Projective spaces in flag varieties

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Projective spaces in flag varieties

E. Strickland

a

a

_{Dipartimento di Matematica, Universitá di Roma "Tor Vergata", Via della Ricerca}

Scientifica, Roma, 00133, Italy

Available online: 23 Dec 2010

To cite this article: E. Strickland (1998): Projective spaces in flag varieties, Communications in Algebra, 26:5,

1651-1655

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COMMUNICATIONS 1N ALGEBRA, 26(5), 1651-1655 (1998)

Projective Spaces in Flag Varieties

Dipartiniento tii Matcrnatica UniversitA di Roma "Tor Vergata", Via della Ricerca Scicntifica, 00133, Roma, Italy.

0. Introduction

Let

G

be a semisimple algebraic group over an algebraically closed field

k

of characteristic zero. Let B be the variety of Borel subgroups. Consider the the projective embedding U -+ B ( H O ( U , L p ) * ) , where Lp is the line bundle associated to the Steinberg weight p, which is the half sum of positive roots. In this paper we show than any positive dimensional projective space contained in the image of U is necessarily of dimension one.

Furthermore we determine exactly these lines. Indeed we show that if

P

c

B

is such a line, then there exists a minimal parabolic subgroup P C

G

such that

t

is the set of Borel subgroups which are contained in P. In particular this implies that the possible homology classes of such lines correspond under the usual identification of the root lattice with H 2 ( U , Z) to the set of simple roots.

The result is obtained as an application of some properties of the intersection of Schubert cycles in the cohomology ring of U.

1 Copyright O 1998 by Marcel Dekker. Inc

STRICKLAND 1. Linear Spaces in U

Given a semisimple sirnply cor~nt:c:ted algebraic group G we want to recall a few facts on the geometry of the projective variety, 8, of thr Borcl subgroups of G.

Let 11s choose a maximal torus I' C G and a Bore1 suhgroup B

>

T . Let

W = N ( T ) /T be the Weyl group. Let f = LicT be t,he corresponding Cartan subalgebra, @ C 1" be the root system associated to LieT anti cP+ be the set of positive roots corresponding to the choice of B and { a , ,

.

. . ,a,} the set of simple

roots. Let us denote by B- the unique Borel subgroup such that B f l B- = T .

Furthermore we shall denote by s, E W the simple refection corresponding to a root a E

+

and, for an element w E W, t ( w ) the length of w with respect to the generators given by the simple refections s, = s a i . As usual we shall denote by wo the longest element in W. We shall also consider the weight lattice A = X S ( T ) ,

which is the dual of the root lattice Q, and denote by { w l , . . . ,w,) the set of fundamental weights defined by ( w , , a j ) =

&,,.

We set p =

CaE9+

a =

CI=l

w,.

Since all Borel subgroups in G are conjugate and a Borel subgroup equals its normalizer, we can identify

L?

with G / B . Once this has been done, we get an explicit cellular decompositiorl by locally closed affine spaces of U, whose cells, indexed by the elements w E W, are the B orbits C, = BrtwB/B, n,,, C N ( T )

being a representative of w . The C,,'s are called Schubert cells and their closures

-

C , Schubert varieties. One has dim C, = e ( w ) . It follows from this

[F],

that the Chow ring A S ( 8 ) has a basis given by the classes

[c,],

dual to the classes of the Schubert varieties and, if 6 = @, coincides with the cohomology (doubling the degrees). We also have that if we consider the pairing on A8(U) given by

(x,

Y)

=

/

xv

tz

for z, y E A*(U), then we haw that

([c,],

[c,,,])

= 6,,,1,,,,,. This is easily seen by

taking the cellular decomposition C , corresponding to B- arid remarking that the intersection of

C,

and Ci, is t3mpt,y unless ww' = wo, while if ww' = wo, C , and

C,, intersect transversally iri a single point.

PROJECTIVE SPACES IN FLAG VARIETIES 1653

As far as A' is concerned, one can give another description. Namely, given a character X

E

X * ( T ) , we can extend, using t,he natural homomorphism B -+ T, X to a character of B, and thus consider the corresponding one dimensional B module

k x . We now define the line bundle L A on

B

= G / B by G X B kx. By taking the fust Chern class c ( L x ) we get a isomorphism between A and A 1 ( B ) which takes the fundamental weights w , to the classes [??,,.i].

