=1 - mathematicsmath.mit.edu/.../library/hitchin_stable_bundles_integrable_systems.pdf · the...

Vol. 54, No. 1 DUKE MATHEMATICAL JOURNAL (C) 1987

STABLE BUNDLES AND INTEGRABLE SYSTEMS

NIGEL HITCHIN

1. Introduction. The moduli spaces of stable vector bundles over a Riemannsurface are algebraic varieties of a very special nature. They have been studied forthe past twenty years from the point of view of algebraic geometry, numbertheory and the Yang-Mills equations. We adopt here another viewpoint, consider-ing the symplectic geometry of their cotangent bundles. These turn out to bealgebraically completely integrable Hamiltonian systems in a very natural way.For :ank 2 bundles of odd degree this appeared as a byproduct of an investiga-tion [5] into certain solutions of the self-dual Yang-Mills equations, a subject onwhich Yuri Manin has had a profound influence.The cotangent bundle T*N of an n-dimensional complex manifold N is a

completely integrable Hamiltonian system if there exist n functionally indepen-dent, Poisson-commuting holomorphic functions on T*N. These functions for thecase where N is the moduli space of stable G-bundles on a compact Riemannsurface M, and G a complex semisimple Lie group are easy to describe. Thetangent space of the moduli space at a point is identified with the sheafcohomology group Hi(M; ) where g is a holomorphic bundle of Lie algebras.By Serre duality the cotangent space is H(M; (R) K). An invariant polynomialof degree d on the Lie algebra then gives rise to a map from this cotangent spaceto the space H(M; Ka) of differentials of degree d on M. Taking a basis for thering of invariant polynomials yields a map to the vector space W=1 H(M; Kd’) where d are the degrees of the basic invariant polynomials.

Somewhat miraculously, the dimension of this vector space is always equal to thedimension of the moduli space N, thus providing the n functions.The Hamiltonian vector fields corresponding to these functions give n com-

muting vector fields along the fibres of the map to W. The system is calledalgebraically completely integrable if the generic fibre is an open set in an abelianvariety and the vector fields are linear. For the system above, at least for the casewhere G is a classical group, this also turns out to be true, the abelian varietybeing either a Jacobian or a Prym variety of a curve covering M. The construc-tion of this curve, and its corresponding Jacobian, parallels the solution ofdifferential equations of "spinning top" type involving isospectral deformationsof a matrix of polynomials in one variable. A point of the cotangent bundle ofthe moduli space consists of a stable vector bundle F (with G-structure) and aholomorphic section H(M; fi (R) K), which gives a holomorphic map "F F (R) K. We form the curve of eigenvalues S defined by the equation

Received November 29, 1986.

91

92 NIGEL HITCHIN

det(X ) 0 in the total space of the cotangent bundle of M. Over S there isa natural eigenspace line bundle L which defines a point in the Jacobian of S.Fixing the values of the invariant polynomials fixes the coefficients of thisequation and hence also the curve. Thus the only further variation possible is tochange the line bundle. For G GL(m, C), the line bundle can take all possiblevalues in the Jacobian. For G--Sp(m,C) or SO(2m + 1, C) the bundle isrestricted to lie in the Prym variety of S with respect to an involution with fixedpoints. For G SO(2m, C), the curve S is singular but the line bundle is definedon its desingularization, which has an involution without fixed points. It is thePrym variety of this curve which is a generic fibre in this case.

Having established the existence of the integrable system we apply it in therealm of algebraic geometry to compute some sheaf cohomology groups of themoduli space of stable bundles of rank 2 and odd degree with fixed determinant.This consists of a generalization of the result of Narasimhan and Ramanan thatH(N; T) 0 and dim Hi(N; T) 3g 3 to consideration of the correspond-ing groups for the k-th symmetric power bundle SgT. A consequence of thisresult is that the only globally defined holomorphic functions on the cotangentbundle of the moduli space, and polynomial in the fibres, are those generated bythe functions defining the integrable system. Thus the global geometry of themoduli space determines the interesting symplectic geometry of its cotangentbundle.The question which we have not treated here is to find explicitly the Hamilton-

ian differential equations which correspond to these systems. Since the modulispaces are unirational (and conjectured to be rational) then ultimately these maybe expressed as equations with rational coefficients. Finding some natural,concrete realization of the integrable systems which arise so naturally in this waymay lead to an application in the other direction--from algebraic geometry todifferential equations. This would be an agreeable outcome, and one consistentwith Manin’s view of the unity of mathematics.

2. Stable bundles

2.1. Let M be a Riemann surface and V a CO complex vector bundle of rankm over M. By a holomorphic structure A on V we shall mean a differentialoperator

d;: v) v)

such that

dJ’( fs ) Of (R) s + fdA’s (2.2)

for s f(M; V) and f Coo(M).The local sections s satisfying dJ’s 0 are defined to be holomorphic and with

this definition one obtains holomorphic transition functions relating local holo-

STABLE BUNDLES AND INTEGRABLE SYSTEMS 93

morphic bases and consequently the usual definition of holomorphic bundle (see[21).

2.3. Let ff denote the group of automorphisms of the vector bundle V. Thegroup acts on the space of holomorphic structures by

d’ --+ g-ld’g.

The space of all holomorphic structures on V is an infinite dimensional affinespace since from 2.1

d" d" B o,1A (M; End V)

and ff acts on z/ via affine transformations.

2.4. If M is compact and of genus g > 1, then there is an open set z’s c z/

of holomorphic structures, the set of stable structures, which is preserved by ffand is such that the quotient zC’s/f is a smooth complex manifold whosetangent space at A is isomorphic to the sheaf cohomology group Hi(M; End V).The condition for stability is that for every proper subbundle U,

deg(U) deg(V)<

rkU rkV

If equality occurs, the bundle is semistable.

2.6. The quotient space .A=z’/ff is the moduli space of stable vectorbundles of rank m over M. By the Riemann-Roch theorem (since stability impliesH(M; End V) C), the moduli space .#’ has dimension

dim ag’= m2(g- 1) + 1. (2.7)

In the case where the rank m and Chern class q(V) are mutually prime, thenis a compact projective variety.

