Neeman, "Topology of Quotient Varieties"

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The Topology of Quotient Varieties Author(s): Amnon Neeman Source: Annals of Mathematics, Second Series, Vol. 122, No. 2 (Sep., 1985), pp. 419-459Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1971309Accessed: 13-09-2015 19:49 UTC

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Annals of Mathematics, 122 (1985), 419-459

The topology of quotient varieties

By AMNON NEEMAN

0. Introduction to Part 1

Suppose X is a scheme acted on by a reductive algebraic group G, and suppose a space X/G exists, together with an affine morphism g: X -4 X/G, and X/G can be covered by open affines such that over them r is given by the map Spec A -4 Spec AG. Then we call X/G the quotient of X by G. All known quotients are of this type. What we try to do here is investigate the relation between the ordinary, complex topology of X and that of X/G.

The key results of Part 1 of the paper are that there exists a closed subset C C Spec A such that

(a) The composite map C -4 Spec A -4 Spec AG is proper and surjective. (b) C is a deformation retract of Spec A, with a deformation retraction that

commutes with g: Spec A -4 Spec AG. (a) may be found in Corollary 1.4 and (b) is Theorem 2.1. The rest of Part 1

of the paper contains various applications of these results. A curious application of (a) is that it can be used to prove that the topology of X/G is the quotient topology it inherits from X; for the Zariski topology this is well-known (see Mumford [6]). But as far as I know the result is new for the complex topology of X/G. A proof may be found in Corollary 1.6 and Remark 1.7.

The rest of Part 1 is a study of some vanishing theorems on the high cohomology of X/G, a study that heavily relies on Theorem 2.1. In Section 4, I prove some vanishing theorems, and in Section 5, I apply the theorems to an example.

Now let us say something about Section 3. When I wrote the paper it was a largely conjectural section, but now I know that both Conjecture 3.1 and Conjecture 3.5 are true. Conjecture 3.5 is a special case of an inequality due to Lojasiewicz, and Conjecture 3.1 can be proved from Lojasiewicz's inequality using estimates similar to those in Section 3. I chose not to rewrite the text, because at present I do not feel I could give an adequate account of the proof of Conjecture 3.1. Although Lojasiewicz's inequality is enough, a stronger inequality should be true; roughly speaking, I conjecture that the correct value for E in



420 AMNON NEEMAN

Conjecture 3.5 is 1/2 (see remark 3.7). For this reason I feel the appendix is still important; it contains evidence for my new conjectures. If I rewrote Section 3 to incorporate my new conjectures, the new section would be too long, and largely unconnected with the rest of the paper. As it stands, Section 3 tells how to improve Theorem 2.1, and the discussion at the end of Section 5 shows how to use this improvement to get better vanishing theorems for the cohomology of X/G. Any changes made now could only detract from the presentation; in a later paper I hope to be able to explain Conjecture 3.1 better.

Part 2 of the paper is largely unrelated to Part 1; the reason they are lumped together is that they should be quite closely related once the problem is better understood. I refer the reader to the introduction to Part 2 and to the sequel where there is an explanation of what I think the relation should be.

Part 2 concerns itself with the study of the Chern and PontrJagin classes of bundles on the quotient. A technique is developed to study these classes. Then I apply this technique to an example, and show that it yields partial results for a certain conjecture due to Ramanan.

Finally, I would like to thank Mumford for valuable discussions. When I initially studied the flow of Sections 2 and 3, I did so by direct estimates. It was Mumford who suggested that I try to exhibit the vector field Y as the gradient of some function.

I also want to thank J. Moore for helpful conversations regarding the topological Section 7. Again, I initially studied the problem too narrowly, looking only at Lie groups. Moore suggested that I try a more general approach.

1. The sets C(a)

Let G be a connected reductive group acting holomorphically on a finite dimensional complex vector space V. Then G acts in a natural fashion on C[V *], the symmetric algebra of polynomial functions on V. It is a standard fact that the G-invariant polynomials in C[V *] form a finitely generated C-algebra, and that the map 7T: V= Spec C[V *] - Spec(C[V *]C) is surjective, and identifies points of V if and only if the closures of their G-orbits intersect (see Mumford [6]). It is also well known that the Zariski topology of Spec(C[V *]C) is the quotient topology inherited from V. What we want to do here is study the ordinary, complex topology of the quotient. In this section, we construct for every real number a ? 0 a subset C(a) C V such that the composite map C(a) - V -> Spec(C[V *]C) is proper (with respect to the ordinary topology) and surjective.

We start with some preliminaries. Let K be a maximal compact subgroup of G, and let (, ) be an inner product on V left invariant by K. Let A and 5 be the real Lie algebras of K and G respectively. 5 has a natural structure of a



TOPOLOGY OF QUOTIENT VARIETIES 421

complex vector space. Let X1,..., Xn be a set of right invariant vector fields spanning A, and put Yj = iXi E 0. All these vector fields naturally induce vector fields on V, and by abuse of notation we will call those Xi and Y1 too. If we identify V with its own tangent space, these vector fields correspond to linear endomorphisms of V. We call these also Xi and Yi. So, depending on the context, Xi and Y, could denote right invariant vector fields on G. vector fields on V, or linear endomorphisms of V. The fact that K respects the inner product translates to say that the Xi's are skew-hermitian, so that the Y1's are hermitian.

Let f: V -4 R be defined by f(v) = (v, v). Let g: V -4 R be given by g = Z n=l(Yif)2. The set on which g vanishes is precisely the critical points of the function f with respect to the action of G. It is a theorem of Kempf and Ness (see [4]) that in any G-equivalence class of points of V there is exactly one orbit of K on which g vanishes. In particular, the map g - '(0) -* V Spec(C[V *]C) is surjective, and has compact fibers.

The map g: V -4 Spec(C[V *]C) is almost never proper; the key result of this section is that if we throw in the map g we get a proper map. Precisely, we have:

THEOREM 1.1. The map

(g , g): V----Spec(C[V *1 G) x R

is proper (in the ordinary topology).

Proof Let P1, ... . Pr be invariant homogeneous polynomials in C[V*] which generate C[V *] C as a C-algebra. It clearly suffices to prove that, for some choice of positive integers a, a2, . .ar+1

(?T, g) 95V l ) ( V--:Spec(C[V*IG) X R Cr X R

is proper, where

(n(X), 5y) = (,P1(X) a, p2(X) a2 p . a,+(X) a I)

So we may choose a,, ... .,ar+? so that the map ?o(g, g): V Cr x R is homogeneous. By this we mean that there is an integer b such that multiplication by X in V will induce multiplication by Xb on the first r factors of Cr x R and multiplication by IX b on the last.

From homogeneity, it is now enough to show that the inverse image of some ball in Cr X R is compact.

First we consider the null cone N of V (i.e., the set of vectors v E V such that 0 E Gv). This is precisely g - 1(g(0)); so it is closed in V. Consider the intersection of N with the unit sphere S of V. On N n S. g does not vanish, by



422 AMNON NEEMAN

Kempf-Ness [4]. So g(N n S) is compact and does not contain 0 E R. It follows that there exists a positive real number a such that g- [0, a] n N does not meet S. Now g- [0, a] n S is compact and does not meet N. So its image by

projection

g9'[0,a] n S -V U x R - .c

is compact, and does not contain 0 E C r. It follows that there is a positive number /3 such that the image does not meet the closed ball in Cr of radius /3. Put y = min(a, /); then it is easy to show that the inverse image of the ball of radius y in Cr x R by the map T o (, g) does not meet S C V. Now homogeneity assures us that it must be entirely inside S, and so it is bounded. It is clearly closed as qp o (s, g) is continuous; hence it is compact. QED

We now derive some easy corollaries, but first a definition:

Definition 1.2. For any non-negative real number a, we write C(a)=

g'[O, a].

Remark 1.3. We observed earlier that the theorem of Kempf-Ness tells us that C(O) " V -4 Spec(C[V*]G) is surjective. So the same is true for any non-negative a.

COROLLARY 1.4. For any a 2 0, the map C(a) - V-4 Spec(C[V*]G) is proper.

Proof: The diagram

C(a) r V { j&OT,g)

Spec(C[V*]G) X [0, a] Spec(C[V*]G) X R

is a pullback diagram; so it follows from this and Theorem 1.1 that C(a) Spec(C[V *]G) X [0, a] is proper. But clearly

Spec(C[V*I G) X [O, a] -4 Spec(C[V*I G)

is proper; hence the corollary is proved.

COROLLARY 1.5. Given any affine variety Spec A of finite type over C on which a reductive group G acts, there are sets CA(a) C Spec A such that the composite CA(a) - Spec A -* Spec AG is proper and surjective.

Proof We may always embed Spec A equivariantly as a Zariski closed subset of a vector space V on which G acts. Put CA(a) = C(a) n Spec A.




COROLLARY 1.6. Let Spec A be of finite type over C and G a reductive group acting on Spec A. Then Spec AG has the quotient topology induced by the map 'r: Spec A -4 Spec AC.

Proof Consider the composite CA(a) -4 Spec A Spec AG. It is a proper map of locally compact spaces, and hence is closed. It is surjective; so it is an identification map. Hence so is aT.

Remark 1.7. The same extends to quotients of arbitrary varieties. The question is local, so that we may cover by invariant open affines and apply Corollary 1.6.

2. The deformation retraction: Part 1

We start with the following, very general setting. Let P: R' -4 R be a homogeneous polynomial, and suppose P ? 0 on all of R'. Let f: R' -4 R be defined by f(v) = v 11V2, where 11 11 is the usual, Euclidean norm. We observe:

(grad(P))f = (grad(P), grad(f))

= (grad(f ))P

and (grad(f))P is the derivative of P in the radial direction. As P is homogeneous, (grad(f))P ? 0, with equality if and only if P = 0. So we immediately have

(1) grad(P) = 0 P= 0

(the implication is the above; to get <= observe that P = 0 =* P is minimized).

(2) (grad(P))f 0. Locally, the vector field grad( P) can be integrated to give a flow. In the negative direction, f decreases. That flow takes the compact ball f < r to itself, and it follows that the grad(P) flow can be continued to - ox. Since the only critical points are where P vanishes, the flow can be used to construct a deformation retraction R -n P'-[0, a], for any a > 0.

Now let us specialize to the case where V, G, K and the Y 's, 1 < i < n are as in Section 1. Put g = Z(Yif)2 as above. We study the grad(g) flow. It can be used to define a deformation retraction V -4 g1 [0, a] = C(a). The key fact we need is that grad(g) is a vector field pointing in the direction of the orbits.