Remark that, using the above considerations, we deduce that, if we identify ~ d i m 5 - I

(B)

with the root lattice

&

by associating to a simple root cr, the class

[c.;],

we get that for z E A 1 ( B ) and y E AdimB-'(B), (z, y) = (2,y). Before we proceed, we need the following simple fact:

Proposition 1.1. Let X C B be a subvariety. Let [XI E A*(B) denote the corre- sponding dual class. Then [XI is a linear combination of the Schuhert classes [Cw] with non negative integer coefficients.

Proof. We can also clearly assume that X is of pure dimension d. The only thing we need to prove is that, when we express

[XI

as a linear combination of Schubert classes, the coefficients are non negative.

Indeed by a result of Kleiman (see [HI p. 268), since

B

is a homogeneous space, we can find an element g E G such that for every Schubert cell C,, of dimension complementary to that of X,

X

n

c,

is contained in C, and consists of a finite number of simple points. This clearly implies that, if we set n, = IX

n

??,I,

then

-

[XI =

Cl(tu,=d

nwow-1[Cw], hence the claim. O

Corollary 1.2. Let a,, .

.

.

,

at be Schubert classes, then the product class a , . .

.

a t is a h e a r combination with non negative integer coefficients of Schubert classes.

- Proof. By an easy induction, we can assume that t = 2. Let a l =

[c,,

1,

a2 = [C,",]. Then again, by the result of Kleiman quoted above, we can find g E G such that

-

a l a z = [C,,

n

g??,,] and the corollary follows from Proposition 1.1. U

Recall now that to any simple root a , we can associate a minin~al parabolic group

B

c

P,

c

G such that the natural map

is a PI-fibration whose fiber over [I] is exactly the Schubert variety

[?,;I.

Also given a weight X E A , the corresponding line bundle L A is very aniple if arid only if (A, a ; )

>

0 for all i = 1,.

.

.

,

r . Furthemiore L A is the pull hack of a line bundle on GIP,, which by abuse of notation we shall denote by the samc name, if and only if (X,ai) = 0, and, if this is the case, L A is very ample on GIP, if and only if (A, u,)

>

0 for all j

# i.

Let 11s now consider the weight p =

4

xaEO+

Q =

C:=,

w i . By the above

discussion the line bundle Lp is very ample on B while the line bundle L p i , with

pi = p - w , is very ample on

GIP,.

Consider now

l?

embedded in

P(HO(D,

L p ) * ) and

Ci,

.

We have the following

Theorem 1.3. Let

S

be a positive dimensional linear subspace in P ( H O ( f ? , L , ) * ) .

Assume S C

B.

Then (1) dim S=l

(2) There is a Bore1 subgroup B

c

G and a simple re1 ection s, ; such that Sis

the S c h b e r t variety F,,reiative to

B

i.e. S = F I B , P being the unique

minimal parabolic subgroup containing B and associated to s,.

Proof. Set t = d i m s and N = = dimB. Denote by c, the Chern class of the line bundle L,,. so that for any X =

XI

niw,, c ( L x ) =

xi

nict.

By definition we have that

Now, by Proposition 1.1 and Corollary 1.2, we immediately deduce t,hat if M is any

It follows, ~lsing the expansion of ( c ,

+

. . .

+

c,)" that there exists a unique z such that.

PROJECTIVE SPACES IN FLAG VARIETIES

while, for any other degree t monomid in cl.. . . , c,,

In particular we deduce that

Consider now the projection a,B -t GIP,. As we have already remarked, the line

bundle L,...,, is very ample on GIP,, so that

implies that dim a , ( S )

<

t .

Now xi is a PI-fibration, so dim x i ( S )

<

t - 1. Moreover xl-'(xi(S)) is an irre-

ducible variety containing S and of the same dimension of S. Hence S = x;' ( x ; ( S ) ) .

We have thus shown that our S is the total space of the P1-fibration S -+ x i ( S ) .

If we consider the line bundle L on S obtained by restricting L p r we deduce that S equals P ( a , , ( L ) ) . Thus H Z ( S ) = H 2 ( x , ( S ) ) H O ( r i ( S ) ) . If t

>

1, then rkH2(xi(S))

2

1, so we would deduce that rkH2(S)

>

2, a contradiction. Therefore the first part of our Theorem follows.

As for the second part, notice that we have proved that S is a fiber of the PI- fibration a,. Thus, if we take a Bore1 subgroup B E S , we deduce that S =

C,,

.

[F] W. Fulton,, lnterseclron Theory, Springer Verlag,, 1984. [HI R. Hartshorne,, Algebraac Geometry, Springer Vedag,. 1977

Received: March 1997

Revised: July 1997