2.8. The notion of stability may be extended from vector bundles to principalbundles ([2], [9]). In this case the space of holomorphic structures on a principalbundle P over M with structure group a complex semisimple group G is aninfinite dimensional affine space with group of translations fl0,X(M; ad P), wheread P is the vector bundle associated to the adjoint representation. For a Liegroup G, semistability of the principal bundle is equivalent to semistability of theholomorphic structure on the vector bundle ad P.

In this more general situation there is a group f of automorphisms of P, withLie algebra given by fl(M; ad P) and a subspace z’ of stable holomorphicstructures on P such that the quotient space zC’,/f /" is a smooth complex

94 NIGEL HITCHIN

quasi-projective variety with

dim dim G ( g 1) (2.9)

for a semisimple group G.All these varieties have very special properties which have been investigated

from many points of view ([2], [7], [8]). The point of view we adopt here is that ofsymplectic geometry in the holomorphic category. We shall consider the totalspace T*4 of the cotangent bundle of the moduli space of stable bundles as acomplex symplectic manifold.

3. Symplectic geometry.try.

We recall some basic facts from symplectic geome-

3.1. Let N be any manifold and T*N its cotangent bundle with projection r:T*N N. The tautological section of r*T*N defines a 1-form 0 on T*N suchthat dO t is a symplectic form on T*N. If (Xl,..., Xn) are local coordinates onN and cotangent vectors are parametrized by the coordinates (Yl,---, Yn)--Ei=lyi dxi, then the form 0 is defined by

0 Yi dxiand

3.2. If f is a smooth function on a symplectic manifold M, then it defines avector field Xf:

df i( Xf).

The Poisson bracket of two functions f and g is defined by the function

( f g ) Xf. g -Xg f -(g,f}.

Two functions are said to Poisson-commute if (f, g } 0. A particular case is if,dy A dxi; and f and g are functions of (yl,..., Yn) alone, for then

df E Yi dyi i( X/)( E dYi A d.x,i)

and so

tgf 9Xf= -E ay tgxi

Thus (f, g} Xf g E( tgf/tgy)( i)g/tgx ) O.


3.3. If the vector fields XA,..., X/ form the basis of a Lie sub-algebra g ofthe Lie algebra of vector fields on a symplectic manifold M, then the functionsdefine a map to the dual of g,

given by

g" Mg*

EZ(x)

where { i } is the dual basis in g*.When the vector fields integrate to give the action of a Lie group G on M, and

if/ is equivariant, then g is called a moment map.Suppose G acts freely on the submanifold/-i(0). Then the symplectic form is

degenerate along the orbits of G in g-l(0) and invariant by G. It defines asymplectic form on the quotient g-I(O)/G which is called the Marsden-Weinsteinquotient.

3.4. A special case of the Marsden-Weinstein quotient is when a Lie group Gacts freely on N, and hence on T*N. A vector field X on N generated by thisaction defines a function from T*N by contraction in the fibres: if X,a(8/Ox), f= Z,ay. The vector field X on T*N is the natural extension of Xto the cotangent bundle.

In this case, the Marsden-Weinstein quotient is the cotangent bundle of theordinary manifold quotient:

_-- s-3.5. Let M be a symplectic manifold with the action of a Lie group G and

moment map g: M g* defined by functions fl,--., fn- If g is a G-invariantfunction on M then it defines by restriction a G-invariant function on -1(0) andhence a function g on the Marsden-Weinstein quotient g-l(0)/G.

Let g, h be two such functions. Then Xy,g 0 since g is G-invariant. But then

x f,= (g,/,} -X ,g o

so that Xg is tangential to f-1(0) for all and thus to -1(0).If { g, h } O, then X. h 0 so h is constant along the orbits of Xg. But the

projection of these orbits to -~(O)/G are the orbits of X. Thus is constant onthe orbits of X and so { if, h } O.Thus if g and h Poisson-commute in M, so do and in -(O)/G.3.6. A symplectic manifold M of dimension 2n is said to be a completely

integrable Hamiltonian system if there exists functions fl,.--, fn which Poisson-commute and for which dft/x /x df is generically nonzero.

96 NIGEL HITCHIN

The map f: M C" defined by these functionsthe moment map for anabelian Lie algebra--has the property that a generic fibre is an n-dimensionalsubmanifold with n linearly independent commuting vector fields Xyx,..., Xf,. Ifthis fibre is an open set in an abelian variety and the vector fields are linear, thenwe shall say that the system is algebraically completely integrable.

We shall see that the cotangent bundle of the moduli space .A/" of stablebundles on a Riemann surface always has n dim ," natural Poisson-commut-ing functions and, at least for the classical groups, these form an algebraicallycompletely integrable Hamiltonian system.The most trivial example is the case of the complex group G C*. An

equivalence class of holomorphic structures on a C*-bundle is defined by a pointin the Picard group Hi(M; d)*), or fixing the degree by a point in the JacobianJac(M). The tangent bundle of the Jacobian is trivial and isomorphic toHX(M; d)), and so by Serre duality the cotangent bundle is

T*,A/’--- Jac(M) H(M; K)

where K is the canonical bundle.The g functions obtained by projecting on the second factor Poisson-commute

and every fibre of this map is a copy of the abelian variety which is the Jacobian.Every holomorphic vector field defined on a compact abelian variety is linear, sothis is an algebraically completely integrable system, though not a very interest-ing one.

{}4. Invariant polynomials. Let 4 be the moduli space of stable holomorphicstructures on a principal G-bundle P. The tangent space at a point of may berepresented by the vector space HX(M; ad P) and hence by Serre duality thecotangent space is isomorphic to H(M; ad P (R) K).The vector bundle ad P is a bundle of Lie algebras each isomorphic to g, and

so if p is an invariant homogeneous polynomial on g of degree d, it defines amap

p: H(M; ad P (R) K) H(M; Kd).

