On the intersection theory of the moduli space of rank two bundles

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1 On the intersection theory of the moduli space of rank two bundles Alina Marian a, Dragos Oprea b,1 a Yale University, P.O. Box , New Haven, CT, , USA b M.I.T, 77 Massachusetts Avenue, Room 2-492, Cambridge, MA, 02139, USA. Abstract We give an algebro-geometric derivation of the known intersection theory on the moduli space of stable rank 2 bundles of odd degree over a smooth curve of genus g. We lift the computation from the moduli space to a Quot scheme, where we obtain the intersections via equivariant localization with respect to a natural torus action. Key words: Moduli spaces of sheaves, Quot schemes, equivariant localization. We compute the intersection numbers on the moduli space of stable rank 2 odd degree bundles over a smooth complex curve of genus g. This problem has been intensely studied in the physics and mathematics literature. Complete and mathematically rigorous answers were obtained by Thaddeus [7] [8], Donaldson [2], Zagier [11], and others in rank 2, and by Jeffrey-Kirwan [3] in arbitrary rank. These answers are in agreement with the formulas first written down by Thaddeus in rank 2 [7], and by Witten in arbitrary rank [9]. We indicate yet another method of calculation which recovers the exact formulas obtained by the aforementioned authors in rank 2. Our approach works in principle in any rank, and we will turn to this general case in future work. To set the stage, we let C be a smooth complex algebraic curve of genus g. We let N g denote the moduli space of stable rank 2 bundles with fixed odd determinant, and we write V for the universal bundle on C N g. N g is a smooth projective variety of dimension 3g 3. We also fix, once and for all, a symplectic basis addresses: Alina Marian), Dragos Oprea). 1 Present address: Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA. Preprint submitted to Elsevier Science 17 October 2005

2 {1, δ 1,..., δ 2g, ω} for the cohomology H C). A result of Newstead [6] shows that the Künneth components of c 2 End V) generate the cohomology ring H N g ). We will use the notation of Newstead and Thaddeus, writing c 2 End V) = β ψ k δ k + 2α ω for classes α H 2 N g ), β H 4 N g ), ψ k H 3 N g ). Thaddeus [7] showed that nonzero top intersections on N g must contain the ψ k s in pairs, which can then be removed using the formula g α m β n ψ k ψ k+g ) p k = α m β n, N g N g p where p = k p k and 2m + 4n + 6p = 6g 6. The top intersections of α and β are further determined: Theorem 1 [7][8][2][11][3] N g α m β n = 1) g 2 2g 2 m! m g + 1)! 2m g+1 2)B m g+1. 1) Here B k are the Bernoulli numbers defined, for instance, by the power series expansion u sinh u = 2 k 2 B k u k. 2) k k! In this paper, we reprove Theorem 1. The idea is to lift the computation from N g to a Quot scheme as indicated in the diagram below. Then, we effectively calculate the needed intersections on Quot by equivariant localization with respect to a natural torus action. The convenience of this approach lies in that the fixed loci are easy to understand; in any rank they are essentially symmetric products of C. In rank 2, their total contribution can be evaluated to the intersection numbers 1). Quot N,d P N,d π M g τ N g J The idea that the intersection theory of the moduli space of rank r bundles on a curve and that of a suitable Quot scheme are related goes back to Witten [10] in the context of the Verlinde formula. Moreover, in [1] the authors have the reverse 2

3 approach of calculating certain intersection numbers on Quot in low rank and genus by using the intersection theory of the moduli space of bundles. In this note however, the translation of the intersection theory of the moduli space of bundles into that of the Quot scheme is really very straightforward. The spaces in the diagram are as follows. M g denotes the moduli space of rank 2, odd degree stable bundles on C. By contrast with N g, the determinant of the bundles is allowed to vary in M g. There is a finite covering map τ : N g J M g of degree 4 g, given by tensoring bundles. Here J is the Jacobian of degree 0 line bundles on C. We will write Ṽ for the universal sheaf on M g C, which is only defined up to twisting with line bundles from M g. For a positive integer N, we let P N,d be the projective bundle π : P N,d M g whose fiber over a stable [V ] M g is PH 0 C, V ) N ). In other words, P N,d = Ppr Ṽ N ). We will take the degree d to be as large as convenient to ensure, for instance, that the fiber dimension of P N,d is constant. One can view P N,d as a fine moduli space of pairs V, φ) consisting of a stable rank 2, degree d bundle V together with a nonzero N-tuple of holomorphic sections φ = φ 1,... φ N ) : O N V considered projectively. As such, there is a universal morphism Φ : O N V on P N,d C. A whole series of moduli spaces of pairs of vector bundles with N sections, each labeled by a parameter τ, were defined and studied in [8] for N = 1 and [1] for arbitrary N. P N,d is one of these moduli spaces, undoubtedly the least exciting one from the point of view of studying a new object, due to its straightforward relationship with M g. One would expect the universal bundle V on the total space of P N,d and the noncanonical universal bundle Ṽ on its base M g to be closely related, and 3

