Bundles of generalized theta functions over abelian surfaces
aa r X i v : . [ m a t h . AG ] D ec BUNDLES OF GENERALIZED THETA FUNCTIONS OVER ABELIANSURFACES
DRAGOS OPREA
Abstract.
We study the Verlinde bundles of generalized theta functions constructedfrom moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type ofthese bundles is expressed in terms of indecomposable semihomogeneous factors. Fur-thermore, Fourier-Mukai symmetries of the Verlinde bundles are found, consistently withstrange duality. Along the way, a transformation formula for the theta bundles is derived,extending a theorem of Dr´ezet-Narasimhan from curves to abelian surfaces. Introduction
Overview.
In this paper, we put forward an analogy between aspects the strangeduality proposal for curves and abelian surfaces. Such an analogy is by no means obvious:we study moduli spaces whose geometries are very different. It is therefore surprising thatthe emerging pictures share common features, some which we point out below.In short, our main results are:(i) we discuss how the theta line bundles over moduli spaces of sheaves on abeliansurfaces depend on the choice of reference sheaf; the case of curves was consideredin [DN];(ii) in degree 0, we determine the splitting type of the Verlinde bundles (defined inSection 1.6) in terms of indecomposable factors. In particular, we determine theaction of a certain group of torsion points on the space of generalized theta func-tions; the curve case was solved in [O2];(iii) via (i), we recast the abelian surface strange duality conjecture as a specific isomor-phism between the Verlinde bundle and its Fourier-Mukai transform, as in [Po].Furthermore, using (ii), we confirm that the relevant bundles of generalized thetafunctions are indeed abstractly isomorphic, in degree 0.As already mentioned, the results above parallel the case of curves. This is indeed oneof the main points, giving credence to the strange duality conjecture in the abelian sur-face context. The proofs however require new ideas, and the results may be of interestindependently of strange duality.We now detail the discussion.
Moduli of sheaves and their Albanese maps.
To set the stage, consider a com-plex polarized abelian surface ( A, Θ), and fix the Mukai type v of sheaves E → A , sothat v = ch( E ) . The Mukai vectors used in this paper are of the form v = ( r, k Θ , χ ) , and the following assumption will be made throughout:( A.1 ) the vector v = ( r, k Θ , χ ) is primitive of positive rank, the polarization Θ is generic ,and furthermore the Mukai self pairing d v := 12 h v, v i := k · Θ − rχ is an odd positive integer.We consider the moduli space M v of Θ-(semi)stable sheaves of topological type v . Themoduli space comes equipped with the Albanese morphism α v = ( α + , α − ) : M v → A × b A, which, up to the choice of a reference sheaf E , takes sheaves E to their determinant anddeterminant of the Fourier-Mukai transform α v ( E ) = (det R S ( E ) ⊗ det R S ( E ) ∨ , det E ⊗ det E ∨ ) . Here R S : D ( A ) → D ( b A )denotes the Fourier-Mukai transform. The Albanese fiber will be denoted by K v , thusparametrizing semistable sheaves with fixed determinant and fixed determinant of theirFourier-Mukai transform. It is known that K v is a holomorphic symplectic manifold ofdimension 2 d v −
2, deformation equivalent to the generalized Kummer variety of the samedimension [Y1].1.3.
Theta bundles.
We consider the natural theta line bundles over the above modulispaces. Assume that w is a Mukai vector orthogonal to v in the sense that in K -theorywe have χ ( v · w ) = 0 . this ensures that the moduli spaces we consider consist of stable sheaves only UNDLES OF GENERALIZED THETA FUNCTIONS 3
Pick a complex F → A representing the Mukai vector w , and following [Li], [LP], constructthe Fourier-Mukai transform of F with kernel the universal sheaf E → M v × A : Θ w = det R p ! ( E ⊗ L q ⋆ F ) − . Generalized theta functions are sections of Θ w over either one of the moduli spaces K v or M v considered above.1.4. Relating different theta bundles.
Over the moduli space M v , the notation Θ w isslightly imprecise, since the bundle Θ F → M v may depend on the choice of representative F . The following result, paralleling the Dr´ezet-Narasimhan theorem for bundles overcurves [DN], controls this imprecision: Theorem 1. If F and F have the same Mukai vector orthogonal to v we have Θ F = Θ F ⊗ (cid:0) ( − ◦ α + (cid:1) ⋆ (cid:0) det F ⊗ det F − (cid:1) ⊗ (cid:0) ( − ◦ α − (cid:1) ⋆ (cid:0) det R S ( F ) ⊗ det R S ( F ) − (cid:1) . By contrast, the notation Θ w → K v is unambiguous.1.5. Verlinde numbers.
The holomorphic Euler characteristics of the line bundles Θ w → K v are calculated in [MO]:(1) χ ( K v , Θ w ) = d v d v + d w (cid:18) d v + d w d v (cid:19) . In order to use these numerics for the study of generalized theta functions, one needs toprove that h ( K v , Θ w ) = χ ( K v , Θ w ) . This occurs for instance when Θ w carries no higher cohomology over K v , or alternativelyover some smooth birational model of K v . This is the case when( A.2 ) Θ w belongs to the movable cone of K v .By [BMOY], the above requirement is satisfied, symmetrically in v and w , for primitiveMukai vectors of the form( A.2 ) ′ v = ( r, k Θ , χ ) , w = ( r ′ , k ′ Θ , χ ′ ) with k, k ′ > χ, χ ′ < . In addition, if ( k, χ ) or ( k ′ , χ ′ ) = (1 , − A, Θ) is not a product of ellipticcurves. or by descent from the Quot scheme in the absence of the universal sheaf DRAGOS OPREA
Bundles of generalized theta functions.
When assumptions ( A1 ) − ( A2 ) aresatisfied, we push forward Θ w → M v via the Albanese morphism α v . We obtain in thisfashion the Verlinde bundle of generalized theta functions E ( v, w ) = ( α v ) ⋆ Θ w over the abelian four-fold A × b A . By (1) its rank equals d v d v + d w (cid:18) d v + d w d v (cid:19) . From the discussion above, it is clear that the Verlinde bundle is well-defined only upto translation. In Sections 2.1 and 2.2, in particular equations (7) and (11), we will fixthe normalization of the Albanese morphism and of the theta bundle Θ w → M v usedthroughout this paper, thus pinning down the Verlinde bundle unambigously.In the context of curves, the Verlinde bundles were introduced and studied in [Po], andwere further analyzed in [O1], [O2]. For abelian surfaces, we point out how the techniquesof [O1] and [O2] need to be changed to the new setup we consider.1.7. Semihomogeneous bundles.
Assume ( A, Θ) is a principally polarized surface ,such that the line bundle Θ is symmetric( − ⋆ Θ = Θ . We showed in [O2], in the context of a calculation for curves, that for any pair of coprimeodd integers ( a, b ) there exists a unique symmetric semihomogeneous bundle W a,b over A such that rank W a,b = a and det W a,b = Θ ab . The bundles W a,b are the higher dimensional analogues of the Atiyah bundles over ellipticcurves. Recall that semihomogeneity is the requirement that all translations of W a,b by x ∈ A are of the form t ⋆x W a,b = W a,b ⊗ y for some line bundle y over A . The bundles W a,b split after pullback(2) a ⋆ W a,b = a M i =1 Θ ab . We may also consider semihomogeneous bundles over the dual abelian variety ( b A, − b Θ),where − b Θ = det R S (Θ) − is the dual polarization. We will use the notation W † a,b for thecorresponding bundles of rank a and determinant b Θ ab . Most of our results also hold without change for nonprincipal polarizations. We focus on principalpolarizations to keep the numerics simple.
UNDLES OF GENERALIZED THETA FUNCTIONS 5
Action of torsion points on generalized theta functions.
We will now considerthe case c ( v ) = 0 i.e. v = ( r, , χ ) , with r and χ odd coprime integers, cf. (A.1) . The dual vector w must have the form w = ( rh, k Θ , − χh )for some integers h and k .We consider the action of ( x, y ) ∈ A × b A on the moduli space M v given by( x, y ) : E t ⋆x E ⊗ y. The action is seen to leave K v invariant provided χx = 0 , ry = 0 . Let us writegcd( χ, k ) = a, gcd( r, k ) = b. If the stronger condition ax = 0 , by = 0is satisfied, the action lifts to the line bundle Θ w → K v , see Section 3. We will assumethis is the case. Write χ ( K v , Θ w ) = X i ( − i H i ( K v , Θ w )for the signed sum of cohomologies of the theta bundle. We show Theorem 2. If ζ = ( x, y ) ∈ A [ a ] × b A [ b ] has order δ , then the trace of the action of ζ isgiven by Trace ( ζ, χ ( K v , Θ w )) = d v d v + d w (cid:18) d v /δ + d w /δd v /δ (cid:19) . The Theorem should hold for arbitrary c ( v ), but we are unable to prove this here.1.9. Explicit expressions for the Verlinde bundles.
We keep the same setup as inSubsection 1.8. In addition, we make the assumption ( A . ) requiring that Θ w belong tothe movable cone of K v . We give an explicit expression for the Verlinde bundles. The situation is easily under-stood when gcd( r, k ) = gcd( χ, k ) = 1 . The strict inequalities in (A.2) ′ are not fulfilled for the numerics we consider. However, ( A . ) isachieved for sufficiently large slope µ ( w ) > µ + . By the argument in Section 6 of [BMOY], an explicitbound is given by µ + = √− χr · r − √ r − , whenever χ = − r > DRAGOS OPREA
Then E ( v, w ) = M (cid:16) W − χ,k ⊠ W † r, − k ⊗ P − h (cid:17) . This much can easily be derived from the representation theory of Heisenberg groups.The difficulty of the calculation lies however in the case when the integers k and rχ are not coprime. Representation theory only gives E ( v, w ) up to torsion line bundles over A × b A of orders dividing ( a, b ) . We will prove the following:
Theorem 3.
We have E ( v, w ) = M ζ (cid:18) W − χa , ka ⊠ W † rb , − kb ⊗ P − h (cid:19) ⊗ ℓ ⊕ m ζ ζ . The sum is taken over torsion line bundles ζ over A × b A of orders dividing ( a, b ) . A linebundle ζ of order exactly ω comes with multiplicity m ζ = 1 d v + d w X δ | ab δ ( ab ) (cid:26) ab/ωδ (cid:27) (cid:18) d v /δ + d w /δd v /δ (cid:19) . The line bundles ℓ ζ → A × b A in the sum are roots of ζ of order (cid:0) − χa , rb (cid:1) : (cid:16) − χa , rb (cid:17) ℓ ζ = ζ. For each ζ , only one such root ℓ ζ is chosen. We will see that the choice of ℓ ζ does notaffect the expressions involved.The Jordan totient { } appearing above is defined in terms of prime factorization.Specifically, for any integer h ≥
2, we decompose h = p a . . . p a n n into powers of primes. We set (cid:26) λh (cid:27) = ( p a − . . . p a n − n does not divide λ, Q ni =1 (cid:16) ǫ i − p i (cid:17) otherwise , where ǫ i = ( p a i i | λ, h = 1.1.10. Fourier-Mukai symmetries.