Let p,..., Pk be a basis for the ring of invariant polynomials on the Lie algebrag, then we obtain a map

k

p" H(M;adP(R)K) ) H(M;Kd’).i=1

PROPOSITION 4.1. Let G be a semisimple complex Lie group and P a stableholomorphic principal G-bundle over a compact Riemann surface M of genus g > 1.


Then

k

dim H (M; ad P (R) K ) dim ]) H (M; Ka’).i=1

Proof If P is stable, then H(M; ad P)= 0 (see [2]) and so by Riemann-Roch, using the Killing form to identify ad P* --- ad P,

dim H(M; ad P (R) K) dimG(g 1).

Since G is semisimple there are no linear invariant polynomials, so each d;and therefore applying Riemann-Roch to the fight hand side

>1

k k

dim ) H(M; Kd,) ., (2d 1)(g- 1).i=1 i=1

(4.3)

But it is well known that

k

dimG= E(2di-1).i=1

From a topological point of view, each invariant polynomial defines an invariant(2d 1)-form on G and the exterior algebra generated by these is the cohomol-ogy H*(G;C) [3]. More algebraically, Kostant’s principal three-dimensionalsubgroup breaks up the Lie algebra g into representation spaces of dimension

2di 1 [6]. From either source, the proposition is proved.From 4.1, we obtain a map

k

p" T*4- ( H(M; Kd’) (4.4)i=1

which provides n holomorphic functions f defined on the 2n-dimensionalsymplectic manifold T*.

PROPOSITION 4.5. The n functions fi on T*4/ Poisson-commute.

Proof. From 3.4 we shall represent T*V’ as a Marsden-Weinstein quotient:

where

is the moment map for the action of f on the affine space ’.

98 NIGEL HITCHIN

Now T*o’s -= s fO(M; ad P (R) K) with the canonical 1-form 0 defined by

0(, ))= fMB(/ A {I}) (4.6)

where ’l(M; ad P) is a tangent vector to at A andf(M; ad P (R) K) a tangent vector to f(M; ad P (R) K) at , and B is theKilling form of the group.

Since f acts on ’ by conjugation of the operator dJ’, the Lie algebra, f0(M; ad P) generates vector fields

A d’+ o,(M; ad P)

for q f(M; ad P).Consequently, from 3.4, #(A, ) 0 if and only if

for all q f(M; ad P). Thus /z(A, ) 0 if and only if dJ’ 0 i.e.,fO(M; ad P (R) K) is holomorphic with respect to the holomorphic structure dJ’.Taking a basis for the invariant polynomials on defines a map

k

q" ’ fl(M; ad P (R) K) -o () f2(M;i=1

simply by evaluating on the second factor. On the submanifold/z-1(0) (i.e., thepoints (A, ) satisfying dJ’ 0) it takes values in the holomorphic sections ofKd, and, being invariant by f, descends to #-l(0)/ff T*/" to define the mapp in (4.4). Now any two of these functions certainly Poisson-commute inT*. s fl(M; ad P (R) K) since they are functions of f(M; ad P (R) K)alone as in 3.2. Therefore, by 3.5, they commute also on the quotient T*.#’,proving the proposition.

We see then that the 2n-dimensional symplectic manifold T*/" admits nPoisson-commuting functions, where W is the moduli space of stable holomor-phic structures on a principal G-bundle, for G semisimple.The same is true for G GL(m, C), i.e., the moduli space of stable bundles of

rank m, the only difference being that the centre gives rise to a linear invariant,so that the dimension of the fight hand side in 4.1 is E/=2(2di 1)(g 1) + g(m2 1)(g 1) + g, but by Riemann-Roch this is also the dimension of the lefthand side. Restricting attention to vector bundles with fixed determinant bundlereduces this situation to that of the semisimple group SL(m, C).


5. Algebraic integrability. We shall show now that the n Poisson-commutingfunctions defined above make T*ff into an algebraically completely integrableHamiltonian system for the classical groups G GL(m, C), Sp(m, C), SO(2m, C)and SO(2m + 1, C). To do this we need to show that the genetic fibre of thefunction p in (4.4) is n-dimensional (to obtain the functional independence ofthe functions f), and further that it is an open set in an abelian variety, with thevector fields Xk linear.

5.1. Let G GL(m, C). A point of T*,A/" is given by a stable vector bundle Vof rank m and a holomorphic section H(M; End V (R) K).A basis for the invariant polynomials on GL(m,C) is provided by the

coefficients of the characteristic polynomial det(x A) of a matrix A gl(m, C).There is one polynomial in each degree < m, so we obtain the map

m

p" T*/z’..- n(m; Ki).i=1

To study the fibre of this map we take sections a; H(M; K i) and consider thevector bundles V and sections H(M; End V (R) K) for which

det(x ) xm "{- alxm-1 + +am.

Let X be the complex surface which is the total space of the cotangent bundleKt of M. The pull-back of Kt to M has a tautological section ?. Consider thesections of the line bundle Kt on X of the form

S km + alkm-1 + +am,

for a H(M; K).As the a vary, the zero set of s forms a linear system of divisors on X, and its

compactification P(KM 1). Any base point of this system must lie in the zerosection of Kt since )m lies in the system. But then it is a base point for Kt onM since a vanishes there for all a H(M; Kin). But K is without basepoints for any Riemann surface, so the above system has no base points either.By Bertini’s theorem the genetic divisor is a nonsingular curve S.The genus of S is given by the adjunction formula

Kx. S + S:= 2g(S) 2.

Now Kx 0 (X is a symplectic manifold!) and S is in the linear system mK,vl.The zero section is in the linear system K and has self-intersection number

K cl(K) 2g- 2, hence

2g(S)- 2 2m2(g- 1)

100 NIGEL HITCHIN

and

g(s) m’(g- ) + . (S.3)

The curve S is an m-fold covering r: SM of M. If det(x-)=xm+alxm-l+ am, then pulling back the bundle V to S, we haveH(S; End V (R) KM) satisfying det( ) O. Thus on S, has the eigenvalueX H(S; KM). Generically the eigenvalues are distinct and so we obtain aone-dimensional eigenspace which defines a line bundle

L c ker(X ) c r*V

on S and hence a point of the Jacobian Jac(S), an abelian variety of dimensiong(S) m2(g- 1)+ 1 dim ./V’ from (2.7).

