poincaré polynomials of the variety of stable bundlesypandey/s.ramanan/papers/75.pdfmath. ann. 216,...

Math. Ann. 216, 233--244 (1975) @ by Springer-Verlag 1975

Poincar6 Polynomials of the Variety of Stable Bundles

Desate Usha V. and S. Ramanan

§1.

In this note, we indicate a few improvements to [3]. Let X be an irreducible, non-singular projective algebraic curve defined over a finite field Fq with q elements, of characteristic p. Let SL(n, d) be the coarse moduli scheme of isomorphism classes of stable vector bundles of rank n and determinant isomorphic to an lFq-rational line bundle L of degree d. [We will assume that (n, d) = 1.] By replacing lFq by a finite extension if necessary, we may assume that SL(n, d) is defined over IFq. As one might expect, it is then indeed true that the IFq-rational points of SL(n, d) are precisely the stable vector bundles on X defined over IFq (see [3]). By the Weil conjectures, it is easy to write down the Poincar6 polynomials of SL(n, d) once the number of its lFq-rational points is known. In order to compute the latter, one first notices that the fact that the Tamagawa number of SL(n) is 1 can be interpreted as follows.

1 1 _ q(,2 _ 1)(o- 1)(x(2)." .(x(n) ' the summation ex- Proposition 1.1. ~ ~AutE ( q - 1)

tending over the isomorphism classes of vector bundles of rank n, degree d with a fixed determinant. Here ~x denotes the zeta function of the curve X.

Proof The correspondence between X-lattices and vector bundles, together with the fact that the Tamagawa number of SL(n) is one leads to the formula (see [3, §2.3] or [2, §2])

1 = q{ ,~ - ~*{o- , ) { x ( 2 ) . . . ~x(n ) =_- z , ~x ~ {x- lkxc~SL(n, K)}

the summation runs over double cosets in k\~SL(n, A)/SL(n, K), K, k denoting respectively the function field of X and the stabiliser in SL(n, ,4) of a fixed lattice. Interpreting x - I kxc~SL(n, K) as the group of automorphisms of the vector bundle E x determined by x, which have determinant one, the L.H.S. of the above formula

reduces to

1

(1.2) ~x ~ {AutE x with detl}"

To complete the proof of the proposition we now need the following simple lemma which generalises the Lemma t of [2].

The second-named author would like to thank the "Sonderforschungsbereich Rir Theoretische Mathematik" of the University of Bonn for its hospitality during the preparation of this paper.

234 U.V. Desale and S. Ramanan

Lemma 1.3. Let E o be a vector bundle of rank n with a f i xed determinant. Then there exist exactly #(iFq*/ImAutEo) double cosets in k\SL(n, A)/SL(n, K) which determine a bundle isomorphic to Eo, where ImAutE o denotes the image o f AutE o in IF* under the determinant mapping.

Using the lemma, the expression (1.2) reduces to

# {F* / ImAu tE} (1.4) ~E # {AutE with detl}"

In view of the exact sequence

0--,AutE with detl--*AutE d--~ IF * ~ I F * / I m A u t E ~ 0

(1.4) becomes

~E q - 1

# AutE

where the summation runs over the isomorphism classes of vector bundles of rk n, with a fixed determinant.

Since stable bundles admit only scalar automorphisms, Proposition 1.1 reduces the problem of computing the number ~L(n, d) of IFq-valued points of SL(n, d) to one of computing the IFq-aUtomorphisms of non-semistable bundles.

Proposition 1.5. Any vector bundle E on X admits a unique flag

0=EoCE, C...CEk=E

satisfying

i) Ei/E i_ 1 is semistable for i= 1 ..... k ii) #(Ei/E , _ 1) > p(Ei + x/E~) for i = 1 ..... k - 1,

where #(F) is the rational number degF/rkF. Equivalently, 1) Ei/E i_ 1 is semistable for i = 1 ..... k 2) For any subbundle F o r e containing Ei we have

l l (e l /Ei- 1)> It(F/Ei- 1), i= 1 ..... k .