We furthermore consider the interaction of the Ver-linde bundles with the Fourier-Mukai transform. Assuming both vectors v and w satisfy( A . ) − ( A . ), we show UNDLES OF GENERALIZED THETA FUNCTIONS 7
Theorem 4.
When c ( v ) and c ( w ) are divisible by their ranks r and r ′ , there is anisomorphism (3) E ( v, w ) ∨ ∼ = \ E ( w, v ) . The isomorphism (3) is obtained by direct comparison of both sides, using Theorem 3.The same result should be true for any vectors v and w satisfying ( A . ) − ( A . ).1.11. Strange duality.
The above symmetry of the Verlinde bundles is related to thestrange duality conjecture . This was observed in the case of curves by Popa [Po].Considering the fibers of (3) over the origin, we obtain isomorphic spaces(4) H ( K v , Θ w ) ∨ ∼ = H ( M w , Θ v ) . As stated, this is merely saying that the dimensions of both vector spaces agree. However,there is a geometrically induced map, called strange duality , which conjecturally yields theisomorphism above. Even stronger, various strange duality maps can be packaged into anexplicit bundle morphism, constructed as a corollary of Theorem 1(5) SD : E ( v, w ) ∨ → \ E ( w, v ) . In many cases, SD should provide a specific geometric isomorphism of bundles, as predictedby Theorem 4. This is proven for generic abelian surfaces in [BMOY] for an infinite classof topological types; the general case is however still open.1.12. Comparison to the case of curves.
To end this introduction, let us remark thata similar picture emerges in the case of curves, cf. [O2]. Indeed, let C be smooth of genus g , and consider the vectors v = r [ O C ] , w = k [ κ ] , for a Theta characteristic κ . The bundles of rank r , level k generalized theta functions E r,k = det ⋆ (cid:16) Θ kκ (cid:17) → Jac( C )can be obtained pushing forward the pluri-theta bundles Θ w = Θ kκ → M v via the deter-minant/Albanese map det : M v → Jac( C ) . The pushforwards E r,k take the form E r,k = M ζ W ra , ka ⊗ ℓ ⊕ m ζ ( r,k ) ζ . for surfaces, strange duality phenomena have first been studied by Le Potier [LP2] DRAGOS OPREA
The torsion line bundles ζ have orders dividing a = gcd( r, k ), and the ℓ ζ s are ra -roots of ζ . The multiplicities m ζ ( r, k ) of ℓ ζ are explicit. Furthermore, the symmetry E ∨ r,k ∼ = b E k,r holds true, and is a manifestation of strange duality.1.13. Outline.
The paper is organized as follows. The next section discusses prelimi-nary results about theta bundles and their behavior under ´etale pullbacks; in particularTheorem 1 is proved there. The third section is the heart of the paper and contains thecomputation confirming Theorem 2. This is the most involved of our calculations, andTheorems 3 and 4 follow from it.1.14.
Acknowledgements.
The author was supported by the NSF through grant DMS1150675 and by a Hellman Fellowship. Some of our results were announced in Fall 2011under more restrictive technical assumptions, which meantime have been relaxed.2.
Theta bundles over the moduli of sheaves
This section collects various observations about theta bundles. The main result hereis Theorem 1, which describes how the theta bundles depend on choices. Over curves,a similar statement was made in [DN], and proved by entirely different methods. Asa corollary of the theorem, we construct the strange duality map between the Verlindebundles.2.1.
Setup.
Let ( A, Θ) be a principally polarized abelian surface, with Θ a symmetricline bundle ( − ⋆ Θ = Θ . Throughout the paper, we will use the Fourier-Mukai transform R S : D ( A ) → D ( b A ) , R S ( E ) = R p ! ( P ⊗ q ⋆ E ) , where P is the normalized Poincar´e bundle over A × b A . If E satisfies the index theorem[M1], we often write b E for the sheaf representing R S ( E ), up to shift. We furthermorerecall the following identities [M1]: R S ( t ⋆x E ) = R S ( E ) ⊗ P − x , R S ( E ⊗ y ) = t ⋆y R S ( E ) , for x ∈ A, y ∈ b A . We set b Θ = det R S (Θ) , UNDLES OF GENERALIZED THETA FUNCTIONS 9 so that − b Θ is the polarization on the dual abelian variety b A . Finally, we writeΦ : A → b A, b Φ : b A → A for the morphisms induced by Θ and b Θ, so thatΦ ◦ b Φ = − , b Φ ◦ Φ = − . Consider two orthogonal Mukai vectors v = ( r, k Θ , χ ) , w = ( r ′ , k ′ Θ , χ ′ ) , satisfying assumptions ( A . ) and ( A . ) . Central for our arguments is the following dia-gram [Y1], [MO]:(6) K v × A × b A τ / / p (cid:15) (cid:15) M vα (cid:15) (cid:15) A × b A Ψ / / A × b A .
Here, the morphism τ : K v × A × b A → M v is given by τ ( E, x, y ) = t ⋆x E ⊗ y. Throughout this paper, we normalize the Albanese map α = ( α + , α − ) : M v → A × b A of the introduction by(7) α ( E ) = (cid:16) det R S ( E ) ⊗ b Θ − k , det E ⊗ Θ − k (cid:17) . As before, we write K v for the fiber of α over the origin. Lemma 4 . x, y ) = ( − χx + k b Φ( y ) , k Φ( x ) + ry ) . It can be seen that Ψ is ´etale of degree d v = ( k − rχ ) , but we will not need this fact.2.2. Properties of the theta bundles.
As already mentioned, we prove the followinganalogue of the Dr´ezet-Narashimhan theorem [DN], originally conjectured in [MO]. Thereader will notice that the argument below applies in higher generality (in particular, inSection 3 we will need this result for nonprincipal polarizations).
Theorem 1. If F , F have the same Mukai vector orthogonal to v , then Θ F = Θ F ⊗ (cid:0) ( − ◦ α + (cid:1) ⋆ (cid:0) det F ⊗ (det F ) ∨ (cid:1) ⊗ (cid:0) ( − ◦ α − (cid:1) ⋆ (cid:0) det R S ( F ) ⊗ det R S ( F ) ∨ (cid:1) . Proof.
We begin by noting that Θ F depends a priori on the holomorphic K -theory classof F . To compare the different theta bundles, we consider the virtual difference f = F − F . By Lemma 1 of [MO] or alternatively by Lemma 17 of [BS], in K -theory we can write(8) f = M − M + O Z − O Z , for line bundles M , M over A , and zero dimensional subschemes Z , Z . (The proof in[MO] is written for sheaves, but the case of complexes is a consequence.) By assumption,the Mukai vector of f equals 0. In particular c ( M ) = c ( M ) = ⇒ χ ( M ) = χ ( M ) . Since χ ( f ) = 0, it follows that the lengths of Z and Z are equal ℓ ( Z ) = ℓ ( Z ) := ℓ. Let us write m = M − M , n = O Z − O Z , and observe that Θ f = Θ m ⊗ Θ n . To prove the Theorem, we showΘ m ∼ = (( − ◦ α + ) ⋆ (det m ) ⊠ (( − ◦ α − ) ⋆ (det R S ( m ))and Θ n ∼ = (cid:0) ( − ◦ α − (cid:1) ⋆ det R S ( n ) . Note that det R S ( n ) = P a ( Z ) − a ( Z ) , where a is the addition morphism. The isomorphismΘ n ∼ = (cid:0) α − (cid:1) ⋆ P a ( Z ) − a ( Z ) will be checked along any test family of sheaves. Indeed, consider a flat family E → S × A of sheaves of type v , inducing a morphism α − : S → b A by taking determinants and twisting. Remark that by the see-saw theoremdet E = ( α − × ⋆ P ⊗ p ⋆ V ⊗ q ⋆ D, UNDLES OF GENERALIZED THETA FUNCTIONS 11 for some line bundle