In the computations that follow, Yj will alternate in meaning. Occasionally Yj will be a vector field, and occasionally it will be a linear map Yj: V -4 V. To distinguish its two possible roles, when it occurs as a vector field I will write brackets around its argument; when it is a linear operator, there will be no



424 AMNON NEEMAN

brackets. Thus Y v is the linear operator Yj acting on v. Yj(K v, v)) is the vector field Yj acting on the function v -- (v, v).

Now

YVfP) = YP(Kv, v))

= Yiv, v) + (v, Yiv) = 2KYiv, v) as Yj is hermitian.

The metric on V is given by Re(, ) where V is identified with its tangent space. Thus

Re(gradY (f), w) = 2Re(grad((Yiv, v)), w)

= 2 Re[(Yiv, w) + (Yiw, v] = 4Re(Y v,w)

Therefore

(grad Y( = 4Yiv,

grad Yi (f) = Y

Now we obtain the formula n

(*) grad(g) = E8Yj(f)Yj, i=1

and summarizing, we get:

THEOREM 2.1. For a > 0, there is a deformation retraction V C(a), which is defined by a flow along the orbits of G.

From this, we deduce:

COROLLARY 2.2. Let A be a finitely generated C-algebra on which G acts. There is a deformation retraction Spec A -4 CA( a) for each a > 0, where CA( a) is as in Corollary 1.5. Furthermore, this retraction commutes with the map Spec A -4 Spec AG.

Proof As in Corollary 1.5, embed Spec A in a vector space V. Then the flow above, being a flow along orbits, respects the G-invariant set Spec A.

3. The deformation retraction: Part 2

For most of what we do in the sequel, the results of Section 2 are enough. The reader should feel free to skip this section. We know now that for a > 0, C(a) is a deformation retract of V, with the flow grad(g) giving the retraction. The problem we address here is whether the flow can be continued to - ox. The




question is: we have a map ( - x, 0] X V -- V, given by integrating grad(g). Can this be extended continuously to a map [ - ox, 0] x V -V?

I do not know the answer to this problem, but would like to conjecture that it is positive.

Conjecture 3.1 (Mumford). Let P: R' -- R be a homogeneous polynomial, and suppose P(v) ? 0 for all v E Rn. Then the grad(P) flow can be continued to - ox; precisely, there is a map [- 0,0] x Rn -* Rn extending the grad(P) flow.

Remark 3.2. If Mumford's conjecture were true, we would clearly be done; g is just a special polynomial satisfying the hypothesis of the conjecture. However, I do not know whether it is true. In any case, for the special case we have in mind, we can make some estimates on the flow.

The key to our estimate is the observation:

LEMMA 3.3. Let f, g be as in Sections 1 and 2; then (grad(g))(f) = 8g.

Proof. By formula (*) of Section 2, grad( g) = 8?n= 1Yi( f)Y1. So n

(grad(g))(f) = 8Z (Yi(f))2 = 8g. i=1

COROLLARY 3.4. Re~grad(g), grad(f)) = 8g.

It follows that

Ilgrad(g)II211grad(f)112 2 64g2.

However, this estimate turns out to be inadequate. So suppose we could prove a slight improvement; precisely, suppose we knew

Conjecture 3.5. For some E > 0, and some c > 0

11 grad( g )11 grad( f)12- E2c2-E

Remark 3.6. The inequality of conjecture 3.5 is homogeneous, so there is a chance that it holds. While I cannot prove the conjecture in general, I can prove it for G a torus; see Appendix.

In the rest of this section I will show how to prove that, given conjecture 3.5, we can extend the flow.

It is easy to verify that 11grad(f) 112 = 4f. So, changing the constant c appropriately, we get the inequality

f1-2E grad(g),grad(g)) ? Cg2-Es



426 AMNON NEEMAN

Now f increases along integral curves, so that we get the inequality, for t < 0,

dg C 2-E

dt (1 + f(O))- -

Note that we may suppose 0 < E < (1/2), because given any E > 0 for which an inequality as in conjecture 3.5 exists, such inequalities exist for all smaller E. So we now integrate the inequality.

d 1 _ c(I-)to dt g 1E (1 + f(o))1-2<

1 1 c(l - _ _t

g - g(0)'- (1 +f0) )

So along the integral curve which at t = 0 passes through v(O) E V, we get an estimate

g < A(- t)' /'-? for t < 0,

where A is a constant for this integral curve, and varies continuously with v(O). (Explicitly,

A = c(1 -1 ) l \ (1+ f(O))1'2E

The essential fact is that (-t )g is integrable on (-o, - 1). In fact, f~1 JA-1 | (- t)Eg ?| A(- t) (11-?)+? < B 00 00

where B = B(v(O)) is just the function A multiplied by the constant _of

1 )- 0(l/l?) +E

?

Now recall that

g(s) = dg dt = 11 grad( g)112 dt. 00 ~~~~00

Then

f'-1 s)Eg(s)ds =f'(-s)j Ilgrad(g) t dtds. 00

cc

The integrand is positive definite, so that we can change the order of integration. The region of integration is t < s < - 1, and the integral becomes

f II grad(g) ( s = f 1grad(g




Then we get

f 'Igrad(g)112 (- t)l+e-1i dt < (1 + c)B.

Now we use H6lder's inequality to estimate the integral of IIgrad(g) I. For any r < - 1,

f 11grad(g)II = j_(Ilgrad(g)II [(- t)+E- 111/2 )[(- t)le- 1/2 -00 __

< ( lgrad(g)11 (_ t)l+E - 1)

X (] [ t)l+e 1]

< (1 + E)1/2B1/2( [( t)1+E -1

Note that II grad(g) is integrable on (-o, 0). More importantly, we have shown that fr 0II grad(g)II -1 0 uniformly on compact sets as r - o - .

Using this, it is easy to extend the flow: We define v( - oc)= v(r) - fr .Lgrad(g). The fact that this extended function [-o, O] X V -* V is continuous comes from the uniform convergence of frO grad(g) to zero as r v-x.

Remark 3.7. I suspect that the correct E in conjecture 3.5 is 1/2. It works for the torus, and it gives the particularly simple estimate that A and B are independent of the starting point v(O).

4. High Betti numbers of quotient varieties

We now apply the results of Sections 1 and 2 to study the cohomology of the quotient. Let X be a variety over C and suppose G is a connected, reductive group acting on X. Suppose also that a quotient space X/G exists, as in the introduction; i.e., the map 'n: X -* X/G is affine, and X/G can be covered by open affines over which 77 is given as Spec A -- Spec AC. We will show that the vanishing of high cohomology groups on X implies the same on X/G. The argument is similar to the one in Seshadri [10]. There Seshadri used the idea to prove the vanishing of high etale cohomology groups of X/G, when the action of G on X was proper and without fixed points. The major improvement on Seshadri's result is that we show how to handle the points on which G does not act properly.



428 AMNON NEEMAN

A point x E X is called stable if x has a finite stabilizer and its orbit Gx C X is closed. It is well known that G acts properly on the set of stable points (see Mumford [6], Proposition 0.8).

Let XS be the set of stable points in X. Put Z = X - XS. Then we have inclusion maps j: XS - X, i: Z -* X. Let Q denote the sheaf of constants on X with values in Q. Let Q I Xs, Q I z be the obvious sheaves on XS and Z. Then we have an exact sequence of sheaves on X,

O -*j(QlXs) -*Q-*i*(Qlz) -*0

where jl(Qlxs) is the extension by 0 of the sheaf QIXs* So there is a corresponding long exact sequence in cohomology, from which we get that

HP '(Z, Q) = HP(X, Q) = 0 HP(X, j!(QIXs)) 0.

Now let k1 be an integer such that 1 ? k1 * H'(Z,Q) = 0, and k2 an integer such that 1 ? k2 H'(X, Q) = 0. Then 1 ? max(k1 + 1, k2) H'(X, j!(QIxs)) = 0.

The sheaf for which we will study the spectral sequence in cohomology is j!(QIxs). First we observe that when we take quotients, we also have inclusions

j: XS/G - X/G, i: Z/G - X/G and X/G - XS/G = Z/G,

where XS/G is open and Z/G is closed. Now we wish to compute the sheaf RZ'TT* !((QXs)). First we observe that when we restrict this sheaf to XS/G, we can compute it. Observe that

[Ri'nT*j!(Qlxs)] KXS/C -R(TIxs)*(QIxs)

and when we restrict to XS, G acts properly and with finite stabilizers; so for any x E XS/G choose x- E XS over x. Then the map X: G -* XS given by g gx- defines a map

HZ(X): RZ(lTIxs)*(QIxs) - H'(G,Q).

This map is independent of x, and it defines a map

RZ(7TIxs)*(Qlxs) > H'(G, Q)& (QIxS,/c)

which can be shown to be an isomorphism of sheaves. So we certainly obtain a map

a: HZ(G,Q) ? j!(QIXs/c) --* R?g*j!(Qjxs)-

I assert:

LEMMA 4.1. a is an isomorphism of sheaves.

For now we defer the proof of Lemma 4.1; it is the only argument in this section that is not trivial.




Suppose we knew Lemma 4.1. Then we could study the spectral sequence of the sheaf j!(QIxs). We get

E ,q = (HPRqG*,j((I = Hq(G. Q) ? HP(X/G, j! (Q I Xs/c))

and this spectral sequence converges to HP ? q( X, j( Q I xs)). Now observe that the E q form a rectangle of (possibly) non-vanishing terms. In particular, if q = dim G, and p is the largest integer for which HP(X/G, !I(QIxs/c)) fails to vanish, then the E ' q term survives to the limit. We could conclude that HP+q(X, jI(QIXs)) # 0. It follows that dimG + p < max(k1 + 1, k2); i.e., p < max(k1 + 1, k2) - dim G. What this means is that if

p > max(k1 + 1, k2) - dimG,

then HP(X/G, ji(Q1xs/c)) = 0. Again, on X/G we have an exact sequence of sheaves

0 ' I1(Q1xS1c) Q i*(QIZIG) 0.

From the exact sequence in cohomology we get that if H '(X/G, j!(Q I xsG)) and H'(Z/G, Q) both vanish, so does H'(X/G, Q). Now if k3 is an integer such that 1 > k3 H'(Z/G, Q) = 0, then we get

1 > max(k1 + 1 - dimG,k2 - dimG,k3) * H'(X/G,Q) = 0.

Let us put this together as a theorem:

THEOREM 4.2. Let k1, k2, k3 be integers such that

1? k1 => H'(ZQ) = 0,

1 ? k2 H'(X, Q) = 0,

1? k3 H'(Z/G,Q) = 0.

Then we have

1 2 max(k1 + 1-dimG, k2 -dimG, k3) = H'(X/G, Q) = O. It remains to prove Lemma 4.1, but now we prove a slightly more general

lemma. LEMMA 4.3. Let m: X -* X/G be as above. Let j: Z -* X/G be an open

immersion, and by abuse of notation j is also the inclusion 77 - '(Z) -* X. Then

RiT *(jgQIs-1z) = jiR'(Lgjr-1z)*(Qj1-,z).