For proof, see I-3, 1.3.9-1 and also [-8, 2.5.1].

Definition 1.6. The length of the unique flag corresponding to E is the length of E and is denoted by l(E). We denote by di(E), ri(E), ~i(E), the numbers deg(Ei/E_ 1), rank (Ei /E i_ 1) and ~(Ei /E i_ 1) respectively.

Proposition 1.5 has two consequences. Firstly, since such a flag is unique, we see by Gatois descent that the flag is actually defined over IFq, when E is IFq- rational. Secondly, this enables one to set up an induction on rank, in order to compute the number

1 ilL(n, d) = ~ Esemistable ~ AutE"

Poincar6 Polynomials of the Variety of Stable Bundles 235

When n and d are coprime, since semistabte bundles are stable, we have (q - DilL(n, d)=~L(n, d). The induction to compute ilL(n, d) is done in [3] working again with the adele group. We will show in this note that once Proposition 1.1 is admitted, one could carry the induction forward only with vector bundles and this seems to lead not only to greater clarity, but also to better results. The key step in our approach is the following

Proposition 1.7. i) ilL(n, d) is independent of L, and hence may be written simply fl(n, d).

1 ii) Denote by tic(n1 ..... nk) the sum ~ ~ A u t E ' where the summation extends

over isomorphism classes of bundles E of rank n, determinant L and of length k with ri(E) = ni. Then

flL(nl ..... rig) = g" #e Jg21 ~1~=1 fl(ni, d~) , Z.,qZQ.:~:)

where the summation extends over (d I ..... dk)eZ k with ~i di = d and dl/nt >. . . > dk/n k,

J ~ denotes the number oflFq-valued points q the Jacobian of X, and Z Ida. ..dk)

denotes

~i < j (dins- djni) + ~,i <,i n i n j 1 - 9 ) .

Proof. We prove both (i) and (ii) simultaneously by induction on n = Zn~. Consider a vector bundle E of length >2 admitting the flag of Proposition 1.5. Write M=E/E~. Notice that p(EO>Sup#(N), the supremum being taken over

N all subbundles N of M. In view of the proof of the Proposition 4.4 [6], there exist no nonzero homomorphisms of E~ to M so that every automorphism of E keeps E~ invariant and hence goes down to an automorphism of the quotient M. Thus we get a map

AutE~AutE~ x AutM

with kernel I + H°(X, Hom(M, E0). Also, AutE 1 x AutM acts on //~(X, Hom(M, E~)) i.e. on equivalence classes

of extensions of M by E~ and two bundles given by such extensions are isomorphic if and only if they belong to same orbit under this action. The isotropy subgroup at E is precisely the image of AutE under the above map and hence is isomorphic to AutE/I + Ho(X, Hom(M, E0. Therefore we get

1 f l L ( n l "" "nk)= 2E,,M ZExtens . . . . orMbye, 4~AutE. #(elts in the orbit of E)

1 = ~E~,M # (AutE1 x AutM)q x(M*®E°

where x(M*®EO denotes the Euler characteristic of M * ® E v The summation extends over all pairs of bundles (El, M) where E 1 is semistable of rkn~, M is of

236 U . V . Desa le a n d S. R a m a n a n

length ( k - 1) and has determinant equal to detE®detE~- 1 and

# ( E 1 ) > p a ( M ) > . . . > p k _ l ( M ) and ri(M)=ni+ 1, i=1 ..... k - 1 .

Denoting by jd, the variety of isomorphism classes of line bundles of degd I on X [nl n - n l 1

and noticing that z ( M * ® L ) = Z \ d l d _ d l j in the notation of the proposition,

we have

fiL(nl""nk)= 2d, sZ 2re.,~. flL®r-t(n2""nk) 2detE,=r 1 d,>e~ qX(]:;-;:) ~AutE1 " tl I n2

Applying the induction to fiE®r -,(n2 ..... rig) and noticing that the last summation

is simply fi,(n, d), since X \ d id_a l l ] +Z dz. . .d k - Z \dl . . .dk }, the R.H.S. of the

above equality simplifies to the required expression in the R.H.S. of the equality in (ii) of the proposition. Thus flE(na...nk) are independent of L and since

fiL(n,d) = Z, EfiL(nt...nk ) q--1

(the summation being over all ordered tuples (nl...nk) with ~ = a ni =n and k=> 2), it follows that fiE(n, d) is independent of L.

