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The Strange Duality Conjecture for Generic Curves Author(s): Prakash Belkale Source: Journal of the American Mathematical Society, Vol. 21, No. 1 (Jan., 2008), pp. 235-258 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/20161365 . Accessed: 15/06/2014 11:05 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Mathematical Society. http://www.jstor.org This content downloaded from 195.34.78.137 on Sun, 15 Jun 2014 11:05:40 AM All use subject to JSTOR Terms and Conditions

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Page 1: The Strange Duality Conjecture for Generic Curves

The Strange Duality Conjecture for Generic CurvesAuthor(s): Prakash BelkaleSource: Journal of the American Mathematical Society, Vol. 21, No. 1 (Jan., 2008), pp. 235-258Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/20161365 .

Accessed: 15/06/2014 11:05

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Journalof the American Mathematical Society.

http://www.jstor.org

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Page 2: The Strange Duality Conjecture for Generic Curves

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 21, Number 1, January 2008, Pages 235-258 S 0894-0347(07)00569-3 Article electronically published on April 25, 2007

THE STRANGE DUALITY CONJECTURE FOR GENERIC CURVES

PRAKASH BELKALE

1. Introduction

Let SUx(r) he the moduli space of semi-stable vector bundles of rank r with trivial determinant over a connected smooth protective algebraic curve X of genus g > 1 over C. Recall that a vector bundle E on X is called semi-stable if for any subbundle V, ?eg(V)/rk(V) < deg(E)/'rk(E). Points of SUx(r) correspond to isomorphism classes of semi-stable rank r vector bundles with trivial determinant

up to an equivalence relation. For any line bundle L of degree g-l on X define 6L =

{E G SUx(r), h?(E?L) >

1}. This turns out be a non-zero Cartier divisor whose associated line bundle C =

?(Ql) does not depend upon L. It is known that C generates the Picard

group of SUx(r) (for this and the precise definition of ? in terms of determinant of cohomology see [DN]).

Let Ux(k) be the moduli space of semi-stable rank k and degree k(g ?

1) bundles on X. Recall that on Ux(k) there is a canonical non-zero th?ta (Cartier) divisor 6^ whose underlying set is {F <E U^(k),h?(X,F) ^ 0}. Put M = 0(?fc). Consider the natural map Tfc?r : SUx(r) x Ux(k)

? i/^(fcr) given by tensor product. From

the theorem of the square, it follows that rjlrM is isomorphic to Ck E3 Mr. The canonical element Qkr ? H?(Ux(kr),M) and the Kunneth theorem give a map well defined up to scalars:

(t) H?(U*x(k),Mry - H?(SUx(r),Ck). The strange duality conjecture asserts that (f) is an isomorphism. It is known that h?(U^(k), Mr) equals h?(SUx(r),Ck) (see for example [B2], Section 8). This

conjecture is known to hold when

k = 1 and arbitrary r [BNR], r = 2, k = 2, and C has no vanishing thetanull [Bl], r = 2, k = 4, and C has no vanishing thetanull [vGP], r = 2, k even, and k > 2g

? 4, and generic C [LJ.

An element F of Ux(k) produces an element 0^ of H?(SUx(r), Ck) well defined up to scalars. The zero locus of Qp is the set of all E E SUx(r) such that h?(X, E?

F) t? 0 (the degree of F is such that x(X, E <S> F) =

0). It is easy to see that (f) is an isomorphism if and only if Op for F e Ux(k) span H?(SUx(r),Ck).

Received by the editors February 23, 2006.

2000 Mathematics Subject Classification. Primary 14H60; Secondary 14D20. The author was partially supported by NSF grant DMS-0300356.

?2007 American Mathematical Society

235

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Page 3: The Strange Duality Conjecture for Generic Curves

236 PRAKASH BELKALE

Let Mg denote the moduli-space of connected smooth projective algebraic curves

of genus g. In this paper, we prove the strange duality conjecture for generic curves:

Theorem 1.1. For generic X G Mg, the sections Op for F G Ux(k) span

H?(SUx(r),Ck). For the history of this problem as well as recent developments we refer the reader

to Beauville [B2], Donagi-Tu [DT], Polishchuk [Po], and Popa [P].

1.1. The main idea. The starting point for this paper is the classical relation

between the cohomology of Grassmannians Gr(r, n) and invariant theory of the

special linear group SL(r) (or equivalently, SU(r)). In [Bel], this relation was fur

ther strengthened by demonstrating how triple intersections of Schubert varieties

geometrically produce a basis of the space of invariants for the associated SL(r) ten sor product multiplicity problem. The next step is Witten's [W] relation between

the (small) quantum cohomology of Grassmannians and structure coefficients in the

Verlinde algebra for SU(r) (these are dimensions of spaces of sections of theta bun

dles on moduli spaces of parabolic bundles on P1). This relation can be geometrized in a similar way. Theorem 1.1 of this paper is a higher genus generalization of this

relation with H?(SUx(r)1Ck) viewed as a representation theoretic object. The

associated enumerative problem needs to be invented. The linearly independent sections coming from the inherent transversality in the enumerative problem will

be shown to have the form Op for F G Ux(k). To help us invent the enumerative problem that should correspond to

H?(SUx(r))Ck), we calculate the rank M(r,k,g) of the latter (it is known that

the rank of H?(SUx(r),Ck) does not vary with X G Mg). Using a factorization

formula, the dimension can be related to the dimensions of conformai blocks for P1.

This reduction uses the Verlinde formula (Beauville and Laszlo [BL], Faltings [Fa2] and Kumar, Narasimhan, and Ramanathan [KNR]), and the factorization formula of Tsuchiya, Ueno, and Yamada [TUY]. By a theorem of Witten (cf. [W], [A]) there

is a relation between conformai blocks for P1 and the (small) quantum cohomology of Grassmannians. Putting all these together, one finds a formula (see Section 8.4

for some examples):

(t) M(r,k,g) = ^(w/i,... ,ul9,u{Iiy,... ,u{l9y)0 -k(9-i)

where the sum is over all sequences of subsets (I1,..., I9) of [r+k] =

{1,2,..., r+k} with r elements, each of which contains 1, and where the Gromov-Witten invariants are "twisted" (see Section 8.2 for the definition and Section A.l for the definition

of the classes ui). The simultaneous appearance of Schubert cohomology classes uji and their duals

ujv in equation (f) leads one to suspect the role of the diagonal in a product of Grassmannians Gr (r,n) x Gr(r,n). However, the restricted nature of the sum

suggests a piece in a partial degeneration of the diagonal (A+ in Section 5). Using the insight given to us by equation (\), we introduce an enumerative

problem that corresponds to M(r,k,g) (see Section 4). After this work was completed, I received a preprint from Takeshi Abe [Ab] in

which he proves the strange duality conjecture for generic curves, for r = 2, arbi

trary k. In these cases he proves the stronger version of strange duality (generalized

degrees) formulated in Donagi and Tu [DT].

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STRANGE DUALITY 237

2. Determinant of the cohomology

Let ? be a line bundle on a (possibly non-complete) variety Z and let s\,..., sm be global sections of C. One way of proving that si,.. .,sm axe linearly independent is to find points z\,... ,zm in Z so that the determinant of the matrix (s?(zj)), i = 1,..., m and j

= 1,..., m, is non-zero. To make sense of this determinant,

note that S?(zj) G CZj and hence the determinant is an element of 0 =! CZj. We view Zj as witnesses to the linear independence of the Si.

In this section we formulate the notion of an "(m, r, fc)-frame". Intuitively an

(m, r, fc)-frame (E\,..., Em; F\,..., Fm) on a possibly singular irreducible curve C of arithmetic genus g is a tuple (E\,..., Em', Fi,..., Fm) where the Ej are vector bundles on C of rank r, degree 0, and with isomorphic determinant line bundles; Fi aie vector bundles on C of rank k and degree k(g?1), so that assuming det(?y -^ ?

(otherwise we will have to twist by a line bundle of degree 0), Ej G SUxfrYs are

witnesses to the linear independence of the sections Qp..