This has Lemma 4.1 as a special case, if we put Z = XS/G (I apologize that here Z switched roles. It has become the open set).

Proof The question is local in X/G, so that we assume X = Spec A, X/G = Spec AG. Then by Sections 1 and 2 there exists a set CA(l) C Spec A



430 AMNON NEEMAN

such that CA(l) is a deformation retract of Spec A, with a deformation retraction respecting fibers. Now we can replace Spec(A) by CA(l) and S by the composite CA(l) -> Spec A -- Spec A' in the homology computation. But the map CA(l)

Spec AG is proper, so that Lemma 4.3 follows from the next, general result:

LEMMA 4.4. Let m: X -* Y be a proper map of locally compact spaces. Let

j: U -* Y be an open immersion, and let H be a sheaf on T- 1U C X. By abuse of notation, let j denote the inclusion r - 'U - X. Then we get

Proof. This is an easy corollary of Corollary 2 to Proposition 3.10.1 in Grothendieck [2].

5. Examples

Let M(n) be the set of n X n matrices, n > 2, regarded as the affine variety Spec C[ x ij]. PGL( n) acts by conjugation so that it also acts on M( n)' by simultaneous conjugation. We study this example.

First we compute the set of stable points. If a point (Ml, M2, ..., Mr) is not stable, there exists a 1-parameter subgroup X of PGL( n) for which (X(t)-1MX(t) ..., X(t)-1MrX(t)) has a limit as t -* 0. (Here, by a 1-parameter subgroup we mean a map C * PGL(n)). With respect to some choice of basis,

tal

we may assume X(t) ( ta ) a1 integers and a, ? a2 > a a tan,

Let M be the matrix mip, X(t) - 1MX(t) = (taj- aimij). This has limit if whenever

a, < ai, mi, = 0. So we get:

Observation 5.1. (M, ..., Mr) is not stable if and only if there exists a non-trivial subspace of C' which is invariant for the action of all the Mi.

It follows that if X = M(n)r,

dim(X - XS) < M2 - r(n - 1) + dimPGL(n) = (r + 1)n2 - 1 -r(n - 1).

This estimate comes about because not being stable forces at least r( n - 1) zeros in some conjugate of (Ml, . . ., Mr). For r large compared with n, this dimension is much smaller than dim X.

If we apply Theorem 4.2 to this space X, we get triviality because X, Z, X/G and Z/G are all contractible. But let us forget for a second that we know that X/G is contractible and use Theorem 4.2, knowing that X, Z, and Z/G




are. Then k = = = k3- 1, and we get

1 2 max(2 - dimPGL(n), 1) H'(X/G) = 0.

Then I > 1 H'(X/G) = O.

The result checks. To get a non-trivial result, we look at subsets of X. For instance, if we

require that each matrix Mi should have a non-vanishing trace, we get a subset X' C X, and in this case it turns out that Z' = X- (X)s = X' Il Z. (In general, it is not true that the stable points of an open subset are stable in the entire space.) The homology of X' is easily computed to be the homology of a product of r circles; so it vanishes in dimension greater than k2 = r + 1. Z' is affine, of complex dimension at most (r + 1)n2 - 1 - r(n - 1). Since it is a Stein space, its homology vanishes above that dimension, and k1 (r + 1)n2 - r(n - 1) will do.

To estimate k3, we observe that any unstable point has, in the closure of its orbit, a point (M1,. . ., Mr) where, after replacing each Mi by A - 'Mi A we can

assume it takes the form ( si). Suppose M' is n1 x n M,' is n2 X n2,

n1 + n2 = n. Then this type of matrix contributes a component to Z'/B whose dimension is at most n 2r + n 2r - dim PGL(n1) - dim PGL(n2) (at least, this is

true if r is large enough so that the map sending points (0( ) (' 0 Ml')) to Z'/G has generically a fiber of dimension dim PGL(n1) + dim PGL(n2)). This gives that the homology of Z'/G vanishes in dimension greater than max nn nO2 = n ((n2 + n2)(r - 1) + 2). Now we have, for r large, the estimate that k3 = ((n - 1)2 + 1)(r - 1) + 3 will certainly do. We get

1 > max[m2 + 2 - r(n - 1), r + 2 - n2, n2(r - 1) -2(n - 1)(r - 1) + 3]

=* H'(X'/G)= 0.

In practice, if r is large compared to n, the first term in the maximum dominates; so we get

1 ? r(n2 - n + 1) + 2 H'(X'/G) = 0.

Note that dim(X'/G) = m2 _ (n2 _ 1). Now as X'/G is affine the trivial vanishing is for 1 ? (r - 1) n2. If r is sufficiently larger than n, we clearly get that the bound of Theorem 4.2 is better.

Now note that the dimension that dominates is dim X' - (X ')S. Suppose we knew conjecture 3.1 or conjecture 3.5. Then we should continue the flow to - o, and get a deformation retraction of X' - (X')s to the critical set C(O). But inside X' - (X')s there is a closed subset F on which G acts with positive



432 AMNON NEEMAN

dimensional stabilizers. The same deformation retraction also takes F onto the intersection of X' - (X')s with the critical set. So F and X' - (X')s have the same homotopy type. The codimension of X' - (X')s is at least (n - 1)r - dim PGL(n), and the codimension of F is at least 2( n - 1) r - dim PGL( n). So our result roughly doubles in accuracy once we are given either of our conjectures.

Appendix: Proof of conjecture 3.5 for the torus

Here I propose to outline a proof of the conjecture when G is a torus. We start with a theorem:

THEOREM A.1. Let Y1,... , Yn be commuting, real, symmetric matrices on a finite dimensional vector space V over R. Let (, ) be the natural inner product on the space. Then there exists a constant c > 0 for which

n ~~2 /f \3/2

(Yi= , V)y1V > C (YiV" V)2

Proof We prove this by induction on the number of Yi's and the dimension of V. Suppose the theorem is known for lower dimensional spaces and the same number of Yi's, and for any dimensional space and a smaller number of Yi's. To start the induction, suppose n = 1. Then we get:

11 (Yv, v)YvII2 = (Yv, v)2(Yv, YV).

Now Y commutes with the orthogonal projection P to the image of Y.

(YPv , Pv) = (Yv, v) and (YPv , YPv) = (Yv , Yv),, so that we can study the inequality on Im Y only. But on Im Y, Y is invertible; so we get the estimate I (Yv, v) I < 11y- 111YVI112, and it follows that

11(yVV, )yV112 > Ily-111 -11(yV V) 13

Thus there is a constant c as above. Now observe first that both i = 1( Yi v, v )Yiv and IKYiv, v)2 are invariant

under an orthogonal change of the Yi's. Furthermore, inequality (1) is homogeneous; so it suffices to prove it on the sphere. Since the sphere is compact, it is enough to show that any non-zero point of V has a neighbourhood on which there exists a constant c for which (1) holds.

Take v0 E V, V0 # 0. After an orthogonal change of the Yi1's, we may assume Yiv0, Y2vO, ... , YkvO are nonzero and orthogonal and Yk ? 1vO = Yk + 2vO = * = Ynvo = 0. If k = 0, then all the Yi's kill vo and we are reduced to studying the problem on vo, which is a smaller dimensional space. So by induction, we may assume k # 0.




Let V1 C V be S[Y1. . . Yj] vo, where S[Y1. . . YJ is the symmetric algebra in the Y 's. Then V1 is clearly an invariant subspace for all the Y, 's; furthermore, on V1, IYk?I1 . . , Yn all vanish. Let P: V -* V1 be the orthogonal projection to V1. Then P(=1K Yiv, v)Yiv) is shorter than YLn1K Yiv, v)Yiv, and we get

||?(yiVV)yiV ||2 PE(YiV)YiV

k 2

= |? Yiv V)YiPv i=1

k

= ? (Yivv)KYiPv, YjPv)KYjv v). i,j=l

But at vo, the k x k matrix A whose entries are a = (YiPvOY1Pv0) = ( Yi v0, Yvo) is positive definite; so the same is true in some neighbourhood of vo. Now we get that for some constant c1 > 0 and for some neighbourhood of vo,

n ~~~2 k

Z Yiv,v )Yv || c1Z?Yiv5V)2. i=l i=1

If k = n, we are clearly done. If k # n, we have that

1k k

? ()is V 5 iA < ? I JYiv, V) I 11 YiV1 || i=1 i=1

Near vo, we get the estimate

(YiVV)YiV |C2 | ?(YiVV)YiV i=1 i=1

for some c2 > 0. Now observe that if

k n

2 ZK(YiVV)YiV 2 ZK(YivV)YiV | i=1 k+1

the above estimate tells us

n ? 1 >3K=

vv) 3 ? i= 1Yvv)~ i~k+1



434 AMNON NEEMAN

On the other hand, if Ek? 1K Yi v, v) Yi v> 2 1KY v,v)Y vI1, the triangle inequality says that

n n k

|E(Yivv)Yiv ? 2|(Yivv)Yiv - LK(Yiv,V)Yiv i=1 ~~~k+1i=

2 34 (YiV)YiV + (YiV, V)Yiv].

Putting all of this together, we get that there exists a constant c3 such that

n ~~2 k + ZKn VV) 2V 1 Y (Yiv,v) Yv |2 C3 )E (2Yi V, + |E (YiV, V2V

i=1 ~ ~ ~ [i=1 k+1J Now we can apply induction, and we get a constant c = c4 such that

KY 2 n)Y3/2 ||E(Yi v, vA ) iv| C4 ( E ( YiV, V ) 2) i=1 ~ ~ ~ U=

for all v sufficiently near v0. The result follows.

6. Introduction to Part 2

Sections 7, 8 and 9 are largely the technical background needed for Section 10, where a method to show how to prove vanishing for Chern or Pontrjagin rings on quotient varieties is indicated. In Sections 11 and 12 this technique, called the "Program", is applied to one example. We obtain partial results on a conjecture of Ramanan's about the vanishing of Pontrjagin classes on the moduli space of stable vector bundles of rank 2 and degree 1 over an algebraic curve.

Let X be a topological space, G a topological group acting on X. Let Y" be a G-bundle on X. The key result of Section 7, Theorem 7.12, allows us to form the quotient bundle Y/'G over X/G when G acts freely, and some other technical conditions are satisfied. Because the theorem is so crucial, I give a relatively careful and complete proof. However, Section 7 is so written that it can easily be skimmed. All the important statements are numbered, and only the statements and definitions are necessary for the rest of the paper.