As mentioned earlier, as a consequence of the Weil conjectures and Artin comparison Theorems (see [3]), we have the following recipe to compute the Poincar6 polynomial Pn,d(t) of Sc(n , d), when n and d are coprime. The zeta function of X is given by [4, Chapter VI, §3]

1-12_ ° , (1--~oiq -s) ~x(S) = (1 -- q - ")( 1 -- q 1 - ,)"

Then the substitution e ) i - - ' - t and q ~ t 2 in the expression for fi(n, d) . ( q - 1 ) gives P.,d(t).

From Propositions 1.1 and 1.7 we obtain

q(n z- I ) (o - 1)

fi(,.d~- (X(2)...(x(n)- ~:C,,=, fiE(n, ..... rig) ( q - 1) k> 2

_ q(,2_ 1)(0- 1) r "VI ¢1 e) ~k- 1 [I~= 1 fi(ni, di) (q--l) (x(2) '"~x(n)- / -"2k '>==z" ' '~ t -" 2 qY(~',i]')

where the last summation is taken over d i6Z , n~eN with ~ d i = d such that dt d2 dk

- - > - - . . . > --. The last step uses the fact that the number of lFq-valued points /'/1 n 2 nk

in the Jacobian of X is given by I-I2=0 ~ (1 -o)~). Since tensoring by a line bundle of degm gives a bijection of the set of iso-

morphism classes of semistable vector bundles of rkn, degd with the set of isomorphism classes of semistable vector bundles of rkn, deg(d+mn) and Aut E ~ A u t E ® L , for any line bundle L, we see that fl(n,d) depends only on the


congruence class of d modulo n. This enables us to express fl(n, d) in the form

q ( n 2 - 1)(o- 1)

(1.8) fl(n, d)= ( q - 1) (x(2)(x(3)...(x(n)

- Z Hi (1 - wl) k l qy,<j,,,,{o-1, Z(6,)e[I~-,Z/niQ~,8 1-I~=1 fl(ni, 6i}

where the first summation extends over all tuples (nl...nk) of integers with ~)= 1 ni = n and k > 2. Q~,~ are numbers independent of the genus g of the curve, gwen by

1 (1.9) Q~,~= Zq£,<~(d,,,-a,n,)

where fi=(bi)~(Z/n~) k and the summation extends over all (di)~2g k such that dk

~i di = d, di =- 6i(ni) and dl > . . . > --. nl nk

It is quite convenient to write the above expressions in the language of partitions.

Definition 1.10. A partion (respectively an ordered partition) ~z of length k of a positive integer n is a (k + 1)-tuple (respectively an ordered (k+ 1)-tuple (nl ..... rig+ 1) such that ~+11 nl = n. A separation H of a partition n = (n 1 ..... rig+ 1) is a partition of length less than or equal to the length of n obtained by bracketing the n;'s into smaller partitions. The smaller component partitions will be called the separates of H with respect to ~.

For example, if ~ =(nl...nT), then H = (n 1 + n 2 -f-n 3, n 4 +ns, n6, nT) is a separation of n with separates (nl, n2, n3), (ha, ns), (n6) and (nO.

Notation 1.11. r(rc)=length of ~. Let 2(~)= ~ i< j n~nj. Note that 2(~) does not depend on the order of the partition 7r. Let z ,=~, . . . z ,~+, and r~(n)(~>z~=the power of ~x(i) in the product z~. ri(~) are determined as follows: Arrange the entries of ~ so that nl <n2 < ... <rig+ 1. Then bracket n as ~=((//1)...(/7~)) where Hj = (mjmj...mj), m l<mE<. . .<m~ , ri(Tr) is given by r i (~)=(k+l) - -~l]=l r(Ilj) for mp <i<mp+ 1. The following additivity properties of 2@) and r(~) will be useful later.