V → S , with p, q being the projections from S × A to S and A . Here,we wrote D for the line bundle used to twist the determinants to reach degree zero (ofcourse, in our case, D = Θ k , but the notation emphasizes that the proof is general). Fora zero dimensional subscheme Z of length ℓ , we calculatedet R p ! ( E ⊗ q ⋆ O Z ) = O z ∈ Z det E z = O z ∈ Z (cid:0) ( α − ) ⋆ P z ⊗ V (cid:1) = ( α − ) ⋆ P a ( Z ) ⊗ V ℓ . Therefore,Θ n = (cid:16) ( α − ) ⋆ P a ( Z ) ⊗ V ℓ (cid:17) − ⊗ (cid:16) ( α − ) ⋆ P a ( Z ) ⊗ V ℓ (cid:17) = (cid:0) α − (cid:1) ⋆ P a ( Z ) − a ( Z ) , as claimed.It remains to prove the first isomorphism. To simplify notation, let us write M = M ⊗ ζ, M = M for ζ ∈ b A a degree 0 line bundle, and M an arbitrary line bundle over A . Set c M = det R S ( M ) . We have det m = ζ, anddet R S ( m ) = det R S ( M ⊗ ζ ) ⊗ det R S ( M ) − = t ⋆ζ c M ⊗ c M − = b Φ M ( ζ )where b Φ M : b A → A is the homomorphism induced by c M . We show(9) Θ m = ( α + ) ⋆ ζ − ⊠ ( α − ) ⋆ Φ c M ( ζ ) − . First, we will verify this equality after pullback by τ . To begin, we find the pullback ofthe right hand side: τ ⋆ (cid:0) ( α + ) ⋆ ζ − ⊠ ( α − ) ⋆ Φ c M ( ζ ) − (cid:1) = pr ⋆ Ψ ⋆ (cid:0) ζ − ⊠ Φ c M ( ζ ) − (cid:1) = pr ⋆ (cid:16) − χx + k b Φ( y ) , k Φ( x ) + ry (cid:17) ⋆ (cid:0) ζ − ⊠ Φ c M ( ζ ) − (cid:1) = pr ⋆ (cid:16) ζ χ ⊗ Φ ⋆ ( b Φ M ( ζ )) − k ⊠ b Φ ⋆ ζ − k ⊗ b Φ M ( ζ ) − r (cid:17) . We claim next that τ ⋆ Θ m = Q ⊠ pr ⋆ (cid:16) ζ χ ⊗ Φ ⋆ ( b Φ M ( ζ )) − k ⊠ b Φ ⋆ ζ − k ⊗ b Φ M ( ζ ) − r (cid:17) for some line bundle Q over K v , where Q is the restriction of τ ⋆ Θ m to K v . For M fixed,the Chern class of Q depends only on the Chern class of ζ , and since for ζ = O we obtainthe trivial bundle, this must be the case for all ζ ’s of degree zero. Since K v is simplyconnected, Q must be trivial. To prove the splitting above, note that for E ∈ K v , the restriction of τ ⋆ Θ m to { E }× A × b A equals L E = det R p ( m ⋆ E ⊗ p ⋆ P ⊗ p ⋆ ( M ⊗ ζ − M )) − , where m : A × A × b A → A denotes the addition on the first two factors, and the p ’s are the projections over thefactors of A × A × b A . We prove that(10) L E = (cid:16) χ ( E ) · ζ − Φ D ( b Φ M ( ζ )) (cid:17) ⊠ (cid:16) − b Φ D ( ζ ) − rank ( E ) · b Φ M ( ζ ) (cid:17) , where D is the determinant of E (of course, D = Θ k , but as before we prefer to write ageneral proof of the Theorem). HereΦ D : A → b A, b Φ D : b A → A are the Mumford homomorphisms induced by the line bundles D and det R S ( D ) respec-tively. We also switched to additive notation, for ease of reading.The idea of the proof is already contained in the above argument. We first note that L E only depends on the holomorphic K -theory class of E . In fact, we argue that L E dependson the rank, determinant and Euler characteristic of E . Indeed, for two sheaves E , E with the same data as above, we form the virtual difference e = E − E . Using equation (8), we can write e = O Z − O W , with ℓ ( Z ) = ℓ ( W ) . Then L e = L E ⊗ L − E is trivial, since for any a ∈ A , we havedet R p ( m ⋆ O a ⊗ p ⋆ P ⊗ p ⋆ ( M ⊗ ζ − M ))= det (( t a × ◦ ( − × ⋆ ( P ⊗ p ⋆ ( M ⊗ ζ − M ))= (( t a × ◦ ( − × ⋆ p ⋆ ζ = pr ⋆ ζ − . Since for fixed rank and Euler characteristic, L E depends only on the determinant of E ,we may assume that E splits into rank 1 and rank 0 factors. Since both sides of (10)change multiplicatively as E splits, it therefore suffices to consider the cases E = line bundle , E = O a , a ∈ A. UNDLES OF GENERALIZED THETA FUNCTIONS 13
Now, note that for E = O a we obtain by the preceding paragraph that L O a = det R p ( m ⋆ O a ⊗ p ⋆ P ⊗ p ⋆ ( M ⊗ ζ − M )) − ∼ = pr ⋆A ζ. The calculation when E = O ( D ) is a line bundle will be more involved. We need to show L = (cid:16) χ ( D ) · ζ − Φ D ( b Φ M ( ζ )) (cid:17) ⊠ (cid:16) − b Φ D ( ζ ) − b Φ M ( ζ ) (cid:17) . Observe that over A × A we have m ⋆ O ( D ) = (1 × Φ D ) ⋆ P ⊗ p ⋆ O ( D ) ⊗ p ⋆ O ( D ) . We compute L − = det R p ( p ⋆ (1 × Φ D ) ⋆ P ⊗ p ⋆ P ⊗ p ⋆ ( M ⊗ ζ ⊗ D − M ⊗ D ))= (Φ D × ⋆ det R p ( p ⋆ P ⊗ p ⋆ P ⊗ p ⋆ ( M ⊗ ζ ⊗ D − M ⊗ D )) . The pullback p ⋆ O ( D ) did not contribute above by the projection formula and the vanishingof the Euler characteristic. Note furthermore that p ⋆ P ⊗ p ⋆ P = (1 × b m ) ⋆ P where 1 × b m : A × b A × b A → A × b A is the addition map. We conclude L − = (Φ D × ⋆ det R p ((1 × b m ) ⋆ P ⊗ p ⋆ ( M ⊗ D ⊗ ζ − M ⊗ D ))= (Φ D × ⋆ b m ⋆ det R p ( P ⊗ p ⋆ ( M ⊗ D ⊗ ζ − M ⊗ D ))= (Φ D × ⋆ b m ⋆ (cid:0) det R S ( M ⊗ D ⊗ ζ ) ⊗ det R S ( M ⊗ D ) − (cid:1) = (Φ D × ⋆ b m ⋆ (cid:0) t ⋆ζ det R S ( M ⊗ D ) ⊗ det R S ( M ⊗ D ) − (cid:1) = (Φ D × ⋆ b m ⋆ b Φ M + D ( ζ ) . Now for a degree 0 line bundle U we have b m ⋆ U = p ⋆ U ⊗ p ⋆ U, hence L − = (Φ D × ⋆ (cid:16)b Φ D + M ( ζ ) ⊠ b Φ D + M ( ζ ) (cid:17) = (Φ D × ⋆ (cid:16) ( b Φ D ( ζ ) + b Φ M ( ζ )) ⊠ ( b Φ D ( ζ ) + b Φ M ( ζ )) (cid:17) = (cid:16) − χ ( D ) · ζ + Φ D ( b Φ M ( ζ )) (cid:17) ⊠ (cid:16)b Φ D ( ζ ) + b Φ M ( ζ ) (cid:17) as claimed in (10). For the last equality, we used that Φ D ◦ b Φ D = − χ ( D ) · , see Lemma4 . Equality (9) is now checked under pullback by τ . To complete the argument, fix M .Observe that the assignment ζ → Θ m ⊗ (cid:0) ( α + ) ⋆ ζ ⊠ ( α − ) ⋆ Φ c M ( ζ ) (cid:1) defines a morphism π : b A → Pic ( M v ) . Note moreover that the above discussion implies that τ ⋆ ◦ π = 0 . Since the kernel of τ ⋆ is discrete, π must be constant. Since π ( O ) = O , we must have π ( ζ ) = O for all ζ ∈ b A , completing the proof. (cid:3) Convention 1.
The theorem above shows that Θ F only depends on the rank, Eulercharacteristic, determinant and determinant of the Fourier-Mukai of the bundle F . TheMukai vectors used in this paper are all of the form w = ( r ′ , k ′ Θ , χ ′ ) . We define the normalized theta bundlesΘ w := Θ F → M v , for complexes F satisfying(11) rank F = r ′ , χ ( F ) = χ ′ , det F = Θ k ′ , det R S ( F ) = b Θ k ′ . Even though the exact choice for F is irrelevant, for concreteness we may take F = ( r ′ − O ⊕ Θ k ′ ⊕ ( χ ′ − k ′ ) O o , with o ∈ A denoting the origin. The normalization we use is aligned with that of theAlbanese morphism in (7). Example 1.
Assume that v = (1 , , − n ) so that M v ∼ = A [ n ] × b A via the isomorphism( Z, y ) I Z ⊗ y. Then, α becomes the morphism( − a,
1) : A [ n ] × b A → A × b A, where as usual a is the addition map. For a sheaf F → A of rank r , we obtainΘ F = det R p ( p ⋆ I Z ⊗ p ⋆ F ⊗ p ⋆ P ) − , UNDLES OF GENERALIZED THETA FUNCTIONS 15 where the projections are considered onto the factors of the product A [ n ] × b A × A and Z is the universal subscheme in A [ n ] × A . This yieldsΘ F = det R p ( p ⋆ F ⊗ p ⋆ P ) − ⊗ det R p ( p ⋆ O Z ⊗ p ⋆ F ⊗ p ⋆ P ) . The second line bundle can be found via the see-saw theorem and Section 5 of [EGL](det F ) ( n ) ⊗ M r ⊗ ( a, ⋆ P r . Here, M is half the exceptional divisor on the Hilbert scheme and ( · ) ( n ) denotes thesymmetrization of a line bundle from A over Sym n ( A ). Therefore, over A [ n ] × b A ,Θ F = (cid:0)(cid:0) (det F ) ( n ) ⊗ M r (cid:1) ⊠ det R S ( F ) − (cid:1) ⊗ ( a, ⋆ P r . This expression is consistent with the statement of the theorem.2.3.
Theta bundles and ´etale pullbacks.
We now return to the ´etale diagram ofSection 2.1: K v × A × b A τ / / p (cid:15) (cid:15) M vα (cid:15) (cid:15) A × b A Ψ / / A × b A .
We continue to refer the reader to [MO], where we showed the splitting of the pullback τ ⋆ Θ w = Θ w ⊠ L . The line bundle Θ w was shown to be independent of choices on the simply connectedmanifold K v . Furhermore, we proved that L = (det R p ( m ⋆ E ⊗ p ⋆ P ⊗ p ⋆ F )) − , for E ∈ K v and for F satisfying (11). Here, the p ’s denote the projections on the corre-sponding factors of A × A × b A, while m : A × A × b A → A is as usual the addition on the first two factors. We moreover calculated the Euler char-acteristic of L in [MO]. In the lemma below, we identify L explicitly. As a corollary weobtain(12) Ψ ⋆ E ( v, w ) = p ⋆ τ ⋆ Θ w = H ( K v , Θ w ) ⊗ L . Lemma 1.
We have L = (cid:16) Θ − χ ′ k − χk ′ ⊠ b Θ − rk ′ − r ′ k (cid:17) ⊗ P r ′ χ + kk ′ . The Lemma will be used in Section 3.4 for k = 0, but for future reference, we write theproof for all k . The result is also valid for nonprincipal polarizations. In this case, theexponent of the Poincar´e bundle should be modified to r ′ χ + kk ′ e where e = χ (Θ). Proof.
