Proc. Indian Acad. Sci. (Math. Sci.), Vol. 103, No. I, April 1993, pp. 41-71. Printed in India.

Determinants of parabolic bundles on Riemann surfaces*

I BISWAS and N RAGHAVENDRA* School of Mathematics, Tata Institute of Fundamental Research, Bombay 400005, India (E-mail: [email protected]) * School of Mathematics, SPIC Science Foundation, 92, G.N. Chetty Road, Madras 600 017, India (E-mail: [email protected])

MS received 23 November 1992

Abstract. Let X be a compact Riemann surface and M~(X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural K/ihler metric on MP,(X).We obtain a natural metrized holomorphic line bundle on M~(X) whose Chern form equals mr times the K/ihler form, where m is the common denominator of the weights and r the rank.

Keywords. Parabolic bundles; Riemann surfaces; K~hler metric; Chern forms; n-bundles; Hermitian line bundles; parabolic determinant bundles.

1. Introduction

In a foundational paper 118] in 1985, Quillen studied determinants of R-operators over a compact Riemann surface. Suppose X is a compact Riemann surface whose genus g is at least 2. If E is a C ~ vector bundle of rank r and degree d on X, the space d of all holomorphic structures (or ~-operators) in E is a complex affine space. For each D6~r the object Z~'D = 2(ker D)* | 2(coker D), where 2 denotes the top exterior power, is a well-defined line and, as D varies in ~r these lines form a holomorphic line bundle Le on ~r which is called the determinant line bundle. Using the theory of analytic torsion, Quillen defined a smooth Hermitian metric in the determinant bundle and proved that its curvature equals a natural K/ihler form on ~r The theorem of Quillen has an immediate application to moduli spaces. The collection ~r of stable holomorphic structures is an open subset of ~r and the complex gauge group ~ = Aut(E)/C* acts freely on ~r The orbit space of this action is the moduli space M~ of stable bundles of rank r and degree d on X. When d = r(g - 1), the action of ~f~ on ~r lifts to an action on ~ and, hence, the determinant descends to a holomorphic line bundle L over Ms. The Quillen metric in L~ induces a metric in the bundle L on Ms. It can be shown, as a corollary to Quillen's theorem, that the curvature of this metric in L is a naturally defined K/ihler form on the moduli space Ms. In particular, the metric in L is positive. In the above argument the

* After this paper was written we came across an announcement of similar results by Witten [22]. 41 42 I Biswas and N Raghavendra restriction d = r(g- 1) is purely technical and can be bypassed. Thus a modification of the above construction yields a positive line L on M s for all arbitrary r and d. When r and d are coprime, Ms is known to be compact, so Kodaira's embedding theorem implies that the positive line bundle L is ample on Ms. There is another interesting interpretation of the above corollary to Quillen's theorem. A well known result of Narasimhan and Seshadri [16] shows that the moduli space is isomorphic to a certain variety R of irreducible projective unitary representations of the fundamental group n 1(X). For each p: n 1(X)--* PU(r), in R let adp:nl(X)~ Aut(g) denote the composition with p of the adjoint representation of PU(r) on fl = Lie PU(r) and let W(adp) denote the local system of real vector spaces on X defined by ad p. Then the real tangent space to R at p is isomorphic to the de Rham group H 1(X, W(adp)). Define

~'l :H~(X, W(adp)) x H~(X, W(adp))--*R by P /* np(~, fl) = jxtr(~ A

As p varies, the family Qp gives a 2-form ~ on R. This 2-form ~ is symplectic structure on R. One can ask wether fl represents an integral cohomology class in H2(R, R). Yes is the answer to this question: under the isomorphism R ~- Ms, ~ corresponds to the Kiihler form on Ms which, in turn, equals the Chern form of the metrized line bundle L on Ms. Since all Chern classes are integral cohomology classes, Quillen's theorem implies that the symplectic form ~ on R is integral. In this paper, we extend these results to the case of stable parabolic bundles on Riemann surfaces. Parabolic bundles are vector bundles over marked Riemann surfaces with weighted flags in the fibres at the marked points. There are two ways to study parabolic bundles: one way is to approach parabolic bundles directly; an alternative approach is through n-bundles. A ~-bundle is an equivariant vector bundle over a Riemann surface provided with an action of a finite group ~. The above results easily go through for n-bundles and define a metrized n-determinant. The more difficult part is to interpret this ~-determinant in terms of the parabolic structure, Reserving the relevant definitions till later, we just state the main results here.

Theorem I.I. Let Y be a compact Riemann surface and n a finite group acting effectively on Y. Denote by M~( Y) the moduli space of stable n-bundles of rank r and fixed local type on Y. Then there exists on M~( Y) a metrized line bundle L ~ whose Chern form equals r times the natural Kiihler form O r. Using a formalism of Deligne, Beilinson and Manin [3, 5] we interpret the above theorem in terms of parabolic structures as follows.

Theorem 1.2. Let X be a compact Riemann surface and J a finite subset of X. Denote by M~(X) the moduli space of stable parabolic bundles of rank r and degree d with parabolic structure concentrated on J and having fixed rational weights, fixed multiplicities and fixed top exterior power. Then there exists on M~(X) a metrized line bundle L p whose Chern form equals mr times the natural Kfihler form, where m is the common denominator of all the weights. Parabolic determinants on Riemann surfaces 43

Now we sketch the outline of the paper. In w2 we recall the basic notions about parabolic bundles and n-bundles, and display the relation between them. The determinants of n-bundles form the topic of w4 and here Theorem 1.! is proved. The n-determinant is expressed in terms of parabolic structures in w The proof of Theorem 1.2 is contained in w5 and, finally, in w6 we give an application of our results.

2. Parabolic bundles and n-bundles

In this section we define parabolic bundles and n-bundles and describe how the two notions are related to each other. The two basic references for this section are Seshadri 120] and, Mehta and Seshadri [13]. Let X be a compact Riemann surface and E a holomorphic vector bundle over X.

DEFINITION 2.1 A quasi-parabolic structure on E at a point x~X is a strictly decreasing flag

Ex= Ft E:,~_F2Ex ...... ~_FkEx~_Fk+ I E=O x of linear sub-spaces in E~. We define

rj = dim FJEx - dim F j+ 1E~.

The integer k is called the length of the flag and the sequence (rl .... , rk) is called the type of the flag. A parabolic structure in E at x is, by definition, a quasi-parabolic structure at x as above, together with a sequence of real numbers 0 ~< 0t t <... < 0~k < 1. The ~tj are called the weights and we set

k

j=t

DEFINITION 2.2 Fix a finite set I of points in X. We say that E is a parabolic bundle with parabolic structure on I if we are given a parabolic structure in E at each point xeI. The parabolic degree, pdeg(E), is defined by

pdeg(E) = deg(E) + ~ d~(E),

where deg(E) denotes the topological degree of E, and we put

p#(E) -- pdeg(E)/rank(E).

The points in I are called the parabolic points.

DEFINITION 2.3 Let E and F be parabolic bundles of the same type over X (i.e. E and F have the same parabolic points, the same types of flags, same weights etc.). A morphism from 44 I Biswas and N Raohavendra

E to F is a homomorphism of vector bundles f:E ~ F, which preserves the flags at the parabolic points.

DEFINITION 2.4 Let E be parabolic bundle with parabolic locus I. A parabolic subbundle of E is a parabolic bundle F with parabolic locus I, such that: (a) F is a subbundle of E; and (b) for each point xel and for every j~{1 ..... kx}, we have aj(F)= oti(E), where i is the largest integer such that FJF~ ~_ FiE~ and k~ is the length of the flag in Fx.

DEFINITION 2.5 A parabolic bundle E is said to be semistable if for every proper subbundle F of E, we have pI~(F) ~ p#(E). (2.1) In case strict inequality holds in (2.1), for all F, we say that E is stable. In [13], Mehta and Seshadri construct the moduli space Mr(X) of semistable parabolic bundles over X with fixed rank, degree and parabolic type, and show that Mr(X) is a projective variety. They also prove that the set M~(X) of stable parabolic bundles is an open smooth subset of Mr(X). We now proceed to define n-bundles. Consider a compact Riemann surface Y and let n be a finite group acting holomorphically and effectively on the left of Y. Then X = n\Y has a natural structure of a Riemann surface such that the canonical projection p:Y--,X is a ramified covering. The ramification points of p in Y are exactly those points y where the isotropy subgroup Ity of y in n is non-trivial. Note that ny is a cyclic subgroup of n for every y~ Y.

