≤ i, j ≤ n)]. We define closed subschemes
(0)
Y0
/ Y1(0)
O
?
(0)
Zn−1
/
/
(0)
...
...
(0)
(0)
/ Yn−1
O
?
/ Z1(0)
(0)
/ Z0(0)
(0)
(0)
of X (0) , by setting Yr := V + (Ir ), Zr := V + (Jr ), where Ir is the
homogenous ideal in Z[x00 , xij (1 ≤ i, j ≤ n)], generated by all (r + 1)× (r + 1)(0)
(0)
subdeterminants of the matrix (xij )1≤i,j≤n , and where Jr = (x00 ) + In−r for
0 ≤ r ≤ n − 1. For 1 ≤ k ≤ n let the scheme X (k) together with closed
(k)
(k)
subschemes Yr , Zr ⊂ X (k) (0 ≤ r ≤ n − 1) be inductively defined as
follows:
X (k) → X (k−1) is the blowing up of X (k−1) along the closed subscheme
(k−1)
(k−1)
(k)
(k)
Yk−1 ∪ Zn−k . The subscheme Yk−1 ⊂ X (k) (respectively Zn−k ⊂ X (k) )
(k−1)
(k−1)
is the inverse image of Yk−1 (respectively of Zn−k ) under the morphism
X (k) → X (k−1) , and for r 6= k − 1 (respectively r 6= n − k) the subscheme
(k)
(k)
Yr
⊂ X (k) (respectively of Zr ⊂ X (k) ) is the complete transform of
(k−1)
(k−1)
⊂ X (k−1) ). We set
⊂ X (k−1) (respectively Zr
Yr
KGln := X (n)
and
Yr := Yr(n) , Zr := Zr(n)
(0 ≤ r ≤ n − 1) .
We are interested in finding a modular description for the compactification
KGln of Gln = Spec Z[xij /x00 (1 ≤ i, j ≤ n), det(xij /x00 )−1 ].
Let (α, β) ∈ Sn × Sn and set
(0)
xij (α, β) :=
xα(i),β(j)
x00
(1 ≤ i, j ≤ n) .
For 1 ≤ k ≤ n we define elements
yji (α, β) , zij (α, β)
(1 ≤ i ≤ k,
(k)
xij (α, β)
i < j ≤ n)
(k + 1 ≤ i, j ≤ n)
Documenta Mathematica 5 (2000) 553–594
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Ivan Kausz
of the function field Q(X (0) ) = Q(xij /x00 (1 ≤ i, j ≤ n)) of X (0) inductively
as follows:
(k−1)
yik (α, β) :=
zkj (α, β) :=
(k)
xij (α, β) :=
xik
(α, β)
(k−1)
xkk (α, β)
(k−1)
xkj (α, β)
(k−1)
xkk (α, β)
(k−1)
(α, β)
xij
(k−1)
xkk (α, β)
(k + 1 ≤ i ≤ n) ,
(k + 1 ≤ j ≤ n) ,
− yik (α, β) zkj (α, β)
(k + 1 ≤ i, j ≤ n) .
Finally, we set t0 (α, β) := t0 := x00 and
ti (α, β) := t0 ·
i
Y
(j−1)
xjj
(α, β)
(1 ≤ i ≤ n) .
j=1
Observe, that for each k ∈ {0, . . . , n}, we have the following decomposition of
the matrix [xij /x00 ]:
9 9_ _ _ _
9
9 9 0
91 _ _ _
xij
= nα
yij (α, β)
x00
In−k
_ _
1
t1 (α,β)
t0
0
_= _ _ _ _
1 =
(α, β)
==zij
0
=
=
tk (α,β)
1 _ _ _ −1
t0
nβ
tk (α,β)
(k)
[xij (α, β)]
0 In−k
0
t0
_ _
0
Here, nα is the permutation matrix associated to α, i.e. the matrix, whose
entry in the i-th row and j-th column is δi,α(j) . For convenience, we define for
∼
each l ∈ {0, . . . , n} a bijection ιl : {1, . . . , n + 1} → {0, . . . , n}, by setting
if 1 ≤ i ≤ l
i
0
if i = l + 1
ιl (i) =
i − 1 if l + 2 ≤ i ≤ n + 1
for 1 ≤ i ≤ n + 1. With this notaton, we define for each triple (α, β, l) ∈ Sn ×
Sn × [0, n] polynomial subalgebras R(α, β, l) of Q(KGln ) = Q(X (0) ) together
with ideals Ir (α, β, l) and Jr (α, β, l) (0 ≤ r ≤ n − 1) of R(α, β, l) as follows:
R(α, β, l)
:=
Ir (α, β, l)
:=
Jr (α, β, l)
:=
tιl (i+1) (α, β)
(1 ≤ i ≤ n), yji (α, β), zij (α, β) (1 ≤ i < j ≤ n) ,
Z
tιl (i) (α, β)
tιl (r+2) (α, β)
if l ≤ r ≤ n − 1 and Ir (α, β, l) := (1) else,
tι (r+1) (α, β)
l
tιl (n−r+1) (α, β)
if n − l ≤ r ≤ n − 1 and Jr (α, β, l) := (1) else.