The proof is similar to that of Theorem 1. Pick two complexes E and F satisfyingConvention 1, not necessarily orthogonal in the Mukai pairing. First, we note that thebundle L E,F = (det R p ( m ⋆ E ⊗ p ⋆ P ⊗ p ⋆ F )) − depends only on the holomorphic K -theory classes of E and F . We will furthermoreremark below that the line bundle only depends on the Mukai vectors v and w , thedeterminant and determinant of the Fourier-Mukai of E and F . In fact, only the statementabout the first argument E will be useful to us, so that we show L E,F ∼ = L E ′ ,F when ( E, E ′ ) have the same Mukai vectors, determinants and determinants of Fourier-Mukai.To prove this isomorphism, consider the virtual sheaf e = E − E ′ . Note that by equation (8), in K -theory we have e = O Z − O W for two zero-dimensional subschemes which have the same length. Since the Fourier-Mukaitransform of e has trivial determinant, we must have a ( Z ) = a ( W ) . We prove L e ,F = L E,F ⊗ L − E ′ ,F is trivial. Over A × b A , we calculatedet R p ( m O Z ⊗ p ⋆ P ⊗ p ⋆ F ) ∼ = O z ∈ Z (( t z × ◦ ( − × ⋆ det ( P ⊗ p ⋆ F )which only depends on a ( Z ) by the theorem of the square. Thus, we obtain the sameanswer replacing Z by W , therefore showing L e ,F is trivial.With this understood, we prove the lemma. We may assume then that E splits as adirect sum of copies of O , Θ and structure sheaves O o E = ( r − k ) O + k Θ + ( χ − k ) O o . In fact, since L E,F is multiplicative as E splits, it suffices to prove the lemma separatelyfor the three sheaves E = O , E = Θ and E = O o . UNDLES OF GENERALIZED THETA FUNCTIONS 17
First, for E = O , we obtain L = det R p ( m ⋆ O ⊗ p ⋆ P ⊗ p ⋆ F ) − ∼ = O ⊠ (det R S ( F )) − = O ⊠ b Θ − k ′ , while for E = O o , we have L = det R p ( m O o ⊗ p ⋆ P ⊗ p ⋆ F ) − ∼ = det ( − , ⋆ ( P ⊗ p ⋆ F ) − ∼ = P r ′ ⊗ Θ − k ′ . The calculation for E = Θ is more involved. We show L = det R p ( m ⋆ Θ ⊗ p ⋆ P ⊗ p ⋆ F ) − = (cid:16) Θ − χ ′ − k ′ ⊠ b Θ − k ′ − r ′ (cid:17) ⊗ P r ′ + k ′ . Observe that over A × A m ⋆ Θ = (1 × Φ) ⋆ P ⊗ p ⋆ Θ ⊗ p ⋆ Θ . We calculate L = det R p ((1 × Φ) ⋆ p ⋆ P ⊗ p ⋆ P ⊗ p ⋆ ( F ⊗ Θ) ⊗ p ⋆ Θ) − = det R p ((1 × Φ) ⋆ p ⋆ P ⊗ p ⋆ P ⊗ p ⋆ ( F ⊗ Θ)) − ⊗ pr ⋆ Θ − r ′ − χ ′ − k ′ , where we noted that χ ( F ⊗ Θ) = r ′ + χ ′ + 2 k ′ . This in turn becomes L = (Φ × ⋆ det R p ( p ⋆ P ⊗ p ⋆ P ⊗ p ⋆ ( F ⊗ Θ)) − ⊗ pr ⋆ Θ − r ′ − χ ′ − k ′ = (Φ × ⋆ det R p ((1 × b m ) ⋆ P ⊗ p ⋆ ( F ⊗ Θ)) − ⊗ pr ⋆ Θ − r ′ − χ ′ − k ′ where we noted again that over A × b A × b A we have p ⋆ P ⊗ p ⋆ P = (1 × b m ) ⋆ P . We continue the calculation L = (Φ × ⋆ det ( R p ((1 × b m ) ⋆ P ⊗ p ⋆ ( F ⊗ Θ))) − ⊗ Θ − r ′ − χ ′ − k ′ = (Φ × ⋆ b m ⋆ det R p ( P ⊗ p ⋆ ( F ⊗ Θ)) − ⊗ Θ − r ′ − χ ′ − k ′ = (Φ × ⋆ b m ⋆ det R S ( F ⊗ Θ) − ⊗ Θ − r ′ − χ ′ − k ′ = (Φ × ⋆ b m ⋆ b Θ − r ′ − k ′ ⊗ Θ − r ′ − χ ′ − k ′ Noting again that b m ⋆ b Θ = ( b Φ × ⋆ P ⊗ p ⋆ b Θ ⊗ p ⋆ b Θ , we obtain the result L = (Φ × ⋆ ( b Φ × ⋆ P − r ′ − k ′ ⊗ (cid:16) Φ ⋆ b Θ − r ′ − k ′ ⊠ b Θ − r ′ − k ′ (cid:17) ⊗ Θ − r ′ − χ ′ − k ′ = ( − , ⋆ P − r ′ − k ′ ⊗ (cid:16) Θ r ′ + k ′ ⊠ b Θ − r ′ − k ′ (cid:17) ⊗ Θ − r ′ − χ ′ − k ′ = P r ′ + k ′ ⊗ (cid:16) Θ − χ ′ − k ′ ⊠ b Θ − r ′ − k ′ (cid:17) . The lemma is now proved.The only detail that still needs clarification is the fact, used above, that(13) det R S ( F ⊗ Θ) = b Θ r ′ + k ′ . Here, we crucially use that Θ is symmetric so thatdet R S (Θ k ) = b Θ k . This statement follows for instance by taking determinants in Lemma 2(ii) in [O2]. Thelemma is stated for odd numerics, but the case k even follows since both sides dependpolynomially in k . For the left hand side, this is a general statement about integraltransforms, which can be proved via induction on dimension. The inductive step consistsin cutting with hyperplanes in the (pluri)-theta series to reduce dimension.To prove (13), let A F = det R S ( F ⊗ Θ) , and observe that A F depends on the rank, Euler characteristic, determinant and deter-minant of the Fourier-Mukai transform of F . Indeed, if F and F are two such sheaves,we write f = F − F = O Z − O W , where a ( Z ) = a ( W ) . But then A F ⊗ A − F = A f = det R S (Θ ⊗ f ) = det R S (Θ ⊗ ( O Z − O W )) = P a ( Z ) − a ( W ) = O . Therefore, it suffices to assume that F = ( r ′ − O ⊕ Θ k ′ ⊕ ( χ ′ − k ′ ) O o . To conclude, note that in this case A F = det R S (Θ) r ′ − ⊗ det R S (Θ k ′ +1 ) ⊗ det R S (Θ ⊗ O o ) χ ′ − k ′ = b Θ r ′ − ⊗ b Θ k ′ +1 = b Θ r ′ + k ′ . (cid:3) UNDLES OF GENERALIZED THETA FUNCTIONS 19
Construction of the strange duality map.
In this subsection, we use Theorem1 to construct the duality map SD mentioned in equation (5) of the Introduction when c ( v ⊗ w ) = 0. A similar construction was achieved in [Po] in the case of curves bypackaging together all the strange duality morphisms relatively over the Jacobian.We assume that c ( v ⊗ w ) · Θ > . The case c ( v ⊗ w ) · Θ < E and F we have H ( E ⊗ F ) = 0 . Furthermore, the locus(14) D = { ( E, F ) : h ( E ⊗ F ) = 0 } ֒ → M v × M w has expected codimension 1. The defining equation of (14) is used to prove that: Lemma 2.
There exists a natural morphism SD : j ⋆ E ( v, w ) ∨ → \ E ( w, v )Here we write j : A × b A → A × b A for the multiplication by ( − , − j will not be necessary.Note that both sides are locally free. For the left hand side, we use (12). For the righthand side, we argue that E ( w, v ) satisfies the index theorem with index 0. This can bechecked after pullback under isogenies. To this end, we invoke (12) and the fact that underour assumptions, L is ample, cf. the criterion used in the proof of Theorem 4, Step 3 . Proof.
To construct SD , we need a natural section of the bundle j ⋆ E ( v, w ) ⊗ \ E ( w, v ) = pr ⋆ ( j ⋆ E ( v, w ) ⊠ E ( w, v ) ⊗ Q )where pr : A × b A × A × b A → A × b A is the projection onto the first two factors, and Q → ( A × b A ) × ( A × b A ) is the Poincar´ebundle on the self-dual abelian variety A × b A . Note furthermore that j ⋆ E ( v, w ) ⊠ E ( w, v ) ⊗ Q = α ⋆ (Θ w ⊠ Θ v ⊗ α ⋆ Q )where α = ( j ◦ α v ) × α w : M v × M w → ( A × b A ) × ( A × b A ) . We will construct a natural section of the line bundleΘ w ⊠ Θ v ⊗ α ⋆ Q → M v × M w , which we will then pushforward by pr ◦ α to complete the proof.For simplicity let us assume that universal sheaves E → M v × A and F → M w × A exist, using quasi-universal families otherwise. We form the bundle D = det R p ! ( E ⊠ L F ) − , obtained by pushforward via the projection p : M v × M w × A → M v × M w . We claim that D = Θ w ⊠ Θ v ⊗ α ⋆ Q . This is precisely the see-saw principle combined with Theorem 1. Indeed, the restrictionΘ E of Θ to { E } × M w equalsΘ E = Θ v ⊗ α ⋆w (cid:16) (det E ⊗ Θ − k ) ∨ ⊠ (det R S ( E ) ⊗ b Θ − k ) ∨ (cid:17) = Θ v ⊗ α ⋆ Q| { E }× M w . The calculation of the restriction to M v × { F } is similar:Θ F = Θ w ⊗ α ⋆v (cid:16) (det F ⊗ Θ − k ′ ) ∨ ⊠ (det R S ( F ) ⊗ b Θ − k ′ ) ∨ (cid:17) = Θ w ⊗ α ⋆ Q| M v ×{ F } . We can now complete the proof. The locus where
T or ( E, F ) =
T or ( E, F ) = 0has complement of codimension at least 2 in the product space by Proposition 0 . . E and F are generically locally free in their corresponding moduli spaces,giving the claim. The only exception occurs for vectors of the form v = r exp( c ( L )) − [pt]for a fixed line bundle L → A . In this case, E is always nonlocally free. In fact, byCorollary 4 . E ’s are obtained as kernels of morphisms H ⊗ L → C x for x ∈ A , and for homogeneous vector bundles H → A , so E fails to be locally free at x . The codimension analysis holds true in this case as well: for each fixed F , the locusof E ’s which intersects the singularity locus of F has codimension 2 in the moduli space.The argument when either E or F have rank 1 also follows by a moduli count. UNDLES OF GENERALIZED THETA FUNCTIONS 21
Now, along the locus of vanishing Tor’s, the pushforward R p ! ( E ⊠ L F ) can be representedby a two step complex of equal rank vector bundles0 → A σ → A → . This yields a section det σ ofdet A ⊗ det A − = det R p ! ( E ⊠ L F ) − = D = Θ w ⊠ Θ v ⊗ α ⋆ Q vanishing precisely along the theta locus (14). (cid:3) Conjecture 1.
The morphism SD : j ⋆ E ( v, w ) ∨ → \ E ( w, v ) is an isomorphism. Remark 1.
Just as in the case of curves, there is a slight asymmetry in the roles of v and w in the strange duality morphism (4): on one side, the determinant and determinant ofthe Fourier-Mukai vary, while on the other side these invariants are kept fixed. However,just as in the case of curves [Pol], the above reformulation makes it clear that Corollary 1.
If the duality morphism (4) is an isomorphism for the pair ( v, w ) , then itis an isomorphism for the pair ( w, v ) . The Verlinde bundles in degree
Setup.
We specialize to the case c ( v ) = 0 i.e. we assume v = ( r, , χ ) , w = ( rh, k Θ , − χh )with r, χ odd, and ( r, χ ) = 1. The integers ( rh, k, − χh ) were previously denoted ( r ′ , k ′ , χ ′ ),but the new notation should make the exposition easier to follow.For these numerics, the usual ´etale diagram K v × A × b A τ / / p (cid:15) (cid:15) M vα (cid:15) (cid:15) A × b A Ψ / / A × b A . takes a simpler form. In particular,Ψ( x, y ) = ( − χx, ry ) . In Subsection 2.3, we calculated the pullback(15) τ ⋆ Θ w = Θ w ⊠ (cid:16) Θ − χ ⊠ b Θ − r (cid:17) k ⊗ ( P rχ ) h = Θ w ⊠ (cid:16) Θ − χ ⊠ b Θ − r (cid:17) k ⊗ Ψ ⋆ P − h . Consequently, we have(16) Ψ ⋆ E ( v, w ) = p ⋆ τ ⋆ Θ w = H ( K v , Θ w ) ⊗ (cid:16) Θ − χ ⊠ b Θ − r (cid:17) k ⊗ Ψ ⋆ P − h . Group actions.