Section 8 shows that even when the quotient bundle fI/G is not defined, one can, with suitable functoriality in X and G. define a generalization of the Chern class of f7G. Again, with suitable functoriality one can compare the ordinary Chern class of fIG, when it exists, to the fancy one constructed here. This section is largely formal and straightforward. Nonetheless, I feel that it is best said in the right categorical framework.




Section 9 contains the real point of the exercise. There I prove special properties for algebraic bundles over a scheme X that allow considerable freedom in moving between spaces. The key theorems are Theorem 9.6 which allows us to resolve G-sheaves, and Theorem 9.11 which permits us to extend G-sheaves from an open set.

Section 10 spells out how we can use all these constructions. Section 11 is the application to the moduli space of vector bundles over a curve. In Section 12, I specialize to vector bundles of rank 2, where I can, using a construction that is less than elegant, get a significant theorem about the vanishing of the Pontrjagin ring. This section is included to show that the technique developed in the paper can be powerful.

In the sequel, I outline some rather wild conjectures. The way to improve Theorem 12.7 should not be by replacing my unsatisfactory construction with an equally disagreeable one. I feel that a better understanding of the earlier, more formal sections of the paper should yield more powerful techniques and then the answers to the conjectures should become evident.

For this reason, Sections 7 through 11 are written fairly completely. Section 12 is not intended to be detailed; there is indication enough of what to do so that the interested reader can reconstruct the work.

7. Topological preliminaries

For compact groups, the K-theory of quotient spaces has been studied; see Segal [8]. However, the groups studied here will rarely be compact so that something must be said about the generalization of Segal's arguments. The noncompact case raises some topological difficulties and this section is devoted to treating them.

The key result is Theorem 7.12, which allows us to construct quotient bundles. I did not attempt to follow Segal's methods which reduce the problem to the case where G is a compact Lie group. For our purposes this is altogether the wrong approach; viewed correctly, Theorem 7.12 is a formal statement in point-set topology, and should have a purely formal proof, as given here.

To make the construction, we work in the category of "countably K-spaces." We shall see below that this is a sensible category in which to construct quotients.

Definition 7.1. A Hausdorff topological space X is called a countably K-space if there is a countable collection { X Ii c N) of compact subsets Xi c X such that

(1) UXi = X, (2) X has the weak topology with respect to the Xi's.



436 AMNON NEEMAN

Remark 7.2. If X is a countably K-space, clearly the Xi's can be taken to be increasing; it will often be convenient to do so in the constructions that follow.

Example 7.3. If X is Hausdorff, locally compact, and has a countable basis for open sets, then clearly X is a countably K-space. So, for instance, a locally compact connected Lie group is a countably K-space.

The proofs of the following lemmas are clear.

LEMMA 7.4. The product of two countably K-spaces is a countably K-space.

LEMMA 7.5. If X -* Y is a quotient map between Hausdorff spaces X and Y, then when X is a countably K-space, so is Y.

PROPOSITION 7.6. If X and Y are countably K-spaces, so is the join.

The join of X and Y is a Hausdorff quotient of X X Y X I.

LEMMA 7.7. If X = Ui NAi, where each Ai is a countably K-space, X is Hausdorff and has the weak topology with respect to the A i's, then X is a countably K-space.

In fact, if Ai c A it is automatic that X is Hausdorff; cf. Theorem 7.10.

PROPOSITION 7.8. Let G be a topological group which is a countably K-space. Then EG has a model which is also a countably K-space.

Proof: Milnor's construction of EG expresses it as a countable union of successive joins of G with itself.

Remark 7.9. Proposition 7.8 is the main reason for introducing countably K-spaces. The proposition allows us to do certain constructions for EG, using successive approximations by compact sets. As an example of how one uses countably K-spaces, we prove:

THEOREM 7.10. Let X be a countably K-space, and let G be a topological group which acts on X properly. Then the topological quotient space X/G is a countably K-space.

Proof. By Lemma 7.5, it suffices to prove that X/G is Hausdorff. Put X =UiENXZ, i c XcX i+ Xi compact for all i, and suppose further that X has the weak topology with respect to the Xi's (cf. Definition 7.1 and Remark 7.2). Let g: X -* X/G be the projection. Pick x, y E X such that 7(x) # 7(y). We have to separate 7(x) from g(y) by open sets; equivalently, we have to separate the orbit of x from the orbit of y by G-invariant open sets. The idea is to do this inductively for the X 's.

Step 1: Suppose A, B C Xi are closed subsets which are G-invariant in the sense that AG n Xi = A, BG n = B. Then we show that there exist U. V




open in Xi, A c U. B c V, U, V Ginvariant (i.e. UG n = U, VG ni =

V). Since Xi is compact, it is normal; so if we do not insist that U. V be

G-invariant there is nothing to prove. Start with any U. V. We "improve" them to G-invariant sets.

The point is that since G acts properly, the multiplication map m: Xi x G X is proper. In particular m`(Xi) is compact. Let 7T1: m`(Xi)

Xi x G Xi be the projection to the first factor. Since 7r '(Xi - U), 7T 1(Xi- V) are closed, they are compact. Now m(7r7 '(Xi - U)), m(7rj1(X -V)) are closed in Xi, and are clearly G-invariant (with G-invari- ance being as above). Replacing U by Xi - - U)) and V by Xi - m (7T 1(X - V)) we have what we need.

Step 2: Pick x, y E X. There exists an i such that x, y E Xi. Suppose ST(x) # ST(y); then we want to separate xG from yG. Since the action is proper, xG and yG are closed in X; so xG n Xi, yG n Xi are closed in Xi. By Step 1, Xi is "normal" with respect to G-invariant sets; so we can choose G-invariant open sets Ui and Vi in Xi and G-invariant closed subsets Ai and Bi in Xi so that xG n Xi c Ui c Ai and yG n Xi c Vi c Bi and Ai n Bi = 0. (One of the oddities is that if Ui c Xi is G-invariant in the sense that UiG n Xi = Ui, it does not follow that Ui is G-invariant. So we cannot take Ai = Ui.)

Step 3: This is the induction step. Suppose there exist G-invariant open sets Ui and Vi in Xi and G-invariant closed sets Ai and Bi in Xi such that xG n Xi c U, c Ai, yG n Xi c V, c Bi and Ai n Bi = 0. We will show that there exist Uit+, Vi+1, A i+ and Bi+1 in Xi+1 with the same properties, and with the extra property that Ui+ n Xi = Ui, V,+ n Xi = V,.

Recall that m: Xi X G -- X is proper. Now m-'(Xi +) is compact, and it follows easily that XiG n Xi+1 is closed in Xi+1 and UG n Xi+,, VG n Xi+, are open in XiG n Xi+,; also AiG n Xi+,, BiG n Xi+, are closed in XiG n Xi+ l So Ai n X i+ Bi G n Xi+ are disjoint, closed, G-invariant subsets of X + 1. They can be separated by G-invariants; in Xi+ 1 there exist Ui+1, Vi+

invariant open sets, Ai+ I, Bi+I1 invariant closed sets with

AiG n Xi+1 c Ui+1 c Ai+15 BiG n Xi+, c 41 c si+1

where A i + 1 n Bi + 1 = 0. We leave the A i + 1 and B + 1 alone; we need to modify Ui+> and Vi+? so that Ui+1 n Xi = U,, V,+, n Xi = V,+1.

Choose any open subsets Ui+ c Uc, X>Vi+ c Zi+l such that Ui+> n XiG = , UG V, V+ l XiG = VG. Then we use the procedure of Step I to shrink these

to G-invariant open sets. The reader may check that, after shrinking, we still have Ui c Ui + 1, Vi c Vi + 1.



438 AMNON NEEMAN

End of Proof. Put U = UUi, V = UV. Now U and V are open (because X has the weak topology with respect to the Xi's). Since they are clearly disjoint and G-invariant, they provide the necessary separation of xG from yG. QED

Remark 7.11. Theorem 7.10 will not be needed in the remainder of this paper. It is included to convince the reader of the usefulness of Definition 7.1.

What is needed is the following:

THEOREM 7.12. Let X be a countably K-space, and let G be a locally compact topological group. Suppose G acts on X freely (= without fixed points and properly). Let f' be a vector bundle on X on which G acts, compatibly with its action on X. (In the future we will call such bundles G-bundles.) Then if v: X -* X/G is the projection, there is a bundle 5"/G on X/G and a canonical isomorphism vr*(5//G) * .

Proof We need to show that Y" has enough invariant sections to locally generate it at every point. The proof is lengthy, so we divide it up into a sequence of steps.

Step 1: Choose Xi C X, Xi compact, Xi c XC + 1 UX3 = X and such that X has the weak topology with respect to the X i's. First we want to show that, for each i, XiG is closed and locally compact.

This is clear: since G acts properly on X, the map m: Xi X G -> X is proper. Xi x G is locally compact (Xi is compact, G is locally compact); so it follows easily that m(Xi X G) = XiG c X is closed and locally compact.

Step 2: Choose a point x E X. We want to construct plenty of G-invariant sections of 1" near x. Pick a neighborhood U of x over which Y" is trivial. Then there exist sections SI, S2,. . ., sn of 1" over U which form a basis for Y" at every point of U. In Step 2 we shrink U sufficiently to make the "averaging" of sections a safe operation.

Restricting attention to the orbit xG, we get for each si a map Ki: G -V, V the n-dimensional vector space which is the fiber of Y'' at x. The map is defined by gi(g) = si(xg)T(g-1), where T(g-1) is translation (of f') by g-1. The 91(e), &2(e),..., &n(e) form a basis for V; so for some sufficiently small neighborhood N of e, we have that if we choose ji in the convex hull of si(N) for each i, then . ., Jn form a basis for V. In this sense, averaging over N is harmless.

Because G acts freely, the map g 4 xg defines a (closed) embedding G - X. Now shrink U to ensure that U n xG C xN.

Step 3: Here we construct a function on X that we will eventually use to average the s5 s.




Suppose x E Xi (this is true for some i). Since XiG is locally compact, we can pick a function (Pi of compact support in Xi G, (Pi 2 0, whose support is contained in U and such that (pi(x) = 0. The Tietze extension theorem allows us to extend (Pi to cPi+l on Xi+ G, where 9?i+l has the same properties. (Caution: Xi+IG need not be normal. But because Xi+lG is locally compact, we can embed supp (Pi in the interior of a compact subset of Xi, G. We apply Tietze to that compact.) In this way, we get a continuous function 9p on X, namely T = Uqi

Step 4: Let so: X X G -> X be the projection, m: X x G -* X the action. The action of G on 'V gives an isomorphism 4: ma Y 1* . If s1,..., Sn are the sections on U as before, then Ts1,..., cpsn are continuous sections on all of X (continuity can be checked on the Xi's). So 0m*(qpsi) are global sections of s71". Let Mi be the right Haar measure on G; we put ai = fGcm*(T si) d1.