2.1. Determinant of cohomology and basic operations. Let ir : X ?> 5 be a relative curve (that is, it is flat, the fibers are proper of dimension 1, and

n*(Ox) =

Os)- If F is a coherent sheaf on X which is flat over 5, we form the determinant of its cohomology T>(F) which is a line bundle on 5. Formally, the fiber of V(F) at a point s G 5 is the one dimensional vector space

det H?(Xs,Fsy ?det#1(Xs,Fs).

More precisely, the push forward in the bounded derived category Rn*(F) is rep resented locally by a complex T? ?> T\ of vector bundles (unique up to quasi

isomorphism). Here V(F) is defined to be det T? ? det T\ (where det J7 denotes the top exterior power of the vector bundle T). This local definition globalizes.

Suppose that Vs G S, x(Fs) ? 0. Locally on S, there exists a complex F$ ?> T\ of

vector bundles representing Rtt*(F) with rk(^o) ?

rk^i). Taking the top exterior

product of V>? we find a canonical section cr(F) G T>(F). It is easy to see that for 5 G S, g(Fs)

= 0 if and only if h?(Xs, Fs) ? 0. The determinant of cohomology and its canonical section satisfy some compati

bility properties:

(1) If a : 0 ?> F\ ?> F2 ?? Fs ?> 0 is an exact sequence of 5-flat coherent sheaves on X, then there is an induced isomorphism

V(a):V(F2)^V(F1)^V(F3).

(2) If, in (1), the relative Euler characteristics of Fi, F2, and F3 are each zero, then

V(a)(*(F2)) = a(F1)?a(F3). In [Fal], G. Faltings makes the following definition.

Definition 2.1. Let / and J be 5-flat coherent sheaves on X. We say J and J coincide generically in if-theory if the following holds: For all s G S such that

depth(?s,s) ?

0, the difference of / and J in the ?T-theory of coherent Os,s flat sheaves on X Xs^pec(Os,s) is represented by a finite complex of sheaves, such that the support of its cohomology is finite over Spec Os,s

If, for example, X and S are reduced and irreducible and / and J axe 5-flat and have the same generic rank (that is, rank at the generic point of X), then they coincide generically in if-theory. This case will be sufficient for this paper.

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Page 5: The Strange Duality Conjecture for Generic Curves

238 PRAKASH BELKALE

Faltings then proves (Theorem LI of [Fal]) that if i" and J are two 5-flat coherent sheaves on X of the same relative Euler characteristic and which coincide generically in if-theory and if E, E' are vector bundles on X of the same rank with a given isomorphism ^ : det(E) ?> &et(E'), there is a natural isomorphism

Qe,e>(iI>) ' V(E ? I) ? V(E' ? J) ̂ V(E' ? I) ? V(E ? J)

with various functorial and compatibility properties (compatible exact sequences in (E,E')). We will always assume that I and J are in addition also of the same

Euler characteristic. This ensures that $e,e' (VO does not depend upon the choice

of the isomorphism if; : det(E') ^ det(E'). We will therefore drop the dependence on i?) in the maps $e,E'

In the remainder of this section we will recall the construction of $e,E' and note its basic properties.

Consider the special case when there exist line bundles L\,...,Lk and filtrations

E = Ex D E2 D D Er D 0 = Er+i,

Ef = E[DE2D"'DEfrD0

= E'r+1

and (given) isomorphisms Ej/Ej+i ??

Lj and Ej/Ej^ ??

Lj for j = 1,... ,r .

Such a situation can be arranged after a suitable flat base change of 5 (for details see [Fal]). If 5 =

Spec(C), no base changes are needed! In such a set up there are

induced isomorphisms r r

A : V(E ? I) -

Yl V{Lj ?I), B: V(E' <g> J) -? JJ V(Lj ? J),

r r

C : V(E' ? J) -> nP(^' ? J)' D : V(E?J) " IIP(LJ 0 J)*

The isomorphism $#,?' in "this good coordinate system" is just the identity map. That is, the diagram below commutes (where the bottom arrow is identity):

V(E ? I) ? V{E' ? J)-?-^ ?>(?' ? /) ? V(E ? J)

A<g>? C?D

re=i ^ ? /) ? n;=i ?(?i ? -/) -^ n;=i ^ ? /) ? n;=1 ^ ? a

Faltings shows that $#,?' is independent of the good coordinates (by this we mean the choice of filtrations of E and E'). Together with [KM], Proposition l(ii), it is easy to see that $e,e' is compatible with exact sequences in (I, J). That is, if

0 -

Ii -> J -+ I2 -

0, 0^Ji-?J-^J2-^0

are exact sequences of 5-flat coherent sheaves such that Ia, Ja have the same relative Euler characteristics for a = 1,2 and coincide generically in if-theory, then the

following diagram commutes (E and E1 are as above):

V(E ? I) ? V(E' ? J) *E,E' > V(E' ? I) ? 2?(JS? ? J)

D(E ? Ji) ? ?>(?*' ? Jx)? ?^ D(?' ? A) ? V(E ? Ji)?

D(?7 ? J2) <8> ?>(?' ? J2) D(Ef ? J2) ? P(J5 ? J2)

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Page 6: The Strange Duality Conjecture for Generic Curves

STRANGE DUALITY 239

where the bottom $e,e' is the tensor product of ?e,e' acting on the two levels.

Faltings does not list this property, and we learned it from the proof of Lemma 1 in [E].

The second observation, which is obvious in good coordinates, is that if I = J, then the isomorphism $e,e' is just the permutation of the two factors (if L is a one dimensional vector space with l\,l2 G L, then t\ ? l2 = t2 ? i\ G L ? L).

The third observation is that if (E\,..., Em; F\,..., Fm) are vector bundles on X such that E\,..., Em are vector bundles with isomorphic determinants and each of rank r, F\,... ,Fm aie vector bundles on X of the same relative degree and rank and if n G Sn (the symmetric group on {1,... ,n}), then there are "natu ral" isomorphisms, compatible with compositions:

V(EX ? Fi) ? V(E2 ? F2) ? ... ? V(Em ? Fm)

h V(E7r(1) ? Fi) ? V(E7c{2) ? F2) ? ... ? V(E<m) ? Fm). To see this, there are such maps for transpositions it viz. $Ei,Er Writing any permutation as a composition of transpositions, we will obtain maps for all 7r. The independence from the choice of the representation of tt as a composition of

permutations (as well as compatibility properties) follows from the following: "in

good coordinates for F?" all these maps are the identity.

2.2. Frames of vector bundles on curves. Let C be a reduced irreducible pro jective algebraic curve of arithmetic genus g with only ordinary double points for

singularities. Consider a tuple (E\,..., Em', F\,..., Fm) where

(PI) Ei,..., Em are vector bundles on C each of degree 0, rank r, and such that for i,j G {1,..., m}, det (?y is isomorphic to det (Ej),

(P2) Fi,..., Fm aie vector bundles on C, each of rank k and degree k(g ?

1). Notice that x(C, Ei? Fj) =0 and hence we have canonical elements cr(F? ? Fj) G

V(Ei ? Fj) (a one dimensional vector space). By the third observation in Section 2.1, the one dimensional vector spaces for

various ir G 5n (the symmetric group)

V(En(1) ? Fi) ? T>(En(2) ? F2) ? ... ? V(E7r{m) ? Fm)

aie all candnically identified. Consider the mxm matrix (<r(F??F:7)), i = 1,..., m, j

= 1,..., m. Because of the remark above it makes sense to speak about the non

zeroness of the determinant of this matrix.

Definition 2.2. A tuple (Ei,..., Em; Fi,...,Fm) satisfying properties (PI) and

(P2) is said to be an (m, r, fc)-frame on C if the determinant of the mxm matrix

(a(Ei ? Fj)), i = 1,..., m, j

= 1,..., m, is non-zero.

We will give a geometric interpretation of the definition of a frame. Let (Fi,..., Fm; Fi,..., Fm) be a tuple as above. Fix a vector bundle Eq of the same rank as

Ft's whose determinant is isomorphic to that of F? (for example one of the F?). Let F0 be a vector bundle of the same degree and rank as the F/s. Choose non zero elements Sj G V(E0 ? Fj) and t G V(E0 ? F0). These choices do not involve

Ei,..., Em. We have canonical isomorphisms

$e,Eo : V(E ? pi) ? ^(Eo ? F0) -^ V(E0 ? F?) ? 2?(F ? F0). The choice of t, sq, ..., sm gives a morphism Aj : V(E ? Fj)

? P(F ? Fq).