We consider actions of the torsion group G = A [ − χ ] × b A [ r ]on the three spaces K v , A × b A and M v appearing in the diagram. The action of G on A × b A is given by translation on both factors, the action on M v is trivial, while the actionon K v is given by ( x, y ) · E = t ⋆ − x E ⊗ y − . There are induced actions of a certain subgroup K of G on the theta bundles, which wenow describe. Writing a = gcd( k, χ ) , b = gcd( k, r ) , we conclude that gcd( a, b ) = 1 and a, b are odd . We set K = A [ a ] × b A [ b ] ֒ → G . The morphisms τ and Ψ are invariant under the action of G , hence also under the actionof K . The bundles τ ⋆ Θ w and Ψ ⋆ P in (15) are naturally K -equivariant. We claim thatΘ w → K v is K -equivariant as well.Certainly, Θ − χk ⊠ b Θ − rk carries a linearization of the Heisenberg group H = H [ − χk ] × b H [ rk ]defined as an extension0 → C ⋆ × C ⋆ → H → A [ − χk ] × b A [ rk ] → . Our convention is that H [ m ] denotes the Heisenberg group of Θ m consisting of pairs ( x, f )where f : t ⋆x Θ m → Θ m UNDLES OF GENERALIZED THETA FUNCTIONS 23 is an isomorphism, while b H [ n ] denotes the Heisenberg group of b Θ n . We have a naturalmorphism ι : H [ a ] × b H [ b ] → H [ − χk ] × b H [ rk ] = H which over the center restricts to ( α, β ) → ( α − χka , β rkb ) . It is now useful to pass to the finite Heisenberg groups e H obtained by restricting thecenters to roots of unity. For instance0 → µ a → e H [ a ] → A [ a ] → , → µ b → e H [ b ] → b A [ b ] → . Since the center of e H [ a ] × e H [ b ] is trivial under ι , we obtain a morphism j : A [ a ] × b A [ b ] → H . Furthermore, since a, b are odd, the identificationΘ − χk ⊠ b Θ − rk ∼ = ( a, b ) ⋆ (cid:16) Θ − χka ⊠ b Θ − kχb (cid:17) is compatible with the action of the group K = A [ a ] × b A [ b ]coming from j on the left, and via the pullback on the right. Using (15), and the natural K -action on line bundles τ ⋆ Θ w and Ψ ⋆ P , we obtain a K -linearization ofΘ w → K v . A similar construction was carried out in Section 3 . . w → K v has the property that each ζ ∈ K acts trivially in the fibersover the ζ -fixed points.3.3. Trace calculations.
As a consequence of the above discussion, the signed sums ofcohomologies χ ( K v , Θ w ) = X i ( − i H i ( K v , Θ w )carries a K -action. We will determine the K -virtual representation χ ( K v , Θ w ) explicitly,under assumption ( A . ) for the vector v . Theorem 2.
Consider ζ = ( x, y ) ∈ A [ a ] × b A [ b ] of order δ . Then the trace of ζ is given byTrace ( ζ, χ ( K v , Θ w )) = d v d v + d w (cid:18) d v /δ + d w /δd v /δ (cid:19) . Proof.
For future use, we prove a slightly more general version of the Theorem, valid inthe context of Bridgeland stability conditions. We work with moduli spaces K v ( σ ) ofBridgeland stable objects, for certain generic Bridgeland stability conditions σ . As it iswell-known, the Gieseker moduli spaces emerge as particular cases. In fact, the spaces K v ( σ ) are all smooth birational models of the Gieseker space K v , see [Y3], Corollary3.33. The objects parametrized by K v ( σ ) are certain 2-step complexes E • in the derivedcategory of A . These complexes have cohomology sheaves only in degree − H − ( E • ) ∈ F and H ( E • ) ∈ T , for a certain torsion pair ( F , T ); in particular Hom( T , F ) = 0. While it is possible to givea more precise description of these complexes, this will not be needed for our argument.The discussion of the previous subsections carry over in this context as well. In partic-ular, the moduli space K v ( σ ) carries an action of ζ ∈ A [ a ] × b A [ b ]: ζ · E • = t ⋆ − x E • ⊗ y − . The action similarly lifts to the theta bundle and to the signed sum of cohomologies. Wewill show that Trace ( ζ, χ ( K v ( σ ) , Θ w )) = d v d v + d w (cid:18) d v /δ + d w /δd v /δ (cid:19) . The proof is an application of the Lefschetz-Riemann-Roch theorem, as in [O2]. Thedetails, both quantitative and qualitative, are however different.
Step 1.
We first find the fixed points of the action of ζ : t ⋆ − x E • ∼ = E • ⊗ y − , E • ∈ K v ( σ ) . Possibly replacing a and b by some of their divisors (thus without insisting on a and b being the gcd of the relevant numerics), we may assume that the order of x is a and theorder of y is b . Of course the following still hold a | χ, b | r = ⇒ gcd( a, b ) = 1 , and ab | k. We have δ = ab .Now, t ⋆ − x E • ∼ = E • ⊗ y − = ⇒ t ⋆ax E • ∼ = E • ⊗ y a = ⇒ E • ∼ = E • ⊗ y a . Since the order of y is coprime to a , we obtain E • ∼ = E • ⊗ y and consequently t ⋆x E • ∼ = E • . Consider the abelian cover p : A → A ′′ = A/ h x i . UNDLES OF GENERALIZED THETA FUNCTIONS 25
The Galois group G of the cover is generated by translations by x . Let b G be the dualgroup, and pick a generator x ′′ of b G . This corresponds to a line bundle x ′′ → A ′′ of order a which determines the cover p . Then E • = p ⋆ E • ′′ for some complex E • ′′ on A ′′ withrank E • ′′ = r, χ ( E • ′′ ) = χa . We have p ⋆ E • ′′ = E • ∼ = E • ⊗ y = p ⋆ ( E • ′′ ⊗ y ′′ )where y = p ⋆ y ′′ . Pushing forward under p , we obtain E • ′′ = E • ′′ ⊗ y ′′ ⊗ x ′′ ℓ for some ℓ . Replacing y ′′ by y ′′ ⊗ x ′′ ℓ we may assume E • ′′ = E • ′′ ⊗ y ′′ . Note that y b = p ⋆ y ′′ b = 1 = ⇒ y ′′ b ∈ h x ′′ i = ⇒ y ′′ ab = 1 . Therefore, the order ∆ of y ′′ satisfies ∆ | ab . We will show shortly that in fact ∆ = b .Now, let π : A ′ → A ′′ be the cover determined by y ′′ which has degree ∆. Let G ′ denote the Galois group of thecover, and let b G ′ be the dual group which is generated by y ′′ . We collect the followingfacts about complexes over arbitrary abelian covers:(i) if E • ′′ ∼ = E • ′′ ⊗ y ′′ , then E • ′′ is the pushforward of a complex E • ′ → A ′ : E • ′′ = π ⋆ E • ′ (ii) π ⋆ E • ′ = π ⋆ E • ′ iff E • ′ = β ⋆ E • ′ for some β ∈ G ′ (iii) π ⋆ π ⋆ E • ′ = M β ∈ G ′ β ⋆ E • ′ (iv) the action of y ′′ ∈ b G ′ on π ⋆ π ⋆ E • ′ leaves each of the factors β ⋆ E • ′ invariant. Theweight of the action on each factor equals h y ′′ , β i . Specifically, (i) is proven in Proposition 2.5 of [BM], while (ii), (iii) and (iv) follow byimitating the proofs of Lemmas 2 . . E • ′′ = π ⋆ E • ′ where rank E • ′ = r ∆ , χ ( E • ′ ) = χ ( E • ′′ ) = χa . In particular, ∆ | r hence ∆ is coprime to a . Since ∆ divides ab , we obtain that ∆ divides b . On the other hand, y ′′ ∆ = 1 = ⇒ y ∆ = p ⋆ y ′′ ∆ = 1 = ⇒ b | ∆ . Hence ∆ = b . Thus all ζ -fixed complexes are of the form E • = p ⋆ π ⋆ E • ′ , rank E • ′ = rb , χ ( E • ′ ) = χa . It is clear that the map p is uniquely determined by ζ = ( x, y ). The same is true about π . To see this, note that the preimage of y under p contains at most one element of order b since the preimage of 0 contains at most one such element. Indeed, p ⋆ (0) consists onlyin powers of x ′′ which all have order dividing a . This fixes y ′′ and therefore π .At this point we briefly discuss stability. Consider the stability condition over A ′ σ ′ = π ⋆ σ ′′ , where σ ′′ over A ′′ is chosen so that p ⋆ σ ′′ = σ. It can be immediately checked from definitions that E • is σ -stable implies that E • ′ is σ ′ -stable, and conversely the stability of E • ′ implies (semi)stability of E • . These choicesof stability conditions will be assumed below, and will be suppressed from the notation.To describe the ζ -fixed loci over K v ( A ), we fix the determinant and determinant ofFourier Mukai of E • . We claim that there are ( ab ) fixed loci all isomorphic to K v ′ ( A ′ )for v ′ = (cid:16) rb , , χa (cid:17) . We first calculate the determinantdet E • = p ⋆ det π ⋆ E • ′ = O = ⇒ det π ⋆ E • ′ ∈ h x ′′ i . Writing det E • ′ = π ⋆ M ′′ UNDLES OF GENERALIZED THETA FUNCTIONS 27 we obtain by the proof of Lemma 3.4 in [NR] thatdet π ⋆ E • ′ = det π ⋆ π ⋆ M ′′ = det b M j =1 M ′′ ⊗ y ′′ j = M ′′ b ∈ h x ′′ i = ⇒ M ′′ ∈ h x ′′ i + b A ′′ [ b ] , where above we used that the order of x ′′ is a which is coprime to b . Thusdet E • ′ ∈ π ⋆ h x ′′ i + π ⋆ b A ′′ [ b ] . There are ab choices for the determinant of E • ′ , since the torsion point y ′′ ∈ b A ′′ [ b ] pullsback trivially to A ′ . The calculation of the Fourier-Mukai determinant is similar. (In thecalculation below, for simplicity of notation, we denote by b the Fourier-Mukai over theappropriate abelian varieties, without assuming that these transforms are represented bysheaves.) We have det c E • = det b p ⋆ b π ⋆ d E • ′ = O . Write det (cid:16)b π ⋆ d E • ′ (cid:17) = b p ⋆ M and observe that this givesdet b p ⋆ b p ⋆ M = M a = O = ⇒ M ∈ A [ a ] . This gives a options for b p ⋆ M since x ∈ A [ a ] pulls back trivially. Finally, this identifiesdet b E • ′ up to the b elements in the kernel of π : π (det b E • ′ ) = p ( M ) ∈ p ( A [ a ]) = ⇒ det d E • ′ ∈ π − ( p ( A [ a ])) . We obtain ( ab )( a b ) fixed loci, one for each choice of determinant and determinant ofFourier-Mukai of E • ′ . However, this answer does not account for repetitions. We observethat p ⋆ π ⋆ E • ′ = p ⋆ π ⋆ E • ′ ⇐⇒ π ⋆ E • ′ = π ⋆ E • ′ ⊗ x ′′ α ⇐⇒ E • ′ = β ⋆ E • ′ ⊗ π ⋆ x ′′ α , β ∈ G ′ . The second statement above follows pushing forward via p and using stability. The thirdstatement is contained in (ii). As a result, we are left with only ( ab ) fixed loci. Indeed,via the above equivalence we only obtain b choices for the determinant, if we requiredet E • ′ ∈ π ⋆ b A ′′ [ b ] . Similarly, the determinant of the Fourier-Mukai dual can be fixed in a ways. It is easyto see there are no other repetitions. This yields ( ab ) distinct fixed loci. Step 2.