DEFINITION 2.6 A n-bundle over Y is a holomorphic vector bundle E over Y such that: (a) the bundle projection p: E--* Y is n-equivariant; (b) if yE Y and ),~n, the map Ey-. Ery given by E~.~, is a linear isomorphism.

DEFINITION 2.7 A n-subbundle of a n-bundle E is a subbundle F of E which is invariant under the action of n on E.

Notation 2.8. If E is a vector bundle over Y (which need not be a n-bundle), we denote degree (V) ~(v)= rank (V) "

DEFINITION 2.9 A n-bundle E is said tobe semistable if for every proper subbundle F of E (note that we do not assume that F is a n-subbundle of E), we have

/~(F) ~< #(E). (2.2) Parabolic determinants on Riemann surfaces 45

We say that a n-bundle E is stable if for every proper n-subbundle F of E, we have a strict inequality

#(F)

Remark 2.10 Given a n-bundle E of rank r over Y and a point yeY, there exists a ny-invariant neighbourhood U of y in Y such that Elu is defined by a representation p:ny ~ GL(r), and p is unique up to ~,quivalence (cf. Seshadri [20], Proposition 2 of w2). The equivalence class of the representation p is called the local type of E at y and is completely determined by: (a) an integer k such that 1 ~< k ~< r; (b) a sequence of integers 0 ~< It ~< 12 ~<... ~< l~ ~< n - 1, where n = In~l; and, (c) positive integers rl, r2 ..... rk such that rt + .... + rk = r. In fact, given the above data, we can define p as the block matrix

exp(2nioq)'I1 . 0 1 ~ p(O = [ .. 0 exp(2ni=l,)'Ik where ~ is a generator ofny, i 2 = -- 1, 0tj - I/n and lj is the identity matrix of order r i. In [20] Seshadri constructs the moduli space M'(Y) of semistable n-bundles of fixed rank, degree and local type over Y and shows that M~(Y) is a projective variety. It is proved by him that the set M~(Y) of stable n-bundles is a smooth open subset of M~(Y). We now describe the connection between n-bundles on Y and parabolic bundles on X = n\Y. Let E be a n-bundle over Y. Consider the invariant direct image p,t~(E) of d~(E). This is the d~x-module defined on each open set U _ X by

p,r = {n-invariant sections of E on p-l(U)}.

Since p is a proper map, p, tP(E) is a coherent d~x-module and is clearly torsion-free. Since dim(X) = 1, it follows that p, d~(E) is a locally free Ox-module of the game rank as E, and we denote the corresponding vector bundle on X by E ~. Denote by J the ramification set ofp in X. For each xeJ choose a point y(x)ep- 1(x) and let n~ = t ny{~l. Let E be a n-bundle of rank r over Y. Suppose that the local type (cf. Remark 2.10) of Y is given by the integers 0 <<, l~. 1 <~ l~, 2 <~... <<, lx.kx ~< nx -- 1 and (r~. 1..... r k~). We then have the following result for which we refer to Bhosle [4] and Mehta-Seshadri [13].

PROPOSITION 2.11 There exists a natural parabolic structure in the bundle E ~ on X with parabolic set I = p(J). For each x~I, the type of the flag in E: is (rxA ..... rx,kx) and the weights at x are (*tx.1 ..... Ct,k~), where ~,j = lx,/n~. Conversely, given a parabolic bundle F on X of the above parabolic type, there exists on Y a n-bundle E, unique up to isomorphism such that E ~ ~- F as parabolic bundles. The correspondence E~--~E ~ takes stable n-bundles on Y to stable parabolic bundles on X and semistable n-bundles to semistable parabolic bundles. The resulting bijection between the semistable moduli spaces M~( Y) and MP(X) is an isomorphism of varieties. 46 I Biswas and N Raohavendra

3. The determinant bundle on M~'~(Y)

Consider a compact Riemann surface Y and let it be a finite group acting holomorphi- cally and effectively on Y. Let ~ be a Hermitian metric on Y which is invariant under the action of n. Denote by A p'q the space of all C ~ (p, q)-forms on Y and set A'= Y.p+q=,Ap'q. Let w~Al"Ibe the fundamental (1, 1)-form of the metric # and assume that g is normalized so that Srco = 1. Fix a C ~ x-vector bundle E of rank r and degree d over Y. Let h be a it-invariant Hermitian metric in E. Denote by AP'q(E) the space of C | (p, q)-forms on Y with values in E and let A'(E)= Zp+q=,AP'q(E). There is a natural finear action of on each of the spaces A p,~ and AP'~(E). We denote the corresponding subspaces of n-invariant forms by (AP'q)~ and (AP'q(E))~.

DEFINITION 3.1 A holomorphic structure in E is a C-linear map D:A~ A~ which satisfies the Leibnitz identity

D(fs) =riD(s) + gf | s for all f eA ~ and s~A~ If D~ and D2 are holomorphic structures, the difference

= D1 - D2:A~176 is an A~ map, i.e., ~ ~A ~ ~(End E). Conversely, if D~ is a holomorphic structure and if aCA~ E), the sum D2 = Dx + a is again a holomorphic structure. Thus the set ~r of all holomorphic structures in E is an altine space modelled after the C-vector space A ~ (End E). Let ~ denote the group of C | automorphisms of the bundle E. Given DzM and g~, we define a new holomorphic structure D'gz~ by the prescription D.g=f-loDog.

Any holomorphic structure D in E induces a holomorphic structure, also denoted D, in End E as follows

D:A~ E)-*A~ E) f~-.[D,f] = Dof -foD.

Having made this definition, we can write D.g - 0-1 o(Dg) + D. Anyway, we now have a map • ~--,~

(D, g)~D-g.

This map is clearly an affine linear right action of ~r on ~. The multiplicative subgroup C* is embedded in ff as the normal subgroup consisting of constant multiples of the identity automorphism of E. This subgroup C* acts trivially on ~. Hence the above action induces an action of the group Parabolic determinants on Riemann surfaces 47

~f~ = q/C* on ~r We now state a well-known integrability theorem, whose proof can be found in Donaldson [6].

PROPOSITION 3.2 Given a holomorphic structure D~/, there exists a unique structure of a holomorphic vector bundle E o in E such that the ~-operator of ED equals D. Now we have the trivial

PROPOSITION 3.3 Let DI and D2 be holomorphic structures in E. Then Eo, and Eo2 are isomorphic as holomorphic vector bundles if and only if D 1 and D2 lie in the same (~-orbit of ~.

Proof. The bundles Eol and Ea2 are isomorphic as holomorphic vector bundles if and only if there is a C ~~ isomorphism g:Em~Eo2 which commutes with the ~--operators of the bundles, i.e., such that goD~ = D2og. This means precisely that g~f~ and D~ = D2.0.

DEFINITION 3.4 A holomorphic structure D in E is called a holomorphic n-structure if the map D: A~(E) ~ A~ 1(E) is n-equivariant. The set ~r of all holomorphic n-structures in E is an affine subspace of ~r in bijection with the subspace A ~ (End E) ~ of A ~ (End E). Let ~ denote the subgroup of f~ consisting of n-automorphisms of E. It is easily seen that ~r is invariant under the action of ~ on ~r Further C* is contained in f#~, hence we get an action of ~= f~c" on ~r

Remark 3.5. If V and W are C ~ n-bundles on Y of the same rank, degree and local type (cf. Remark 2.10), then V and W are isomorphic as C | n-bundles. Thus the set zr ~ can be thought of as the set of isomorphism classes of holomorphic n-bundles on Y of rank r, degree d and of the local type as E.

DEFINITION 3.6 We say that a holomorphic n-structure D is stable if the corresponding holomorphic n-bundles Eo on Y is stable. Let ~r denote the subset of ~r consisting of stable holomorphic n-structures. Then it is obvious that ~r is a f~-invariant subset. We know from Seshadri !20] that the isotropy subgroup of fg~ at any De~r is exactly C*. Thus the group ~f9 ~ acts freely on .~r Let M~ (Y)- ~r We state without proof.

PROPOSITION 3.7 The canonical projection q~:s/'~~ M~( Y) is a holomorphic principal ~(~'-bundle.

Remark 3.8. We have not defined any topology or complex structure on ~r and M'~(Y), so Proposition 3.7, as it stands, is meaningless. But using Sobolev spaces it is possible to give a natural complex manifold structure to M~(Y) and to make sense of Proposition 3.7. Since there are many accounts now of this kind of argument using 48 I Biswas and N Raohavendra

Sobolev spaces, we omit all the details and make a blanket reference to Narasimhan and Ramadas [23], Donaldson [6], Liibke and Okonek [12], Kim [10] and, Atiyah and Bott [1].