Conversely, the line bundle L determines V and on M. We consider thedirect image sheaf r,L which is the sheaf of sections of a vector bundle of rankm on M. The fibre of the bundle r,L at x M is by definition

(,L) , =-

where o,-,(x is the ideal sheaf of q’/’-l(x). If x is not a branch point then this is

y r-(x)Ly. If x is a branch point, then the fibre is

Jk(y)(L)yy=r-X(x)

a direct sum of jets of sections of L of degrees k(y) given by the degree oframification of y.Each k-jet has a natural map to the O-jet

J(L) - G- o

and hence dually a map

0 -, L - J(L).

There is thus a natural sequence for all x

0 - z; - (,z)* - *x J(y)(L ) y/Ly ---> O.yr-X(x) y-(x)

This defines a sheaf map on M

o(,L)* -y 0


where 6a is a sheaf supported at the branch points of r: S M in M. Thekernel of s is a locally free sheaf (.0(W) of rank m.

If L was obtained from a vector bundle V and H(M; End V (R) K) asabove, then the vector bundle W is V*, from the natural map Vy* y ,,-l(x) Linduced by the inclusion L c V.

This way, we recover V from the line bundle L. The operation of multiplica-tion by 2, H(S; Kt) on L yields a section of End r,L (R) Kt which takes Wto W (R) Kt and so defines H(M; End V (R) Kt). We see therefore that thegenetic fibre of p lies in an abelian variety of dimension n dim /. Since pmaps to an n-dimensional space this proves the functional independence of thePoisson-commuting functions fl,..-, f, and shows that the fibre is an open set inthe abelian variety. Not all vector bundles produced from line bundles on S thisway will be stable and it is the stable ones which determine the open set.What remains is to show that the vector fields Xf, are linear on the Jacobian.

This is equivalent to showing that they extend as holomorphic vector fields to thewhole compact Jacobian.Our construction of V and from a line bundle L on S was, however, natural

in the sense that if a holomorphic isomorphism takes V and to V’ and O’, thenit takes the corresponding line bundle L to L’. Thus the Jacobian is a space ofequivalence classes of points in f(M; End V (R) K) with dJ’ 0, underthe action of the group of automorphisms ft. This space is strictly larger thanT* which we used in [}4, but by the Marsden-Weinstein construction stillgives rise to a symplectic quotient. The cotangent bundle T*V" therefore lies asan open set in a larger symplectic manifold which contains the whole Jacobianand hence allows the vector fields Xf, to extend and therefore be linear. We shallconsider this larger moduli space in more detail in the next section for the caseG GL(2, C).The Hamiltonian system is thus in this case algebraically completely integra-

ble.Finally note that the sheaf map (5.4) gives rise to an exact sequence of sheaves:

which determines the degree of the line bundle L.From the Grothendieck-Riemann-Roch theorem,

ch(vr,L)td(M) r,(ch(L)td(S))so

ch(rr,L) (1 + (g- 1)[Ml)r,(1 + (degL + (1 g(S)))[SI)

and therefore

degr,L m(g- 1) + deg L + (1 g(S)). (5.6)

102 NIGEL HITCHIN

On the other hand, the ramification divisor on S is defined by the points whosetangent space projects to zero on M and this is the divisor of a section of KsKwhich has degree

2) 2).

Thus from (5.5), since 6a is supported on the branch points,

Hence

deg F* deg( ,r,L)* deg 6a

m(g-1) (g(S) 1)-degL

-m(m- 1)(g- 1)- degL from (5.3).

deg L -m(m 1)(g 1) degF*. (5.9)

5.10. Let G Sp(m, C). A point of T*o/f/" now consists of a stable vectorbundle V of rank 2m with a symplectic form ( ), together with a holomorphicsection H(M; End V (R) K) which satisfies (v, w) (v, Ow).

Suppose A sp(m, C) has distinct eigenvalues Ai with eigenvectors v; on C 2m,then

ki(Oi, Oj) (Aoi, oj) -(oi, Aoj) -kj(oi, oj)

so (vi, vj) 0 unless j -)ti. Since (vi, vi) 0 it follows from the nondegen-

eracy of the symplectic inner product that if )ki is an eigenvalue so is -)t .Thus the characteristic polynomial is of the form

det(x A) x2m + a2x2m-2 + +a2m.

The polynomials a2,... a2m in A form a basis for the invariant polynomials onthe Lie algebra sp(m, C).As in the previous section, we consider the linear system of divisors of the

sections ,2m + a2A2m-2 + +azm of Km on the complex surface X and byBertini’s theorem see that the genetic divisor is a nonsingular curve S of genusg(S) 4m2(g 1) + 1 from (5.3). We obtain again a line bundle

L c ker()t ) c r*V

and thus a point on the Jacobian which determines V and .In this case, however, the curve S with equation

)k2rn + a22m-2 + +a2m 0


possesses the involution o(,)=- with fixed points the zeros of a2mH(M; K2") on the zero section X 0. The involution acts on the Jacobian ofline bundles of degree zero. The subvariety of line bundles such that o*L L* isan abelian variety--the Prym oariety. Its dimension is g(S) g(S/o).Now o has 4m(g 1) fixed points since they are the zeros of a section of gEm

on M, so

2- 2g(S)= 2(2- 2g(S/o)) -4m(g- 1)

and thus

dim Prym(S) g(S ) g( S/o )

1/2g(S) +m(g-g(S)

=2m2(g- 1) +re(g-l) from (5.3)

m(2m + 1)(g- 1). (5.11)

In our situation the eigenspace bundle L c ker(X ) actually defines a pointin the Prym variety. This is because of the nondegenerate pairing above ofeigenvectors corresponding to eigenvalues and -X. If L is the eigenspacebundle corresponding to ,, then o*L corresponds to -, and so the symplecticform defines a section of L* (R) o’L* which is nonvanishing if the eigenvalues aredistinct i.e., away from the ramification locus of S.

Since V is symplectic, deg V 0 and so from (5.7) and (5.8),

deg L 1/2deg(KsK).