We apply the Lefschetz-Riemann-Roch theorem to calculate the trace in thetheorem by summing the contributions from the fixed loci K v ′ ( A ′ ). We obtainTrace( ζ, χ ( K v , Θ w )) = ( ab ) Z K v ′ ( A ′ ) Todd ( K v ′ ( A ′ )) · i ⋆ ch Z (Θ w )( ζ ) · Y z =1 (cid:0) ch −h ζ − ,z i N ∨ z (cid:1) − . Here Z is the cyclic group generated by ζ . The notation we used is as follows: i ⋆ ch Z (Θ w ) ∈ H ⋆Z ( K v ′ ( A ′ )) = Rep( Z ) ⊗ H ⋆ ( K v ′ ( A ′ ))is the restriction of the equivariant Chern character via inclusion i : K v ′ ( A ′ ) → K v ( A ) , followed by the evaluation against ζ ∈ Z in the representation ring. The normal bundleof the inclusion i splits into eigenbundles N z indexed by elements in the character group z ∈ b Z . Finally, we write ch t ( N ) = Y i (1 + te x i )for any bundle N with Chern roots x , . . . , x ℓ . The prefactor ( ab ) comes from the factthat all fixed loci will have identical trace contributions. Step 3.
We evaluate the integral above explicitly. We begin by computing the normalbundles N z in the expression above. We will compute the eigenvalues of the action of ζ on T E • K v . We first consider the similar action on T E • M v and use the morphism α : M v → A × b A to find the eigenvalues on the fiber.The tangent space to M v at a fixed point E • was calculated in [I]: T E • M v = Ext ( E • , E • ) = Ext ( p ⋆ π ⋆ E • ′ , p ⋆ π ⋆ E • ′ ) = Ext ( π ⋆ E • ′ , π ⋆ E • ′ ⊗ p ⋆ O )= a − M α =0 Ext ( π ⋆ E • ′ , π ⋆ E • ′ ⊗ x ′′ α ) = a − M α =0 Ext ( π ⋆ E • ′ , π ⋆ ( E • ′ ⊗ π ⋆ x ′′ α ))= a − M α =0 Ext ( π ⋆ π ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′ α ) = a − M α =0 M β ∈ G ′ Ext ( β ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′ α ) . We claim that T α,β = Ext ( β ⋆ E • ′ , E • ′ ⊗ x ′′ α )are the isotypical components of the tangent space. Indeed, by (iv) in Step 1 , y acts oneach summand via the root of unity h β, y ′′ i of order b . The action of x also leaves thesubspace invariant, and the action has weight h x, x ′′ α i which is root of unity of order a .As α, β vary, we obtain all the ( ab )-roots of unity as eigenvalues. UNDLES OF GENERALIZED THETA FUNCTIONS 29
We calculate the Chern roots of the eigenbundles T α,β . Clearly, by Hirzebruch-Riemann-Roch, the Chern character of the virtual bundles X i =0 ( − i Ext i ( β ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′ α )must stay constant as β and π ⋆ x ′′ α vary in the abelian varieties A ′ and b A ′ . (The index i ischecked to have the correct range. Indeed, vanishing of the ext’s for indices i ≤ − i ≥ α, β not both trivial we haveExt ( β ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′ α ) = Ext ( β ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′ α ) = 0while for trivial α and β the two dimensions are 1, by stability of E • ′ . Indeed, for nontrivial( α, β ), we calculate by dualityExt ( β ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′ α ) = Ext ( E • ′ ⊗ π ⋆ x ′′ α , β ⋆ E • ′ ) = Ext (( β − ) ⋆ E • ′ , E • ′ ⊗ π ⋆ x ′′− α )so it suffices to prove the statement about Ext . This is immediate since any non-zeromorphism β ⋆ E • ′ → E • ′ ⊗ π ⋆ x ′′ α is an isomorphism inducing a map π ⋆ β ⋆ E • ′ = π ⋆ E • ′ → π ⋆ E • ′ ⊗ x ′′ α . Comparing determinants, we must have x ′′ αr = 0 = ⇒ x ′′ α = 0 since ( r, a ) = 1. Therefore, α = 0. Now, using the isomorphism β ⋆ E • ′ → E • ′ and letting q : A ′ → A ′ / h β i be the projection determined by β , we obtain that E • ′ is a pullback from the quotient.Evaluating Euler characteristics, we obtain that χ ( E • ′ ) = χa is divisible by the order of β .But ord( β ) | b and ( b, χ ) = 1, hence ord( β ) = 1 and β = 1.We are now in the position to calculate the normal bundles N z . The isotypical com-ponents correspond to nontrivial pairs ( α, β ). Each isotypical component has dimension2 rχab := ℓ + 2 . We just argued the Chern roots of T α,β equal0 , , x , . . . , x ℓ , with x i the roots of the tangent bundle of K v ′ ( A ′ ). This follows by comparison with α, β trivial; the two trivial roots come from the four dimensional base of the Albanese map, after canceling two trivial factors corresponding to infinitesimal automorphisms andobstructions over M v ′ ( A ′ ). Step 4.
With this understood, we calculate Y z =1 ch −h ζ − ,z i N ∨ z − = Y ξ =1 (1 − ξ ) ℓ Y i =1 (cid:0) − ξe − x i (cid:1)! − = 1 δ ℓ Y i =1 − e − x i − e − δx i , where ξ = h ζ − , z i runs through the non-trivial δ -roots of 1.We now claim that(17) i ⋆Z Θ w ∼ = Θ δw ′ , where the last bundle carries a trivial Z -linearization and the vector w ′ is specified below.Indeed, let Θ ′′ → A ′′ be the symmetric polarization such that p ⋆ Θ ′′ = Θ a = ⇒ χ (Θ ′′ ) = a. Also, write Θ ′ = π ⋆ Θ ′′ , hence χ (Θ ′ ) = ab . Let w = ( r ′ , k Θ , χ ′ ) where r ′ = rh, χ ′ = − χh .We introduce the Mukai vectors w ′′ = (cid:18) r ′ , ka Θ ′′ , χ ′ a (cid:19) over A ′′ , and w ′ = (cid:18) r ′ b , kab Θ ′ , χ ′ a (cid:19) over A ′ . Observe that p ⋆ w = aw ′′ , π ⋆ w ′′ = bw ′ . These equalities imply that(18) i ⋆ Θ w = Θ aw ′′ , i ⋆ Θ w ′′ = Θ bw ′ , which together then give (17). Here, we factored i : M v ′ ( A ′ ) → M v ( A ) , i = i ◦ i where i : M v ′′ ( A ′′ ) → M v ( A ) , i : M v ′ ( A ′ ) → M v ′′ ( A ′′ )are induced by pullback by p and pushforward by π respectively, and the correspondingMukai vectors are v ′ = (cid:16) rb , , χa (cid:17) , v ′′ = (cid:16) r, , χa (cid:17) . For instance, to justify the first identity in (18), note that tautologically we have ι ⋆ Θ F = Θ p ⋆ F , UNDLES OF GENERALIZED THETA FUNCTIONS 31 for any complex F representing the vector w . Equation (18) follows by recalling thenormalization conventions for theta bundles in Section 2. Specifically, if F is suitablynormalized as in (11), then p ⋆ F represents the vector aw ′′ and also satisfies convention(11).The statement about the Z -action in (17) will be proved in Step 5 below.Substituting into Lefschetz-Riemann-Roch, we findTrace( ζ, χ ( K v , Θ w )) = δ Z K v ′ ( A ′ ) ℓ Y i =1 x i − e − x i · δ ℓ Y i =1 − e − x i − e − δx i · ch(Θ δw ′ )= δ Z K v ′ ( A ′ ) ℓ Y i =1 x i − e − δx i · ch(Θ δw ′ )= δχ ( K v ′ , Θ w ′ ) = δ d v ′ d v ′ + d w ′ (cid:18) d v ′ + d w ′ d v ′ (cid:19) . The last line follows from the backward application of Hirzebruch-Riemann-Roch and (1).The proof is completed by observing that d v ′ = d v δ and d w ′ = d w δ . Step 5.
