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

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