We can now make the following assertions: (1) The ai are continuous sections of I". (2) The ai are G-invariant. (3) At x, the as1, . ., an form a basis for the fiber of I". (2) is clear, and (3) follows easily from Step 2. (1) can be checked on XiG

for each i. Because XiG is locally compact and G acts properly, locally in XiG the integration takes place really only over a compact subset of G. The details are left to the reader.

Therefore since the a1. . ., an form a basis at x, they form a basis in some neighborhood of x, which can, of course, be taken to be G-invariant. QED

Remark 7.13. It follows from Step 1 of the proof that, under the hypotheses on X and G in Theorem 7.12, X is a union of countably many, G-invariant, locally compact closed sets. Moreover, X has the weak topology with respect to them. In fact, this is all we use about the topology of X. So this could have been a starting definition instead of Definition 7.1. This was the main reason to include a proof of Theorem 7.10: namely, to persuade the reader that Definition 7.1 is better and more natural.

Remark 7.14. From Example 7.3, Proposition 7.8 and Remark 7.13, it follows that if G is a locally compact group with a countable neighborhood basis, EG has a model which is a countable union of locally compact closed sets as in Remark 7.13. This is not completely transparent from Milnor's construction; note that the join of locally compact spaces is not necessarily locally compact.

Before concluding this section, I should state two more results needed below. The first is trivial, so I include a proof. The second is less trivial, but seems



440 AMNON NEEMAN

to be well-known. Since a good reference for the proof is unknown to me I give the barest sketch.

PROPOSITION 7.15. Suppose X and Y are G-spaces, all Hausdorff, and that G acts properly on Y; then the diagonal action of G on X X Y is also proper.

Proof. We need to show that the map X X Y X G -* X X Y X X X Y given by (x, y, g) -* (x, y, xg, yg) is proper. We are given that m: Y X G

id xmr Y x Y is proper. Multiplying by X X X, we get that (X X X) X (Y X G) 0

(X X X) X (Y X Y) is proper. The map X X Y X G -* X X Y x X x Y de- id xmr

composes as X X Y X G -* X X X Y x G *X x Y x X x Y; the second map is proper, as we have just said. The first is a closed immersion, being the graph of a map X x Y x G -* X.

Finally we will need:

PROPOSITION 7.16. Suppose G is a Lie group acting differentiably on a manifold M. Suppose that the action is proper, and the stabilizers of points are finite subgroups of G. Now G acts on M x EG by the diagonal action; let M x EG be the quotient. The map M x EG -* M/G (projection to the first

G G factor) induces an isomorphism on rational cohomology.

Sketch of Proof. By the spectral sequence relating the cohomology of the Cech cover with the cohomology of the space, the problem is local in M/G. The idea is to find a suitably small G-invariant open in M. What one does is take a transverse section to the G-orbit at x. Call this space N. Then N x G -* M, for suitably small N. is a Galois covering map. The Galois group is naturally isomorphic to GX, the stabilizer of x, and its action commutes with the action of G. The implications of this for the rational cohomology yield the proposition.

8. Equivariant characteristic classes

We first define the categories studied in this section.

Definition 8.1. The category <,enormous has for its objects all the pairs (X, G) where X is a topological space, G a topological group, and G acts on X (on the right). A morphism (X, G) -* (Y, H) is a pair of maps f: X -* Y and g: G -* H such that, with the induced G structure on Y. f is a G map.

Definition 8.2. The category Wbig is the full subcategory of enormous whose objects are pairs (X, G) where X and G are both countably K-spaces and G is locally compact.




Definition 8.3. The category %'little is the full subcategory of %big' where the objects are pairs (X, G) such that G acts on X freely (= without fixed points and properly).

Remark 8.4. The category we are most interested in is %big' But it is sometimes convenient to do the constructions over the other two.

There are two functors we want to consider. We define V: 5enolous

JVv6 lea I (,/6 le I is the category of (small) abelian categories) to be the functor which takes (X, G) to the category of G-bundles on X. We define V: 567enormous

Th/ Wa to be the functor which takes (X, G) to the category of all bundles on X/G (the topological quotient). There is a natural transformation 4: V -* V which takes a bundle on X/G to its pullback. Clearly ?((x ) is an exact functor for every object (X, G).

Note. The category of vector bundles on a space is not an abelian category; it is only a full, thick subeategory of an abelian category. Nothing we do will be influenced by this point. So in a misuse of the language, I use the words " abelian category" to refer to "full, thick subcategories of abelian categories".

The significance of Theorem 7.12 is that on 51ittle 4 is a natural isomorphism; it admits an inverse 4: V -*V which takes a bundle J/ to its quotient --G.

Now we define the corresponding Grothendieck groups.

Definition 8.5. There is a functor k: 5eenormous -* -16 (j/6 being the category of abelian groups) which takes a pair (X, G) to the Grothendieck group of the category V(X, G) (concretely, the free group on bundles on X/G divided by the relations generated by f- i' - fr", where 0 -' -* '"-* i"" -* 0 is an exact sequence).

Definition 8.6. There is a functor K 5eenormous -* J2/6 which sends (X, G) to the Grothendieck group of V(X, G).

Remark 8.7. Definition 8.6 is not new; see Segal [8]. He denotes K(X, G) by KG(X).

Definition 8.8. Being exact, 0 induces a natural transformation k -* K. By abuse of notation we call it 4 also. On Wlittle 4 is a natural isomorphism, its inverse induced by 4: V -*V.

The real point of this section is that we want to construct an analogue of the inverse of 4 on Wbig' Since the group does not act freely, we cannot always construct quotient bundles. We set about "correcting" that problem.



442 AMNON NEEMAN

Let G be a topological group; let EG denote the total space over the classifying space of G, as constructed by Milnor. We have to pick a model for EG because we do not work in the homotopy category. The properties of EG that we use are:

Property 8.9. EG is functorial in G: given a map G -- H there is an induced map EG -* EH which respects composition (this is immediate from Milnor's construction).

Property 8.10. If G is a countably K-space, so is EG (see Proposition 7.8).

To fix ideas, we take EG to be a left G-space (to be consistent with the literature). If X is a right G-space, X x EG is a right G-space with the action (x, a)g > (xg, g -a). With this definition of the action we get:

Construction 8.11. (X, G) 4 (X x EG, G) defines a functor F: W7big

-* Whale'

Proof We have to show that, for (X, G) an object of 5ebig) (X X EG, G) is an object of 0'little. By definition, X and G are countably K-spaces. By Property 8.10, EG is a countably K-space. So by Lemma 7.4, X x EG is a countably K-space. Now at least we know that (X x EG, G) is in Wbig'

The action of G on X x EG is free because G acts freely on EG. The properness comes from Proposition 7.15. The fact that there are no fixed points is obvious. So, in fact, (X x EG, G) is an object of rlittle*

Fact 8.12. There is a functor E: 567enormous -* 4J> (g being the category

of topological spaces) where E(X, G) = X/G. The functors V and k can be viewed as composites V o E, K o E of E with the natural analogue of V and K, done on 9ol. 4: K o E -* K is a natural transformation.

We also have the functor F defined on Wbig in 8.11 above. There is a natural transformation of functors p: F -* 1 given by projection to the first factor; p(X,G): (X x EG,G) -* (X,G) is the projection. So we get a commutative diagram of natural transformations, which, on objects, is

Kp K(X, G) K(X x EG, G)

I KEp. I (F K(X1 'G)K((X x EG)/G) = K(X xGEG)

where the commutativity is just the naturality of 4.




Construction 8.13. We define K to be K o E o F (so that K(X, G) =

K(X xGEG)). On W6big we have a natural transformation A: K -> K given by (OF) - o Kp (OF is an isomorphism on Wbig because F takes Wbig to r1ittle) Then the commutative square above yields a commutative triangle:

K

(*)-/_ K pK

kEp

Usually, by abuse of notation, we will write p for KEp; it is the map K(X/G) -* K(X x GEG) induced by the projection.

Remark 8.14. This is the sense in which we construct an inverse to 4. The point is that 4 is close to being an inverse when we pass to the cohomology.

We really want to compute characteristic classes. On 37ol4, there are natural transformations Chem: K -* H* (resp. Pontr: K -* H*) which send a vector bundle to its Chem (resp. PontrJagin) class. They send K(X) to (the center of) the group of units of the ring H *(X), and as such are maps of (abelian) groups.

Now the commutative triangle (*) gives a commutative diagram:

K

Chern H IChern

H*E HE EpH*EF

Let H = H*E, H H*EF, and for H*Ep just write p. (It is the map induced on cohomology by the projection X X GEG -* X/G.) Summing up, we get:

PROPOSITION 8.15. There are natural maps Chern: K -* H (resp. Pontr: K H) which make the following diagrams commute:

- 0

Chern j Chern

Up



444 AMNON NEEMAN

(resp. the same with Pontr in place of Chern). In the notation above, Chern: 4' -Chern-

K -- H is the composite K --K H.

Remark 8.16. We have constructed a way to assign characteristic classes to an arbitrary element of K(X, G), for (X, G) an object of Wbig' in a way that is comparable to the natural assignment of characteristic classes on the quotient. The value of this elaborate construction is that p: H -- H is often an isomorphism. What we will need is that it is an isomorphism for the rational cohomology if G is a Lie group, X is smooth, G acts differentiably, properly and with finite stabilizers (Proposition 7.16).

9. K for smooth algebraic varieties

We again start by defining the category to be considered.

Definition 9.1. Let V~agebraic be the category of pairs (X, G) where X is a separated scheme of finite type over Spec C, G is a reductive, complex algebraic group, and G acts on X on the right. A morphism (X, G) -- (Y, H) is a pair of morphism of schemes f: X -- Y, g: G -- H such that g is a homomorphism of groups and, with the induced G-structure on Y, f is a G-map.

In complete analogy with Section 8, we can define a functor V: 'Valgebraic -6 Val by sending (X, G) to the category of algebraic G-bundles on X. What is new here is that we want to consider another functor W: ralgebraic At '7r which sends (X, G) to the category of all coherent sheaves acted on by G.

Definition 9.2. There is a functor Kalgg: ralgebraic -A -E, which sends (X, G) to the Grothendieck group of V(X, G).

Definition 9.3. There is a functor L: ralgebraic REX, which sends (X, G) to the Grothendieck group of W(X, G).

Note. The " functor" L is not a functor on all of 'ealgebraic' If (fg): (X, G) - (Y, H) is a morphism in (ealgebraic, L(f, g) is defined only when f is flat. To be correct we should define a new category 'eflat on which L is a functor. I only use the functoriality of L for open immersions; I believe that my lack of rigor in fact clarifies the constructions, whereas excessive correctness would obscure them.