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Page 7: The Strange Duality Conjecture for Generic Curves

240 PRAKASH BELKALE

Therefore one obtains m tuples of complex numbers well defined up to non-zero

scalars 6(F) =

[Ai(a(E ? Fi)),..., Am(a(F ? Fm))]. But 0(F) (up to non-zero

scalars) does depend upon the choice of (Fq, Fq, t, Si,..., sm). For example, if we

change s?, then the entry i of 0(F) gets scaled. It is easy to verify that (Ei,..., Fm; Fi,..., Fm) is an (m, r, fc)-frame if and only

if O(Fi),..., 6(Fm) are linearly independent. The main properties of (m, r, A

frames are

Proposition 2.3. Let X G Mg and let (Ei,..., Fm; Fi,..., Fm) 6e an (m, r, &)

,/rarae on X with det(F?) -^ ?*. TTien 6Fj G H?(SUx(r),Ck), j = l,...,m, are linearly independent. Ifm> rki?0(5?7x(0>?fc), these sections give a basis of

H?(SUx(r),Ck) (and hence m = rkH?(SUx(r),Ck).)

Proof. From the determinantal condition in the definition of the frame, we see that

for each i there exists a j (similarly for each j there is an i) so that cr(F? ? Fj) ^ 0.

Therefore by Lemma A.l, F? and Fj aie semi-stable vector bundles on X for all i

and j . For the linear independence we can replace SUx(r) by a space Z that carries a universal family of semi-stable vector bundles of trivial determinant and rank r

on X. The same determinantal condition now implies the linear independence of

QFj. ?

Lemma 2.4. Suppose that an (m, r, k)-frame (Ei,..., Em', Fi,..., Fm) exists on a

reduced irreducible protective curve C with at most double point singularities. Then

there exists such an (m,r,k)-frame on C with det(F?) isomorphic to Oc

Proof. The group of line bundles of degree 0 on C is a divisible group. Suppose that Lr ^ det(Ei) (recall that we are assuming that F? is of degree 0). Then we

consider the modified frame

(Ei ? IT1,..., Em ? L'1', Fi ? L,..., Fm ? L). D

Lemma 2.5. Consider a protective flat family X ? 5 of curves. Let so G 5 be

such that XSo is a reduced irreducible curve with at most ordinary double point

singularities . Suppose that an (m, r, k)-frame exists on XSQ. Then there is an open subset U ? 5 containing sq such that Xs has an (m, r, k)-frame for each s ?U.

Proof. Let (Fi,..., Fm; Fi,..., Fm) be an (m, r, fc)-frame on XSQ. Using Lemma 2.4, assume that det(F?) ^> ?xs for i = 1,..., m.

Using Lemma A.2, find an ?tale base change Sf ?> 5 so that the image contains

so and (Fi..., Fm; Fi,..., Fm) lift to vector bundles on Xg' satisfying properties

(a) and (b) of the frame. The non-vanishing of the determinant is an open condition

and this concludes the proof. D

3. Line bundles and vector bundles on rational nodal curves

Let us fix a rational nodal curve C of arithmetic genus g together with a nor

malization P1 ?> C. Let ri,..., rg be the nodes of C and let f~x(vj) =

{pj, qj} for

j = 1,... ,g. This notation will be fixed throughout the paper. Let L be a line bundle of degree 0 on C (that is, the degree of f*L is zero).

It is clear that f*L is trivial; that is, there is a isomorphism unique up to scalars

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Page 8: The Strange Duality Conjecture for Generic Curves

STRANGE DUALITY 241

A : Opi ?? /*L. The canonical map f*(L)Pj ?>

f*(L)qj gives us a well defined scalar c? which makes the following diagram commute:

'cj c??*c

Y

f*LPj ?=+

/*L?.

We therefore obtain a well defined morphism of groups

{line bundles of degree 0 on C} ?

(C*)^, L i-> (ci,..., cp).

It is easily checked that this is a group isomorphism.

3.1. 5-bundles on rational nodal curves.

Definition 3.1. An 5-bundle on C is a pair (V, r) where V is a vector bundle on

P1 together with a p-tuple r = (ri,..., rg) where Tj : VPj

-^ Vqj for j = 1,..., g aie

morphisms of vector spaces.

Consider the surjective morphism f*V ?

0? Vg?lr? 0^lr? is the skyscraper sheaf supported at rj with fiber Vq. ) corresponding to the map VPj 0 Vq.

?> Vq.

given by -Tj on the first factor and the identity on the second. Let V be the kernel. It is a coherent sheaf on C and will be called the coherent sheaf underlying (V,r). It is easy to see that if the Tj are isomorphisms for j

= 1,... ,g, then V is the vector bundle on C obtained by gluing VPj and Vq. via the maps Tj, with a given

isomorphism f*V -^ V. Let (V, t) and (Q, 0) be 5-bundles as above. A morphism (V, r) ?> (Q, 0) is a

morphism r : V ?? Q on P1 such that for j = 1,..., g the diagram

QPj ?^ <39i

commutes. The (vector) space of such homomorphisms is denoted by Hom((y,r), (Q,0)).

Let (V.ri1)) and (V,t^) be 5-bundles on C with V = 0?r such that rj%)

are

isomorphisms for i = 1,2, j = 1,..., g. These give rise to coherent sheaves V\ and

V2 on C which are locally free sheaves. We view t> ' and r- as r x r matrices rj15 and ,f using the identification V ?

C?? . The following lemma is immediate:

Lemma 3.2. As line bundles on C, det(Vi) and det(V^) are isomorphic if and

only if det(rf]) =

det(rf}) eC* for j

= 1,... ,g.

4. Outline of the argument

Lemmas 2.5, 2.4 and Proposition 2.3 give a strategy for proving the strange duality conjecture for generic X G Mg.

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Page 9: The Strange Duality Conjecture for Generic Curves

242 PRAKASH BELKALE

Initial strategy. Construct an (m, r, fc)-frame on the rational nodal curve C of arithmetic genus g with m > M(r, k, g) (M(r, k, g) was defined in the introduction).

Here we use the fact that there exists a family of projective curves degenerating into C, such that the general member of this family is a curve in Mg.

An (m, r, fc)-frame on C will be constructed by working on the normalization P1 of C. We will first recall some relevant definitions and properties of "evenly split" bundles on P1.

4.1. Evenly split bundles on P1. A vector bundle T on P1 is said to be evenly split (ES) if W =

0"=1 Opi (ai) with | a? -

?j |< 1 for 0 < i < j < n. Let D and n be integers with n > 0. It is easy to show there is a unique, up

to isomorphism, ES-bundle of degree ? D and rank n on P1 (the negative sign is

introduced for conformity with notation used in quantum cohomology). Let T be a vector bundle on P1 of degree

? D and rank n. Define Gr(d, r, T) to be the moduli space of subbundles of T which are of degree

? d and rank r. This can be obtained as an open subset of the quot scheme of quotients of T of degree d? D and rank n ? r. In the notation of [Pot], we are looking at the open subset

of B.ilhn~r,d~D(T) formed by points where the quotient is locally free. The proof of the following standard result may be found in [Be2].

Proposition 4.1. Let T be an ES bundle of degree ?D and rank n. Then Gr(d, r, T) is smooth and irreducible of dimension r(n

? r) + dn ? Dr. Moreover, the subset of

Gr(d, r, T) formed by ES-subbundles V ? T such that T/V is also ES is open and dense in Gi(d,r,T).

4.2. The enumerative problem and the resulting frame on C. Let T be an

ES-vector bundle of rank n = r -\-k and degree %-l) onP1. Fix a generic tuple g

(7i,--.,7?i) e nHom? i(TPj,Tqj)

3=1

where Homn_i(TPj, Tqj) denotes the set of maps of vector spaces TPj ?+ Tq. of rank

n ? 1 (therefore the kernel of 7j is one dimensional.) The enumerative problem is the following: Count the number of "singular S

subbundles" (V,r) of (T,*y) so that deg(V) = 0 and rk(F)

= r. More precisely, we want to count subbundles V of T of degree 0 and rank r so that for j = 1,..., g,

lj(VPj)cVqj, the induced map tj :

VPj ?>

Vqj is singular.