Thus

deg(L* (R) o’L*) deg(KsKt)

and since L* (R) o’L* has a section which vanishes on the ramification locus, adivisor of KsK, we have

o*L L* (R) (KsK) . (5.12)

Choosing a holomorphic square root (KsK)1/2 of KsK we obtain

L U(KsKt) -1/2 (5.13)

where U is a line bundle such that o*U -= U*, i.e., U lies in the Prym variety.

104 NIGEL HITCHIN

Now from (5.11)

dim Prym(S ) m (2m + 1) ( g 1)

dimSp(m)(g- 1)

Hence, the generic fibre of p is in this case an open set in an abelian varietymaPrym variety.Note that in the converse construction of the vector bundle V from the line

bundle L, the involution o produces a bilinear form on ,r,L away from thebranch locus by taking o H(p-l(U); L) and setting

<v, w>s o*v w H( p-X(u), KsK)from (5.12), where s H(S; KsK) is the canonical section iven by thederivative of r" S --, M.The form is skew because o has real codimension 2 fixed points and is

therefore of odd type [1] i.e., it lifts to an action on K/ of order 4. From (5.13)this means that o - -1 on L.

5.14. Let G SO(2m, ). A point of T*V’ is now a stable vector bundle Vof rank 2m with a nondeenerat symmetric bilinear form ( ), toether with aholomorphic section H(M;EndV (R) K) which satisfies (v, )=-(v, ew).Now suppose A so(2m, C) has distinct eigenvalues ’i with eigenvectors v

on z,,. Then

Xi(Vi, Vj) (Avi, vj) -(vi, Avj) -Xj(vi, vj)

so if h #: 0, (vi, vi) 0 and, as in the previous case, if k is an eigenvalue so is-h i. Thus the characteristic polynomial is of the form

det(x A) xTM -Jr" a2x2m-2 + ....+.a2m.

In this case, the polynomial az,. det A of degree 2m is the square of apolynomial p,mthe Pfaffian--of degree m. A basis for the invariant polynomialson so(2m, C) is then given by the coefficients a2,..., azm_z and Pm"NOW let Pm H(M; K’) be a section with simple zeros and consider the

linear system of divisors of the sections ,2, + a22m-2 + +a2m_2t2 + p2for a2i H(M: K2i), 1 < < m 1. Each curve of this system has singular-ities where , 0 and p,, 0. These are the base points of the system and so byBertini’s theorem a genetic divisor has these as its only singularities. Since p,, is asection of K ’, there are deg K"* 2m(g 1) singularities which are geneticallyordinary double points.


The curve S in X given by the generic divisor then has virtual genus given bythe adjunction formula (5.3)

p=4m2(g-1) + l.

The genus of its nonsingular model q is thus

g() 4m2(g- 1) + 1 2m(g- 1)

2m(2m- 1)(g- 1) + 1. (5.15)

The involution o(X) -2 on S has as fixed points the singularities of S whichare double points and so extends to an involution on q without fixed points.We consider the Prym variety of S. Its dimension is g(ff) g(q/o). Since o

has no fixed points,

and thus

2- 2g()= 2(2- 2g(q/o))

dim Prym() g(q) g(q/o)

g(d) }_g(d)

m(2m- 1)(g- 1). (5.16)

Because q is smooth we obtain an eigenspace bundle L c ker(, tI)) inside thevector bundle 17 pulled back to q, and as in the previous case, by applying thesymmetric form, a section of L* (R) o’L* which is nonzero if the eigenvalues aredistinct. It remains nonzero in fact when an eigenvalue tends to zero as may beseen by considering the eigenvectors of a 2 x 2 skew-symmetric matrix. Thus thissection is nonvanishing away from the ramification points of the map r" q --, M.Arguing analogously to the symplectic case, we find

o*L L* (R) (KgK) -1

and setting

we obtain a point of the Prym variety ofNow from (5.16),

dimPrym(q) m(2m- 1)(g- 1)

dimSO(2m)(g- 1)

106 NIGEL HITCHIN

We therefore obtain algebraic complete integrability in this case too. Since theinvolution o on S now has no fixed points it is of even type and this is the reasonthat it provides a symmetric bilinear form on the corresponding vector bundleover M.

5.17. Let G SO(2m + 1, C). We now consider a stable vector bundle Vover M of rank (2m + 1) with a nondegenerate bilinear form (,) and aholomorphic section H(M;EndV (R) K) which satisfies (v, w)=-(o, Cw).By specializing from SO(2m + 2), the characteristic polynomial of A

so(2m + 1) acting on C 2m+ is

det(x A) x(x2m "-[- a2x2m-2 + +a2m )

and the coefficients form a basis for the invariant polynomials on SO(2m + 1).As in the case of Sp(m), we choose a2i H(M; K) for 1 < < m such that

the curve S defined by

.2m + a22m-2 + +a2m 0

is nonsingular, and consider V and such that

det(h- *) h(h2m + a2h2m-2 + +a2,,).

The vector bundle homomorphism : V V (R) KM now always has an eigen-value zero, so we automatically obtain a line bundle V0 c V defined by

Vo c ker .The bundle V on M is therefore naturally an extension

o-) Vo---, v vo. (5.18)

The zero eigenspace can be canonically identified for if we use the inner producton V to convert the skew map to a holomorphic section H(M; A2V (R)

KM), then

m ( HO(M; A2mv (R) K) H(M; V (R) K)

defines a homomorphism from Km to V whose image is the line bundle Vo.Thus Vo --- Ktm and hence from (5.18) since V V*, we have

A2"V1 -= K. (5.19)

Since Vo lies in the kernel of we obtain an induced homomorphism "


V1 ---’ V1 (R) Kt and we proceed as in 5.1 to obtain a line bundle L on S definingthe eigenspace of .The inner product ( ) on V defines a skew form on V with values in KM, by

(o, w) (o, w) (5.20)

for representative vectors v and w in V. Since (Co, w) -(v, Cw) the form isclearly well defined on V but is singular when : V ---, V (R) KM is singularwhich is when a2m 0. It nevertheless plays a role analogous to the symplecticform in 5.10 and just as in that case, if we use the involution o(,) -, on S,we obtain a section of

L* (R) o’L* (R) Km

from the pairing of the eigenvectors with eigenvalues + ,. Now from (5.9),

deg L 2m(2m 1)(g 1) deg VI*

-2m(2m-1)(g-1) +m.2m.2(g- 1) from (5.19)

2re(g- 1)

1/2deg K.