tιl (n−r) (α, β)
Proposition 4.1. There is a covering of KGln by open affine pieces X(α, β, l)
((α, β, l) ∈ Sn × Sn × [0, n]), such that Γ(X(α, β, l), O) = R(α, β, l) (equality
as subrings of the function field Q(KGln )). Furthermore, for 0 ≤ r ≤ n − 1
the ideals Ir (α, β, l) and Jr (α, β, l) of R(α, β, l) are the defining ideals for the
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Compactification of the General Linear Group
closed subschemes Yr (α, β, l) := Yr ∩X(α, β, l) and Zr (α, β, l) := Zr ∩X(α, β, l)
respectively.
Proof. We make the blowing-up procedure explicit, in terms of open affine
coverings. For each k ∈ {0, . . . , n} we define a finite index set Pk , consisting of
all pairs
p0
q0
(p, q) = : , : ∈ {0, . . . , n}k+1 × {0, . . . , n}k+1
pk
qk
with the property that pi 6= pj and qi 6= qj for i 6= j and that pi = 0 for some i,
if and only if qi = 0. Observe that for each k ∈ {0, . . . , n} there is a surjection
Sn × Sn × {0, . . . , n} → Pk , which maps the triple (α, β, l) to the element
α(ιl (1))
β(ιl (1))
,
:
:
(p, q) =
α(ιl (k + 1))
β(ιl (k + 1))
of Pk . (Here we have used the convention that α(0) := 0 for any permutation
α ∈ Sn ). Furthermore, this surjection is in fact a bijection in the case of
k = n. Let (p, q) ∈ Pk and chose an element (α, β, l) in its preimage under
the surjection Sn × Sn × {0, . . . , n} → Pk . We define subrings R(k) (p, q) of
(k)
(k)
Q(xij /x00 (1 ≤ i, j ≤ n)) together with ideals Ir (p, q), Jr (p, q) (0 ≤ r ≤ n),
distinguishing three cases.
First case: 0 ≤ l ≤ k − 1
R(k) (p, q)
:=
Z
"
tιl (i+1) (α, β)
tιl (i) (α, β)
(1 ≤ i ≤ k), yji (α, β), zij (α, β)
1 ≤ i ≤ k,
i
Documenta Math.
A Modular Compactification
of the General Linear Group
Ivan Kausz
Received: June 2, 2000
Communicated by Peter Schneider
Abstract. We define a certain compactifiction of the general linear
group and give a modular description for its points with values in arbitrary schemes. This is a first step in the construction of a higher rank
generalization of Gieseker’s degeneration of moduli spaces of vector
bundles over a curve. We show that our compactification has similar properties as the “wonderful compactification” of algebraic groups
of adjoint type as studied by de Concini and Procesi. As a byproduct we obtain a modular description of the points of the wonderful
compactification of PGln .
1991 Mathematics Subject Classification: 14H60 14M15 20G
Keywords and Phrases: moduli of vector bundles on curves, modular
compactification, general linear group
1. Introduction
In this paper we give a modular description of a certain compactification KGln
of the general linear group Gln . The variety KGln is constructed as follows:
First one embeds Gln in the obvious way in the projective space which contains
the affine space of n × n matrices as a standard open set. Then one successively blows up the closed subschemes defined by the vanishing of the r × r
subminors (1 ≤ r ≤ n), along with the intersection of these subschemes with
the hyperplane at infinity.