In Proposition 6.1, we will show that there are only finitely many such S-subbundles and that the natural parameter scheme of such objects is reduced. The number m

of these will be shown to be > M(r, k, g) (Proposition 6.2). Denote them by V^ for a = 1,..., m. Let QW

= T/V^ for a = 1,..., m. Both

V^ and Q^ have natural S-bundle structures. We therefore obtain S-bundles

(y(a)jT(a)) and (qC^ #(<*)). We will show that the gluing maps 0^a) for Q^ aie

nonsingular (Lemma 6.4(h)). Hence the coherent sheaf underlying (Q^a\0^) is a

vector bundle Q^ of degree k(g ?

1). We perturb the maps r^ (and leave maps 0^ unchanged) so that they become

non-singular and such that the corresponding vector bundles (on C) V^ have

isomorphic determinant line bundles for a ? 1,..., m. This perturbation is possible

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STRANGE DUALITY 243

because we are starting from singular maps r- . Finally we will show, using the

transversality in the enumerative problem, that the tuple

((V^)\...,(V{m)y;Q{1\...,Qim))

is an (m, r, fc)-frame on the rational nodal curve C, using the fact that in the limit, an appropriate matrix is diagonal with non-zero entries on the diagonal. This will

conclude the proof.

5. Reformulation in terms of the diagonal

The enumerative problem given in Section 4.2 is a counting problem of the number of "singular" subbundles of a bundle on C with singular gluing maps. However recall that if tj were non-singular, then the set of subbundles of the induced

bundle T (on C) of degree 0 and rank r can be described as follows: Consider the natural map

7T : Gr(0,r,T) - f[(Gr(r,TPj) x Gr(r,T,.)).

Since there is a given isomorphism jj : TPj

?> TQj, we can define a "diagonal"

A,- = {(A,B) Gr(r,TPi) x Gr(r,T?) | 7j(A)

= B}.

The set of subbundles of T of degree 0 and rank r is just 7r_1 n?=i Ar In Section 6, we show that the enumerative problem in Section 4 has a simi

lar description, but we need to replace A? by a singular "diagonal". To obtain

transversality in the enumerative problem and to make use of it, we will need to

study tangent spaces as well.

5.1. Degeneration of the diagonal. Let W be a vector space of rank n and 0 < r < n an integer. We will consider partial degenerations of the diagonal A = {(A, B) G Gr(r, W) x Gr(r, W) \ A = B}.

If $ G End(W), then define

A* = {(A, B) G Gr(r, W) x Gr(r, W) | $(A) ? B}. We therefore obtain a subscheme

A ? Gr(r, W) x Gr(r, W) x End(W) such that the fiber of the projection T : A ?? End(W) over $ is A$. Clearly T is

flat over Aut(W). The map A -> Gr(r, W) x Gr(r, W) given by (A, B, $) ^ (A, B) is Zariski locally, a trivial fiber bundle with smooth fibers. Therefore A is smooth. We claim that T is flat over endomorphisms of rank > n ? 1. It suffices to

show that if $ is singular with kernel L of rank 1, then A$ is equidimensional of dimension r(n

? r). If (A, B) G A$, then either L C A or B C im($). These two

(irreducible) components each have the correct dimension (by a small calculation, for example, the second dimension is dim(Gr(r, n ? 1)) 4- dim(Gr(r, r + 1))).

Let L = Ci ? W and K ? W be subspaces of ranks 1 and n ? 1, respectively, so that the natural map L ? K ?> W is an isomorphism. Let /? : W ?> VF be the

corresponding projection to K. Here A^ is a union A+ U A_ where

A+ = {(A, B) G Gr(r, W) x Gr(r, W) \ ?(A) ?B,L? A}, A_ = {(A, B) G Gr(r, W) x Gr(r, W) \ ?(A) ?B,B? K}.

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244 PRAKASH BELKALE

Let K = Span{&2, , kn}. Define complete flags on W:

F.:0?Fi?F2?.--?Fn = W,

F'.:O?F??F??---?Fn = W,

as follows: F\ = L, Fi = L ? Span{fc2,... ,fc?} for i = 2, ...,n, and F[

=

Span{fcn,..., kn-i+i} for i = 1,..., n ? 1 and Fn = W. Define 4>a(t) : W ?? W for

a = 1,..., n as follows:

(?i (t) is multiplication by ? on L and identity on K, for a = 2,..., n, (?)a(t) is multiplication by t on Fa and <j>a(t)(ki)

= ki for ? > a.

For a subset / of [n] of cardinality r, let I' = {n +1

? ?, z G /} (see Section A.l). It

is well known that the cycle class (in cohomology) of the diagonal A ? Gr(r, W) x

Gr(r, W) is the sum of cycle classes of ?li(F0) x ?ip(F^), each with multiplicity one. There is actually a geometric degeneration of A into this union of products of Schubert varieties [BP]. The degenerations below of A, A+, and A_ are built in a similar manner.

Proposition 5.1. Consider the following limits in the Hubert scheme o/Gr(r, W) x

Gr(r,W): ? = lim (1 x (?n(tn)), lim (1 x <t>n-\(tn-i)) . . lim (1 x 01 (ti))A, tn? () tn_i?>0 ti? o

5+ = lim (1 x <l>n(tn)) hm (1 x (?)n-i(tn-i)) . lim (1 x (?2(t2))A+, tn?>0 tn_i?*0 t2?*o

6- = lim (1 x (j)n(tnj) hm (1 x 0n_i(tn-i))... lim (1 x 02(t2))A_. tn? O ?n_i? O ?2? O

Then, 5, ?+, and?- coincide set-theoretically with the schemes (J7 ??j(F.)x??j/(i^),

(J/:le/f2/(F#) x ??//(F^), and

U/:i^j^/(^?) x ^/'C^.)> respectively. This iden

tification is scheme-theoretic over the smooth open dense subschemes (in each)

U/n?(F.) x ?UF?), IW/?W) x WAK), and U/:i*/??(*".) x 0?,(F.'), re

spectively.

Remark 5.2. By Lemma 6 in [BL], the class [Oa] of A in the X-theory of coher ent sheaves on Gr(r, W) x Gr(r, W) coincides with the class of the reduced scheme

\JjQ,i(Fm) x f2//(F#/). It now follows from the above proposition (by consider

ing Hubert polynomials) that 8 equals (J/?2/(F.) x Qjf(F^) as schemes. Similar statements hold for ?+ and <5_. We will however not use this refinement of Propo sition 5.1.

Proof. We proceed by induction on n. First degenerate, as before,

lim (1 x </>i(?i))A =

A+ U A_. t\?>o

It suffices therefore to prove the statements about ?+ and S-. The case of <5_ is similar to that of S+, so we will prove only the statements for ?+. For any subscheme

X of Gr(r -

1, K) x Gr(r -

1, K), let T(X) be the Pn"r bundle over X given by

T(X) = {(A, B, A', B') G Gr(r, W) x Gr(r, W) x X :

(A',B')eX, A = ??L,BD B'}.

The natural map T(X) ?

Gr(r, W) x Gr(r, W) is birational onto its image.

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STRANGE DUALITY 245

Clearly T(A(r -

l,K)) -^ A+, where A(r -

1,K) is the diagonal in

Gr(r ?

1,K) x Gr(r ?

l,K). The operator T can be applied to families of sub

schemes of Gr(r -

1, K) x Gr(r -

1, K). Now apply the operator T to the induc

tively obtained degeneration of A (r ?

1,K) which is set-theoretically the subset

\Jj ?j(G.) x ?lj>(G'm) of A(r -

1, K) (here G. and G'm are the induced flags on K, and J runs through all subsets of [n

? 1] of rank r ? 1). The natural map

T([j?j(G.) x ?j<(G'.)) - Gr(r,W) x Gi(r,W)

j

has image U/:iej ^i(E*) x ̂r(K) and is birational over U/:ig/ ̂j(^) x ̂/'C^.') ^ Gr(r, W) x Gr(r, W) (the association between J and J is I = {1} U {a +1 : a G J}). The proof is now complete. D

5.2. Tangent spaces. Let

AS

It is easy to see that A+ is a smooth open dense subset of A+. Let (A, B) G A+. Recall that ? : W ?> W is the projection to K corresponding

to the decomposition W = L 0 K. Then ?(A) C B and ? induces a morphism

W/A -> wyB.