PROPOSITION 3.9 I rf ~o.sg. It ~ M sIt ( Y) denotes the quotient projection and if D ~ s4~, then the tanoent space to M~(Y) at the point q~(D) is naturally isomorphic to H x (Y, ~, End ED).

Remark 3.10. If V is a ~-sheaf on Y, define H~ V) to be the set of all ~-invariant sections of V on Y. The correspondence

V~ H~ Y, ~, V) is then a left exact additive covariant functor from the abelian category of n-sheaves on Y to the category of abelian groups. For i/> 1, the cohomology group Hi(Y, ~, V) is, by definition, the ith right derived functor of the above functor. It is known (el Seshadri [20]) that Hi(Y,~, V) is naturally isomorphic to Hi(X,p,(~(V)), where X = n\ Y and p: Y--, X is the canonical projection.

Proof of Proposition 3.9. Fix O~d~ and define

s gv--, D.g.

The Lie algebra of ~(r is ~@" = A~ E)'. The differential of a at the identity element of ~f#" is given by

D": A~(End E)" --, A~ 1(End E)" f~--*Df. Thus the tangent space along the fibre of q~ at D is precisely Im(D'). The Hodge theory implies that

Ao. 1(End E) ~ = Im (D ") ~ ker(D*"), where D *'' is the Hermitian adjoint ofD ~. Hence the tangent space to M~(Y) at ~o(D) is given by

Tan. space to ~'at D = A ~ (End E)" ker(D*.~). Vertical tan. space Im(D ")

As usual, let X = it\ Yand p: Y--*X the canonical projection. Let 0tP'q(End E) denote the sheaf of C~176 on Y with values in End E. Consider the sequence

D ~ 0 ~ p~ (End Ea) --* p, ~o (End E) ---* p, ~t~ (End E) --* 0.

By averaging over It, this sequence is easily seen to be a fine resolution of p, t~(End Ea) Parabolic determinants on Riemann surfaces 49 on X. Hence the sheaf theory implies that

HI (X, p. t~ (End E a)) = coker (D ~: A~(End E)" --} A~ 1(End E)") = kerD *.~ = T,w,(M~(Y)).

On the other hand, we know (cf. Remark 3.10) that

H x(X, p~ ~ (End E D)) ~ H t ( y, ~, End Eo ).

Thus we conclude that

T~m(M *~y ()) = H t ( Y, ~t, End Ea).

We recall now the definition of the determinant bundle on M as in Quillen [18]. For each D~M, define a complex line -~a by

~a = A(ker D)* | 2(coker D), where 2 denotes the top exterior power. Since each D is an elliptic operator, the kernel and cokernel of D are finite dimensional, so -~a is well-defined. We thus have a family

-~= U -~a (3.1) z~d of lines parametrized by z/. It is a fact, proved by Quillen [18], that .~ is actually a holomorphic line bundle over ~t. There is a right (~-action on .~ given by

(D,(s~ A ... A s~)| A... A t,)) x O ~--~(D'o,(O-I(s *) A ... A g-l(s~))| A ... A g(tl)) where (s* ..... s~) is a basis of (ker D)* and (tl ..... h) is a basis of coker D. If g = 2.id~ where A~C*, then D'g = D and g acts on -~a as

r =(s~ A ... A s~')@(tl A ... A t,) ~((;t- Is *) ^... A (Jr- ~s*))| A... A 0,t~)) = ,t~-k. r = jt-r~}.r where x(E)= index(D) which by Riemann-Roch, equals d + r(1- O), d being the degree of E, r the rank of E and g the genus of Y. We would like to modify the line bundle ~ on M in order to obtain a line bundle .~ on which the action of C* is trivial. To do this, fix a point yeY such that the isotropy subgroup 7tr of y in n is trivial. Define

.~ = ~'|162 x Er)X{E}, (3.2) where A~ denotes the rth tensor power of A~ and ~ x E r is the trivial vector bundle 50 I Biswas and N Raghavendra on d with fibre E~. The computation of the previous paragraph shows that C* acts trivially on ~a and, hence, ~ becomes a ~f~-line bundle on ~r We denote the restriction of ~ to ~' by the symbol ~ itself. Since the natural projection C--, M (Y) is a holomorphic principal ~f~-bundle, the ~(~-line bundle -~ on ~/~ descends to a holomorphic line bundle L ~ over M~(Y).

DEFINITION 3.11 The line bundle L" is called the determinant bundle on the moduli space M~(Y).

Remark 3.12. If z(E)= 0, then the above argument shows that C* acts trivially on the initially unmodified line bundle Ae (cf. equation (3.1)). Thus La itself descends to M~(Y) when the index of E is zero. In the earlier part of this section we have constructed certain holomorphic objects. We would now like to study their metric properties. To do this we need the space of unitary connections and the unitary gauge group. Recall that we have fixed a rc-invariant Hermitian metric h in the Coo vector bundle E.

DEFINITION 3.13 A unitary connection in E is a connection

V:A ~ (E) --, A I (E) such that for all s, t~A~ we have d(h(s, t) ) = h(Vs, t) + h(s, Vt). Let End(E, h) denote the C oo R-vector bundle of skew-Hermitian endomorphisms of (E, h). It is easily checked that the set ~ of all unitary connections in E is an affine space modelled after the R-vector space A t (End (E, h)). If V is a unitary connection in E, the (0, 1)-part Dv of V is a holomorphic structure in E. Let ~ denote the subgroup of ~ consisting of unitary automorphisms of (E, h). Given V~f and 0~d, we define V'g~ by V.g=g-loVog.

This gives a right action of q/on ~. There is a natural embedding 2~-~2"id~ of U(1) in ~//. This subgroup U(1) of ql acts trivially on g and hence we get an action of ~ql = ~1/v(1) on qr The proof of the following simple proposition can be found in Kobayashi [11].

PROPOSITION 3.14 The (0, 1)-part map

V~-*Dv --- (0, 1)-part of V is a ~-equivariant affine R-linear isomorphism. Parabolic determinants on Riemann surfaces 51

DEFINITION 3.15 A unitary connection V in E is called a unitary n-connection if the map V:A ~ ~ A ~(E) is n-equivariant. The set c~ of all unitary n-connections in E is an affine subspace of ~ isomorphic to the real subspace Al(End (E,h)) '~ of A~(End (E,h)). Let q/~ denote the subgroup of q/ consisting of unitary rr-automorphisms of (E, h). It is easily seen that c~ is invariant under the action of ~ on c~. Further U(1) is contained in q/~, hence we get an action of ~q/~ = ~n/U(1) on c~.

DEFINITION 3.16 A unitary n-connection V in E is said to be reducible if there exists proper C ~ n-subbundles E~ and E 2 of E together with unitary n-connections V~ and V2 such that E = E~ ~) E 2 and V = V1 ~ V2. IfV is not reducible we say that V is irreducible.

DEFINITION 3.17 A unitary n-connection V is said to be Einstein-Hermitian if the curvature R(V) of V satisfies the identity 2n.~(E) R(V) = "~| r i'lnl where ~ is the K~ihler form on (Y,#), Inl is the order of the group n and

degree(E) ~(E)- rank(E)

DEFINITION 3.18 A holomorphic n-structure Deal ~ is said to be Einstein-Hermitian if the corresponding connection is so. A holomorphic n-structure is called indecomposable if there are no proper holomorphic n-subbundles E 1 and E 2 of E o such that E D = E 1 ~ E2. The following theorem is a special case of an important theorem of Simpson [21]. This also follows from Seshadri [20].

Theorem 3.19. An indecomposable holomorphic n-structure De~" is stable if and only if there exists an Einstein-Hermitian holomorphic,n-structure in the ~f~-orbit of D. Further if D 1 and D E are two Einstein-Hermitian holomorphic n-structures in the ~(~-orbit of D, then DI and D 2 lie in the same ~ql~-orbit. Let ~ denote the subset of ~ consisting of irreducible Einstein-Hermitian n-connections. Then it is obvious that c~ is a ~/~-invariant subset. It can be easily checked (cf. Kobayashi [11]) that the stabilizer in q/~ at any Ve~ is exactly U(1).