Thus L* (R) o’L* (R) Kt is of degree zero with a nontrivial section and is hence atrivial bundle. Therefore

o*L-- L* (R) KM.

/1/2Choosing a square root "’M we obtain

where U is a line bundle in the Prym variety of S. From (5.11)

dim Prym S rn (2m + 1)( g 1)

dim SO(2m + 1)(g 1)

The methods of the previous cases allow us to recapture the rank 2m bundle 1/1from the line bundle L. It comes equipped with a skew form ( ) with values inKM and a homomorphism : V --> V (R) KM satisfying (Co, w) -(v, Cw).We need to reconstruct the vector bundle V of rank (2m + 1).

108 NIGEL HITCHIN

To do this we consider how F arose and duaiize the exact sequence (5.18) toobtain

0 VI*-) V--) V0*- 0. (5.22)

The composition V0 V Vo* defines a section of Vo* (R) Vo* =- Km which iseasily seen to be the coefficient a2m. On the zero set D of this we have aninclusion Vo c VI*.Now consider the exact sequence of sheaves

0 --) Hom( Vo*, Vx* ) Hom(V0 V* ) Hom(Vo, V* )1D -- 0

and its exact cohomology sequence. The extension (5.21) is defined by thecoboundary map

8" H(D; Hom(Vo, VI* )) --> HI(M; Hom(Vo*, V* ))

applied to the inclusion Vo c V*. An analogous construction occurs in thetwistor approach to magnetic monopoles [4].The advantages of this construction of V is that it depends only on the bundle

Vx* and the symmetric form induced on it from V via the inclusion V* c V. Weconsider a bundle V* of rank 2m with a symmetric form. The form degenerateswhere the corresponding homomorphism g: VI* --) V is singular i.e., on adivisor D of the line bundle (A2mV). Over D there is a 1-dimensional subbundleVo c Vt*, the kernel of g. The bundle Vt*/Vo has a nondegenerate symmetricform and so over D,

Vo-- hmVl*.In the exact sequence of sheaves

0 -o Hom( A2mv1, VI* ) det___g Hom( A2mv1* VI* ) --> Hom(A2mv1* VI* )l O --> 0

the inclusion of V0 -= A2mvI* in VI* over D generates an extension

0 -’> Vl* "-> V -’> A2mv1 -->0

by the coboundary map.We use the above applied to the bundle V arising from a point of the Prym

variety of S. It has the skew form to H(M; V* (R) VI* (R) KM) and thehomomorphism H(M; End V (R) Ku). We obtain a symmetric form byconsidering

tom-1 no(M; A2m-Iv1, (R) A2m-IvI, (R) Kllm-1)

.o(M; V1 (R) V1 (R) (A2mV1$ )2(R) gllm_l)H(M; V (R) V1 @ Kt)


and applying to obtain

(I)0)2m--1 H(M; V (R) V).

Hence, a generic fibre of the map p is an open set in the Prym variety of S andthe system is integrable.

Remark. Note the similarity between the cases of G Sp(m,C) andSO(2m + 1, C). In both cases the projection p maps to the same vector space ofdifferentials on M and the generic fibre is the same Prym variety: locally the twosystems are equivalent. Globally, the difference in identifying the fibres as Prymvarieties (compare (5.13) and (5.21)) will affect the structure of the moduli spaces.This may be a convenient context in which to bring in the duality theory of Liegroups by means of which Sp(m, C) and SO(2m + 1, C) are viewed as dualgroups.

6. An application to moduli spaces. We consider now an application of thesesymplectic methods--we use the integrable system to derive information aboutthe sheaf cohomology groups of the moduli space of stable vector bundles ofrank 2 and odd degree over a Riemann surface of genus g > 1. This space is acompact smooth projective variety. If we fix the determinant line bundle A2V ofthe stable vector bundle V, then it has complex dimension 3g 3.

In [7], Narasimhan and Ramanan proved that if 1 is the moduli space ofstable bundles of rank m and degree k, with (m, k)= 1 and with fixeddeterminant bundle, then the sheaf cohomology groups Hi(4/; T) of the holo-morphic tangent bundle T vanish if :/: 1 and dim H1(o6/’; T) 3g 3. In fact,the vanishing for > 1 is a consequence of the vanishing theorem of Akizuki andNakano, so the cases 0 and I are the only nontrivial ones. We shall generalizetheir result to deal with the k-th symmetric power bundle SkT.We restrict attention to the case of vector bundles of rank 2 because here, by

analytical means, we have a precise description of the symplectic manifoldalluded to in 5, in which T*I/" lies as an open set [5]. We say that a pair(A,) T* consisting of a holomorphic structure A and a sectionf(M; ad P (R) K) satisfying dJ’ 0 is a stable pair if for each q)-invariant

holomorphic line bundle L c V, deg L < 1/2deg V. Note from (2.5) that if theholomorphic structure is itself stable, then (A, ) is stable for all .The quotient of the set of stable pairs with fixed determinant by the group f of

automorphisms of V of determinant one is, from [5], a complex symplecticmanifold ’ of dimension 6g 6 with the following properties:

PROPOSITION 6.1. (i) ’ is a Kiihler manifold.(ii) The map det: ’ H(M; K2) defined by applying the invariant poly-

nomial det on sl(2, C) to dp is proper and surjective.(iii) The fibre of a point q H(M; K2) consisting of a section with simple

zeros is isomorphic to the Prym variety of the double covering S of M defined byXz q 0 in the total space X KM of the cotangent bundle of M.

110 NIGEL HITCHIN

(iv) The cotangent bundle T*dV" of the moduli space of stable vector bundles liesnaturally in /[ as the complement of an analytic set of codimension at least g.