Lemma 5.3. For (A,B) G A+, ?/ie induced map ? : W/A ?

W/B is an isomor

phism. The induced map ? : A ?> B is singular with kernel of rank 1.

Proof. Consider the natural map p : K ?> W ?> W/jB (inclusion followed by

projection). The rank of the kernel is rk(K f) B) =

ik(B) ? 1 (since (A, B) $. A_,

B (?L K), and hence the rank of the image is rk(K) + 1 - rk(JB) =

xk(W/B). Therefore the composite K ?> W/B is surjective. But ? induces a surjective map

3 v W ?> K. The composite W ?> K ?+ W/B is surjective and A is in its kernel. So ? induces a surjective (hence isomorphism) map W/A ?> W/B.

Since ker(/3) = L and LcA, the second assertion is clear. D

Lemma 5.4. The tangent space to the scheme A? at (A, B) is the vector subspace

of the tangent space of Gr(r, W) x Gr(r, W) at (A, B) given by pairs of maps A -4

W/A, B -5 W/B such that the following diagram commutes:

A-?^B

TA Tb Y Y

W/A?t^W/B.

Proof Choose splittings W = A 0 W/A and W = B 0 W/B. For i/j G W/A, write

?(0 0 ip) = &W0 ? /?W0 (t'ie ̂ ast term arises from the natural map W/A ?

W/B). Suppose r^ : A ?> W/A and r# : ? ?> W/B are deformations of A and 5,

respectively. This means that the deformed A (similarly B) is the C[e]/(e2) span of elements of the form a 0 er^(a). The condition that this deformation stays inside A? is that /?(a0er^(a)) should be in the C[e]/(e2) span of terms of the form

60eF?(6). A typical element in this span looks like (b+eb')?eTB(b) with 6, bf G B. Write ?(a 0 er^(a))

= (b + e&') 0 eF?(6). This forces b = /3(a), and hence reading

the e term and its component in W/B, we get the desired commutativity. D

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246 PRAKASH BELKALE

6. 5-BUNDLES AND THE DIAGONAL

Let T be an ES-vector bundle of rank r + k and degree k(g ?

1) on F1. Fix

isomorphisms Gj : TPj

?> W and Hj : Tqj

?> W. We therefore have maps

g g

Gr(0,r,T) -+ f[(GT(r,TPj) x Gr(r,Tqj))

- f[(Gr(r,W)

x Gr(r,W)). 3=1 3=1

Call the composite $ = $(G, H). Fix W = L&K and the projection ? :W -^W

as in Section 5. Define jj : TPj

?? Tq. using the diagram

(6.1) TPj -^ w

Tqj-^W

Proposition 6.1. (a) For generic (G, H),

g g

^(G,H)-1(llA+) =

^(G,H)-1(l[Al). 3=1 3=1

(b) $(G, H)~1(Yl9j==1 A+) is in bisection with the set of singular subbundles of

degree 0 and rank r of (T,j) considered in Section 4.

(c) For generic (G,H), $(G, ff)~1(IIf=i A+) ?5 reduced and of the expected dimension 0.

Proof. Item (b) follows easily from the definitions. For (c), the expected dimension is

dimGr(0,r,T)-#dimGr(r,WO = rk + k(g

- l)r

- grk = 0.

Note that GL(W) x GL(W) acts transitively on Gr(r, W) x Gr(W). Therefore the

group nj=i(GL(^) x GL(W)) acts transitively on

II?=i(Gr(r, W) x Gi(r,W)). We now recall that A+ is smooth and connected. Hence, by Kleiman's transver

sality theorem (cf. [Kl], [Fu] ?B.9.2), for generic r G Uj=AGL(W) x GLW)>

^~1r~1(Y[9j=i A+) is reduced and of the expected dimension = 0. Notice that the

action of r just modifies the maps G and H. Therefore the assertion (c) follows. The equality in (a) holds because if S is a non-empty subset of {l,...,g}, the

expected dimension of (any component of) $(G, ^0_1 (11^=1 A?) *s negative where

Aj =

A+ifj(?S and Aj =

A+ n A_ if j G S. D

Now let G and H be generic and let

m=i^-^n a;)i, {W,..., 7<m>}=?-^n a;). 3=1 3=1

Let Q(fl) =

T/V{a) for a = 1,... ,m. We will now endow V^ and Q^ with 5 bundle structure using the diagrams, obtaining ?>-bundles (V^a\r^) and

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STRANGE DUALITY 247

(Q(a),0(a)), respectively:

(6.2) vPj

r<?)

Pj "

7j

^W

K9j *T? #i

-*w

and

(6.3)

L9j -

Q$ ?(a)

?ff

Proposition 6.2. The inequality m > M(r,k,g) holds.

The proof of Proposition 6.2 will appear in Section 8. It will eventually be shown that m =

M(r,k,g).

Lemma 6.3. The tangent space at V^ to $_1(r?j=i A^) is

Hom((yW,rW),(QW,?'a?)).

Proof. The tangent space to Gr(0,r,T) at V^ ? T is Hom(^?),T]/V^) by Grothendieck's theory of quot schemes.

For j = 1,..., g, let $? be the composition (where the last map is the projection

to the jth factor): g

Gr(0,r,T) - Y[Gi(r,TPj) x Gi(r,Tqj)

3 = 1

g

3 = 1

An element Y G Hom(V^a\T/F(a)) is in the tangent space of *_1(n?=i A+) at

V^ if and only for each j = l,...,g, ($j)*(T) is in the tangent space of A+ at (Gj(VPj),Hj(Vqj)). By Lemma 5.4, this condition implies that, as desired,

rGHom((^,rW),(gW,^)). D

Lemma 6.4. (i) RomdV^Kr^), (Q&KqW)) ? 0 if and only if a ?b. (ii) The coherent sheaves underlying the S-bundles (Q^a\0^), a = 1,... ,m,

are vector bundles on C of degree k(g ?

1).

(iii) For a = 1,..., m and j

= 1,..., g, the linear map of vector spaces r- :

VPj ? >

Vqj is singular with a rank 1 kernel.

(iv) The vector bundles V^ (resp. Q^) for a = 1,... ,m on P1 are ES and

are hence isomorphic. In particular, V^ -^ 0?r.

Proof. This proof is modeled after a similar proof in [Bel]. By Lemma 6.3, the

tangent space at V^ to $-1(A+) is JJomdV^Kr^), (Q(o),0(a))) which is conse

quently 0, because of Proposition 6.1.

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248 PRAKASH BELKALE

If a t? b, consider the non-zero (!) composite V^ ?? T ? Q^ (inclusion fol

lowed by projection). This composite belongs to Hom((V(a), rW), (Q(6), 0(fe))) (the property of being a morphism of S-bundles follows from diagrams (6.2) and (6.3)). This proves (i).

Now V^ G ?"1^), and therefore 0Ja) : Q$

-? Qqf is an isomorphism (see

Lemma 5.3). So (ii) follows immediately. Assertion (iii) follows because L ? ker(/3). Assertion (iv) follows from Proposi

tion 4.1 and Kleiman's transversality theorem. D

7. S-BUNDLES AND THE DETERMINANT OF COHOMOLOGY

Let V = Opi and let Q be an ES-bundle of degree k(g

? 1) and rank k. Let

(V, r) and (Q, 0) be 5-bundles on C so that 0j is an isomorphism for j = 1,..., g.

It follows that the coherent sheaf underlying (Q, 0) is an honest vector bundle Q on C of rank k and degree k(g

? 1).

Define A = A(r, 0), a coherent sheaf on C, by the following exact sequence of

sheaves: g

0^A-> f*Hom(V,Q) Z 0Hom(Fp.,Qq.)\r?

- 0 3=1

where the last sheaf is a skyscraper sheaf at Tj and

s(r) =

e, o rp.

- r?

o Tj e Kom(vPj,QPj). Since #7 are assumed to be isomorphisms, the map S is surjective. If r and

0 vary in families so that 0j aie isomorphisms (on each fiber over the parameter

space), then A(r, 0) is flat over the parameter space. This is because the kernel of a surjective map of flat modules is flat (long exact sequence in Tor!).