There is a natural transformation V -* W which sends a bundle to itself, viewed as a sheaf. It induces a natural transformation r: K9 -- L. We will show that, on some subcategory of 'ealgebraic) -T is an isomorphism of functors.

To prove this, we need to resolve coherent sheaves by vector bundles. Even in the case when G is trivial, to do this one has to impose the condition that X




be quasi-projective. It is natural enough to assume that X admits an ample line bundle Y' on which the action of G linearizes. But this does not seem to suffice. The "classical" condition that would usually be imposed is some stability assumption. But if X is not proper, this is very unnatural. Stability depends not only on Y', but also on a choice of a finite dimensional C vector subspace of H0(Y-) which embeds X into projective space. Put more concretely, it depends on a choice of compactification for X.

For this reason, we introduce a better, more intrinsic notion of what it should mean for ? to be "nice".

Definition 9.4. Let (X, G) be an object of Walgebraic' An ample line bundle ? on X for which the action of G linearizes is called nice if for every point p E X there is a G-invariant section s E H0(Y) such that Xs is affine and p E Xs.

Remark 9.5. If X is embedded equivariantly in Pn, and Y is the pullback of 0(1), then we can ask how Definition 9.4 compares with stability. If every point of X is stable in Pn, Y' is nice. However, if one only requires that every point of X be semistable, ?f need not be nice.

The first result needed is the following:

THEOREM 9.6. Let X be a scheme of finite type over Spec C, G a reductive algebraic group acting on X. Let ? be a nice line bundle on X. Then for any coherent G-sheaf 9Y on X (i.e., a coherent sheaf that admits a G action), there exists a surjection A'm-* J, where Y'" is a locally free G-sheaf and the map is G equivariant.

Proof. Suppose first that X = Spec A is affine and ? is trivial. Then we have to show that if M is a finite A module acted on by G = Spec R, then there exists a locally free finite A module F, also with a G action, and a surjection F M which is a G map.

The essential point is well-known and may be found in Mumford ([6]). Given any element m c M, mG generates a finite dimensional C vector space. This is clear because the action is algebraic, which gives a map yt: M -* M ?CR. If i(m) = ml ? r1 + . +?m n ? rn, then for every g c G, mg lies in the vector space generated by ml,..., mn.

Let ml, m2,. .., mk generate M over C. Let V be the vector subspace of M spanned by mlG U m2G U ... UmkG. Then V is a finite dimensional G-invariant subspace of M, which spans M as a G module. The required map is just the natural map V ?CA -* M.

(There is a small point here that should be mentioned: Because G is a variety, the fact that VG C V can be checked at the closed points of G.)



446 AMNON NEEMAN

How do we globalize the construction? We cover X by finitely many open affines Xsi, 1 < i < r, si G-invariant sections of YS (this can be done because ? is nice). On Xsi we have, by the above, a vector space Vi C f(Xs, S") with the following properties:

(1) Vi is finite dimensional. (2) Vi is G-invariant. (3) Vi generates 9? on Xsi. Twisting the sections by si, we may assume that the map Vi - f(Xs8,

Y' Y') factors through f X, S"? ?'n) (i.e., the global sections in Vi lift). However, the map Vi -- F(X, S? X Yn) will not necessarily be G-equivariant. After all, we have just lifted sections.

The extent to which the map fails to be G-equivariant is measured on Xs. by the non-commutativity of the square

a

Vi ) Vi X9 R

f(XS, 9? Yn) -_ f(X, S g n) ? R,

and on Xs n = X Xssi the diagram is commutative. Twisting some more by si, we can kill all the maps (,/ X 1)a - y,8 (i.e. kill them on each XS). So if we increase n, we may assume that Vi -4 (X, 5"? ?n) is a G-equivariant map.

Choosing an n that works for each i, we have a map ( f ?) X ?1n -*

It is clearly G-equivariant and surjective. QED

Definition 9.7. Wnice is the full subcategory of Walgebraic whose objects are pairs (X, G) such that X is smooth and admits a nice line bundle.

COROLLARY 9.8. For (X, G) an object in 6nice, the inclusion V(X, G) W( X, G) induces an isomorphism on the bounded derived categories.

Proof Let n be the maximal dimension of any component of X (which exists; X is of finite type). Then every element of W(X, G) has a resolution of length no more than n by objects of V(X, G), and the result follows from Hartshorne [3, Lemma 4.6, part 2, p. 42 and Prop. 3.3, p. 32]. 0

In particular it follows that induced natural map Kalg(X, G) -4 L(X, G) is an isomorphism.

What turns this elaborate construction into a useful tool is that if U - X is an open immersion of G-schemes, the corresponding map L(X, G) -* L(U, G)




has a chance of being surjective; we could reasonably expect to be able to lift G-sheaves on U to G-sheaves on X. But I only see how to do this under certain assumptions. We make the following definition:

Definition 9.9. Let U -- X be a G-equivariant open embedding of G-schemes. The embedding is called good if for every p E U, U contains the entire G-equivalence class of p. That is, if q E X and Gq fl Gp / 0, q E U.

Similarly, (U, G) -* (X, G) is called a good morphism if U -* X is a good open immersion.

Example 9.10. If (X, G) is in Walgebraic' ?f is a nice line bundle on X, and U = XS for some invariant section s in f(X, ?f), then (U, G) -* (X, G) is good. The reason is that, because ? can be trivialized locally by invariant sections, locally everywhere Xs is given by the vanishing of invariant functions. Since the functions are invariant, they are constant on G orbits, so that we cannot have an orbit along which s tends to zero.

THEOREM 9.11. Let (U, G) -* (X, G) be a good morphism of objects in W'algebraic' Suppose (X, G) admits a nice line bundle Y. Then any coherent G-sheaf Y on U extends to a coherent G-sheaf on X.

Proof. The idea is to do the construction one affine at a time. So suppose X = Spec A is affine, and Y is trivial.

The fact that U contains the G-equivalence class of every point says precisely that U = Ur. Spec Af, each f being a G-invariant element of A. (See Mumford [6].) For each i, we have an Af module Mi = r(Spec Af, Y?). Mi is finite over Af because SY is coherent.

For each i, there exists a finite dimensional G-invariant vector space Vi c Mi such that AfV2 = Mi. (See the proof of Theorem 9.6.) Twisting by f enough, we may assume v c r(u, S") and, twisting some more by fi, we may assume that the inclusion Vi - r(u, SY) is G-equivariant (again, the argument is as in Theorem 9.6).

Now put F1 = @1Vi ? 4cA. Then F1 is a G-bundle, and over U we get a suriection F1 I U Y,

Repeating the argument for the kernel of this surJection, we get another G-bundle F2 over A, and an exact sequence

F F Y~~9--- 0 o2fsUeluas oU

of sheaves on U.



448 AMNON NEEMAN

If we twist 4 by if for n sufficiently large, we will "cancel the denomina- tors" in the matrix 4, so that finc is actually a map F2 - F1. If we increase n some, the map will be G-equivariant (see the proof of Theorem 9.6).

Setting n large enough, we can make it work for every i. Define a map ED i 1F - F1 by (x1 * Xr) + ffnX1? + * ?frnXr. The map is G-equivariant, and it is clear that the cokernel restricts to 9' on U.

How do we globalize? Since ? is a good line bundle on X, we can cover X by invariant affines of the form XS) s a section of Y. We first show that Y' can be extended to U U Xs.

The embedding U n Xs XS is clearly good (this does not use any property of Xs). Thus YI u ,x can be extended to all of Xs by the affine case just discussed. Now we get a sheaf Y' on XS, with , i cl nX = 'I5l U n Xs

We define 9Y" to be the kernel of the sequence

0 ---y" -* i*tY' e j*Y- k* lunx,

where i: Xs - U U Xs, j: U-* U U Xs and k: U n Xs -> U U Xs are the embeddings. We clearly have 5? " = Y?, 5"' x - ', so that 5 " is coherent and extends 9Y.

To be able to carry the construction further we need to know that U U Xs > X is a good embedding. By Example 9.10, Xs X is good. By hypothesis, U - X is good. Then U U Xs - X is also good. QED

We get immediately:

COROLLARY 9.12. Let (U, G) -- (X, G) be a good morphism in walgebraic* Then if X admits a nice line bundle, the induced map L(X, G) -- L(U, G) is surJective.

This much comes easily, but we want to know more. Knowing the maps quite explicitly, we can get, with only infinitesimally more care:

COROLLARY 9.13. Let (U, G) -- (X, G) be a good morphism, and assume further that (X, G) is in Vnice Then the induced map Kal (X, G) -- Kalg(U, G) is surJective.

Proof We have a commutative diagram

Kalg(X, G) -L alg(U, G)

+

L(X, G) pL(U, G).




Note that 4 is an isomorphism by Corollary 9.8, and p is surjective by Corollary 9.12. Suppose we pick FYe Kalg(U, G). Take x E L(X, G) so that p(x) = ((f'). From the diagram alone it is not quite clear that po - 1(x) = f'. It will become clear after Corollary 9.18. Nevertheless, there is a virtue in being quite explicit about our maps; it makes the constructions less mysterious.

The sheaf p( l) is the vector bundle fY viewed as a sheaf, and x can be taken to be a coherent sheaf Y? extending Y'. Also - 1(J) is obtained by constructing a resolution of 5Y, and p restricts that resolution to U; so we end up with a resolution of Y'. QED

Before concluding this section, we summarize some facts about good mor- phisms and nice bundles.

Property 9.14. Suppose f: U V, g: V X are good immersions of G-schemes. Then g o f is good.

Property 9.15. Suppose f: U V, g: X > Y are good immersions of G-schemes. Then so is f X g: U X X -> V X Y.

Property 9.16. Suppose f: U -- X is a good immersion of G-schemes, and Y? is a nice line bundle on X. The U = USESXS, where each XS is affine and s E S are G-invariant sections of H0(Y'?), for some n > 0. (cf. the beginning of the proof of Theorem 9.10).

Remark 9.17. In property 9.16 we use the fact that if X admits a nice bundle it is quasi-projective, hence noetherian.

COROLLARY 9.18. If (U, G) -- (X, G) is a good morphism, and if (X, G) is i 'nicen then (U, G) is in W'nice

Property 9.19. The following statements are equivalent: (1) X admits a nice line bundle. (2) There exists a G-equivariant, affine morphism X -, Pn, where G acts

trivially on pn.

Property 9.20. Suppose X admits a nice line bundle. Then the quotient X/G exists, and the morphism X -) pn in Property 9.18(2) factors through X/G. Both the maps X -* Xc and X/G -> p are affine. Since Pn is separated, it follows that X/G is separated.

Property 9.21. Let X admit a nice line bundle. Then U -> X is a good open immersion if and only if there exists a scheme U/G and an open immersion



450 AMNON NEEMAN

U/G -- X/G such that U -. X is the pullback

U +X

I~ It U/G -+X/G.