We explain now that the action of ζ in the fiber of Θ w → K v over each ζ -fixedpoint is trivial, as claimed in Step 4 . The idea of the proof is similar to that of Remark1 in [O2]. Since the details are different, we include the argument for completeness. Let E • = p ⋆ π ⋆ E • ′ be a ζ -fixed point of K v ( A ). Consider t : A × b A → M v ( A ) , t ( λ, µ ) = t ⋆λ E • ⊗ µ the restriction of the morphism τ to { E • } × A × b A. By the construction in Subsection 3.2,it suffices to explain that the identification(19) t ⋆ Θ w ≃ ( a, b ) ⋆ (cid:16) Θ − χka ⊠ b Θ − rkb ⊗ P h · rχab (cid:17) obtained by restricting (15) to { E • } × A × b A is ζ -equivariant.To this end, consider the fiber diagram A + π + / / p + (cid:15) (cid:15) A p (cid:15) (cid:15) A ′ π / / A ′′ . The constructions in
Step 1 show that π + : A + → A is the ´etale cover determined by p ⋆ y ′′ = y , so that c π + : b A → c A +2 DRAGOS OPREA is the quotient by the translation action by y . The natural map( p, c π + ) : A × b A → A ′′ × c A + is the quotient by the action of ζ = ( x, y ). Since t is ζ -invariant, we obtain a morphism t : A ′′ × c A + → M v ( A ) , t ◦ ( p, c π + ) = t. Let N : A ′′ → A, b N + : c A + → b A denote the two norm maps corresponding to the morphisms p and c π + respectively. Then N ◦ p = a, d N + ◦ c π + = b = ⇒ ( N, d N + ) ◦ ( p, c π + ) = ( a, b ) . To establish that (19) holds ζ -equivariantly, we first factor out the action of ζ , and provethat over the quotient A ′′ × c A + we have(20) t ⋆ Θ w ≃ ( N, d N + ) ⋆ (cid:16) Θ − χka ⊠ b Θ − rkb ⊗ P h · rχab (cid:17) . The isomorphism (20) will be shown using the see-saw principle. We verify it over A ′′ × { µ + } for all µ + ∈ c A + . The restriction to { λ ′′ } × c A + is similar and will be omitted.We show(21) ( t ′′ ) ⋆ Θ w ≃ N ⋆ (cid:16) Θ − χka ⊗ d N + ( µ + ) h · rχab (cid:17) , where t ′′ : A ′′ → M v ( A ) is the restriction of t, that is t ′′ ( λ ′′ ) = t ( λ ′′ , µ + ) . We first determine the morphism t ′′ . Write µ + = q ⋆ µ ′′ for some µ ′′ ∈ b A ′′ , where q = p ◦ π + : A + → A ′′ . It follows from the definitions that t ( λ ′′ , µ + ) = t ( λ, µ ) = t ⋆λ E • ⊗ µ whenever p ( λ ) = λ ′′ , µ + = ( π + ) ⋆ µ. In our case, we can take µ = p ⋆ µ ′′ , so that t ′′ ( λ ′′ ) = t ( λ ′′ , µ + ) = t ⋆λ E • ⊗ p ⋆ µ ′′ = t ⋆λ p ⋆ E • ′′ ⊗ p ⋆ µ ′′ = p ⋆ (cid:0) t ⋆λ ′′ E ′′ ⊗ µ ′′ (cid:1) . Here, we used that the ζ -fixed points take the form E • = p ⋆ E • ′′ with E • ′′ = π ⋆ E ′• . In conclusion, t ′′ = ι ◦ τ ′′ UNDLES OF GENERALIZED THETA FUNCTIONS 33 where ι : M v ′′ ( A ′′ ) → M v ( A )is the morphism induced by the pullback p ⋆ already encountered in Step 4 , and τ ′′ : A ′′ → M v ′′ ( A ′′ ) , τ ′′ ( λ ′′ ) = t ⋆λ ′′ E • ′′ ⊗ µ ′′ is the translation map.With this understood, we compute the left hand side of (21) making use of equation(18): ( t ′′ ) ⋆ Θ w = ( ι ◦ τ ′′ ) ⋆ Θ w = ( τ ′′ ) ⋆ ( ι ) ⋆ Θ w = ( τ ′′ ) ⋆ Θ aw ′′ . The pullback of the theta bundle Θ w ′′ under τ ′′ , all the way to the product A ′′ × c A ′′ , wasdetermined in Lemma 1. (We remarked above that the Lemma also holds for nonprincipalpolarizations.) Restricting to A ′′ × { µ ′′ } , we find( τ ′′ ) ⋆ Θ w ′′ = Θ ′′− χka ⊗ (cid:0) µ ′′ (cid:1) h rχa = ⇒ ( t ′′ ) ⋆ Θ w = Θ ′′− χka ⊗ (cid:0) µ ′′ (cid:1) hrχ . For the right hand side of (21), we use the standard identities N ⋆ Θ = Θ ′′ a , N ⋆ d N + q ⋆ ( µ ′′ ) = ( µ ′′ ) ab . Hence, N ⋆ (cid:16) Θ − χka ⊗ d N + ( µ + ) h · rχab (cid:17) = Θ ′′− χka ⊗ (cid:0) µ ′′ (cid:1) hrχ . This establishes (21), and completes the argument. (cid:3)
The calculation of the Verlinde bundles.
We are now in the position to deter-mine the Verlinde bundles in degree 0. Assuming ( A . ) and ( A . ), we prove Theorem 3.
We have E ( v, w ) = M ζ (cid:18) W − χa , ka ⊠ W † rb , − kb ⊗ P − h (cid:19) ⊗ ℓ ⊕ m ζ ζ . The sum is indexed by torsion line bundles ζ → A × b A of order dividing ( a, b ) . An element ζ of order exactly ω comes with multiplicity m ζ = 1 d v + d w X δ | ab δ ( ab ) (cid:26) ab/ωδ (cid:27) (cid:18) d v /δ + d w /δd v /δ (cid:19) . Recall that in the above, for each line bundle ζ of order ( a, b ) over A × b A , we fix one root ℓ ζ such that (cid:16) − χa , rb (cid:17) ℓ ζ = ζ = ⇒ ( − χ, r ) ℓ ζ = 0 . Each ℓ ζ corresponds to a character of A [ − χ ] × b A [ r ] which is uniquely defined only up tocharacters of A [ − χ/a ] × b A [ r/b ] . Remark 2.
A more general class of semihomogeneous vector bundles W ( P ) → A × b A, depending on a triple P of rational numbers, is constructed and studied in [O3]. Forinstance, for a triple written in lowest terms P = (cid:18) ba , dc , h (cid:19) , where ( a, c ) are odd positive and h ∈ Z , we have W ( P ) = W a,b ⊠ W † c,d ⊗ P h . For general triples P , the bundles W ( P ) do not admit such simple expressions. Con-jecturally, for arbitrary numerics, the Verlinde bundle can be written in terms of theirreducible building blocks W ( P ) and torsion points with explicit multiplicities, in a fash-ion compatible with the Fourier-Mukai symmetry of Theorem 4. This will be investigatedin more detail in [O3]. Proof.
The proof of the theorem follows the strategy laid out in [O2], with a few modifi-cations. We give the relevant details here.Let us assume first that Θ w → K v carries no higher cohomology. We noted in (16) that( − χ, r ) ⋆ E ( v, w ) = H ( K v , Θ w ) ⊗ (Θ − χ ⊠ b Θ − r ) k ⊗ (cid:16) ( − χ, r ) ⋆ P − h (cid:17) . This identifies E ( v, w ) up to ( − χ, r )-torsion line bundles. In Section 4 . . − χ ) ⋆ W − χa , ka = (cid:0) Θ − χ (cid:1) k ⊠ R and r ⋆ W † rb , − kb = (cid:16) b Θ r (cid:17) − k ⊠ R for R a representation of H [ − χ ] of dimension ( χ/a ) and central weight − k , while R isa representation of b H [ r ] of dimension ( r/b ) and central weight k . Therefore,( − χ, r ) ⋆ (cid:18) W − χa , ka ⊠ W † rb , − kb (cid:19) = (cid:16) Θ − χ ⊠ b Θ − r (cid:17) k ⊠ R where R = R ⊠ R is the product representation of H [ − χ ] × b H [ r ] . It suffices to explain that H [ − χ ] × b H [ r ]-equivariantly we have(22) H ( K v , Θ w ) = R ⊗ M ζ ℓ ⊕ m ζ ζ UNDLES OF GENERALIZED THETA FUNCTIONS 35 where ℓ ζ runs over the characters of A [ − χ ] × b A [ r ] modulo those of A [ − χ/a ] × b A [ r/b ].We make use of the morphism of Theta groups H [ a ] × b H [ b ] → H [ − χ ] × b H [ r ]which restricts to ( α, β ) → ( α − χ/a , β r/b )over the center C ⋆ × C ⋆ ֒ → H [ a ] × b H [ b ] . Furthermore, passing to the finite Heisenberg, two e H [ − χ ] × e H [ r ]-modules with the centralweight ( − k, k ) are isomorphic if and only if they are isomorphic as representations of theabelian group A [ a ] × b A [ b ], see [O2]. Therefore, it suffices to establish the identification(22) equivariantly for the action of A [ a ] × b A [ b ] on both sides.Crucially, it was explained in Section 3 . R and R are the trivial repre-sentations of A [ a ] and b A [ b ]. Same as in [O2], for ζ of order exactly ω dividing ( a, b ), weuse (22) to compute m ζ = 1dim R · a b X π ∈ A [ a ] × b A [ b ] h ζ, π − i Trace (cid:0) π, H ( K v , Θ w ) (cid:1) = 1 a b X δ | ab d v + d w (cid:18) d v /δ + d w /δd v /δ (cid:19) X ord ( π )= δ h ζ, π − i Theorem 2 was used here to evaluate the trace. Lemma 4 of [O1] gives the sum X ord ( π )= δ h ζ, π − i = δ (cid:26) ab/ωδ (cid:27) . This confirms that the multiplicities m ζ of equation (22) agree with the expressions statedin the Theorem.We can now remove the assumption on vanishing of higher cohomology. By ( A . ),Θ w → K v belongs to the movable cone, hence it is big and nef on a smooth birationalmodel of K v , by Theorem 7 in [HT]. Furthermore, the smooth birational models of K v are obtained as moduli spaces of Bridgeland stable objects K v ( σ ) for stability conditions σ of the type we considered in the proof of Theorem 2; see [Y3], Corollary 3.33 for details.The proof of [Y3] moreover yields the diagram M v j / / α (cid:15) (cid:15) M v ( σ ) α σ (cid:15) (cid:15) A × b A A × b A where α σ : M v ( σ ) → A × b A is the Albanese morphism normalized as in equation (7), and j is birational, regular incodimension 1, and given by the identity on the common open locus. The two theta linebundles Θ w agree under j .Define E σ ( v, w ) = ( α σ ) ⋆ Θ w . The first part of the argument applies verbatim to the moduli of σ -stable objects. Inparticular, since Θ w carries no higher cohomology over K v ( σ ), we have E σ ( v, w ) = M ζ (cid:18) W − χa , ka ⊠ W † rb , − kb ⊗ P − h (cid:19) ⊗ ℓ ⊕ m ζ ζ . The above diagram shows however that E ( v, w ) ≃ E σ ( v, w ) , completing the proof. (cid:3) Example 2.
Rank . Let v = (1 , , − n ) , w = (1 , k Θ , n )with n odd and k ≥
1. Then, by Example 1, we have M v ∼ = A [ n ] × b A, α = ( − a, , and Θ w = (Θ k ) [ n ] ⊠ b Θ − k ⊗ ( a, ⋆ P . Therefore,(23) E ( v, w ) = a ⋆ (cid:16) (Θ k ) [ n ] (cid:17) ⊠ b Θ − k ⊗ P − . Theorem 2 is equivalent to the calculation of the pushforward a ⋆ (cid:16) (Θ k ) [ n ] (cid:17) = M ζ W na , ka ⊗ ℓ ⊕ m ζ ζ , where ζ are line bundles over A of order ω dividing a = gcd( n, k ), ℓ ζ is a root of ζ of order na , and m ω = 1 k X δ | a δ a (cid:26) a/ωδ (cid:27) (cid:18) k /δn/δ (cid:19) . To apply the theorem, we invoke the result of Scala who proved the vanishing of highercohomology of the tautological bundle (Θ k ) [ n ] → K n − , under the assumption that k ≥ . . UNDLES OF GENERALIZED THETA FUNCTIONS 37
Fourier-Mukai symmetries.
We can now prove the following Fourier-Mukai com-parison, which may be seen as evidence for the strange duality conjecture for abeliansurfaces; cf. Conjecture 1 of Section 2. Under the assumptions ( A . ) − ( A . ) madethroughout the paper, we establish: Theorem 4.
When c ( v ) and c ( w ) are divisible by the ranks r and r ′ , there is an iso-morphism \ E ( v, w ) ∼ = E ( w, v ) ∨ . Proof.