Since / is a finite morphism,

X(C, A) = x(C, UHom(V, Q)) - grk

= x(P\ Hom(V, Q))

- grk = kr(g

- 1) + kr - grk = 0.

So there is then a canonical section (note that for a coherent sheaf A on P1, V (A) =

V(fU)) g

(7.1) <t(t, 0) G V(A(r, 0)) ̂ V(Hom(V, Q)) 0 (g)det(Hom(Fp,, Qqj))*. 3=1

We note the following properties of this construction.

Lemma 7.1. (1) H?(C,A) = Egm((V,r),(Q,0)). (2) a(r, 0) ? 0 if and only t/Hom((V; r), (Q, 0)) = 0. (3) // the Tj are isomorphisms for j = 1,... ,g and V the locally free coherent

sheaf underlying (V,r), then we have a natural isomorphism of sheaves on

C: A-+Hom(V,Q).

7.1. Geometric S-bundles. Now consider the S-bundles obtained from geometry in Section 6 (Lemma 6.4). By Lemma 6.4(iv), the vector bundles V^ on P1 are all

isomorphic to the vector bundle V = 0?r, and the bundles Q^ aie all isomorphic

to Q where Q is the unique ES-bundle of the degree and rank of Q^. We choose

isomorphisms (V^, tW) ^ (V, f^) and (Q^a\ 0^) ^ (Q, 0^) for a = 1,..., m, for suitable f and 0.

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STRANGE DUALITY 249

From Lemma 6.4 we know that the 0?

' aie isomorphisms for j = 1,..., g. Con

sider,

g

a(f<a>,0<6>) eP(?^ 3=1

We may think of this as an m x m matrix with entries in the one dimen sional vector space V(Hom(V,Q)) ?

?9j=idet(?om(VPj,Qq.))*. By Lemma 6.4 and Lemma 7.1(2), the (a,b) entry is non-zero exactly when a = b. Hence the

matrix is non-singular.

Find maps AJa)(?) G Rom(VPj,Vqj) = Hom(Cr,Cr) (using the identification

V -+ Ofxr)

so that for j = 1,... ,g,

(l)X(ja)(0) =

f?a\ (2) det(\f(t))=t.

(we may find such matrices using Jordan canonical forms because the kernel of

fj is one dimensional for a = 1,..., m and j = 1,..., g). Therefore, for a, a' G

{1,... ,m}, det(Xj(t)) =

det(A^a '(t)) and these numbers are non-zero if t ̂ 0. Consider the mxm matrix formed by

a(\M(t),0M) G V(A(\(a)(t),0^)) ^ V(Hom(V,Q))?<$det(Kom(VPj,Qqj))\

3=1

This matrix is going to be non-singular for values of t in a sufficiently small Zariski neighborhood of t = 0. Let e ̂ 0 be one such value for t. Let V^ be the

locally free coherent sheaf underlying (V, A(a)(e)). Let Q^ be the coherent sheaf on

C underlying (Q,0^). From Lemma 6.4, we see that Q^ is locally free of degree k(g

? 1). According to Lemma 3.2, V^ have isomorphic determinant line bundles

of degree 0 for a = 1,..., m.

Lemma 7.2. The tuple ((F(1))*,..., (V(m))*; Q(1),..., Q(m)) is an (m, r, k)-frame on the rational nodal curve C.

Together with Lemma 2.5 and Proposition 2.3, this would conclude the proof of the strange duality conjecture for generic curves if we can show that the identifica tions that we made here are compatible with the determinant operation that went

with the definition of the frame.

7.2. The basic compatibility verification. Let (V, r) and (V, r?) be 5-bundles such that V is of degree 0 and rank r. Let V1 and V2 be the underlying coherent sheaves on C. Assume that Tj and rjj are isomorphisms for j = 1,..., g (so V1 and

V2 aie vector bundles on C). Assume further that det(F1) -^ det(V"2). Let (Q,0) and (Q, S) be 5-bundles such that 0j and ?j aie isomorphisms, j = 1,... ,g, and

<2i and Q2 are the corresponding vector bundles on C. Assume that Q is of rank

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250 PRAKASH BELKALE

k and degree k(g ?

1). We need to verify that the following diagram commutes: (7.2)

?W 0 Qi) 0 ?>(V2* ? Q2)-?-> P(V2* 0 Qi) 0 P(Vi* 0 Q2)

I 1 P(ftom(V, Q)) 0 Ugj=1 det(Hom(yp., Qqj ))* ?=^ ?>(Hom(V, Q)) 0 ]J?=i det(Hom(Vp., Qq. ))*

0 0

V{Hom{V, Q)) 0 nj=i det(Hom(Vp., Qq. ))* ?>(7ten(V, Q)) 0 nf=i det(Hom(Vp., Qqj ))*

For this we first consider two natural exact sequences associated to Qi and Q2. The one associated to Qi is

g

(7.3) 0 - Qi -> /*Q A 0Qg. I ri - 0

where a? : QPj 0 Q9j

?> Q9j takes (qi, q2) ?> q2

? 0j(qi). Notice that for a = 1,2

there are isomorphisms

UQ ? K* - /*(^* ? Q) = f*nom(v, Q),

and

(Q* I ^) ? K = Qqj ? ̂ I r; = Hom(yp.,Qq.) | r?.

From Section 2.1, recall the second observation and the compatibility under exact

sequences in (I, J). We now see that diagram (7.2) commutes.

8. Proof of Proposition 6.2

8.1. Conformai blocks. We recall the following.

(1) Irreducible polynomial representations of U(r) are parameterized by weakly

decreasing sequences of non-negative integers A = (Ai > A2 > > Ar) G Zr. These restrict to irreducible representations ? of SU(r). For such a A, set |A|

= Zra=i Aa e Z.

(2) (Ai, A2,..., Ar) and (/ii,/i2,... ,/xr) restrict to give the same irreducible

representation of SU(r) if Aa ?

/xa is a constant for a = 1,... ,r. The

congruence class of \?\ =

Y,a Va (mod r) is therefore a well defined Z/rZ "invariant" of the representation ? of SU(r).

(3) The dual of a representation ? of SU(r) is denoted by ?* and equals (?ii ?

/?r,/?i -/ir_i,...,_/xi -/?2,0).

(4) A representation A of SU(r) with A = (Ai, A2,..., Ar) is said to have level < k if Ai

? Ar < k.

Given irreducible representations ?1,... ,?s of level < k of SU(r), we obtain the

dimensions of spaces of conformai blocks Ng \?x,...,?s) (as in [B3]). It is known

that ivf?(0) = M(r,k,g) (cf. [BL], [Fa2] and [KNR]). There is a similar identifi

cation [Pa] of Ng (?1,..., ?s) with the rank of the space of sections of the fc-fold

tensor product of a certain line bundle on a moduli space of parabolic bundles over a smooth projective curve of genus g (with s marked points and associated

parabolic data ?1,... ,?s).

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STRANGE DUALITY 251

The factorization theorem of Tsuchiya, Ueno, and Yamada (see [TUY] and [B3]) gives

V

(where v rims through all irreducible representations of SU(r) of level < fc). This formula corresponds to a degeneration of smooth genus g curves with 5 marked

points into a nodal curve of arithmetic genus g and its subsequent normalization to a genus g

? 1 curve (this adds two new marked points). Repeated application of this formula gives

(8.1) iV?(0) = Y, N^\?\(?l)\?\(?*)\...,?9,(?3y)

where ?l,...,?9 vary over all irreducible representations of SU(r) of level < fc. The Verlinde algebra for SU(r) at level < fc is given by the rule

where v runs through all irreducible representations of SU(r) of level < fc.

8.2. Generalized Gromov-Witten numbers. Fix positive integers r and fc and set n = r + fc. Suppose I1,..., Is aie subsets of [n]

= {1,..., n} each of cardinality

r.

Let pi,..., ps be a set of distinct points on P1. Let T be an evenly split (ES) bun dle of degree -D and rank n (recall the definition of ES-bundles given in Section 4). Choose generic complete flags EC* on the fibers TPj for j = 1,..., s.