4 indeed they coincide up to a twist of O1) V = π Ṽ O P N,d 1). 3) Finally, Quot N,d is Grothendieck s Quot scheme parametrizing degree d rank 2 subsheaves E of the trivial rank N bundle on C, We let 0 E O N F 0. 0 E O N F 0 be the universal sequence on Quot N,d C. For large d relative to N and g, the Quot scheme is irreducible, generically smooth of the expected dimension Nd 2N 2)g 1) [1]. Then, Quot N,d and P N,d are birational and they agree on the open subscheme corresponding to subbundles E = V where φ : O N V is generically surjective. The universal structures also coincide on this open set. For arbitrary d, Quot N,d may be badly behaved, but intersection numbers can be defined with the aid of the virtual fundamental cycle constructed in [5]. We now define the cohomology classes that we are going to intersect. We consider the Künneth decomposition c 2 EndṼ) = β ψ k δ k + 2 α ω of the universal endomorphism bundle on M g C. In keeping with the notation of [3], we further let c i Ṽ) = ã i bk i δ k + f i ω, 1 i 2, be the Künneth decomposition of the noncanonical) universal bundle Ṽ on M g C. Then f 1 = d, β = ã 2 1 4ã 2, α = 2 f 2 + bk 1 bk+g 1 dã 1. It is an easy exercise, using that N g is simply connected, to see that τ α = α, τ β = β, τ where θ is the class of the theta divisor on J. bk 1 bk+g 1 ) = 4θ, 4) 4

5 On P N,d, we let ζ denote the first Chern class of O1) and let c i V) = ā i bk i δ k + f i ω, 1 i 2, 5) be the Künneth decomposition of the Chern classes of V. Finally, we consider the corresponding a, b, f classes on Quot N,d : c i E ) = a i b k i δ k + f i ω, 1 i 2. We now show how to realize any top intersection of α and β classes on N g as an intersection on Quot N,d. Let m and n be any nonnegative integers such that With the aid of 4) we note M g α + m + 2n = 4g 3, m g. bk 1 bk+g 1 ) m βn = 1 4 g m 4θ) g α g J N m g β n. 6) g A top intersection on M g which is invariant under the normalization of the universal bundle Ṽ on M g C can be readily expressed as a top intersection on P N,d as follows. We assume from now on that N is odd, so that the fiber dimension of π : P N,d M g is Then M g α + bk 1 bk+g 1 2M = Nd 2g 1)) 1. ) m βn = M g = ζ 2M 2 f P N,d = ā M 2 2 f P N,d Here we used 3) to write 2 f ) m bk 1 bk+g 1 dã 1 ã 2 1 4ã 2 ) n = bk 1 bk+g 1 dā 1 bk 1 bk+g 1 dā 1 ) m ā2 ) m ā2 1 4ā 2 ) n = 1 4ā 2 ) n. 7) ā M 2 = ζ 2 + ã 1 ζ + ã 2 ) M = ζ 2M + lower order terms in ζ, 5