Thus the group ~'~ acts freely on c~S" Let N $~ = c~/~q/~. Then we have

PROPOSITION 3.20 The canonical projection ~k "~ --* N '~ is a C ~~ principal ~ll~-bundle. S S 52 I Biswas and N Raghavendra

Remark 3.21. The comments made in Remark 3.8 apply to the above Proposition too. We will assume Proposition 3.20 as proved and quote the same references as in Remark 3.8. Let V be an Einstein-Hermitian holomorphic n-connection in E. Then the curvature R(V): A~ h))~ A2(End(E, h)) of the induced connection V in End(E, h)is given by

R(V)(f) = [R(V),f] = [ct | id~,f], oteA I"1 = ~ | [ide,f] = O.

Thus V is a fiat n-connection in the real vector bundle End(E, h). We denote V also by V. The vector bundle End(E, h) together with the flat connection V = V becomes a local system of real vector spaces on Y which we shall denote by End By. In Seshadri [19], the de Rham and Hodge theories are developed for C ~ forms with values in local systems. We denote the de Rham cohomology groups of Y with values in End 8 v by H~( Y, End o~v). These cohomology groups are finite dimensional vector spaces. The n-invariants in Hia( Y, End d'v) are denoted by H~( Y, n, End By). We now compute the tangent space of the space N~.

Lemma 3.22: The tangent space to c~ at a point VeC~'~ equals the group of cocycles

Z~( Y, n, End By) = Ker {V~: A* (End(E, h))~ --* A 2(End(E, h)) ~}

Proof. Let ~ denote the set of irreducible n-connections, c~o is an open set in cr Let V be the R-subspace of A2(End(E,h)) given by

V = {or| idE/i~eA~'* }, where A~'1 denotes the space of real (1, 1)-forms on Y. Let

An(End(E, h))~ B: A2 (End(E, h)) ~ V denote the natural projection and define

0:~ ~ A2 (End(E' h))" V V o~--~B(R(Vo) ).

We now have the Hodge decomposition

A 2(End(E, h))" = Im(V~) ~ ker(V~'~).

We claim that ker(V*'~)=V. To see this, suppose V*'~(~)=0 for some ~e A 2(End(E, h))~. Then Vo o. (~t)= 0. But the irreducibility of Vo implies that the parallel sections of Vo in End(E, h) are exactly the scalars. Therefore ,(~t)= 2.id~ for some ).eC. Hence .2(ct)= 2.t9 | id~ i.e., ~t = -2.09 | idn. Since ct is skew-Hermitian, i2eR. This proves that ker(V*'~)= V. Therefore, the Hodge decomposition above Parabolic determinants on Riemann surfaces 53 implies that

A2(End (E, h)) ",, Im(V~). V

We conclude from this that 0 is a submersion on c~o.It Clearly ~ It- 0- ~(0). Thus the tangent space to c~ at any point V is given by the kernel of dO at V, i.e., equals ker(W). This is precisely Z~ ( Y, n, End By). Recall that r N~ is the canonical quotient projection.

PROPOSITION 3.23 The tangent space to N~ at a point ~,(V), Vec~, is given by H)(Y,n, EndSv).

Proof. As in the proof of Proposition 3.9, we can check that the vertical tangent space at Ve~ is

B] (Y, n, End r = {V0t/0teA~(End(E, h))it }.

Thus the tangent space to N~ at if(V) is

Tan. space to c~ at V Z1 (Y, n, End d~v) Vertical tan. sp. at V B] (Y, n, End r = H~(Y,n, End~v).

We shall identify the real tangent space of M~(Y) with its holomorphic tangent space. Theorem 3.19 implies that if VeC~, then the (0, 1)-part D v of V belongs to ~r and the map c~r V~-*Dv induces a bijection I:N'~--.M'~(Y) between the quotients. Clearly I is C% The differential of I at ~k(V), Ve~, is given by

dI:H) ( Y, n, End Ev) ~ H 1(Y, n, End Eov) = ker(D*'it),

al--~ ~" where a is the harmonic representative of the class a and ~" is the (0, 1)-part of ~t. Clearly dI is an isomorphism. Thus I:N~--*M~(Y) is a local diffeomorphism, hence a diffeomorphism. We now define a Hermitian metric on M~(Y). If q~:~C~M[(Y) denotes the canonical projection and De ~its, define a Hermitian inner product

(.,.)~,(m:T~(m(Ms())Ity x T(m(M~(Y))~C II II ker(D*'it) x ker(D*'it)~C by (e, fl) =~ tr(e A fl*).

As q~(D) varies <.,.)~(o) gives a smooth Hermitian metric on M'~(Y). We shall denote its fundamental (1, 1)-form by | 54 I Biswas and N Raghavendra

Define a real 2-form ~ on ~ by

fly(,,, = f. tr( A ,6), where Ve~ ~ and ~,fl~Al(End(E,h)) ~. It is easily seen that ~ is a ~//~ invariant symplectic form on c~. Restricted to the submanifold ~, the form ~ vanishes along the vertical direction of ~. This can be seen as follows: any vertical vector ct at V~ is of the form ct = Vf where feA~ h)~; so if fleZ)(Y, r~, End r we have

fl) = fv tr(Vf A #) =f trV(fVfl)-frtr(fVfl)

= fvdtr(ffl)-O (since Vfl=0)

= 0 (by Stokes theorem)

Thus ~ descends to a symplectic form ~) on N"$-

PROPOSITION 3.24 The Hermitian metric on M~( Y) is Kiihler.

Proof. It is easily seen that I*(O)= fl, hence O is closed, as fl is closed and I is a diffeomorphism. Consider the determinant bundle L~' and sr (cf. equation 3.1). For each D ~ ~r let

0~<2o, 1 ~<2o, 2 ~< .... be the eigenvalues of the Laplacian

[]o: = D*oD:A~ ~ A~ and define the zeta function

Co(s)= Y. 2o,i> 0

This series converges for Re s > 1 and the function (o(s) extends to a meromorphic function on C which is holomorphic at 0. Define a Hermitian inner product

he:~ o x -~q~o--*C, by h e = exp(- (~(0)). (., .)~2 where (., .}I.2 is the L2-inner product in L~'o induced by the metric in E. It is proved in Quillen [18] that h o is a smooth Hermitian metric in ~. There is also a natural Hermitian inner product on the affine space ~r = A ~ (End E) Parabolic determinants on Riemann surfaces 55 which is given by

(~,fl) = ~" tr(~ A fl*) for e, fl~A~ E). This gives rise to a K/ihler metric in ~ whose K~ihler form is denoted here by ~. We state the following theorem of Quillen [18].

Theorem 3.25. The Chern form of the canonical connection in the holomorphic Hermitian line bundle (.L~',hQ) on d equals the Ki~hler form ~). Consider the two principal bundles ?

N~ ~ M~(Y). (3.3) I

On the line bundle '~ (cf. equation 3.2) we have a metric given by the tensor product of the Quillen metric h e in ~ and the given metric in E r Since 2(Ey) is a fiat unitary bundle on ~r the Chern form of the metrized line bundle ~' is r times the Chern form of Z~' and, hence, by Theorem 3.25, equals r.~). Let ~ be the pull-back of .~ to c~ by the map I'. Then ~/~ is a C~ #~//"-line bundle on cr and the pull-back metric and the pull-back connection in the line bundle ~,g are invariant under the action of ~" on cr Thus ~r descends to a C ~ Hermitian line bundle M on N~ with a Hermitian connection. This bundle M is naturally isomorphic to I*(L') (cf. definition 3.11) as a Hermitian line bundle. It follows from this that L" naturally gets a metric and a connection. This connection is easily seen to be compatible with the metric and the holomorphic structure in L ~. We denote the metric in L ~ by O.

DEFINITION 3.26 We call g the Quillen metric in L ~.

Theorem 3.27. The Chern form of the Quillen metric in the holomorphic line bundle L ~ equals r times the Kiihler form @~ on M~(Y).

Proof. Consider the commutative diagram (3.3). We have

(I o ~b)* c I (L n, g) = C 1 ((I o ~b)*L ~, (t o ~k)*g) = c1(~, I"*V), where V = canonical connection in .~

= ~'*c~(~) = 7*(r.~) (cf. the remarks following the commutative diagram 3.3)

=r.~

= (Ior 56 I Biswas and N Raohavendra

Since lo~b is a submersion, this implies that

c1(L,9) = r.O.