Define (??ji,l?i2, ... ,uis)^d to be the number of subbundles (0 if this number

is infinite) V of T of degree ? d and rank r, such that for j = 1,..., 5, the point

Vp. G Gr(r,Tp.) lies in the Schubert cell ?l0I?(E^). By Kleiman's transversality theorem the space of such V is equidimensional of dimension (using Proposition 4.1)

s s

dim(Gr(d, r, T)) ?

\^ codim(u;j?) =

r(n ?

r) + dn ? Dr ? \J codim(u;j?).

j=i 3=1

If D = 0, it is easy to see that the above definition gives the structure coefficients in the small quantum cohomology of Gr(r,n) (for example see [FP], Section 10).

Here we use the standard bijection between subbundles of 0?n of rank r and

degree ? d and maps P1 ?> Gr(r, n) of degree d. The numbers when D ^ 0 can be

recovered from the small quantum cohomology structure constants by using shift

operations (see Proposition A.3). One also notes that if L = {n

? r + 1, n ? r + 2,..., n}, then

(8.2) (k>ji,<*;j2,...,<*>i?)d,?> =

(w/i>W/2,...,w/?,a;L)d,?).

This is because we are imposing an open dense condition at the (s + l)th point.

8.3. Proof of Proposition 6.2. We return to the notation and setting of Propo sition 6.2. We introduce a new piece of notation relating Schubert cells in Gr(r, n) and representations of U(r) (equivalently, GL(r)). To an r-element subset / =

{?i < < ir} of [n] =

{1,... ,n} we associate a weakly decreasing sequence of

non-negative integers

(8.3) / ̂ A(7) =

(Ai > A2 > - > Ar) G Zr>0,

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252 PRAKASH BELKALE

where Aa = n ? r -f a

? ia, a =

1,..., r. We note the obvious:

Lemma 8.1. (1) As I runs through all r-element subsets of [n] with 1 G I,

X(I) runs (exactly once) through all irreducible representations of SU(r) of level < fc = n ? r.

(2) For I as in (1), ?(J') = X(I)\

We return to the proof of Proposition 6.2. Recall that by Proposition 5.1, A+ degenerates into a subscheme with support

(J (lj(F.) x ilriFl) ? Gr(r,W) x Gv(r,W). i.iei

By Proposition 5.1, the degeneration agrees as a scheme with the above (reduced) subscheme over the smooth open dense (Ji:iei ^/(^?) x ^r(Fi)

We may assume by modifying the maps Gj, Hj in diagram (6.1) that any

$~1(n|=1i?jj(F#) x

?^jjy(F^)) is smooth of the expected dimension zero (using Kleiman's transversality theorem). We may assume that any point in it is actually in ^(n^i 0}AF.) x %?y(K)) (m feet if U ? U9j=1 il?,(F.) x ilfrj,(J?) is any non-empty open subset, then for generic (G, H), ^~1(U) equals $_1

(11^=1 ^J* C^)

X?{IJy(F:)).) The number of points in ̂~1(]lf=i ^?ij (F*) x

^?(iJy(Fi)) is clearly

(Wji, . . . , Ui9 , W(liy,

. . . , UJ(jgy)o,-k(g-l)

This imphes that for some (t2,..., tn) with U ^ 0,

*-\\ X (?>n(tn))(l X 0n_i(tn_i)) ... (1 X <j)2(t2))A+

has at least

m0 = 2^(c*;ji,.

. . ,Ui9,UJ(iiy,... ,CJ(lgy)o,-k{g-l)

smooth and reduced points, where the sum is over all sequences of subsets (I1,..., I9) of [n] with r elements, each of which contains 1.

Hence for generic G and H (by absorbing 1 and <f>n(tn) o ...<j)(t2) in the maps G and H), we see that $(G, H)~1(Y[^z=1 A+) has at least ra0 isolated and reduced

points (the passage from A+ to A+ is clear because the intersection is over smooth

points of A+). So m > mo. The following proposition follows from a theorem of Witten and will be proved

in Section 8.5.1.

Proposition 8.2. Suppose I1,..., I9 are r-element subsets of [n] and pP ?

X(P) forj

= l,...,g. Then

From Proposition 8.2 and Lemma 8.1(1), it follows that

m0= Yl 4k\?\(?iy,?2,(?2r,...,?9,(?9y) A1,...,/**

where ?1,...^9 vary over all irreducible representations of SU(r) of level < fc. By

equation (8.1), one sees now that rao = M(r, k,g) and this concludes the proof of

Proposition 6.2.

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STRANGE DUALITY 253

8.4. Examples. We first consider the case fc = 1. In this case in the formula (|) n = r + 1 and I1,..., I9 aie subsets of [r +1]

= {1,..., r +1} each with r elements

and containing 1. There are r choices for each P. It is an easy calculation using the shift operations from Proposition A.3 that each of the summands of (t) is 1 and hence we obtain sum r9 as expected.

We move on to a more non-trivial example. We compute the sum in ({) to see that it agrees with 2fl,~1(2flr +1) (see [B3], Section 9). Here we notice that there are three allowable J's, i.e., {1,2}, {1,3}, and {1,4}. By a small calculation we obtain: If a summand in (J) has ? {1,3}'s, then that summand is 1 if ? = 0 or 1 and it is 2?~l otherwise. So the sum is

( o )29 +

( i )29_1 + ? ( ? )

2i~l29~? = 2ff"1(29 +1} The number h?(SUx(r),?k)/r9 is known to be symmetric in fc and r. It is

possible to see this as a consequence of Grassmann duality

Gr(r, r + k) ̂ Gr(fc, r + fc). The factor of r9 in the denominator comes from the requirement that I1,..., I9 in

(t) each contain 1 (this condition is not symmetric under Grassmann duality, but it is symmetric in a cyclic sense).

8.5. Witten's theorem and consequences. Recall that the small quantum co

homology of Gr(r, n) is an associative ring QH*(Gr(r, n)) whose underlying abelian

group is ?f*(Gr(r, n), Z) <S> Z[g] and the product structure is given by

ui*uj = ^2(u;i,c?j,u;K)djo qd^K' K,d

(see Section A.l for the notation) where K runs through all subsets of [n] of car

dinality r and where d runs though all non-negative integers and ujk' is the dual of u)k\ see Section A.l. We note the following formula for the (small) quantum product of Schubert classes:

K,d

Recall that n = r + k. Witten's theorem [W] gives an isomorphism

(8.4) W : QH*(Gr(r,n))/(q -

1) - R(U(r))k)U from the quantum cohomology of Gr(r, n) at q = 1 to the Verlinde algebra of U(r) at SU(r) level < fc and U(l) level < n. Here R(U(r))k,n is additively generated by sequences A = (Ai > A2 > > Ar) G Zr such that Ar > 0 and Ai < fc, where

Xa = n ? r + a ?

ia, a = 1,.. .,r. The isomorphism W takes ujj to the partition

X(I). R(U(r))k,n is related to the tensor product of the Verlinde algebra R(SU(r))k

for SU(r) at level < fc and the Verlinde algebra R(U(l))rn of U(l) at level < nr. We will make this relation precise. R(SU(r))k is a ring with an additive basis given by irreducible SU(r) representations of level < fc, and R(U(l))rn is generated as a

ring by x with relation xnr = 1. Inside R(SU(r))k <8>z R(U(l))rn consider the subspace R spanned by ? <g) xa so

that a = \X\ (mod r). Clearly R is a unital subring of R(SU(r))k <8> i?(?7(l))rn.

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254 PRAKASH BELKALE

On R consider the operator

(8.5) T(X 0 xa) = fj 0 xn+a

where fj is related to ? by the cyclic shift rj = (fc + Ar > Ai > A2 > > Ar_i) and a = \X\ (mod r). It is easy to check that T(u)

? u ((fc, 0,..., 0) 0 xn). The

Z-submodule of R generated by elements T(u) ? u for u G R is an ideal I of R.

Lemma 8.3. (1) Each orbit {Tb(??xa)rh=1} where a = \?\ (mod r), contains

a unique element of the form X 0 x'Al G R where the sequence X satisfies Ai < fc and Xr > 0 (and \X\ =

^?=1 A<* ? Zj. (2) lf?