Lemma 1.12. L e t / / b e a separation of a partition n with separates H i, i = 1 ..... p. Then

1) r(~) = r(n) + YL1 r(H,), 2) 200 =2(H) + ~P=I 2(Hi).

Proof. The first is trivial to check. For the second notice that by induction it is enough to prove 2) in case p = 2.

Theorem 1. In the above notation, the Poincard polynomial P,,d(t) of SL(n, d) takes the form

22(n)(g- 1) r(r0 3 r2(n) -t- t 2 n - -- d , t { ( l + t ) ( l + t ) ...(1 1)r,(,)}2, (1.13) P,,a(t)= L~ ~o~(t)- ( t ~ Z _ ~ + ~ ~ V ; ; ~ . . . ( t 2 , _ 1),2,(~)

where ~o~(t) are functions of t independent of g depending on ~, d, which can be determined inductively and the summation extends over all partitions of n.


Proof . We shall prove by induct ion on n that

(1.14) ( q - 1)fl(n, d )= ~ <p~. q~(~)(o- 1) H i (l --(Di) r(n) (q-1)"(~) r~,

where the summat ion runs over all part i t ions n of n,

(1.15) q)~= - Z n Z6~IIWI ,+'(Z/N,) Qn,6q~,~

w i t h / / v a r y i n g over all ordered separations of 7~ and

(1.16) ~6 _ v W[~(m+l ~, "F//,~ - - / ~{H,}, t l [ i = 1 ~0 / / , ) ,

the summation running over the set of all ordered separates of H with respect to 7~.

(For g of length zero, we define <pd to be 1) (1.13) is simply a restatement of (1.14) by using the recipe given by Weil's conjectures.

We have from (1.8),

(1.17) ( q - 1)fl(n, d) = z , - ~,~ l-[i (1 - ~oi)rt~) q ;'l~'(°- ' )

On repeated applicat ion of induction, one sees that the R.H.S. is of form y ' j ~ % where the coefficients f~ of z~ are functions of 1-[~ (1 -co~) and q; it is also clear from the above expression that a term in f , ~ is contr ibuted only by the ordered separations of 7z. The cont r ibut ion to (q-1) f l (n ,d) by an ordered separat ion 17 = ( N , .. . . . Nk) o f ~ is

- ( q - I) [ L (I - c o y ( m q ~(m(o- ') ~,(6,)~,~H)+, z/N, Qm6 H~ (=r/l) + 1 f l (gi , 6i)

in which the term involving ~, is

- (q - 1) [ I i (I - coy(mq ~(m(°- ')

.2g" ) iFIr(H)+I I~O~q)"II')'g-l>Hi(l--(Di)r(11') } "~ '-~rt,0 ~,1 1i = 1 [ ' ( ~ 1)r(r/0 ( q_ l)r(t/)+ 1

where the summat ion runs over all 6 = ( 6 1 ) e l - [ ~ ( = m + ' Z / N i and also over all sets {H~}~ of ordered separates o f / / w i t h respect to ~z. Note that I-L z n , = z , and we have applied induct ion to fl(N~, 5i). Using the Lemma (1.12) the above expression simplifies to

(1.18) - [ I i ( 1 - ( g i ) ~(n) (q _ 1) "'~) qZ(~)(o- 1)r. Z6~H[(_/.II ) + i Z/N, Qr~,6~OJH,.

6 ~on,~ is obtained simply by picking up the coefficient of

q(Z(~)- z(m)(o- ~) q - 1 F[2= ° 1 (1 - ~oi) '(~)- r(n)%

in the expression for l~.(= m+ i fl(Ni ' 6i ) which is determined inductively. Substi tuting from (1.18) in (1.17) now gives (I.14), (1.15), (1.16).