4. The relation between the n-determinant and the parabolic determinant

In this section we show that the isomorphism from the moduli space of n-bundles to the moduli space of parabolic bundles carries the n-determinant L ~ (cf. Definition 3.11) to a multiple of an "intrinsic" line bundle on the parabolic moduli. By "intrinsic" we mean that the line bundle depends only on the parabolic data and not on the choice of the n-structure. We refer to this intrinsic line bundle as the parabolic determinant bundle. Here we consider only the bundle structures and leave the comparison of the metric structures to the next section. The main results we use here are the Grothendieck-Riemann-Roch theorem and a theorem of Deligne et al ([5] and I-3]) on the determinant of the tensor product of two families of line bundles. Consider a compact Riemann surface Y and let n be a finite group acting effectively and holomorphically on Y. Let X = ~\Y be the quotient Riemann surface and p: Y~ X the canonical projection. The map p is a ramified covering. Let I be the set of ramification points of p in Y and let J = p(I) be the set of critical values of p in X. Both I and J are finite sets. For each xEJ, choose a point yxep-l(x) and let nx be the order of n~, the isotropy subgroup of n at y~. Fix once and for all, for each xeJ, a set C~ _ rt of left coset representatives of rr~.

DEFINITION 4.1 A holomorphic family of compact Riemann surfaces is a proper surjective holomorphic submersion f:3f--, T of complex manifolds such that for each teT, the submanifold X, =f-l(t) is a Reimann surface. Let f:Y'~ T be a family of Riemann surfaces and let ~ be a coherent ~rmodule. Then, by the Proper Mapping Theorem, all the higher direct images R~f,~ are coherent d~r-modules so,

= Y.(- 1)'R'f,Y is a well defined element of K(T). Therefore

is a holomorphic line bundle over T, called the determinant bundle of ~-. The fibre of d(~') at t~ T is canonically isomorphic to

2(HO(X,, ~r )), | 2(H 1(Xt, ~-,)), where ~, = jr Ix,.

Remark 4.2. Suppose f: Y" --* T is a family of Riemann surfaces. Then, the determinant construction defined above enjoys several nice properties. For instance, if

0-~ ~'-+ ~-+ ~"-) 0 Parabolic determinants on Riemann surfaces 57 is an exact sequence of coherent d~r-modules, then

d(.~) = d(.~") | d(~") = d(.~ (~.~").

Hence, when we consider the determinants d(~) we will assume that all short exact sequences of d~r-modules split.

DEFINITION 4.3 A family of vector bundles on the Riemann surface Y parametrized by a complex manifold T is, by definition, a holomorphic vector bundle over Y • T. A family of n-bundles on Y parametrized by T is a holomorphic n-bundle on Y • T, where the action of n on T is trivial.

DEFINITION 4.4 A family of parabolic bundles on X parametrized by a manifold T is a holomorphic vector bundle F on X x T together with the following data: for each x6J, we are given a filtration of the vector bundle F X__-Fl~x r by subbundles

F ~ = F 1 F ~ D_ F2F x ~... ~_ Fk~F ~ D_ F kx+ 1F ~ = O.

We denote Gt4F ~ = F~F~/F J+ ~F x. Suppose E is a family of n-bundles on Y parametrized by a manifold T. Let r be the rank of E. For each te T, the local type of the 7~-bundle E t = Elf ~ t on Y at each point yx is independent of t and is given by integers

0 ~< lx, l <... < l~.kx ~< n~-- 1 and (r~,1 ..... r~.~x) such that kx E rxj=r. jr1

Define ~tj = l~.j/n~ and n~,j = n.0t .j where n = Iltl. Let F--,X x T be the family of parabolic bundles induced by E (of. Proposition 2.11), namely, F=E ~ is the holomorphic vector bundle on X x T corresponding to the locally free r • r "m~ (p x lr),~(E ) together with an induced parabolic structure. We now compute the determinant of E in terms of F and the parabolic structure of F:

PROPOSITION 4.5 Consider a family E of It-bundles on Y parametrized by a complex manifold T and let F = E ~ be the induced parabolic family on X x T, as above. Then as line bundles on T, we have

d(F)=d((p x lr)*F)| x.jQ 2(Gr'FX) -'x'j)

j = 1,...,kx

Proof. We first note that for each x~J andj = 1..... kx, Gr~F x -~ E~x as vector bundles kx on T where E ~--Elyxxr and F-Fl~rX__ and E~= ~ EjX is the eigenspace jffil 58 I Biswas and N Raghavendra decomposition of E x, i.e, on each El., nx acts as a character of weight l j. We prove this by producing a natural homomorphism

hi:GrJF x ~ E~..

The map h s is defined as follows: Given 0 ~ v e GrSF x, let ~eF ~ represent v. Now take any section s of F over some open set containing x which satisfies the condition that s(x) = ~. Let g be the invariant section E over an open set containing y~, which represents s. If g is the image of t| 1~ where t is a section of E such that t(y~)50 and 1~ is the canonical section of (9(- iy~), under the morphism of sheaves O--} E | ~(- iy~)--} E, define h s = t(yx).

From the correspondence between n-bundles and parabolic bundles as described by Mehta and Seshadri [13] it follows that h s is an isomorphism. Therefore to prove the proposition, we need to verify that

d(E)=d((p x l)*F)( x~s(~ 2(E~')-"x's)" (4.1)

j = 1,...,kx

We have a canonical Cr • r "m~ monomorphism

0~(p x 1)d~(F)~dg(E).

The cokernel ~'- of this monomorphism is a torsion d~r • r-module supported on I x T where I denotes the ramification locus in Y. We thus have an exact sequence of d~y• r'm~

0~(p x 1)*tR(F)~d~(E)--}o#---}0. (4.2)

By Remark 4.2, we get

d(E) = d((p x 1)*F).d(~-)

= d((p x 1)*F).(~))\,~, 2(3-') -z (4.3) where 3"Y = ~'lyx r . But vj- is a n-sheaf on Y x T, hence for each yeY'and yen, ~Y is canonically isomorphic to 3 -ry on T. If xeJ and yep-~(x), there is a unique 7eCx such that y = 7.yx (recall that we have fixed a section C~ ___ n of n~n/n~); hence we have a canonical isomorphism ~-Y ~ 5 yx. Thus for each xeJ, Q ~(~-~)=,~(~-~)~/~ y~p- x (x) so that | = | ~(~,,x).,,.x. yeJ xeJ Parabolic determinants on Riemann surfaces 59

Using this (4.3) becomes

(4.4)

From (4.1) and (4.4) we see that it suffices to show that for each xeJ,

kx 2(J-Yx) "I~" = (~) 2(E~.)"",~. (4.5) .i=I

Since n j = n'l~,Jnx, this amounts to verifying that

kx 2(y yx) = (~) 2(Ef.)'~,L (4.6) j=l

Proving (4.6) for a fixed x~J is a local problem at y~. By Remark (2.10), using a u-compatible coordinate chart at yx we are reduced to the following situation: Let n be the group of nth roots of unity acting on the unit disc D by multiplication and let p: ~ D = ~\D be the quotient map p(z) = z". Let V be a vector bundle on the variety T and define an action of n on V by ((v) = (Z'v for each (en and v~ V, where 0 ~< l ~< n - 1 is a fixed integer. Consider the n-bundle W = p~. V on D x T where Pr: ~ T is the projection map and let ~'- denote the cokernel of0~(p x 1)*(p x 1).dg(W)~dg(W). Now, prove that

2(~ ~ = 2( V)-', (4.7) where o is the centre of D. Let us check that (4.7) is true. We can easily see that (p x 1)*(p x 1).d?(W) =ml'r where m is the ideal sheaf of o x T in D x T. Therefore the sheaf J- is isomorphic to the cokernel of 0~m'.r

Now we have an exact sequence

0~m~(9o• r~a~0, where 6e = (9D • Tim. Tensoring this short exact sequence with m j- 16( W),j = 1..... l, we get a system of exact sequences

O~mJ(9(W)~m~-t(9(W)~mJ-l(~(W)| ~O, j= 1,...,I.