= 0, then the T orbit o//20l does not contain any element of the form v 0 1 with v t? 0.

Proof. Find a subset K of [2n] so that X(K) = ?. Let c = a ? |A|. So ? 0 xa =

?(?T) 0 a^+IM^OI. Let J be the subset of [2n] obtained by subtracting 1 from

elements of K and replacing 0's by n. Then it is easy to see that X(J) 0#c+lA(J)l~r

equals ?(if) 0 zc+lAWI if 1 ? R and it equals T(A(K) 0 xc+lAWI) if 1 G if. Now by assumption c = 0 (mod r) and hence there is a unique ? G [n] so that c ? Ir = 0

(mod nr). We repeat this process ? times (the sought for b is the number of elements

in K that are less than or equal to ?). This proves (1). The proof of (2) follows

from the definition of T. D

The linear map R ?> R(U(r))k,n sending ? 0 x'A' G fi to ? G R(U(r))k,n is a

ring homomorphism with kernel / and induces an isomorphism of rings

R/I ̂ R(U(r))k,n.

Clearly elements of the form W(ui) where J runs over all r element subsets of [n] forms an additive basis for R(U(r))k,n

Let K = {1, fc + 2, fc + 3,..., fc + r = n) C [n]. Then using equation (8.5) on

A = (0,0,...,0)GZr,

(8.6) W(ujK)xr = 101.

Remark 8.4. The reader can verify the form of Witten's theorem needed in this

paper by just taking R/I to be the definition of R(U(r))k,n and deducing that the

map W is a ring homomorphism from knowing that Pieri's rule holds on both sides

of equation (8.4) (using [Ber] on the quantum cohomology side and [G] for the

Verlinde algebra for SU(r) at level < fc).

8.5.1. Consequence of Witten's theorem. Let I1,..., Is he subsets of [n] each of car

dinality r. Assume that there is an integer D so that X)j=i codim(u;js)

= rfc ? Dr.

Then we have the following proposition which implies Proposition 8.2 immediately:

Proposition 8.5.

(8.7) (u;Ii,ujI2,...,u;Is)0^D =

N^k)(?1,...,?s)

where ?3 =

X(P) are the associated representations of SU(r).

Proof. For simplicity assume D < 0 (this is the case when we need this proposition). Let K =

{1, fc + 2, fc + 3,..., n}. We claim

(8.8) (^Ji,^j2,...,^js)o,?> =

(l?Ii,UI2,...,U3Is,Uk,UK,...,UK)-D,G

with ujk repeated ? D times. To see this, apply the shift operation from Proposi

tion A.3 to each of the ? D uj^s appearing on the right hand side of equation (8.8).

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STRANGE DUALITY 255

According to Witten's theorem the two sides of equation (8.8) equal the coeffi cient of W(u;{iv..jr}) in the product

s

(8.9) W(ujK)-D fi W(u;jj) G R(U(r))k,n. 3 = 1

First using equation (8.6), we note W(uK) = 1 0 xrn~r. Also W(uI?)

= X(P) 0

xcodim(a;/J ) Then the U(l) component of the product in R corresponds to x raised to the number (mod rn)

2_. codim(-P) + Dr = r(n

? r).

3

Therefore using Lemma 8.3(2), the coefficient of W(uj{i^,^r}) in

8

W{u)k)'d^{W{u)u) 3=1

equals the coefficient of the identity representation in the product ]flj=i ^(^) in

the Verlinde algebra of SU(r) at level < fc ; which equals the right hand side of

equation (8.7) as desired. D

Appendix A. Some results on vector bundles on curves AND GROMOV-WiTTEN THEORY

A.l. Schubert cycles. Given a subset i" of [n] of cardinality r, we will assume

that it is written in the form {ii < i2 < < ir}. Let

E. : {0} =

E0?Ei?---CEn = W

be a complete flag in the n-dimensional vector space W. Define the Schubert cell

noj(E.)?Gi(r,W)hy

i??(E.) =

{Ve Gi(r, W) | vk(V DEu)=a for ia < u < ?a+1, a = 0,..., r}

where i$ is defined to be 0 and ?r+i = n. The cell Q^(Em) is smooth. Its closure

will be denoted by fij(E.), and the cycle class of this subvariety (in H* Gr(r, W)) is denoted by u;j. For a fixed complete flag on W, it is easy to see that every r-dimensional vector subspace belongs to a unique Schubert cell.

The dual of lji under the intersection pairing is uor where V = {n +1

? i, i G /}.

This means that if coding/) + codim(c<;j) =

r(n ?

r), the intersection number

uji. U?J in iJ2r(n~r)X = Z is 1 if J = V and it is 0 otherwise.

A.2. Vector bundles on curves. Let X be a smooth curve of genus g and let

E, F be vector bundles of ranks r and fc, respectively, such that x(X, E 0 F) = 0.

The following lemma is due to Faltings (see e.g. [S]).

Lemma A.l. If H?(X, E 0 F) = 0, then E and F are semi-stable.

Lemma A.2. Let X ?> 5 be a protective, flat family of curves and let sq G 5.

Suppose Eo is a vector bundle on XSo. Then there exists an ?tale map (T,to) ?

(5, so) and vector bundle ET on Xt so that Er,t0 = ^o- Ifdet(Eo) is trivial, then

there exist such (T,?q,Et) so that det(I?t) is trivial.

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Page 23: The Strange Duality Conjecture for Generic Curves

256 PRAKASH BELKALE

Proof (Standard; for example, see [SGA], Proposition l?.l, page 40). We need to extend Eq as a vector bundle. We will extend formally first, over successive thick

enings of the central fibers. Assume that E has been lifted to X& ?? Spec A (the base change of X ?> 5 to Spec A) and that B is an Artinian local ring with an ideal I such that I2 = 0 and B/I

= A. We want to extend E to XB -> Spec(?). This is clear because (formally) the obstruction to the extension over Artin local rings is in H2(Xa, Mn(I)) (nxn matrices with entries in I) which vanishes on a curve.

We now apply Artin approximation theorem [Ar] to obtain (T,t$). If det(l?o) is trivial and we want the extension to preserve this, the extension

problem over successive Artin rings is controlled by H2 of traceless matrices with coefficients in /, which is again 0. D

A.3. Shift operations in Gromov-Witten theory. Let I1,..., Is be subsets of

[n] each of cardinality r and let d, D be integers. Suppose I1 = {ii < < ir}.

Define J a subset of [n] of cardinality r and an integer d as follows:

(1) if ii > 1, let J = {ii ? 1 < < ir

? 1} and d = d,

(2) if ii = 1, let J = {i2 - 1 < < ir

- 1 < n} and d = d - 1.

The following proposition is proved in [Be2] (Proposition 2.5).

Proposition A.3.

(CJ/1, CJ/2, . . . , Uls)d,D =

(<*>J, CJ/2, . . . , Vl')?iD_i>

We recall the reason for the equality (see [Be2] for more details). We first verify that the expected dimensions of both the intersections in the above equality of Gromov-Witten numbers are the same. Let S = {pi,... ,ps} be a set of (distinct points) on P1 as before. Let T be an ES-bundle of degree

? D and rank n. Choose

generic complete flags Epj on the fibers TPj for j = 1,..., s.

Let t be a uniformizing parameter at pi G P1. Define T to be the vector bundle which agrees with T in P1 ? {pi} and whose sections in a small neighborhood U of

pi are sections s of T on U ? {pi} such that ts is a holomorphic section of T on

U whose fiber at pi lies in E{x (the first element of the flag). As coherent sheaves, TDT.lt can be shown that f is ES as well.

T inherits complete flags from T on its fibers at each point of {pi,..., ps}. There is a bijection between the set of subbundles of T and those of T both restricted to P1 ? {Pi}- Proposition A.3 follows from this bijection and a calculation at pi (this gives an inequality between the two Gromov-Witten numbers in Proposition A.3 to start with, but this is a cyclic process so we obtain equality).

Acknowledgments

I thank A. Boysal, P. Brosnan, M. V. Nori, and M. Popa for useful discussions. I acknowledge help from Mihnea Popa in correcting an initial naive idea.

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Department of Mathematics, University of North Carolina-Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, North Carolina 27599

E-mail address: belkaleQemail.unc.edu

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