We were led to the problem of finding a modular description of KGln in the
course of our research on the degeneration of moduli spaces of vector bundles.
Let me explain in some detail the relevance of compactifications of Gln in this
context.
Let B be a regular integral one-dimensional base scheme and b0 ∈ B a closed
point. Let C → B be a proper flat familly of curves over B which is smooth
outside b0 and whose fibre C0 at b0 is irreducible with one ordinary double
point p0 ∈ C0 . Let C˜0 → C0 be the normalization of C0 and let p1 , p2 ∈ C˜0
Documenta Mathematica 5 (2000) 553–594
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Ivan Kausz
the two points lying above the singular point p0 . Thus the situation may be
depicted as follows:
(C˜0 , p1 , p2 )
kkk
kkkk
k
k
k
u kk
k
C˜0
SSSS
SSSS
SSSS
S)
C0
/C
b0
/B
where the left arrow means “forgetting the points p1 , p2 ”. There is a corresponding diagram of moduli-functors of vector bundles (v.b.) of rank n:
v.b. E on C˜0
together with an
∼
isomorphism E[p1 ] → E[p2 ]
v.b. on
C˜0
RRR ∼
R=
RRR
f2
(
l
lll
vlll f1
v.b. on
C0
/
b0
v.b. on
C/B
/B
where E[pi ] denotes the fibre of E at the point pi (cf. section 3 below). The
morphism f1 is “forgetting the isomorphism between the fibres” and f2 is “glueing together the fibres at p1 and p2 along the given isomorphism”. The square
on the right is the inclusion of the special fibre. It is clear that f is a locally trivial fibration with fibre Gln . Consequently, f1 is not proper and thus
{v.b. on C/B} is not proper over B. It is desirable to have a diagram (∗):
generalized
v.b.-data on
˜
(C0 , p1 , p2 )
v.b. on
C˜0
uu
uuuf
u
zuu 1
KK
KK
KK
%
f
2
generalized
v.b. on
C0
/
generalized
v.b. on
C/B
b0
/B
where the functors of “generalized” objects contain the original ones as open
subfunctors and where {generalized v.b. on C/B} is proper over B or at least
satisfies the existence part of the valuative criterion for properness. The motivation is that such a diagram may help to calculate cohomological invariants
of {v.b. on Y } (Y a smooth projective curve) by induction on the genus of Y
Documenta Mathematica 5 (2000) 553–594
Compactification of the General Linear Group
555
(notice that the genus of C˜0 is one less than the genus of the generic fibre of
C/B).
In the current literature there exist two different approaches for the construction of diagram (∗). In the first approach the “generalized v.b.” on C0 are
torsion-free sheaves (cf. [S1], [F], [NR], [Sun]). The second approach is by
Gieseker who considered only the rank-two case (cf. [G]). Here the “generalized v.b.” on C0 are certain vector bundles on C0 , C1 or C2 , where Ci is built
from C0 by inserting a chain of i copies of the projective line at p0 . (Cf. also
[Tei] for a discussion of the two approaches). Of course, this is only a very
rough picture of what is going on in these papers since I do not mention concepts of stability for the various objects nor the representability of the functors
by varieties or by algebraic stacks.
In both approaches the morphism f2 is the normalization morphism (at least on
the complement of a set of small dimension) and f1 is a locally trivial fibration
with fibre a compactification of Gln . In the torsion-free sheaves approach this
compactification is Gr(2n, n), the grassmanian of n-dimensional subspaces of a
2n-dimensional vector space. In Gieseker’s construction the relevant compactification of Gl2 is KGl2 . An advantage of Gieseker’s construction is that in contrast to the torsion-free sheaves approach, the space {generalized v.b. on C/B}
is regular and its special fibre over b0 is a divisor with normal crossings.
Very recently, Nagaraj and Seshadri have generalized Gieseker’s construction
of the right part of diagram (∗), i.e. the diagram
generalized
generalized
/
v.b. on
v.b. on
C0
C/B
b0
/B
to arbitrary rank n (cf. [NS], [S2]). Nagaraj’s and Seshadri’s “generalized
vector bundles” on C0 are certain equivalence classes of vector bundles on one
of the curves C0 , . . . , Cn , whose push-forward to C0 are stable torsion free
sheaves.