Poincar~ Polynomials of the Variety of Stable Bundles 239

2. Methods for Computation of Q,,6

Since the determination of the Poincar6 polynomial reduces to the computation of Q~,~ which in general seems quite difficult, we derive a few explicit formulae for Q~,~ [viz. (2.5) and (2.11)].

For an ordered partition n = ( n l ..... nr+ 0 and 6=(6i)i~IJ'i+=IZ/ni,Q~,~ is defined by (1.9) as

(2.1) Q~,~= ~ q - X , ~ , , , - d j , , ) ,

the summation extending over (di) ~7/r + 1 satisfying ~ di = d, d i =_ 6i(ni), and d 1/n 1 > dz/n2 >. . . > dr+ 1/nr+ 1.

Let F...dini+ 1 - di+ 1hi, i = 1 ..... r. Using the condition ~ + 11 di = d, di's are obtained in terms of Ffs as:

dn~

Let L~ denote the affine form on the R.H.S. of (2.2). In the new coordinates F/'s the expression (2.1) for Q~,~ takes the form

(2.3) r~ -V ,~ -x ;~ , ¢~,~ "~n,6 -- Z.. "1

where the summation runs over (F)E:g ~ such that Fj>0, Lj(r-)-c~j is integral, nj

for all j and

Choose integers mt such that mt is an integral affine form, l = 1 ..... r. Then \ nl /

~ L,~Z/m, =0 otherwise.

by orthogonality relations for characters on finite groups. Hence

Q~,6(t)= I-I-q+ 1 Er 'EZr ~ ; £ ) 1 [I'=1 ~"J'~z,", e q I U = I r o t \ ¢,>o

1 1 2~ij~(dn- ~ " ) H k = l EF=I t 2qkre- 2rti(~ljtfl~jF -- E(jt)e[l~(Z/mt) ¢ r ~ _ l-ITm,

where fl~, denotes the coefficient of Fk in Ldnt. Computing the summation over F and substituting for/3~, from (2.2) we obtain

1 5,j,,~llr+l,Z/m,, e , i ~ = , [(

' / t i v , + , ( Z , : ),)" where A~ = z.,t: ~ =


The general fo rmula looks c u m b e r s o m e but simplifies considerably in part icu- lar cases. We give one example which is of interest viz. n = (1, 1...,1) where r(n) = n - 1.

In case n = ( 1 , 1 ..... 1), 6i...OVi. Not ing that dr+l integer ~ d i integer Vi= 1 ..... r and mr+l can be chosen to be n, we can replace the set of condit ions {Lift) is an integer Vi} by a single condi t ion viz. Lr+ 1(~1...72) is an integer. Hence put t ing Jl...0V1 = 1 ..... r and replacing jr + 1 by j, mr+ 1 by n and m i by 1, i = 1 ..... r, the formula (2.5) reduces in this case to the simple formula

1 _ _ . d . . 2nv--~i. . (2.6) Q~,,...1),6(t)= ~ ~ z / . e2~-1J~I~ ' - - -~ { t 2 a " - O e ~ " J - 1}-1

Another Method for Calculation oJ Q~,6

For (b i )~I~+~ Z/(ni), we choose a representat ive for each 6i~Z/ni lying between 0 and n i - 1, which we again denote by (5 i. Since di=bi(ni), di can be wri t ten as

(2.7) di=6i+qini , q i~Z .

(2.8) Let m i = q z - q i + l i = 1 ..... r .

Tak ing q~+ 1 = b and solving back for qi's we get

q~= ~,j>=i mj=b, i= 1,...,r.

Hence, given (mi)~Z ~ we can obta in (q/)~2U +1 satisfying the condi t ion ~i d i=d iff

(2.9) ~ - 1 mi(~j<=i n i ) - ( d - ~.i 6i) modu lo n .

