18
MARTIN ARKOWITZ AND GREGORY LUPTON
the rational cohomology of X :
with lxi,j I
=
2j and IYi I
=
2i -
1.
The differential
d :
r
X -+
r
X
is given on
generators by
d(xi,j)
=
0 and
d(yi)
=
I:
Xl,j
1

Xk,jk,
j, +·+jk=i
with the convention that Xi,O
=
1 and Xi,j
=
0 for
j
ni or
j
0.
To determine the minimal model Mx we first consider the maximal rank case
n
1
+ · · · +
nk
=
n. In this case Mx is obtained by "eliminating" the generators
Xk,I, ... , Xk,nk together with the generators Y1, ... , Ynk whose differentials contain
linear terms. This yields the minimal model
where lvi,j I
=
2j and lwi I
=
2i -
1.
This is a 2-stage minimal model with
As is well known, the cohomology of Mx in this case is evenly graded.
In the general case, n
1
+ · · · +
nk :::; n and we writer·
=
n- (n
1
+ · · · +
nk)·
Restricting the differential
d
of
r
X
to the generators Yl' ...
'Yn, +··+nk'
we obtain
the same formula as in the maximal rank case
r
=
0. Furthermore, d(yi)
=
0 for
n
1
+ · · · +
nk
+
1 :::; i :::; n. This gives the following lemma.
6.1 Lemma If X= U(n)/(U(ni) x · · · x U(nk)) with
n1 :::; · · · :::;
nk, then
(i) there exists a rational equivalence
X
c::::
U(m)/(U(ni)
X ... X
U(nk))
X
S
2(m+l)-l
X ... X
S2n-l'
where m
=
n1
+ · · · +
nk.
(ii) Mx is a 2-stage minimal model A(V
0
,
V1;
d), where
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