I By general functionality, it follows that 2(~-~ = | 2(3-~.) where o~j= mi-lt~(W)| e j=l and ff~. = IIo • r" But ~--~. = (L~ j- 1 | V where L is the line bundle on D x T associated to the sheaf m and L ~ = LIo x T" Since L ~ is trivial on T, we get ffj~ - V forj= 1..... l. Thus 2(ff ~ = 2(V) l which is precisely (4.7). This proves the proposition. We now construct parabolic determinants. For a more detailed description of these determinant bundles we refer the reader to Narasimhan and Ramadas i-15]. Let X be a compact Riemann surface and J ___ X a finite set. Fix integers k~ and 60 I Biswas and N Raffhavendra

(r~,l ..... r~,k,) such that Z~ ~r~,j = r. Let 0 ~< ~t~,1 ~< .... ~< %.k, ~< 1 be rational numbers for each xeJ. Let M~.(X) denote the moduli space of stable parabolic bundles of rank r and degree d with weights ~j, and multiplicities r~j, Suppose F ~ X x T is a family of parabolic bundles of type {~t,,~j}~j, i.e., rk(GrdF ~) = r~j and the weights are ~y Then a possible candidate pd(F) for the parabolic determinant is the usual determinant minus a factor involving the flags and the weights; thus, we define pd(F) to be the element 0f Pic(T) o =Pic(T) | O given by l

pd(F) = d(F) - ~, %j-~.(Gr~F~). x~J j = 1 ,...,kx

Now if L is a line bundle on T and G = F | p~.(L) where pT:X x T is the projection, then F and G are, by definition,equivalent families.Therefore we would like to have pd(F) = pd(G) in Pic(T) o .But d(G) = d{f) - x(E)'L and A(Gr~ G ~) = 2(__GrJF~) + r~,fL in Pic(T)owhere z(E)- x(Ef) is independent of t. Hence pd(G)= pd(F)- ~(p(F).L, where

x(r) + r :t~J j= l,...,kx

We thus see that the definition of pd(F) has to be modified to get an invariant determinant. Fix once and for all a point xoeX\J.

DEFINITION 4.7 Let F ~ X x T be a family of parabolic bundles of type_ {r~j, ~xj} on X parametrized by a variety T. Then the parabolic determinant pd(F), of F is the element of Pic(T)o = Pie(T) | Q defined by the formula

p-d(F)= d(F) - ~ ~.;~(GrJF ~) + z'(F)A(F~~ xGJ r J: 1,...,kx where

x,(v) = x(F,) + Z x,j

The above discussion shows that if F and G are equivalent families of parabolic bundles on X x T, then pd(F)ffi pd(G) in Pie(7") o . We would like to think of the parabolic determinant as a concrete line bundle representing an element of Pie(T), rather than as an abstract entity in Pie(T) o . To achieve this we have to multiply the pd(F) of Definition 4.7 by an integer to remove the denominators from the rationals ~tj and ;~p(F)/r. Write %,j = Pxj/qxj where p~.# and q~,j are relatively prime positive integers if ~x,j > 0 and qxj = 1 if ctj -- 0. Denote by m the l.c.m of {qx,j/x E J, 1 ~

DEFINITION 4.8 Let F--, X x T be a family of parabolic bundles of type {~,j, r~,j}. Then the parabolic determinant bundle, pd(F), of F is the line bundle on T defined by pd(F)=d(E)".{(~r 2(Gr~FX)-mx.J"}'2(F~~m'zp~r). j= l,...,kx

Note that in Pic( T)o, pd(F) = rm.pd(F).

Remark 4.9. The above arguments again imply that pd(F)= pd(G) if F and G are equivalent families of parabolic bundles on X x T. We recall a proof from Narasimhan and Ramadas [15, w2.c] that there is a parabolic determinant bundle on the moduli space M~(X). By tensoring with a fixed line bundle we may as}ume that the degree d is large enough to ensure that for every E6M~(X), H~ generates E and Hi(E)=0. Let Q denote the Quot scheme of coherent ~x-modules which are quotients of O~t, k = d + r(1- O), with Hilbert polynomial P(m) = d + r(m - O + 1). Then there is a Poincar6 family of sheaves on X x Q. Denote by R the open subset of Q consisting of locally free sheaves E such that H~176 is an isomorphism. Let .~ denote the restriction of the Poincar6 family to X x R. For each x e J, let Flag 3~TM denote the flag bundle of type (rx. 1..... rx.l ) associated to the vector bundle ~x on R; Flag 3~:~ is a fibre bundle over R. Let/~ denote the fibre product/~ = H Flag ~-~. Then/~ parametrizes a family .~ of parabolic R xcJ bundles on X. Let /~, denote the subset of/~ corresponding to stable parabolic bundles. There is a natural action of PGL(k) on/~s and the canonical map/~s-* M~(X) is a holomorphic principal PGL(k)-bundle. Let pd(f) denote the determinant bundle (cf. Definition 4.8.) on R~ defined by the family ~" ~ X x R r Clearly the action of GL(k) on /~, lifts to an action on pd(.~). Let us see how the scalars C*___ GL(k) act on pd(@). From Definition 4.8 and the properties of R, we see that

j = 1,...,kx

Thus a scalar 2eC* acts on pd(~=) as multiplication by 2", where

C = -- rink - r ~, mx.j'rxd + rmZp(.~); xcJ j= l,....kx on simplifying the right side, we see that c = O. Therefore pd(~) is a PGL(k)-line bundle on/~s and hence descends to a holomorphic line bundle L p on M~(X).

DEFINITION 4.10 The above line bundle L p on MP(X) is called the parabolic determinant bundle on M~(X). Consider a family of parabolic bundles on X, F ~ X x T, parametrized by a variety 62 I Biswas and N Raohavendra

T. Choose a compact Riemann surface Y and a group n of order n acting on Y such that (i) the quotient ,\Y is biholomorphic to X; and (ii) if p: Y~X denotes the quotient projection, then for each x~J and yEp-l(x), the order of the isotropy subgroup of n at y equals m. Then n is a multiple of m and we let a = n/m. Define

n~,~=n'otj=a.mx, i forx~Jandj=l ..... k~.

By Proposition 2.11, the family F induces a family of n-bundles on Y, E~ Yx T, parametrized by T. Let red(E) denote the determinant bundle of E as in w3. Fix a point yep(x). Since xEX\J, y is not a ramification point of p and

rid(E) = d(E) | 2(EY~ x(E) where z(E) = g(Et) is independent of t. The two line bundles pd(F) and rid(E) on T are related by

PROPOSITION 4.11 Let F and E be as above. Then as elements of Pic(T)~ = Pic(T)|

{~d(E).pd(F) --a~: 2 : (~{~.(Fx)~(FXo)- l }ra(ra- l) x~J

Before proving the above proposition we make a simple observation whose proof is due to V. Balaji:

Lemma 4.12. Let f :3f -~ T be a family of compact Riemann surfaces and suppose that F and G are holomorphic vector bundles on ~ with ranks r and n, respectively. Then d(F | G)d(F)-"d(G)-" = d(2(F). 2(G))d(2(F))- ' d(2(G))- ~ in Pic(T) | ~.

Proof. Denote by A*(T)= ~ Ai(T) the Chow ring of T and for any ~A*(T), let i=0 {~}i denote the degree i component of ~ in Ai(T). Also let f, :A*(X)~ A*(T) denote integration along the fibre. Then the Grothendieck-Riemann-Roch (G-R-R) theorem implies that for any vector bundle V on ~, ca(d(V)) = f,({ch(V)'td(T:)}2), where 7": denotes the tangent bundle of s along the fibres of f. Note that T/is a line bundle on s Let ~1 .... , ~, be the Chern roots of F and fll ..... ft. the Chern roots of G. Then (~i + flj) are the Chern roots of F | G. Using the splitting principle (cf. Fulton [8], page 54) we formally compute

{ch(F)'td(Ty)}2=rla--~2+~ or, ~+ ~2 12 i=l 2,= where/~ = c1(T:).Similarly.

{ch(F)'td(T:)}2 = ~2 + (,=~ fl,) 2 + ~ ~=lfl~" Parabolic determinants on Riemann surfaces 63

Finally,

12 , s=t + ~(,~1 j~l (~/+ flj))2 -nr"2+12 2,

r 2 + ~/~j + (z~,)(z~ A i.e., {ch(F | G).td(Tr = {n'ch(F)'td(T s) + rch(G)'td(Ts) }2

nrl~ 2 + ci (F)'c, (G) - -- 12

Applying G-R-R and noting that f,(#2)= 0, we get c, (d(F | G)) = n'cl (d(F)) + rc, (d(G)) + f, (c, (F)" c, (G)), i.e., c I (d(F | G))" d(F)-"d(G)-') = f, (c I (F)'c t (G)). Replacing F by 2(F) and G by ;t(G) in the above relation, we get cx (d(,~(F).,~(G)).d(;4F))- i d(;~(G))- i ) = f, (cl (F). cI (G)) since cl (2(F)) = cl (F). Thus

c, (d(F | G)'d(F)-n'd(G) =r) = c1 (d(2(F)2(G) )d(2(F) - ~d(2(G)- 1).

Since c t : Pic(T) | Q -) A* (T) | Q is an isomorphism (Fulton [8], page 294) our lemma is proved. We now recall a formalism due to Deligne [5]; we quote below Proposition 2.5 and Lemma 2.6 from Beilinson and Manin [-3] as a brief summary ofthis formalism.