The proof is by direct computation of both sides, using the expression for theVerlinde bundles in Theorem 3.To begin, write v = ( r, rℓ Θ , χ ) , w = ( r ′ , r ′ ℓ ′ Θ , χ ′ ) . The requirements χ ( v · w ) = 0 and ( r, χ ) = ( r ′ , χ ′ ) = 1 give2 rr ′ ℓℓ ′ = − rχ ′ − r ′ χ = ⇒ r | r ′ and r ′ | r = ⇒ r = r ′ . While in our setting the numerics impose the restriction r = r ′ on the ranks, the symmetryin the Theorem is expected to hold true for general numerics satisfying ( A . ) − ( A . ). In[O3] we will offer evidence for this more general statement, but in a partially conjecturalsetup. Step 1.
We will reduce to the case ℓ = 0 using twists by line bundles. To this end, wecheck that the symmetry in the statement of the Theorem is invariant under such twists.Specifically, letting v = v exp( − ℓ Θ) , w = w exp( ℓ Θ) , we consider the two natural isomorphisms given by tensoring i : M v → M v , E E ⊗ Θ − ℓ and j : M w → M w , F → F ⊗ Θ ℓ . It is easy to see that i ⋆ Θ w = Θ w , j ⋆ Θ v = Θ v . Equation (13) is used here to conclude that tensorization by Θ does not change the nor-nalization convention (11) used in the definition of the theta bundles. Next, the Albanesemorphisms α : M v → A × b A, β : M w → A × b A, are respectively given by E (det R S ( E ) , det E ) , F (det R S ( F ) ⊗ b Θ − r ( ℓ + ℓ ′ ) , det F ⊗ Θ − r ( ℓ + ℓ ′ ) ) . We claim that(24) α ◦ i = ρ ◦ α, β ◦ j = ρ ′ ◦ β where ρ ( x, y ) = ( x − ℓ b Φ( y ) , y ) , ρ ′ ( x, y ) = ( x + ℓ b Φ( y ) , y ) . For instance, the first identity in (24) is equivalent todet R S ( E ⊗ Θ − ℓ ) = (cid:16) det R S ( E ) ⊗ b Θ − rℓ (cid:17) ⊗ P − ℓ b Φ(det E ⊗ Θ − rℓ ) . When E is in the Albanese fiber over the origin, so thatdet E = Θ rℓ and det R S ( E ) = b Θ rℓ , this follows from repeated application of equation (13). For the general case, we make useof the diagram (6) to write E = t ⋆x E ⊗ y, with E in the Albanese fiber. The identity above continues to hold by the usual propertiesof the Fourier-Mukai under tensorization and translation.From the first identity in (24) we find ρ ⋆ E ( v , w ) = E ( v, w ) . In a similar fashion, the second identity in (24) gives ρ ′ ⋆ E ( w, v ) = E ( w , v ) = ⇒ E ( w, v ) = ρ ⋆ E ( w , v ) , using that ρ ′ = ρ − .Assume now that we established the Theorem for the pair ( v , w ). The same resultwill then follow for the pair ( v, w ). Indeed, it is not hard to see that ρ = ρ t , and hence E ( w, v ) ∨ = ρ ⋆ (cid:0) E ( w , v ) ∨ (cid:1) = ρ ⋆ (cid:16) \ E ( v , w ) (cid:17) = \ ρ ⋆ E ( v , w ) = \ E ( v, w ) . Step 2.
As a consequence of
Step 1 , we may assume c ( v ) = 0. After clearing primesfrom the notation, we write v = ( r, , χ ) , w = ( r, rℓ Θ , − χ ) . By assumption, we have χ = − d v r is an odd negative integer and χ + rℓ = d w r is an odd positive integer . To establish(25) \ E ( v, w ) ∼ = E ( w, v ) ∨ , UNDLES OF GENERALIZED THETA FUNCTIONS 39 we explicitly calculate the Verlinde bundles. First, by Theorem 3 we have E ( v, w ) = M ζ (cid:16) W − χa , rℓa ⊠ b Θ − ℓ ⊗ P − (cid:17) ⊗ ℓ ⊕ m ζ ( v,w ) ζ , where a = ( χ, rℓ ) = ( χ, ℓ ) , (cid:16) − χa , (cid:17) ℓ ζ = ζ. In the sum, ζ runs over the ( a, A × b A . Henceforth, we willregard ζ and ℓ ζ as line bundles pulled back from A , without explicitly stating this fact.To find E ( w, v ), we use calculations similar to those in Step 1 to reduce to degree 0. Inthe new notation, we consider the isomorphism k : M w → M w , F → F ⊗ Θ − ℓ , w = w · exp ( − ℓ Θ) , v = v · exp ( ℓ Θ) . Note that w = ( r, , − χ − rℓ ) , v = ( r, rℓ Θ , χ + rℓ ) . Then, as before, we have E ( w, v ) = ρ ⋆ E ( w , v ) . Again by Theorem 3, we have E ( w , v ) = M ζ (cid:18) W χ + rℓ a , rℓa ⊠ b Θ − ℓ ⊗ P − (cid:19) ⊗ e ℓ ζ ⊕ m ζ ( w ,v ) . Here, we used that ( χ + rℓ , rℓ ) = ( χ, ℓ ) = a. As above, the sum also runs over the( a, ζ → A × b A , and(26) (cid:18) χ + rℓ a , (cid:19) e ℓ ζ = ζ. Step 3.
To simplify notation, set W = (cid:16) W − χa , rℓa ⊠ b Θ − ℓ (cid:17) ⊗ P − , W ′ = (cid:16) W ∆ a , rℓa ⊠ b Θ − ℓ (cid:17) ⊗ P − , where we wrote ∆ = χ + rℓ . Then, by the discussion above we have E ( v, w ) = M ζ W ⊗ ℓ ⊕ m ζ ( v,w ) ζ , E ( w, v ) = M ζ ρ ⋆ (cid:16) W ′ ⊗ e ℓ ζ ⊕ m ζ ( w ,v ) (cid:17) . It is clear from the specific expressions for the multiplicities m ζ given in Theorem 3 thatthese quantities are symmetric in v and w , and are invariant under twists, so that m ζ ( w , v ) = m ζ ( w, v ) = m ζ ( v, w ) . To complete the proof of (25), it suffices to give a correspondence ℓ ζ → ˜ ℓ ζ such that(27) ρ ⋆ (cid:16) W ′ ⊗ e ℓ ζ (cid:17) ∨ = R S ( W ⊗ ℓ ζ ) , up to shifts by the index. We first consider the case ζ trivial, proving that (up to shifts)(28) ρ ⋆ W ′∨ = R S ( W ) . To this end, note first that W satisfies the index theorem [M1] with index 0 if ℓ > ℓ <
0. This can be checked after pullback. In our case, using (2), we find (cid:16) − χa , (cid:17) ⋆ W = (cid:16) Θ − χa · rℓa ⊠ b Θ − ℓ ⊠ P χa (cid:17) ⊗ C ( χa ) . Suppose ℓ >
0. The claim about the index follows since the latter line bundle is ample.In turn, this is a consequence of the special form of the Nakai-Moishezon criterion in thecontext of abelian varieties [BL]. Indeed, a direct calculation as in the last section of[BMOY] shows that the line bundleΘ α ⊠ b Θ − β ⊗ P γ → A × b A is ample whenever α > , β > , α · β − γ > . These requirements are satisfied for the numerics we consider.We now turn to the proof of (28). We present here a direct argument, referring thereader to the note [O3] for similar but more involved statements. We observe first thatboth bundles in (28) are simple and semihomogeneous, as both properties are preservedby pullbacks under isomorphisms and Fourier-Mukai. A direct calculation shows that theyhave the same rankrank W ′ = (cid:18) ∆ a (cid:19) , rank R S ( W ) = χ ( W ) = (cid:18) ∆ a (cid:19) . The slopes of ρ ⋆ W ′∨ and R S ( W ) are also directly calculated and seen to match ρ ⋆ (cid:16) Θ − rℓ ∆ ⊠ b Θ ℓ ⊗ P (cid:17) = Θ − rℓ ∆ ⊠ b Θ − ℓχ ∆ ⊗ P χ ∆ . In the line above, we used the following identities derived via the see-saw principle ρ ⋆ Θ = Θ ⊠ b Θ − ℓ ⊗ P ℓ , ρ ⋆ b Θ = b Θ , ρ ⋆ P = b Θ − ℓ ⊗ P . Finally, both bundles in (28) are invariant under the involution ( − , −
1) over A × b A . Thesame argument as in Section 2.2 of [O2] shows uniqueness of simple symmetric semiho-mogeneous bundles with equal rank and determinant, proving (28). UNDLES OF GENERALIZED THETA FUNCTIONS 41
Step 4.
Finally, to establish (25), we match the contributions of the torsion points ineach irreducible summand (27). Fix ζ ∈ b A an a -torsion line bundle over A , viewed as aline bundle over A × b A by pullback. Let ℓ ζ ∈ b A be chosen so that − χa ℓ ζ = ζ. The bundle W = (cid:16) W − χa , rℓa ⊠ b Θ − ℓ (cid:17) ⊗ P − is semihomogeneous of rank (cid:0) − χa (cid:1) with de-terminant D = (cid:16) − χa (cid:17) (cid:18) − rℓχ Θ − ℓ b Θ − P (cid:19) . By Lemma 6 . α ∈ A × b A we have that t ⋆ ( − χa ) α W = W ⊗ φ D ( α ) . In the above equation, the line bundle D induces over the abelian fourfold A × b A theMumford homomorphism φ D : A × b A → A × b A, ( x, y ) (cid:16) − χa (cid:17) (cid:18) − x − ℓ b Φ( y ) , − rℓχ Φ( x ) − y (cid:19) . Pick y ∈ b A such that − ∆ a · χa y = ℓ ζ and define α = ( − ℓ b Φ( y ) , y ) ∈ A × b A and e ℓ ζ = (cid:18) , (cid:16) χa (cid:17) y (cid:19) ∈ A × b A. Our choices guarantee two crucial identities φ D ( α ) = (0 , ℓ ζ )and ρ ⋆ e ℓ ζ ∨ = − (cid:16) χa (cid:17) α. From here, to demonstrate (27), we match the contributions of each ζ to the two sides R S ( W ⊗ ℓ ζ ) = R S ( W ⊗ φ D ( α )) = R S ( t ⋆ ( χa ) α W ) = R S ( W ) ⊗ − (cid:16) χa (cid:17) α = R S ( W ) ⊗ ρ ⋆ e ℓ ζ ∨ = ρ ⋆ (cid:16) W ′ ⊗ e ℓ ζ (cid:17) ∨ . To complete the argument, it suffices to note that − χa ℓ ζ = ζ = ⇒ ∆ a · (cid:16) χa (cid:17) y = ζ = ⇒ ∆ a · e ℓ ζ = ζ, as required by (26). (cid:3) References [BM] T. Bridgeland, A. Maciocia,
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