Also, di di+ 1 (~i+ 1 (~i - - > iff m~> . Hence, subst i tut ing in (2.1), we get n~ n~+l n~+l nl

(2.10) Q~,6(t)=t-2Y,~jo,.~-6~.,) ~t-2E~- ... . . ,

the s u m m a t i o n running over all (mi)eZ ~ satisfying

6i+1 6i, m i > E [ : I rni(~j<=i nj)=(d - ~'.ibi)modn, and ai= ~k<=i<inknj.

ni + 1 ni

Let mi=p~+nv~, O~l l i<n . Then

Q~.~=t- 2x,.,o,.,-~,.,) ~.~.,)~.~ t - 2E7-, u,a, Z(v,ieZ~ t - 2nEf . . . . . .

where #i run over integers satisfying 0 <= #~ < n, ~.~_1 Pi(~.j <=~ n j) = ( d - ~ i 61)modn,

(, ,_./t ummin last series v~ run over integers satisfying v~ > \ n i + l / nil ] n

over v~ we get

1 ~-~ f-2yr=,u,a,+a,nO, (t 2 . . . . (2.11) Q~,o(t)= tk~)/_.tu,)ez.~ ' H r = l 1) -1

=t-k~)l--[7= 1 ( t 2 " ' - - 1) -1 ~(u0~z~ t - 2yr,=, u,~, +..,o, ,

the summat ion running over all (#i) with y '~=lp i (~s<=,ns ) - (d -~ ,6 , )modn , O<=Pi<n, O<=(Si<ni, k (n )=2 ~ i < j (6inj-bjni), ai = ~i<=k<j nknj, Di= the integral


partr +' "' )!] - kt i . O i has value - 1 except in the case when p i = 0 and [\ni+ 1 ni

•i+1 > oi; in this case the value is 0. The formula (2.11) is less cumbersome and hi+ 1 El i

hence is handier for practical purposes while the formula (2.5) has some technical advantages.

Expression for (p~ for rc=(nl, n2)

In this case, (p~ is simply Z , Q~ ..... ,,0 ifnx = El2 and ~0~ = Z , Q( ..... ),0 + Za Q( ..... ),~ i fn I + n 2. Now,

1 S~Q( ..... ),~ ( t )= 2dl;d2dd t2(aln2-a2ni)

nl n

1 dnl +

= 2 1 =[~-] 1 t2 (d ln -dn D

y.,-2.[~,] t 2n- 1

Hence,

(2.12) d q5 ..... )(t)=

and

t~ , . , - =,,[,,,,,] + t ~ '"=- = . [ ~ , ] t 2 n - 1

if n 1 4= n 2

t"-2"[-~] (2.13) _dq,(L~)(t)_ t 2 " - 1

where [m] denotes the integral par t of the rational number m.

§3

Finally, we solve the recurrence formula (1.13) in the cases n = 2, 3, 4 to obtain

Theorem 2. (1) (a) PZ,d(t) where d is odd is given by

(t 3 + 1) 2̀ o _ t2ò(1 + t) 2̀ o

(t 2 - 1 ) ( ¢ - 1)

(b) The sum of Betti numbers is 22`0-2g. (2) (a) P3,d(t) where d = 1 or 2 (3) is given by

(t 5 + 1)2ò(t 3 + 1)2o_ (t 2 + 1)2t 4ò- 2(1 + t)2ò(1 + ta) 2̀ o + (1 + t 2 + t4)t 6ò- 2(1 + t) 4ò

(t 2 - 1)(t 4 - 1)2(t 6 - 1)

(b) The sum o f the Betti numbers is 24~-4(zf).


(3) P4,e(t) for d = - 1 or 3 (4) is given by 1 {(1 + t3)2g(1 + ts)2g(1 + tv) 2a

(t 2 -- 1)(t a - 1)2(t 6 -- 1)2(t s -- 1)

- - (1 + t 2 + t4)(1 + t4)t6g-4(1 + t)2°(l + t3)2g(1 + t5) 2g

- (1 + t 2 + t4)2ts°-4(1 + t)z°(1 + t3) 4°

+ (1 + t2)4(1 + t4)t 1 oo- 6(1 + 040(1 -k- t3) 20 -- (1 + t 2 + t4)/(1 + t4)t 12o- 6(1 + t)Oq.