PROPOSITION 4.13 Let f ::~ --) T be a holomorphic family of compact Riemann surfaces. Then for each pair L, M of line bundles on ~?, there is a line bundle (L, M) on T with the following properties: (i) There are canonical isomorphisms (L, tP~r ) = (gr, (L 1 | L 2, M) = (L 1 , M) (L 2, M), (L,M)=(M,L), (L,M-1)=(L,M) -1

(ii) If M = ~ (D), where D is a relatively positive divisor on 37, locally free on T, then

( L, M) = d(L| C)o)- t'd((~o).

(iii) There is a canonical isomorphism

( L, M)- i = d(L| M)'d(t~:r )'d(L) - ~d(M)- 1. 64 I Biswas and N Raghavendra

Now we proceed to the Proof of Proposition 4.11. Using Proposition 4.5 we see that

diE) = d((p x Ir)*F )" ~) 2(Gr~FX) -"'J. (4.3) xEJ

But H~(X,p, V)= Hi(Y, V) canonically, so

d((p • lr)*F) = d((p • 1),(p • 1)*F) (4.4)

Now, by the projection formula

(p x 1).(p x 1)*F=F| where G = (p x 1),d~r x r = P*P*t~r| and px:X x T~X and pr:X x T~ T are the projections. Thus (4.4) becomes

d((p x 1)*F) = d(F| G). (4.5)

We now prove from Lemma 4.12 that

d(F| G).d(F)-".d(G)-" = (2(F), 2(G))- '.d(2(F))- ~d(2(G))- ' in Pic(T)o. Using Proposition 4.13 (iii), we write this as

d(F | G).d(F)-".d(G)-" = (2(F), 2(a)) - "d(O x ~ r)-"

But d(G) and d(O x ~ r) are trivial on T. Thus

(2(F), 2(G))- 1 = d(F| G).d(F)-" (4.6) in Pic(T) o. Squaring both sides of (4.6) and using Proposition 4.13, we get

(2(F), ~.(G)2 ) = {d(F | G)'d(F)-" }2. (4.7)

But 2(G) = 2(p'p. tPr)'p*(tPr) n. Therefore, ().(F), 2(G) 2 ) --- (,~(F), ,~(p*p. dgr)2 "(p~.~T) 2n > -- <~(F), ~(p~p, ~,)~ > <~(F), p~r> 2" -----(2(F), p~: 2(p. (~rP) z > because ( 2 (F), p* r ) is trivial by virtue of Proposition 4.13. Now 2 ( p, ~ r )2 = ~x (D) where D = - ~ a(m - 1).x (cf. Beauville et al [2], w Therefore, by Proposition xEJ 4.13(ii)

<~.(F), ~(G) 2 ) = (,~(F), p~ ~)x(D) ) -- d(,~(F) | p* {~~)- 1dip* d)D) i.e., <~(r), ~(G) ~ > = | "~'-" (4.8) Parabolic determinants on Riemann surfaces 65

where we drop d(p~(~D) as it is trivial on T. Combining (4.7) and (4.8), and using (4.5),

{d((p x 1)*F)'d(F)-'} 2 = (~)2(F~) ~t'- 1). xcJ

Now apply (4.3) we obtain

d(E). @2(GrJY'~'~.'.d(Y) -" = @2(y~r(m- x) l xeJ ) xeJ

Raising the above equation to the rth power and noting that n~.~ = a'm~.j and n = a.m, we see that

{nd(E).2(EyO)-Z(~).(pd(F)-o.2(F:,O)Zpu,).,,)}2 = (~)2(FX),ot, - 1). (4.9) xcJ

Since Yo is not a ramification point ofp and p(yo) = Xo, we have po ~ EX.. So we can write (4.9) as

{rcd(E).pd(F)~a}2.~(FXo)2,xvte)-2xfE) = (~)~,(FX)ratm - 1). (4.10) xeJ

The Riemann-Hurwitz formula gives

2nz,(F) - 2;((E) = IJI ra(m - 1).

So we may express (4.10) as

{~d(~).pd(F)-.}2 = | 1),o-, xeJ

in Pic(T) o. This completes the proof of Proposition 4.11. Let us see what Proposition 4.11 implies for our moduli spaces. Fix a line bundle 6 of degree d on X and let MP(X, 6) be the smooth subvariety of M~(X) consisting of parabolic bundles F such that 2(F)g 6. Similarly, let M:(Y,6) be the smooth submanifold of M:(Y) consisting of n-bundles E such that 2(E~)~6. Then Proposition 2.11 gives a natural isomorphism O:M~(X,6)--,M'~(Y,6). In w we constructed a ~-determinant L ~ on M'~(Y,6) using Quillen's theorem. In this section we defined the parabolic determinant bundle L p on M~(X, 6). These determinants are related to each other by

PROPOSITION 4.14 The two line bundles O*(L~) and (LP)~, where a is the integer defined in the paragraph following Definition 4. tO, are isomorphic on M~(X, 6).

Proof. Consider the variety /~s mentioned earlier on; /~, sits as a principal PGL(k)-bundle over MP(X) and parametrizes a family .~ of parabolic bundles on X. Let ~" be the corresponding family of 7c-bundles on Y parametrized by/~s. For each 66 I Biswas and N Raffhavendra pair of points x, y~X, let G~,y be the line bundle on/~s given by

We easily see that (~,y is a PGL(k)-line bundle and descends to a line bundle G~.y on M~(X). By Proposition 4.11,

(~d(~')'pd(~-~)-~ -- (~(G .... ),o~,- x~ (4.11) xEJ in Pic(/~s)| Q. But (cf. Drezet and Narasimhan [7], Lemma 4.2) Pi%(/~s)~ Pic(/~s) is injective, where Pi%(/~,) is the group of PGL(k)-isomorphism classes of PGL(k)-line bundles on Rs. Therefore, since all the bundles in (4.11) lie in Pico(/~), we see that (4.11) holds in Pi%(/~)| Q. By descending to M~(X), we get

(O, L*(Ln)-a)2 = | (Gx.xo),a{m- 1) (4.12) xr

Consider a fixed x~X. As y varies in X, Gx,y varies in a continuous family; since PicoM~(X,6) is trivial, we conclude that Gx,y is trivial on M~(X,6)Vx, y6X. Thus (3.12) implies that (O* Ln(LP)-a) 2 = 1 in Pic(M~(X, 3))| Q. Since PicM~(X, 3) is torsion-free, we conclude that O* L ~ = (L p)a on Mn(X, 6).

5. The metric on the parabolic determinant

In this section we define a Hermitian metric on the parabolic determinant bundle and give a proof of Theorem 1.2. Let MP,(X) and M~(X, 3) be as in w Again, choose a covering p: Y--}X of degree n. We continue using the notation of w4. Suppose E~M~. Then EndE is local system on X o = X\J. We denote this local system by End E. Let H~(X o, End E) denote the compactly supported de Rham cohomology of Xo with values in the local system End E. The real tangent space to M~(X) at E is canonically isomorphic to the image Ie of the natural map

H i (X0, End E) -} Hi (Xo, End E), where H i denotes usual de Rham cohomology. We shall prove this fact in Lemma 5.5 below; for the moment let us assume it. We define a sympletic form f~ on Te(MP,(X)) by

f~(a, b)= f xo tr(~t A fl), Parabolic determinants on Riemann surfaces 67 where a, bel e and ct, 8 are compactly supported forms representing a and b. As E varies over M~(X), fFe vary to define a C = symplectic form fF on M~(X). Recall that we have an isomorphism O:M~(X)~M~(Y). In w we defined a symplectic form, which we now denote as f~", on M~(Y). The two forms f~" and fF are related as follows.

Lemma 5.10*(f~ '~) = n.f~ p.

Proof. Recall that the tangent space to M'~(Y) at a point EeM~(Y) is given by H i (Y, n, End E) where End E is the local system defined by the Einstein-Hermitian connection in E. Let F = E ~. Then the differential of 0 at E is described as follows. Let aeTe(M~(Y)). Choose a n-invariant 1-form ~t on y with values in EndE representing a. Then Ctlr\t descends to a form on X0 = X\J which is cohomologous to a compactly supported form ~# on Xo with values in End E. Thus ~# defines an element a# in IF = Tr(M~(X)). The differential of 0-1 at E is given by

dO- 1: TE(M~(y) ) ~ TF(M~(X) )

a ---I. a#.