6 the latter summand being zero when paired with a top intersection from the base M g. Finally, the last intersection number in 7) can be transfered to Quot N,d using the results of [4]. It is shown there that the equality ā M 2 Rā, b, f) = P N,d a M 2 Ra, b, f) Quot N,d 8) holds for any polynomial R in the a, b, f classes, in the regime that N is large compared to the genus g, and in turn d is large enough relative to N and g, so that Quot N,d is irreducible of the expected dimension. Moreover, the equality [Quot N,d] vir am 2 Ra, b, f) = am N vir 2 Ra, b, f) [Quot N,d 2] 9) established in [5] allows us to assume from this moment on that the degree d is much smaller than N. The trade-off is that we need to make use of the virtual fundamental classes alluded to above and defined in [5]. Putting 6), 7), 8) together we obtain N g α m g β n = m g)! m! [Quot N,d] vir am 2 To prove 1), we will show that [Quot N,d ] vir a M 2 2f f ) m b k 1b k+g 1 da 1 a 2 n 1 4a 2. ) m b k 1b k+g 1 da 1 a 2 n 1 4a 2 = = 1) g 2 m 1 2 2g 1 m! ) m 2g + 1)! B m 2g+1. 10) Equation 10) will be verified by virtual localization. The torus action we will use and its fixed loci were described in [5]. For the convenience of the reader, we summarize the facts we need below. The torus action on Quot N,d is induced by the fiberwise C action on O N with distinct weights λ 1,..., λ N. On closed points, the action of µ C is [ i E O ] [ ] N E µ i O N. The fixed loci Z correspond to split subbundles E = L 1 L 2 6

7 where L 1 and L 2 are line subbundles of copies of O of degrees d 1 and d 2. Thus Z = Sym d 1 C Sym d 2 C. The fixed loci are labeled by distinct degree splittings d = d 1 + d 2 and the choice of a pair of copies of O from the N available ones. Note further that E Z = L 1 L 2 where we let L 1 and L 2 be the universal line subbundles on Sym d 1 C C and Sym d 2 C C. We write c 1 L i ) = x i 1 + k y k i δ k + d i ω, 1 i 2. We set the weights to be the N th roots of unity. The equivariant Euler class of the virtual normal bundle of Z in Quot N,d was determined in [5] to be 1 e T N vir ) = 1)g λ 1 h + x 1 ) λ 2 h + x 2 )) 2ḡ x i ) di ḡ λ i h + x i ) N h N Nλi h + x i ) N 1 exp θ i λ i h + x i ) N h 1 )). 11) N x i Here ḡ = g 1, h is the equivariant parameter, and θ i are the pullbacks to Sym d i C of the theta divisor from the Jacobian. Moreover, it is clear that on Z a 1 = x 1 + λ 1 h) + x 2 + λ 2 h), a 2 = x 1 + λ 1 h)x 2 + λ 2 h), 12) f 2 = 2 b k 1 = y k 1 + y k 2, 13) y k 1y k+g 2 + d 2 x 1 + λ 1 h) + d 1 x 2 + λ 2 h). 14) In the equation above, the superscripts are considered modulo 2g. We collect 11), 12), 13), 14), and rewrite the left hand side of 10), via the virtual localization theorem, LHS of 10) = 1) g d 1,d 2,λ 1,λ 2 I d1,d 2,λ 1,λ 2 15) where the sum ranges over all degree splittings d = d 1 + d 2 and pairs of distinct roots of unity λ 1, λ 2 ). The summand I d1,d 2,λ 1,λ 2 is defined as the evaluation on Sym d 1 C Sym d 2 C of the expression λ 1 h + x 1 ) λ 2 h + x 2 )) 2n 2ḡ 2θ 1 + 2θ 2 + d 2 d 1 )λ 1 h + x 1 ) λ 2 h + x 2 ))) m ) di ḡ λ i h + x i ) M x i Nλi h + x i ) N 1 exp θ λ i h + x i ) N h N i λ i h + x i ) N h 1 )). N x i 7