10. The program

Suppose we want to show that some characteristic classes of a bundle vanish, for a specific bundle Y'* on X/G, where (X, G) is in Wbig' Then an approach that might work is this.

(1) Prove that p: fi(X, G) -- H(X, G) is an isomorphism. (See Proposition 8.15. In more standard notation this means that the natural cohomology map induced by the projection X XGEG -* X/G is an isomorphism H*(X/G) H*(X xGEG).)

(2) Exhibit 7T * 1 E K(X, G) as the pullback of some x E Kalg(Y, H), for some (Y, H) in Walgebraic, Y smooth.

(3) Find a good morphism (Y, H) -- Z, H) where (Z, H) is an object of 'enice

(4) Prove a vanishing result for the equivariant cohomology HH*( Z) =

H*(Z XHEH). (In our previous notation, this is H(Z, H).) The reason this would work is that we have a diagram

k(x, G) Kalg(Y, H) Kal9(Z, H)

I i I K(X, G) K(Y, H)+ K(Z, H)

where the right-hand square commutes. (2) permits us to pull back 7T * 1 to K alg( Y, G). (3) together with Corollary 9.13 tells us that a is surjective, so IT* pulls all the way back to Kalg(Z, H). So it certainly pulls back to K(Z, H).

Since Z. H are separated schemes of finite type over Spec C (because (Z, H) is in 'algebraic)5 Z, H are Hausdorff, have a countable neighborhood basis and are locally compact. So they are countably K-spaces in the complex topology (see Example 7.3). If we forget the algebraic structure, (Z, H) is an object of Wbig'

But on W'big there are natural transformations Chern (resp. Pontr) K -3 H (see Proposition 8.15). The naturality tells us that Chern( 7 * ') is a pullback of a class




in H(Z, H). The vanishing we prove in (4) implies the same vanishing for Chern ( *5/).

The commutativity of the diagram in Proposition 8.15 proves that p[Chern( Yr)] = Chern(7*1/). The fact that (1) proves p an isomorphism implies the same vanishing for Chern( */). Finally, every statement has a true Pontr analogue.

Remark 10.1. The power of the technique is that Z is really quite arbitrary. Even in the case where H is trivial, the statement is surprising. Consider for a moment just what we have proved.

Suppose H is trivial. Then for any quasi-projective Z, any ample line bundle is nice. Furthermore, any open embedding is good. So what we know is that if Y -3, Z is an open embedding of smooth quasi-projective algebraic varieties, any algebraic bundle on Y has its Chern class supported in H*(Z). Consider the following example.

Example 10.2. Choose a line bundle ?9 on S' x S' with a nonzero c1(?9). (Such bundles exist; identifying S' x S' with an elliptic curve, just take O(9) for some point on the curve.) S 1 x S 1 is a deformation retract of C * x C *. So ?9 extends to a line bundle, which we also call ?9, on C * x C *, and also on C * x C*, ? has a non-vanishing first Chern class. Let C* x C* _3 C2 be the embedding. It is an open immersion of algebraic varieties. Now any algebraic line bundle on C * x C * must have its Chern class supported on C2, a contractible space. It follows that the line bundle ?9 is not algebraizable.

11. A pleasant construction

Let /#g(n, k) be the moduli space of semistable vector bundles fY over a curve C of genus g, such that Y has rank n and degree k. We wish to study the Pontrjagin ring of these spaces, at least in the case where they are smooth (i.e., n and k are relatively prime). Recall that the Pontrjagin ring of a smooth manifold is the ring generated by the Pontrjagin classes of the tangent bundle.

The point is that we know .11g(n, k) quite concretely as a differentiable manifold. The following description is due to Narasimhan and Seshadri (see [7]).

Let U(n) be the group of unitary n x n matrices. Consider the map f: U(n)2g -> SU(n) given by

al,' bl,' a2, b ...,agbg) = (a-lb-lalbj) ... (a-1bg-1agbg). Then if X is a primitive nth root of unity, [7] tells us: (1) XI E SU(n) is a regular value for f. (2) SU(n) acts on U(n)2g and SU(n) by simultaneous conjugation. Clearly

f is an SU(n) map, and equally clearly XI E SU(n) is a fixed point for the action



452 AMNON NEEMAN

(being in the center). What is not so clear, but is true, is that the action of SU(n) of f -1(A I) is almost free: only the center of SU( n) has fixed points. So PSU( n) acts without fixed points (and, of course, properly), where PSU(n) is the quotient SU( n )/center.

(3) f 1( X I )/PSU( n) =_ g( n, k) as differentiable manifolds.

Remark 11.1. Our construction is quite "robust", and is completely independent of the way k depends on A in (3). Philosophically this may seem reasonable. Because SU(n) is connected, the manifolds f- '(AI) are cobordant. In fact, if we knew that the set of regular values of f is connected, we could deduce that, for different A, the manifolds f l(AI) are actually diffeomorphic. Nevertheless, this is only a philosophical point, because it does not follow that the quotients f-'(AI)/PSU(n) are cobordant (resp. diffeomorphic) for different A.

We propose to apply our Program (see ?10) to (f-(AI), PSU(n)), which is actually an object of Wlittle

(1) We have to show that the natural map

H*[ f -'(XI)/PSU(n)] -->H* [f-'(XI) X pSu(,)E PSU(n)]

is an isomorphism. In fact, it is easy to show that the projection fj'( Il) f 1(AI)/PSU(n) is a principal PSU(n) bundle (by an argument similar to the proof of Proposition 7.16), and so the map f-'(AI) Xpsu(n)EPSU(n) f '(A I)/PSU(n) is a fiber bundle with a contractible fiber.

(2) is harder. Let r be the tangent bundle of f -'(AI)/PSU(n). Then first remember that Pontr( 1) = Chern( Y? RC). If we apply the Program to Y/ RC and the natural transformation Chern, we will be proving results about Pontr(YV).

The idea is to complexify everything. We define a map f: GL(n, C)2- SL(n, C) by

J(ab,. .., ag, bg) = (a'lbjla bi) ... (a;'bg'agbg).

If PGL(n, C) acts on GL(n, C)29 by simultaneous conjugation and on SL(n, C) by conjugation, then f is a PGL( n, C) map. Also XI E SL( n, C) is fixed, being in the center; so PGL(n,C) acts on f '(AI).

We have a diagram of objects in C big:

( f -1(XI ),PSU(n))(U(n)2g, PSU(n)){(SU(n), PSU(n))

(whi(cI)hPGL(n)) (GL(n )2g, PGL(no))m(SL(n), PGL(ns))

which clearly commutes.




Let Tx denote the tangent bundle of X. On f `(XI) we get an exact sequence of PSU(n) bundles

0 -O Tf -(xI), g*TU(n2g *f*TSU(n) 0

But the natural maps

TU(fn )2g ? RC - 1 TGL(n), TSU(fn) ? RC - 2 TSL(fn)

are isomorphisms (because U(n), resp. SU(n), are the maximal compact subgroups of GL(n), resp. SL(n)). We have expressed Tf- '(XI) ? RC as the pullback Of TGL4n)2g -f*TSL(n) in Kalg(GL(n)2g,PGL(n)).

If 1/, as above, is Tf-1(xI)/PSU(n), then on f-'(XI) we get a sequence of PSU(n) bundles

0 -- E -- Tf-1(x I) 7lf- 0

where e is the bundle of vector fields parallel to the fibers. Another description, which permits us to lift E ? RC, is that it is the bundle of vector fields in the direction of the action of PSU(n).

The way to obtain an algebraic lifting of 0 ?RC to GL(n )2g is easy. Consider the map ,u: GL(n)2g X PGL(n) -* GL(n)2g (the action). It is a smooth map; in fact, after an automorphism of GL(n)2g X PGL(n), it is just a projection. Let 0 be the subbundle of the tangent bundle of GL(n)2g X PGL(n) parallel to the fibers. Let s: GL(n)2g -- GL(n)2g X PGL(n) be a section. Then s*@ is a bundle on GL(n)2g lifting 0 ?RC.

Now vT * f' RC E K(f'- 1(X I), PSU(n)) is a pullback of

TGL ( n f TSL(n) - s* e K1 g(GL(n) ,PGL(n)).

This achieves (2). Step (3) of the program is more difficult, but the problem is really (4): In

this case the only way to prove vanishing of the equivariant cohomology is under the condition that H *(X/G) -- H *( X > EG) be an isomorphism; e.g., under the hypotheses of Proposition 7.16.

Still, something can be salvaged. We have a map (GL(n )2g, PGL(n)) (M(n )2g, PGL(n)), M(n) being the variety of all n x n matrices. Now M(n )2g is affine, so that the trivial bundle is nice. The morphism is good because GL(n )2g is given by the vanishing of an invariant section, namely the product of the determinants. (See Example 9.10.) Hence T * Y? RC lifts to an element of K(M(n)2g, PGL(n)), and it follows that Chern(v*? RC) is a pullback from H*(M(n)2g X>PCL(n)EPGL(n)) H*(BPGL(n)). From this we get:

THEOREM 11.2. The Pontrjagin ring of v* l is a quotient of a subring of H *(BPGL( n)).



454 AMNON NEEMAN

We can obtain better results when n = 2, but for a general n, more work must be done.

12. An unpleasant construction

To treat the case n = 2, we need to consider a minor deviation from the Program.

12.1: Variation on the Program. We do Steps 1, 2 and 3 as in the Program, but then we alter it to:

(4') Construct (R, K) in Wbig so that (X,G) -* (Z, H) factors through (RI K).

(5) Prove vanishing for the equivariant cohomology H(R, K) =

H*(R XKEK). The reason this would also work is clear. We have already pulled ir*(Yl*)

back to K(Z, H), and the factoring in (4') allows us to pull it back to (R, K). The rest of the argument in Section 10 is unaltered.

In Section 11 we showed how 7T * Y" can be pulled back to Kalg(M(n)2g, PGL(n)) where Yf is the tangent bundle of f-'(XI)/PSU(n). For n = 2, we want to use Step 3 of the Program to pull it back further.

We start by constructing M(2), a variety containing M(2) as an open subset. The idea is to allow the difference of the eigenvalues of a matrix to go to infinity.

Let U C M(2) be the open subset of matrices with distinct eigenvalues. There is a map T \ PGL(2) X C X C * -* U, where T is the maximal torus of diagonal matrices in PGL(2). The map is given by (A, Xi ? X 2' X+ - A A-' X1 A. The map is a double cover, with covering transformation

(A,A1 Xi+ A 2 X 1- 2) 1((? o)A, X 1 + X 2, X 2 - 1 ). Note that multiplication on the left by (? 1) is well-defined on T \ PGL(2) because (? 1) is in the normalizer of T. Because (? ?) T, multiplication on the left by (? 1) has no fixed points on T\ PGL(2).