Without worrying about stability I have recently (and independently from Nagaraj and Seshadri) constructed the full diagram (∗) at least at the level of
functors (details will appear in a forthcoming paper) and I have reasons to
believe that the fibres of the corresponding morphism f1 should be represented
by KGln . The present paper is the first step in the proof of this fact.
The compactification KGln of Gln has properties similar to those of the “wonderful compactification” of algebraic groups of adjoint type as studied by De
Concini and Procesi (cf. [CP]). Namely:
1. The group Gln × Gln acts on KGln , extending the operation of Gln × Gln
on Gln induced by right and left multiplication (cf. 5.6).
2. The complement of Gln in KGln is a divisor with normal crossings with
irreducible components Yi , Zj (i, j ∈ {0, . . . , n − 1}) (cf. 4.2).
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Ivan Kausz
3. The orbit closures of the operation of Gln × Gln on KGln are precisely
the intersections YI ∩ ZJ , where I, J are subsets of {0, . . . , n − 1} with
min(I) + min(J) ≥ n and where YI := ∩i∈I Yi , ZJ := ∩j∈J Zj (cf. 9.4).
4. For each I, J as above there exists a natural mapping from YI ∩ ZJ to the
product of two flag varieties. This mapping is a locally trivial fibration
with standard fibre a product of copies of PGlnk (the wonderful compactification of PGlnk ) for some nk ≥ 1 and of one copy of KGlm for some
m ≥ 0 (cf. 9.3).
Our main theorem 5.5 says that KGln parametrizes what we call “generalized isomorphisms” from the trivial bundle of rank n to itself. A generalized
isomorphism between vector bundles E and F is by definition a diagram
⊗
E = E0
o
E1
⊗
o
E2 . . . En−1
⊗
o
En
∼
/ Fn
⊗
/ Fn−1
. . . F2
⊗
/ F1
⊗
/ F0
= F
with certain properties, where the Ei and Fj are vector bundles of the same
⊗ / indicates a morphims of the
rank as E and F and where the arrow
source into the target tensored with a line bundle to be specified. Cf. 5.2 for a
precise definition.
The wonderful compactification PGln of PGln is contained as an orbit closure
in KGln , in fact Y0 ∼
= PGln . Therefore theorem 5.5 implies a modular description of PGln . One of the reasons why I decided to publish the present
paper separately from my investigations on the degeneration of moduli spaces
of vector bundles on curves is the fact that PGln has been quite extensively
studied in the past (cf. [Lak1] for a historical overview and also the recent paper [Tha2]). Although some efford has been made to find a modular description
for it, up to now only partial results in this direction have been obtained (cf.
[V], [Lak2], [TK]). In section 8 we explain the connection of these results with
ours. Recently Lafforgue has used PGln to compactify the stack of Drinfeld’s
shtukas (cf. [Laf1], [Laf2]).
Sections 4 and 5 contain the main definitions: In section 4 we give the construction of KGln and in section 5 we define the notion of generalized isomorphisms.
At the end of section 5 we state our main theorem 5.5. Its proof is given in
sections 6 and 7. In section 8 we define complete collineations and compare our
notion with the one given by previous authors, in section 9 we study the orbit
closures of the operation of Gln × Gln on KGln and in section 10 we define an
equivariant morphism of KGln onto the Grassmannian compactification of Gln
and compute its fibres.
My interest in degeneration of moduli spaces of bundles on curves has been
greatly stimulated by a workshop on conformal blocks and the Verlinde formula, organized in March 1997 by the physicists J¨
urgen Fuchs and Christoph
Schweigert at the Mathematisches Forschungsinstitut in Oberwolfach. Part of
this work has been prepared during a stay at the Mathematical Institute of
the University of Oxford. Its hospitality is gratefully acknowledged. Thanks
are due to Daniel Huybrechts for mentioning to me the work of Thaddeus, to
M. Thaddeus himself for sending me a copy of part of his thesis and to M.
Documenta Mathematica 5 (2000) 553–594
557
Compactification of the General Linear Group
Rapoport for drawing my attention to the work of Laksov and Lafforgue. I
would also like to thank Uwe Jannsen for his constant encouragement.
2. An elementary example
This section is not strictly necessary for the comprehension of what follows.
But since the rest of the paper is a bit technical, I felt that a simple example
might facilitate its understanding.