Proof. The formulae for P,,a(t), n = 2, 3, 4 are results of s t ra ightforward com- putations. We give the p roo f of (2) (b). (1) (b) is obta ined exactly on the same lines by using a similar recurrence formula.

(2) (b): It is easy to see that there is a recurrence relat ion

P~,~2(t)- {t6(1 + 0 4 + (t 5 + 1)2(r 3 + 1)2}P~,~l(t) + {t6(] + t)4(t 5 + 1)2(t 3 + 1)2}p~(t)

= (t 2 + 1)2t40+ 2(1 + t)2o+ 2(1 + t3)20+ 2

where P°3, a denotes P3,a(t) for X of genus g. Put t ing t = 1, we get

(3.1) n~+,az(1)- 25P°3+,a1(1)+ 28p~,a(1)= 24°+6 ,

(2) (b) trivially holds for g = 1. Actual c o m p u t a t i o n of Betti numbers for g = 2 shows that (2)(b) holds for g = 2 (see the appendix). The result now follows by induct ion f rom (3.1), not ing that

2 ( 2 g f 2 ) - ( 2 2 ) + 4 = ( 2 g ; 4 ).

Remark (1). The Betti numbers for n = 2 have been obta ined by Newstead [7]. (1) (a) in the above theorem can easily be seen to agree with the results of [7].

(2) The sum of Betti numbers in case n = 4 is likely to be 260-4 (3g). 1

Determination of the Betti Numbers in Low Dimensions

For any rank n, the Betti numbers in some low and high dimensions can be determined easily wi thout the actual calculat ion of the Poincar6 polynomial . Since we have a l ready compu ted Poincar6 po lynomia ls for 2, 3, and 4, we assume henceforth that n > 5.

- J ( t ) The expression on the R.H.S. of(2.5) shows that Q~,6 are of the form [ L (1 - 2it i)

with 2 i~c and f(t) a polynomial . Hence the same s ta tement is true for ~o~ as can be seen by induction. Therefore a t e rm in the expression for the Poincar6 polynomial , cor responding to a par t i t ion ~ is of order at least t 2~) .

Claim: Fo r Tc = (nl ..... ni), (~i ni >= 5), we have

(5.2) 2 ( ~ ) > 2 ( n - 2 ) e x c e p t f o r ~ = ( 1 , n - 1 ) , n = ( 2 , n - 2 ) .

The s ta tement is obvious for r = 2 . Fo r rc=(1,1, n - 2 ) , 2 ( T r ) = 2 ( n - 2 ) + l . For 7r = (n l , n2, n3) with no two of the ni's equal to one, we have

2(~) = nl(n 2 + n3) + n2n 3 >= 2(n 2 + n 3 -- 2 )+ (n 2 + n3)n 1 (by s ta tement for r = 2)

= 2(n - 2) + (n 2 + n 3 - 2)n l

> 2 ( n - 2) as n 2 + n 3 > 2 .


The claim now follows easily by induct ion on r: We have for r > 3 ,

2 (~)=n r Z ~ n,+ El<i<j<r ninj=n ~ Z~2~ ni+ 2(Z~2 { n,-2), by induction

= 2 ( n - 2)+ n,(E~=-~ n i - Z )

> 2 ( n - 2).

F rom (2.12) and (1.11), for the term corresponding to ==(1 , n - I ) in the Poincar6 polynomial ,

,,t :,-' t (3.3)... (P~(t)t2~4=)(°-l)=t2(n-1)(a- t -[2nZ- l J"

Since (n, d) = 1 and Poincar6 polynomials for degree d and n - d are same, we may assume 1 < d < ~. Then

so that

2 ( n - 1 ) d - 2 n I ~ ] = 2 ( n - d ) > 2d.