Proof For parts (i) to (iii), see [5], 8. In fact (iii) is essentially the descriptionfor SL(2, C) -= Sp(1, C) in 5.10. The fact that V is of odd degree only changes theisomorphism with the Prym variety.

For part (iv), if V is unstable, then it has a canonical subbundle L c V withdeg L > 1/2deg V. Thus V is defined as an extension

0 L V L* (R) A2V---> 0

and hence by an element of Hi(M; L (R) A2V*).Now the stability of the pair (V, ) implies from [5], (3.12), that

0<deg(L2(R)A2V*) <2g-2

(with strict inequality in this case since deg(V) is odd). This means thatdim Hi(M; L2 (R) A2V*) < g 1 with equality only if L is special.We need now another property of the symplectic manifold .t’. Recall that a

submanifold of a symplectic manifold is Lagrangian if its tangent spaces areisotropic of maximal dimension. The moduli space rig’ is foliated by Lagrangiansubmanifolds (of dimension (3g- 3)) each of which is obtained by fixing theequivalence class of the complex structure and allowing to vary (see [5], 8).On the open set T*.Azc t’ these manifolds are just the fibres of the cotangentbundle.The moduli space dg is stratified by the degree of the destabilizing subbundle

L, so each stratum (for degree > 0) is a bundle of (3g 3)-dimensional spacesover equivalence classes of extensions of line bundles. Since L is special there isat most a (g 1)-dimensional choice for L and since dim Hi(M; L2 (R) A2V*) isless than g, at most a (g- 2)-dimensional family of extensions for a fixed L,giving the maximum dimension of stratum as (3g- 3) + (g- 1) + (g- 2)=5g 6, which is of codimension g in ’.

Using these facts we prove the following:

THEOREM 6.2. Let ./ff be the moduli space of stable vector bundles with fixeddeterminant, rank 2 and odd degree over a compact Riemann surface of genusg > 1. If T is the tangent bundle of uV" and SkT the k-th symmetric power of T,then

H(4; SkT) 0 if k is odd

H(/’; S2eT) SeHI(M; TM).

Proof. Let s be a holomorphic section of SkT on ,z,. By contraction itdefines a holomorphic function on the cotangent bundle T’W, which is a

polynomial of degree k on each fibre.

STABLE BUNDI,ES AND INTEGRABLE SYSTEMS 111

Since ’, the moduli space of stable pairs, is obtained by adjoining an analyticset of codimension g > 2 to T*./ff’, then g extends by Hartog’s theorem to aholomorphic function on

Consider the map det: t’---> H(M; K). The generic fibre is compact andconnected--a Prym variety--and hence, since det is proper and H(M; K2)smooth, every fibre is compact and connected. Thus the function g is constant oneach fibre and so

f det

for some holomorphic function f: H(M; K) C.Now the map induces a C* action on and since s was homoge-

neous of degree k, we have

(h. m) Xk(m) form’.

But det(h, rn) ,2det(rn), and so f satisfies

f(X2x) f(x). (6.3)Now a holomorphic function on a vector space W which is homogeneous ofdegree ’ defines a holomorphic section of (O(’) on P(W) and hence a poly-nomial of degree . Since (.0(1) generates H(P(W), (.0") there are no homoge-neous functions of noninteger degree, so if k is odd, f 0. If k 2’, then f is apolynomial of degree ’ on W H(M; K2) and hence an element of the vectorspace SW*. Thus, since H(M; K2)* H(M; g-1) by Serre duality, we have

0 if M is odd and f odet for f SeH(M; K-) for rn 2/’ and hencethe theorem.

Remark 6.4. (1) Theorem 6.2 gives H(vff’; S2T) H(M; K-l). This showsthat the 3g- 3 functions which make T*vr into an algebraically completelyintegrable Hamiltonian system are globally determined by v4/’ itself: they are theonly holomorphic functions on T*vf’ which are homogeneous of degree 2 in thefibres. Thus if is given in some way other than as the moduli space of stablevector bundles, it is still possible to describe T* as an integrable system simplyby finding the global sections of S2T. An example would be the case of g 2where .,4/" is the intersection of two quadrics in p5 [8].

(2) If we consider the ring of all holomorphic functions on T*vf’ which arepolynomial in the fibres, then Theorem 6.2 shows that this ring is generated bythe 3g 3 functions of degree 2. Since they Poisson-commute by 4.5, the wholePoisson algebra of polynomial functions is commutative.

In a similar manner we prove now the following:

THEOREM 6.5. Let be as in 6.2, and suppose that g > 2. Then(i) H(o/ff; SkT) 0 if k is even,(ii) HX(; T) =- H(M; TI),(iii) H(ohr; $2+1T) =- HX(o/ff’; T) (R) H(/’; SeT).

112 NIGEL HITCHIN

Proof. We begin as in Theorem 6.2. Each cohomology class a H1(’; SkT)defines by contraction a class t HI(T*.A/’; 0) which is homogeneous of degreek under the C*-action.We may use the cohomological version of Hartog’s theorem and extend this

over an analytic set of codimension > 3, provided by Proposition 6.1 to a classa H(’; O) [10]. This class is determined by its restriction to the complementof an analytic set of codimension > 2 and so is determined by its restriction toT*vV" or even to the complement of the zero section in T*.A/’. Thus anyfl H(’; ), homogeneous of degree k is determined by its restriction to thecomplement of the zero section in T*vV’ and hence by the class inH(P(T*.Af); O(k)) which it defines, where (9(1) is the tautological hyperplanebundle along the fibres.