Let A be a discrete valuation ring, K its field of fractions, m its maximal ideal,
t ∈ m a local parameter and k := A/m the residue class field of A. Let E and
∼
F be two free A-modules of rank n and let ϕK : EK → FK be an isomorphism
between the generic fibers EK := E ⊗A K and FK := F ⊗A K of E and F .
We can choose A-bases of E and F such that ϕK has the matrix presentation
diag(tm1 , . . . , tmn ) with respect to these bases, where mi ∈ Z and m1 ≤ · · · ≤
mn . Now let a0 := 0 =: b0 and for 1 ≤ i ≤ n set ai := − min(0, mn+1−i ) and
bi := max(0, mi ). Note that we have
0 = a0 = · · · = an−l ≤ an−l+1 ≤ · · · ≤ an
and
0 = b0 = · · · = bl ≤ bl+1 ≤ · · · ≤ bn
for some l ∈ {0, . . . , n}. Let
En ⊆ · · · ⊆ E1 ⊆ E0 := E
be the A-submodules defined by
tai+1 −ai In−i 0
Ei
Ei+1 :=
Ii
0
and Fn ⊆ · · · ⊆ F1 ⊆ F0 := F
Fi+1 :=
,
Ii
0 t
0
bi+1 −bi
In−i
Fi
,
where Ii denotes the i × i unit matrix. Then ϕK induces an isomorphism
∼
ϕ : En → Fn and we have the natural injections
Ei ֒→ mai −ai+1 Ei+1
Fi+1 ֒→ Fi
,
m
,
bi −bi+1
Ei ←֓ Ei+1
Fi+1 ←֓ Fi
.
Observe that the compositions Ei+1 ֒→ Ei ֒→ mai −ai+1 Ei+1 and Ei ֒→
mai −ai+1 Ei+1 ֒→ mai −ai+1 Ei are both the injections induced by the inclusion A ֒→ mai −ai+1 . Furthermore, if ai − ai+1 < 0 then the morphism of
k-vectorspaces Ei+1 ⊗ k → Ei ⊗ k is of rank i and the sequence
Ei+1 ⊗ k → Ei ⊗ k → (mai −ai+1 Ei+1 ) ⊗ k → (mai −ai+1 Ei ) ⊗ k
is exact. This shows that the tupel
(mai −ai+1 , 1 ∈ mai −ai+1 , Ei+1 ֒→ Ei , mai −ai+1 Ei+1 ←֓ Ei , i)
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Ivan Kausz
is what we call a “bf-morphism” of rank i (cf. definition 5.1). Observe now
that if ai − ai+1 < 0 and (f, g) is one of the following two pairs of morphisms:
g
f
E ⊗ k −→ (m−ai Ei ) ⊗ k −→ (m−ai+1 Ei+1 ) ⊗ k
g
f
Ei ⊗ k ←− Ei+1 ⊗ k ←− En ⊗ k
,
,
then im(g ◦ f ) = im(g). The above statements hold true also if we replace the
Ei -s by the Fi -s and the ai -s by the bi -s. Observe finally that in the diagram
0
ker(En ⊗ k → E0 ⊗ k)
WWWWW
WWWW+
/
∼
/
/
k)
im(Fn ⊗ k → F0 ⊗ k) / 0
En ⊗ k = Fn ⊗ k
0 ker(Fn ⊗ k → F0 ⊗
WWWWW
WWWW+
im(En ⊗ k → E0 ⊗ k)
0
the oblique arrows are injections.
All these properties are summed up in the statement that the tupel
Φ :=
((mbi −bi+1 , 1), (mai −ai+1 , 1), Ei ֒→ mai −ai+1 Ei+1 , Ei ←֓ Ei+1 ,
∼
Fi+1 ֒→ Fi , mbi −bi+1 Fi+1 ←֓ Fi (0 ≤ i ≤ n − 1), ϕ : En → Fn )
is a generalized isomorphism from E to F in the sense of definition 5.2, where
for a ≤ 0 we consider ma as an invertible A-module with global section 1 ∈ ma .
Observe that Φ does not depend on our choice of the bases for E and F . Indeed,
it is well-known that the sequence (m1 , . . . , mn ) is independent of such a choice
and it is easy to see that En = ϕ−1
K (F ) ∩ E, Fn = ϕK (En ) and
Ei = En + mai E
,
Fi = Fn + mbi F
for 1 ≤ i ≤ n−1, where the +-sign means generation in EK and FK respectively.