Hence (3.3) becomes

(3.4) ~ g 2[(n-1)(O-1l+d](t-~'t2(n-2d)) q~(1,.- l)(t) t2~(1'"- 1)= t 2 " - 1

Similarly, for n = (2, n - 2) we obtain

(3.5) ,,d ~2~2,,- 2) _ [ (1 + t 2(n - 4 a ) ) t Z [ 2 ( n - 2 ) ( 0 - 1) + 2d] n " q ' ( Z ' n - - 2 ) ~ - - - - t 2 n - 1 if d < ~-

/ (1 + t 2(4a - n)) t 2 1 2 ( n - 2 ) L q - 1 ) + n - 2 d ] n n

= t 2"- 1 if ~ > d > ~.

Clearly (n-1)(g-1)+d<2(n-2)(g-1) for n > 6 and also for n=5, d = l . Fo r d = 2, 1, n = 5, the R.H.S. of (5.10) is of order t 212{"-2)~g- t)+ 1] and 2 ( n - 2)( 9 - 1)+ 1 > ( n - 1)(g- 1)+d.

Thus

(3.6)... (n-1)(g-1)+d<2(n-2)(g-1)+l ( n>5 ) .

The formulae (1.13), (3.2), (3.4) and (3.5) give the

Theorem 3. The dimensions of the l-adic cohomology groups of SL(n, d) for low dimensions are given by

/ a i for 0 _ < i < 2 [ ( n - 1)(g- 1 )+d]

dimH'(S, QO= a,+bi, for 2 ~ n - 1) (g- 1 )+d] <i<4(n-2)(g- 1)+ 1

[ ai+bi+c~, for 4 ( n - 2 ) ( g - 1 ) + 2 < i < 2 ( g - 1) (2n-3) ;

(l, d)= 1 = (l, n).


where ~ a,t', ~, b,t' and ~, ci ti are respectively the terms corresponding to the partitions n = ( n ) , (1, n - 1 ) and (2, n - 2 ) in the PoincarO polynomial after expanding all the denominators in the expression ( 1 . 1 3 ) f o r the Poincark polynomial in formal power series.

C o r o l l a r y . I f d f# d'(n), then SL(n, d) and - ' SL(n, d ) are not topologically equivalent.

Appendix

Betti numbers for n = 3 for genus 2 and 3

g = 2 g = 3

b o = 1 1 b I 0 0 b2 1 1 b 3 4 6 b4 3 3 b 5 8 12 b 6 9 19 bv 12 24 b 8 20 58 b 9 - - 62 blo 104 bll 170 blz 194 bx3 292 b14 - - 344 b15 - - 394 bl6 - - 472

References

1. Deligne, P.: La Conjecture de Weil, I. Publ. Math. I.H.E.S. 43, 261--307 (1974) 2. Harder, G.: Eine Bemerkung zu einer Arbeit von P. E. Newstead, Jour. fiir Math., 242, 16~25 (1970) 3. Harder, G., Naras imhan, M. S.: On the cohomology groups of moduli spaces of vector bundles on

curves. Math. Annalen 212, 215--248 (1975) 4. Lang, S.: Abelian Varieties New York: Interscience Publications 1959 5. Mumford, D.: Geometric Invariant Theory, Berlin Heidelberg New York: Springer 1965 6. Narasimhan, M.S., Seshadri, C.S.: Stable and unitary bundles on a compact Riemann surface,

Ann. of Maths. 82, No. 3, 5 4 ~ 5 6 7 (1965) 7. Netstead, P.E.: Topological properties of some spaces of stable bundles, Topology, 6, 241--262

(1967) 8. Tjurin,A.N.: Classification of vector bundles over an algebraic curve of arbitrary genus, Amer.

Math. Soc. Transla. 63, 245 279 (1967) 9. Weil, A.: Adels and algebraic groups, Lecture notes. The institute for Advanced Study, Princeton

(1961)

U. V. Desale S. R amanan Tata Institute of Fundamenta l Research School of Mathematics Colaba, Bombay 400005, India

(Received December 18, 1974)

poincaré polynomials of the variety of stable bundlesypandey/s.ramanan/papers/75.pdfmath. ann. 216,...

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