Since Hi(p"; 0(k)) 0 for > 0, the Leray spectral sequence identifies thisgroup with HI(.Af; SkT). We may therefore simply consider classes flnl(./g’; 0) homogeneous of degree k.We consider the map det: ’ --) H(M; K2) V which defines the integrable

system and the first direct image sheaf Rldet.(.0. Since det maps to a vector spaceV, all the higher cohomology groups of direct image sheaves vanish, and theLeray spectral sequence gives

H(/t’; (9) _-- H( V; Rldet,(.0 ). (6.6)

We shall prove that, outside an analytic set of codimension 2 in V, the sheafRldet,(.0 is isomorphic to the sheaf of holomorphic functions with values inH(M; K-). Since elements of HI(/’; 0) are determined by their restriction tocomplements of codimension 2 sets this will, using Hartog’s theorem, providefrom (6.6) an isomorphism

H(,/t’; ) -_- H(V; (0) (R) HX(M; K-l). (6.7)

To obtain this isomorphism we use the Kihler form to define a class w inHi(.///’; T*). Contraction with a holomorphic vector field gives a homomorphismfrom H(/’; T) to Hl(/t’; (.0). We apply this to the vector space of Hamiltonianvector fields arising from linear functions on V: the basic fields of the integrablesystem. This defines a linear map from V* Hi(M; K-1) to HI(/’; (9).

Let q H(M; K2) have simple zeros, then the fibre .//t’q of det over q isisomorphic to a Prym variety, and the vector fields generated by V* are tangentto t’q and span the space H(’q; T) of all holomorphic vector fields on thisabelian variety. Since the Kihler form restricts on ’q to a Kihler form itdefines an isomorphism"

n(,/fq T) - Hl(,v’/’q; 0) c3g-3. (6.8)


This gives an isomorphism of Rdet,(9 with the functions with values in V* onthe open set of q H(M; K2) with simple zeros. This is the complement of acodimension 1 analytic set.We need to consider more degenerate quadratic differentials. If g > 3, then the

q H(M; K2) which have at most one double zero is the complement of acodimension 2 set, so we must consider the singular fibre .//q for q with a doublezero at x M and simple zeros elsewhere. We consider therefore a stable pair(V, ) on M such that det has a double zero at x.The curve S X Kt defined as in 5 now has an ordinary double point

over x, but we obtain an eigenspace line bundle L over its desingularization S,which has genus (4g- 3)- 1 (4g- 4). By arguments similar to those in(5.10), L --- U (R) L0 for some fixed L0 where U lies in the (3g 4)-dimensionalPrym variety of S. If (x) is nonzero, it is nilpotent and has a unique eigenspacein Vx. This provides an isomorphism between L and L,, on S (and hence infact a point on the Prym variety of the singular curve S).

Since o*L o*U (R) Lo U- (R) Lo --- L* (R) L2o, an isomorphism between Land Lo is a nonzero section of (L (R) LZ)x But the Poincar6 bundle overM Jac(M) is trivial on { x} Jac(M), so if (x) is nonzero, it corresponds toan open set of ’q isomorphic to C* Prym(S).The remaining case is if (x) vanishes. This must occur since the fibre of det is

compact. In this case we can put sI, where s is a section of the line bundle(9(x) vanishing at x and H(M;adP (R) K(-x)). A repetition of 5.10shows that such a stable pair is determined by a point in the Prym varietyPrym(S), since the nonsingular curve S is the double cover of M branched overthe simple zeros of q det . Thus each C* is compactified by a single pointand we find

,/gq E X Prym(S)

where E is a rational curve with an ordinary double point" a degenerate ellipticcurve.NOW HX(’q; (_,0) HX(E; (,0) HX(Prym(ff); 0) c3g-3 SO Rldet,0 is lo-

cally free on the open set of q with at most one double zero. Also, the productwith w HI(’; T*) maps the vector fields tangent to Prym(ff) isomorphicallyto HX(Prym(S); (9) as before, and by direct calculation certainly maps the uniqueholomorphic vector field on the singular curve E to a nonzero element inHi(E; (9) _---- C.Thus the isomorphism of Rldet,O with the sheaf of functions extends to the

complement of a codimension 2 set, and we obtain the isomorphism (6.7).It remains to detect the classes in HI(/’; (9) which are homogeneous of degree

k. Now the canonical symplectic form o on T*uV" is of degree 1 (see (5.11)).Since a function f generates a Hamiltonian vector field X by i(X)o dr, thenthe functions f V* HX(M; K-) which are homogeneous of degree 2 gener-ate vector fields homogeneous of degree 1. It follows that the subspace of

114 NIGEL HITCHIN

nl(,/g; (9) of degree k is isomorphic in (6.7) with the tensor product ofHi(M; K-1) with holomorphic functions on V of degree 1/2(k 1). The theoremthen follows, using for part (iii) the result of Theorem 6.2.

Remarks. (1) The natural isomorphism HI(V’; T) H(M; Tt) is almostcertainly the Kodaira-Spencer map which associates to a deformation of thecomplex structure of the Riemann surface M, a deformation of the moduli spaceob/" (see [7]).

(2) The third part of the theorem says that k nx(dff; SkT) is a free moduleover H(,A/’; SgT) generated by H1(4/’; T).

REFERENCES

1. M. F. ATIYAH AND R. BOTT, A Lefschetz fixed point formula for elliptic complexes: II.Applications, Ann. of Math. 88 (1968), 451-491.

2. The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A308(1982), 523-615.

3. C. CnEVA.LEV, Proc. International Congress of Mathematicians, Cambridge, Mass. 1950, II,(1952) 21-24.

4. N.J. HrrcmN, Monopoles and geodesics, Commun. Math. Phys. 83 (1982), 579-602.5. The self-duality equations on a Riemann surface, to appear.6. B. KOSTANT, The principal three-dimensional subgroup and the Betti numbers of a complex simple

Lie group, Amer. J. Math. 81 (1959), 973-1032.7. M.S. NARASIMHAN AND S. RAMANAN, Deformations of the moduli space of vector bundles over an

algebraic curve, Ann. of Math. 101 (1975), 39-47.8. P.E. NEWSTEAD, Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology

(1968), 205-215.9. S. RAMANAN, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973),

69-84.10. G. SCHEJA, Riemannsche Hebbarkeitsatze fiir Cohomologieklassen, Math. Ann. 144 (1961),

345-360.

ST. CATHEIINE’S COLLEGE, OXFORD OXl 3UJ, ENGLAND.

=1 - mathematicsmath.mit.edu/.../library/hitchin_stable_bundles_integrable_systems.pdf · the...

Documents