Let U be the affine variety T\ PGL(2) X C X C, and write C2\ U for the quotient of U by the cyclic group C2, where the action is given by (A, x, y) (( O)A, x, - Y). Now C2\ U is clearly smooth as C2 acts without fixed points.

We attach C2\ U to M(2) to form M(2) as follows. Take the subset of C2 \ U given by y # 0. It is an open affine variety, and is isomorphic to

U c M(2). We attach by the isomorphism (A, x, y) - A1( Y 1)A. So Y




corresponds to 1/(AX - X2), and in this sense M(2) is constructed by allowing i- X2 to go to infinity.

PGL(2) acts on M(2) by conjugation, and on U by its action on the first factor. The action on U commutes with the action of C, so that we get an action of PGL(2) on C2\ U. The actions are compatible on the intersection; hence PGL(2) acts on M(2).

Observe that (X1 - X2)2 extends to a morphism h: M(2) -- P'. Since C2\ U = h'-(P1 - (0)), M(2) = h'-(P1 - {cx}c), h is actually an affine morphism. Now h is separated, and because P' is separated so is M(2).

Note that h is also PGL(2) equivariant, where PGL(2) acts trivially on P'. So we get an equivariant map (M(2) x M(2))g -* (p1)g. Let Y be the pullback of ir1 (9(1) ? rYT2*9(1) _ ... * * (g*1(1) on (p1)g. Then by Property 9.19, Y is a nice bundle on (M(2) x M(2))g, and (M(2) x M(2))g c (M(2) x M(2))g is the vanishing of an invariant section of Y. So

((M(2) x M(2))g, PGL(2)) -),((M(2) x M(2)) , PGL(2))

satisfies the hypotheses of (3) in the Program. Next we carry out (4') in the variant of the Program. In our case, R will be

an open subset of (M(2) X M(2))g, invariant under the action of PGL(2). Define first S C C2 \ U X M(2) by the following construction.

PGL(2) X C X C X M(2) maps onto C2 \ U X M(2). Define a function P on

PGL(2) X C X C X M(2) by (A, x, y, B) - /312/321, where ABA- =

( f 1 fl2 ). It is easy to show that P is invariant under the action of T and C2

/321 f922/2

so it factors as a function on C2 \ U X M(2). It is also PGL(2)-invariant. Now we get that S = (C2 \ U X M(2))p is a PGL(2)-invariant open subset of M(2) x M(2). The inclusion S > C2 \ U X M(2) is good, being given by the vanishing of P. The inclusion C2 \ U X M(2) -* M(2) X M(2) is also good because C2 \ U c-+ M(2) is given by the vanishing of a section, and Property 9.15 says that C2\ U X M(2) -)_M(2) x M(2) is a good immersion. By Property 9.14, the composite S - M(2) X M(2) is a good open immersion.

Using Property 9.15 again, we have that the map S x (M(2) x M(2)) g-

(M(2) X M(2))g is a good immersion. Put

= Image S x (M(2) x M(2)) M(2) x (2)

where S goes to the ith factor. Then Ri c (M(2) X M(2))g is a good open subset, so that R = U Ii R is good. We want to put (R. K) = (R. PGL(2)) in Step (4').



456 AMNON NEEMAN

We must show that f I)--I)- (M(2) x M(2)) g factors through R. Take any (al, bl, ... ., ag, bg) E U(n)2g such that

(allb-la b,) ... (a-lbg-lagbg) = - I.

Then for some i, a i and bi do not commute. Since unitary matrices are diagonalizable and a is not in the center, the eigenvalues of ai are distinct, so that ai E U = M(2) n C2 \ U c M(2). Simultaneously conjugating ai and bi with a unitary matrix, we may assume ai diagonal. Then bi does not commute

with ai; so if bi =(Al 12 ), either /12 or 821 must be nonzero. But a unitary

matrix cannot be of the form (g b) with b # 0; hence both 8B12 and /821 are nonzero. Now / =12/A21 + 0, and (ai, bi) e S. Therefore f(-1 I) c U9 Ri = R. as required. This completes Step (4'), and we have also obtained a fact that we will need in Step (5):

Fact 12.1. The immersion R - (M(2) X M(2))g is good.

(5). There is a map (PGL(2) X C X C) X C X C X C* -- S given by

(A5 X5 Y5 15 P25b) ((A. x.y), A-'8 ' )A)

This turns out to be a double cover, with covering transformation

(A x, YP1, 2 b) ((1 o)Ax, - Y5M25plb)

The PGL(2) action on the cover is just the action on the first factor, and it has no fixed points. So every orbit in S is three dimensional, and the same is true for

g

Ri= S X(M(2) X M(2)) , or R-= URi1 i=1

Fact 12.1, together with the fact that (M(2) x M(2))g admits a nice line bundle and Property 9.21, tell us that R admits a separated quotient. The fact that every PGL(2) orbit in R is 3-dimensional implies that PGL(2) acts properly on R; this may be found in Mumford [6, Prop. 0.8, p. 16]. Now we are in the situation of Proposition 7.16: PGL(2) acts on R properly and with finite stabilizers, and the natural map H *(R/PGL(2)) -- H *(R x PGL(2)EPGL(2)) is an isomorphism for the rational cohomology.

By Corollary 1.6 and Remark 1.7 in Part 1 of this paper, the geometric quotient R/PGL(2) has the complex topology of the topological quotient




R/PGL(2). By Theorem 4.2 in Part 1, vanishing for the cohomology of R implies vanishing for the cohomology of R/PGL(2). To compute the cohomology of R, we use the open cover R = U 9 R. We need to compute the cohomology of finite intersections of the Ri's. A typical one is homeomorphic to Sk X (M(2) X M(2)) g-k

Hence we are reduced to the following two lemmas:

LEMMA 12.2. H'(S, Q) = 0 if 1 > 4.

Proof. Recall that we have a double cover PGL(2) X C4 X C* S, hence an injection H'(S, Q) -* H'(PGL(2) X C4 X C*, Q). But PGL(2) X C4 X C* has the homotopy type of RP3 x S1.

LEMMA 12.3. H'(M(2)) = 0 if 1 > 4.

Proof M(2) is the union of contractible space M(2) and

C2\ (T\ PGL(2) X C X C) = C2\ U.

So H*(C2 \ U) injects into H*(T\ PGL(2) X C X C) and T\ PGL(2) X C X C has the homotopy type of S2. The intersection C2 \ U n M(2) is doubly covered by T\PGL(2) X C X C*, which is homotopic to S2 x S'. When all this is combined with the Mayer-Vietoris exact sequence, the lemma follows.

COROLLARY 12.4. For any intersection U of Ri's, if I > 4g, H'(U, Q) = 0.

PROPOSITION 12.5. H'(R. Q) = 0 if 1 ? 5g.

Proof: There is a spectral sequence which abuts to HP+q(R, Q), whose El term is

p+1 EP q = 3 Hq n 5Q 1 n si 1 1 <'I <2 < ..<ip+l <g =l 1

Er q is zero except when q < 4g and p < g - 1. It follows that for 1 5g, H'(RQ) = 0.

COROLLARY 12.6. It follows from Proposition 12.5 and Theorem 4.2 that for 1 ? 5g - 3, H'(R/PGL(2), Q) = 0.

This completes Step 5 of the Program. Now we arrive at:

THEOREM 12.7. The Pontrjagin ring of ffg(2, 1) vanishes in dimensions higher than 5g - 3.

Remark 12.8. A conjecture that may be due to Newstead but is usually attributed to Ramanan asserts that the ring vanishes above dimension 4g (see



458 AMNON NEEMAN

Atiyah-Bott [1 Chap. 9]). J#g(2, 1) is a compact manifold of real dimension 8g - 6, but it admits an action by a 2g-dimensional, real torus (the Jacobian). The tangent bundle has a 2g-dimensional, trivial subbundle, and the quotient may be expressed as the pullback of the tangent bundle of a 6g - 6-dimensional, real manifold. Hence the trivial vanishing is in dimension 6g - 6. My method goes "half-way" to proving the conjecture.

Sequel

The construction of Section 12 is not quite right. The key difficulty is Step (5) of the Program: I do not know how to prove vanishing for the equivariant cohomology of a space except in the case where it agrees with the ordinary cohomology of the quotient.

Consider the following, tenuous argument. We know that the Pontrjagin ring of 7T**, f* the tangent bundle of f '(XI)/PSU(n), is supported on any open subset of M(n )2g that contains f ̀ (XI). In other words, the Pontrjagin ring really lives in the cohomology of any such open subset. Section 5 showed, once you grant the conjectures of Section 3 (which are true), that if U is an open subset of M(n )2g which is sufficiently nice, the cohomology of U/PGL(n) must vanish above dimension roughly 2gn2 - 4g(n - 1). This estimate is independent of U, provided H*(U) vanishes above a suitable dimension. If we could somehow realize what this means for the equivariant cohomology, or if we had a construction that altogether avoided the use of the equivariant cohomology, then this might tell us something about Chern classes. For n = 2, the vanishing would be above dimension 8g - 4g(2 - 1) = 4g. For n larger, this frivolous numerical estimate leads us to generalize Ramanan's conjecture. The Pontrjagin ring should vanish above dimension 2gn2 - 4g(n - 1) (approximately; precisely, above dimension 2gn2 - 4g(n - 1) + 2).

PRINCETON UNIVERSrrY, PRINCETON, NJ

BIBLIOGRAPHY

[1] M. F. ATiYAH and R. Borr, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Royal Soc. London 308 (1983), 523-615.

[2] A. GROTHENDIECK, Sur quelque points d'algebre homologique, Tohoku Math. J., 2nd Series, 9 (1957), 119-221.

[3] R. HARTSHORNE, Residues and Duality, Lecture Notes 20, Springer-Verlag, 1966. [4] G. KEMPT and L. NESS, The length of vectors in representation spaces, Springer Lecture

Notes 732 (1978), 233-244. [5] S. LojAsIEWICZ, Ensembles semi-analytiques, IHES preprint, 1965. [6] D. MUMFORD, Geometric Invariant Theory, Ergebnisse, Springer-Verlag, Heidelberg, 1965.




[7] M. S. NARASIMHAN and C. S. SESHADRI, Stable and unitary vector bundles over an algebraic curve, Ann. of Math. 82 (1965), 540-567.

[8] G. SEGAL, Equivariant K-theory, IHES Pub. Math. 34 (1968), 129-151. [9] J-P. SERBE, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier, Grenoble 6

(1956), 1-42. [10] C. S. SESHADRI, Quotient spaces modulo reductive algebraic groups, Ann. of Math. 95 (1972),

511-556.

(Received October 13, 1983) (Revised May 24, 1985)



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