Now, let a, be TE(M'~(Y) ). Then

(0-1)*(ftP)(a, b) = fF(dO- l (a), dO- X(b)) = ~P(a#, b#)

( tr(~ A 8#) ,/x O 1 f tr(ct A 8) =n r

= -1n"(a, b). /1 Thus (0-1), tap = 1~., /1 i.e., fF = 10" f~". 1l Let us now state the obvious.

Lemma 5.2. Let V and W be two complex manifolds and suppose f: V~ W is a biholomorphism. Let 8: T(W)c x T(W)c ~ C be a real positive (1, 1,)-form on W and let 8a denote 81rov~ • r(w)" Suppose ct: T(V) x T(V) --* R is a real 2-form on V such that f*Sa = ~. Then the complexification o~c of ~ is a real, positive (1, 1)-form on V. Thus the complexification | of the 2-form f~P on M~(X) is real, positive and of type (1, 1), so it defines a K~ihler structure on M~(X). In w3, we defined a natural Hermitian metric h ~ in the determinant bundle L ~ on 68 I Biswas and N Raghavendra

M~(Y). This metric induces a metric O*h ~ in the pull-back O*L ~ over M~(X). But Proposition 4.14 implies that O*L~= (Le)~ on M~(X,3). Thus we have obtained a metric in (Le)~ on M~(X, fi). This defines a metric h e in L e. We now have Theorem 1.1 of the Introduction.

Theorem 53. The Chern form of the metrized line bundle (Le, h e) equals mr times the Kf~hler form 6)e on M~(X, 6).

Proof. Since both cl(Ln, h e) and mr.O e are real (1, 1)-form, it suffices to show that they are equal on the real tangent space of M~(X, 6). But as real forms

a" C 1 (Le, M)a = cl ((Le)~ (M)~ ) = cl (0" L ~, 0* h~)a = 0* c 1(L ~, h~)a = O*(r'~') = rn.t) e = rn.O[,

using Theorem 3.27. Dividing by a and noting that m'a = n, we get

c a(L n, he)a = rm'O~.

This proves the Theorem.

The metric a priori depends on the choice of the covering (Y, n). But we now see that in fact it is independent of n up to a positive constant.

PROPOSITION 5.4 Let (Yl,nl) and (Y2,n2) be two ramified coverings of X with the properties of Y as above. Let h~ and h~ be the induced metrics in L e. Then there is a constant c > 0 such that h~ = c.h~ in L e over M~(X,6).

Proof. Since h~ and h~ are two Hermitian metrics in the same line bundle L p, there exists a positive c ~ function f:M~(X,f)--,R + such that hi =f'h2. By theorem 5.3, h 1 and h 2 have the same curvature. So, locally, ddlogh a =ddlogh2, and hence, d~logf = 0. Thus logf is a plurisubharmonic function on M~(X, 6). Now the moduli space MP(X, 6) of semistable parabolic bundles is a normal projective variety and codim of the complement of M~(X, 6)/> 2, so logf extends to a plurisubharmonic function on the whole of MP(X, 6). Since Mn(X, 3) is compact, logf = a is constant, and hi = ch 2 where c = e ~

In the rest of the section we prove.

Lemma 5.5. The real tangent space to M~(X) at E is canonically isomorphic to the image I E of the natural map

H~ (X,, End E)-* n~ (Xo, End E).

Proof. It is enough to prove the lemma in the case where parabolic degree is zero. If the parabolic degree is not zero then choose sufficiently large number of points aa ..... an~X o and include them among the parabolic points and give on the fibre of each at the trivial quasi-parabolic structure; i.e. the only flag is the whole fibre (cf. Parabolic determinants on Riemann surfaces 69

Mehta and Seshadri [13], Remark 1.17). Assign non-zero weights to the flags so that the new parabolic degree is zero. This gives an isomorphism of M~(X) to some other moduli of parabolic stable bundles with parabolic degree 0. Let V denote the image of E under this isomorphism. Stable bundles of parabolic degree 0 arise as representations of the fundamental group of the complement of the parabolic points in U(n). So, as V is a degree 0 stable bundle End V has a structure of a local system. But around each ai the monodromy Is trivial; scalars are in the kernel of the adjoint representation. So End E and End V are naturally isomorphic on Xo; in particular End E has a structure of a local system on X o. Now onwards we will assume the parabolic degree to be zero. So, let X be a compact Riemann surface of genus g/> 2 and J ~ X a finite set. Fix integers de~Z and r>~ 1. Let us recall our earlier notation. For each xed, let 0 ~< ct~ < ~t, 2 < --- < Ct~.k=< 1 be rational numbers and let (rx, 1..... r~.kx) be positive kx integers such that ~ r i = r. Assume that j=l

d "Jr- E O~x,j'rx, j = O. xe~d j= 1,...,r

Let M~(X) ~ denote the moduli space of stable parabolic bundles of rank r, degree d and weights ~.j with multiplicities r j. For each xeJ, define a function cx:{1 .... ,r} ~ [0, 1] by

cx(i)=~.j, ifr,:+..-+rx, j_t +l~

Let W~ denote the conjugacy class in U(r) of the diagonal matrix

e2=/,:=(r)

Express the ctx,J with a common denominator m, 0t, i = mx4/m. Then there exists a discrete subgroup F of PSL2(R) acting on the upper half-plane H with the following properties: (i) F has no parabolic fixed points; (ii) F\H = X; (iii) F is generated by 2g + IJI elements {ax ..... %, bl ..... bg, c~, x~J} with the relations

Let R denote the manifold of irreducible unitary representations 19:F---, V(r) such that p(c~)~ W~ for all xeJ. Then by Mehta and Seshadri [13], R "~ M~(X) ~ The real tangent space to R at a representation p is equal to H~(F, adp) where ad: U(r)~ Lie U(r) is the adjoint representation. Using an argument of Prasad [17], we shall identify this group cohomology with compactly supported cohomology. From Prasad [17, Lemma 3 and w we get Hl(F, adp) = Hx(X,A) where A is the invariant direct image of the constant sheaf Lie U(r) on H. Let Xo = XkJ and let i:Xo--*X denote the inclusion map. It is easily seen that i.(Alxo)= A. Therefore the 70 I Biswas and N Raghavendra edge homomorphism associated to the Leray spectral sequence of i implies that the canonical map g: H 1(X, A) ~ H1 (Xo, A) is injective. On the other hand Theorem 4.10.1 of Godement [9] applied to the closed set J gives an exact sequence

.. +H~(Xo, A)~Hi(X,A)~HI(J,A)~ ... where H i denotes cohomology with compact supports. But d being a finite set, we get H i (J, A) = 0. Thus the mapf:H~ (X o, A)-+ H 1(X, A) is surjective. Let h:n~ (Xo, A)-. nl(Xo, A) denote the canonical map. Then we have a commutative diagram

f n~x (X, A) ~ t7' (X, A) h~ ~.O HI (Xo, A) identity H t (Xo, A)

Since f is surjective and 0 injective, we conclude that HI(X,A)~-Im(h), i.e., Tp(R) is the image of H~(X o, A) in HX(Xo, A) under the canonical map h. This completes the proof of the Lemma.

6. An application

In this section we deduce a finiteness statement from our earlier results. Let Y be a compact Riemann surface of genus at least 2. Fix two relatively prime integers r and d with r/> 1. Let M(Y) denote the moduli space of stable bundles of rank r and degreed on Y. Suppose a finite group ~ acts holomorphically and effectively on Y. Let M'~(Y) denote the moduli space of stable ~-bundles of rank r, degree d and fixed local type on Y. Then both M(Y) and M'~(u are projective algebraic manifolds and there is a canonical forgetful map ct:M~(Y)~ M(Y).

PROPOSITION 6.1 The morphism ~t is a finite map.

Proof. As in w3, we can construct a determinant line bundle L on M and show that L is positive (cf. Theorem 3.27). In fact, to see this we put ~ to be the trivial group everywhere in w3. Since M(Y) is compact, the Kodaira embedding theorem implies that L is ample. But the determinant bundle L ~ on M~(Y) is exactly equal to ~*(L). Thus ct pulls back the ample line bundle L on M(Y) to an ample line bundle on M~(Y). As a result ct is a finite morphism.

Acknowledgements

We thank M S Narasimhan and C S Seshadri for suggesting us the problems studied in this paper, and for giving us constant encouragement and guidance. We also take Parabolic determinants on Riemann surfaces 71 this opportunity to thank V Balaji, T R Ramadas and P A Vishwanath for many useful discussions. The first author (IB) would like to thank the SPIC Science Foundation for hospitality while this work was being done.

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