Observe furthermore that by pull-back the generalized isomorphism Φ induces
a generalized isomorphism f ∗ Φ on a scheme S for every morphism f : S →
Spec (A). Of course the morphisms f ∗ Ei+1 → f ∗ Ei etc. will be in general no
longer injective, but this is not required in the definition.
3. Notations
We collect some less common notations, which we will use freely in this paper:
• For two integers a ≤ b we sometimes denote by [a, b] the set {c ∈ Z | a ≤
c ≤ b}.
• For a n × n-matrix with entries aij in some ring, and for two subsets A
and B of cardinality r of {1, . . . , n}, we will denote by detAB (aij ) the
determinant of the r × r-matrix (aij )i∈A,j∈B .
• For a scheme X we will denote by KX the sheaf of total quotient rings of
OX .
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559
Compactification of the General Linear Group
• For a scheme X, a coherent sheaf E on X and a point x ∈ X, we denote
by E[x] the fibre E ⊗OX κ(x) of E at x.
• For n ∈ N, the symbol Sn denotes the symmetric group of permutations
of the set {1, . . . , n}.
4. Construction of the compactification
Let X
(0)
:= Proj Z[x00 , xij (1 ≤ i, j ≤ n)]. We define closed subschemes
(0)
Y0
/ Y1(0)
O
?
(0)
Zn−1
/
/
(0)
...
...
(0)
(0)
/ Yn−1
O
?
/ Z1(0)
(0)
/ Z0(0)
(0)
(0)
of X (0) , by setting Yr := V + (Ir ), Zr := V + (Jr ), where Ir is the
homogenous ideal in Z[x00 , xij (1 ≤ i, j ≤ n)], generated by all (r + 1)× (r + 1)(0)
(0)
subdeterminants of the matrix (xij )1≤i,j≤n , and where Jr = (x00 ) + In−r for
0 ≤ r ≤ n − 1. For 1 ≤ k ≤ n let the scheme X (k) together with closed
(k)
(k)
subschemes Yr , Zr ⊂ X (k) (0 ≤ r ≤ n − 1) be inductively defined as
follows:
X (k) → X (k−1) is the blowing up of X (k−1) along the closed subscheme
(k−1)
(k−1)
(k)
(k)
Yk−1 ∪ Zn−k . The subscheme Yk−1 ⊂ X (k) (respectively Zn−k ⊂ X (k) )
(k−1)
(k−1)
is the inverse image of Yk−1 (respectively of Zn−k ) under the morphism
X (k) → X (k−1) , and for r 6= k − 1 (respectively r 6= n − k) the subscheme
(k)
(k)
Yr
⊂ X (k) (respectively of Zr ⊂ X (k) ) is the complete transform of
(k−1)
(k−1)
⊂ X (k−1) ). We set
⊂ X (k−1) (respectively Zr
Yr
KGln := X (n)
and
Yr := Yr(n) , Zr := Zr(n)
(0 ≤ r ≤ n − 1) .
We are interested in finding a modular description for the compactification
KGln of Gln = Spec Z[xij /x00 (1 ≤ i, j ≤ n), det(xij /x00 )−1 ].
Let (α, β) ∈ Sn × Sn and set
(0)
xij (α, β) :=
xα(i),β(j)
x00
(1 ≤ i, j ≤ n) .
For 1 ≤ k ≤ n we define elements
yji (α, β) , zij (α, β)
(1 ≤ i ≤ k,
(k)
xij (α, β)
i < j ≤ n)
(k + 1 ≤ i, j ≤ n)
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Ivan Kausz
of the function field Q(X (0) ) = Q(xij /x00 (1 ≤ i, j ≤ n)) of X (0) inductively
as follows:
(k−1)
yik (α, β) :=
zkj (α, β) :=
(k)
xij (α, β) :=
xik
(α, β)
(k−1)
xkk (α, β)
(k−1)
xkj (α, β)
(k−1)
xkk (α, β)
(k−1)
(α, β)
xij
(k−1)
xkk (α, β)
(k + 1 ≤ i ≤ n) ,
(k + 1 ≤ j ≤ n) ,
− yik (α, β) zkj (α, β)
(k + 1 ≤ i, j ≤ n) .
Finally, we set t0 (α, β) := t0 := x00 and
ti (α, β) := t0 ·
i
Y
(j−1)
xjj
(α, β)
(1 ≤ i ≤ n) .
j=1
Observe, that for each k ∈ {0, . . . , n}, we have the following decomposition of
the matrix [xij /x00 ]:
9 9_ _ _ _
9
9 9 0
91 _ _ _
xij
= nα
yij (α, β)
x00
In−k
_ _
1
t1 (α,β)
t0
0
_= _ _ _ _
1 =
(α, β)
==zij
0
=
=
tk (α,β)
1 _ _ _ −1
t0
nβ
tk (α,β)
(k)
[xij (α, β)]
0 In−k
0
t0
_ _
0
Here, nα is the permutation matrix associated to α, i.e. the matrix, whose
entry in the i-th row and j-th column is δi,α(j) . For convenience, we define for
∼
each l ∈ {0, . . . , n} a bijection ιl : {1, . . . , n + 1} → {0, . . . , n}, by setting
if 1 ≤ i ≤ l
i
0
if i = l + 1
ιl (i) =
i − 1 if l + 2 ≤ i ≤ n + 1
for 1 ≤ i ≤ n + 1. With this notaton, we define for each triple (α, β, l) ∈ Sn ×
Sn × [0, n] polynomial subalgebras R(α, β, l) of Q(KGln ) = Q(X (0) ) together
with ideals Ir (α, β, l) and Jr (α, β, l) (0 ≤ r ≤ n − 1) of R(α, β, l) as follows:
R(α, β, l)
:=
Ir (α, β, l)
:=
Jr (α, β, l)
:=
tιl (i+1) (α, β)
(1 ≤ i ≤ n), yji (α, β), zij (α, β) (1 ≤ i < j ≤ n) ,
Z
tιl (i) (α, β)
tιl (r+2) (α, β)
if l ≤ r ≤ n − 1 and Ir (α, β, l) := (1) else,
tι (r+1) (α, β)
l
tιl (n−r+1) (α, β)
if n − l ≤ r ≤ n − 1 and Jr (α, β, l) := (1) else.
tιl (n−r) (α, β)
Proposition 4.1. There is a covering of KGln by open affine pieces X(α, β, l)
((α, β, l) ∈ Sn × Sn × [0, n]), such that Γ(X(α, β, l), O) = R(α, β, l) (equality
as subrings of the function field Q(KGln )). Furthermore, for 0 ≤ r ≤ n − 1
the ideals Ir (α, β, l) and Jr (α, β, l) of R(α, β, l) are the defining ideals for the
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closed subschemes Yr (α, β, l) := Yr ∩X(α, β, l) and Zr (α, β, l) := Zr ∩X(α, β, l)
respectively.
Proof. We make the blowing-up procedure explicit, in terms of open affine
coverings. For each k ∈ {0, . . . , n} we define a finite index set Pk , consisting of
all pairs
p0
q0
(p, q) = : , : ∈ {0, . . . , n}k+1 × {0, . . . , n}k+1
pk
qk
with the property that pi 6= pj and qi 6= qj for i 6= j and that pi = 0 for some i,
if and only if qi = 0. Observe that for each k ∈ {0, . . . , n} there is a surjection
Sn × Sn × {0, . . . , n} → Pk , which maps the triple (α, β, l) to the element
α(ιl (1))
β(ιl (1))
,
:
:
(p, q) =
α(ιl (k + 1))
β(ιl (k + 1))
of Pk . (Here we have used the convention that α(0) := 0 for any permutation
α ∈ Sn ). Furthermore, this surjection is in fact a bijection in the case of
k = n. Let (p, q) ∈ Pk and chose an element (α, β, l) in its preimage under
the surjection Sn × Sn × {0, . . . , n} → Pk . We define subrings R(k) (p, q) of
(k)
(k)
Q(xij /x00 (1 ≤ i, j ≤ n)) together with ideals Ir (p, q), Jr (p, q) (0 ≤ r ≤ n),
distinguishing three cases.
First case: 0 ≤ l ≤ k − 1
R(k) (p, q)
:=
Z
"
tιl (i+1) (α, β)
tιl (i) (α, β)
(1 ≤ i ≤ k), yji (α, β), zij (α, β)
1 ≤ i ≤ k,
i