8 To carry out this evaluation, we use the following standard facts regarding intersections on Sym d C: x d l θ l = g! g l)! for l g, and xd l θ l = 0 for l > g. 16) We will henceforth replace any θ l appearing in a top intersection on Sym d C by g! g l)! xl. Then θ l Nλh + x) N 1 l! exp θ λh + x) N h 1 )) = N x k θ l+k Nλh + x) N 1 l! k! λh + x) N h 1 ) k 17) N x = g l k=0 g! x l+k Nλh + x) N 1 l! g l k)! k! λh + x) N h 1 N x ) k g = N g l l x g λh + x) N 1)g l) λh + x) N h N ). g l We further set x i = x i λ i h and j = λ 11 + x 1 ) λ x 2 ). In terms of the rescaled variables, I d1,d 2,λ 1,λ 2 becomes, via 17) and the binomial theorem, the residue at x 1 = x 2 = 0 of l 1 +l 2 +s=m We expand m! s! d 2 d 1 ) s j 2n 2ḡ+s 2 l i N g l i g l i λ M+l i ḡ i d j 2n 2ḡ+s = 1) α 2 k λ α 1 1 λ α 2 2 z N s, k) x α 1 1 x α 2 2 k=0 α 1 +α 2 =k α x i ) N 1)g l ) i)+m 1 + x i ) N 1) d i l i. +1 where we set 2n 2ḡ + s z N s, k) = λ 1 λ 2 ) 2n 2ḡ+s k. k In [5], we computed the residue { x a 1 + x) N 1+b } Res x=0 = x) N 1) c+1 N a b+p 1) a p N p=0 c ) a. p 8

9 Therefore, I d1,d 2,λ 1,λ 2 = d l 1 +l 2 +s=m k=0 α 1 +α 2 =k p 1,p 2 2 l i N ḡ l i g l i λ M+l i+α i ḡ i { m! s! d 2 d 1 ) s 1) α 2 k z N s, k) α 1 M+N 1)ḡ li )+p i N d i l i ) αi p i 1) α i p i. 18) Now 15) computes an intersection number on M g, hence it is independent of N. Thus it is enough to make N in the above expression. Writing λ 2 = ζλ 1 we have z N s, k) λ M+l1+α1 ḡ 1 λ M+l 2+α 2 ḡ 2 = N 2n 2ḡ + s λ 1 λ 2 2 k ζ 1 ζ M+l 2+α 2 ḡ 1 ζ) 2ḡ 2n s+k. Lemma 2 below clarifies the N dependence of the sum over ζ. Lemma 3 takes care of all the other terms in 18). Together, the two lemmas show that in the limit N the sum over the roots of unity of the terms 18) equals the coefficient of x α 1 1 x 2 ) α 2 in 1 m! 2 s! d 2 d 1 ) s 2 l i g x1 + d k! z s, k) l ) 2 1 x2 + d l ) 2 2 l i d 1 l 1 d 2 l 2 with 2n 2ḡ + s B 2ḡ 2n s+k z s, k) = k 2ḡ 2n s + k)! 1 22n 2ḡ+s k+1 ). Summing over α 1 + α 2 = k first, we obtain 1 m! 2 s! d 2 d 1 ) s 2 l i g k! z s, k) Coeff l x k i x + d l ) 2 1 x + d l ) 2 2. d 1 l 1 d 2 l 2 We sum next over d 1 + d 2 = d. An easy induction on m shows that generally b 1 +b 2 =a a1 a2 b 1 b 2 t + b 1 b 2 ) m = t + a 1 a 2 ) m 19) holds whenever a 1 + a 2 = a. The base case follows from the following identity which will be used repeatedly below a1 a2 a =. b b 1 +b 2 =b b 1 b 2 9

10 Via equation 19), our expression simplifies to 1 m! 2 s! 2 l i g k! z s, k) Coeff l x k 2x) s = 2 m 1 m! g g 1) s z s, s) i l 1 l 2 = 2 m 1 m! B 2ḡ 2n 2ḡ 2n)! 1 22n 2ḡ+1 ) g 1) s = 2 m 1 m! m 2g + 1)! B m 2g m+2g ) ) g 2n 2ḡ + s s ) m 2g l 1 l 2 g g = 2 m 1 m! m 2g + 1)! B m 2g m+2g ). This last equality and 15) complete the proof of 10), hence of the theorem. l 1 l 2 s Lemma 2 For all integers a and k we have lim N 1 N k ζ 1 ζ N 1 2 +a 1 ζ) k the sum being taken over the N th roots ζ of 1. = 1 2 k+1 ) Bk k!, Proof. When N is large compared to a, the sum to compute is 0 for k < 0, and 1 for k = 0. We may thus assume that k 1. We introduce the auxiliary variable z, and evaluate Setting z = u N z k 1 ζ N 1 2 +a = ζ 1 1 ζ) k ζ 1 u k N k ζ 1 ζ N 1 2 +a 1 z ζ = 1 z and making N we obtain ζ N 1 2 +a The lemma follows. = 1+u 1 ζ) k 1 u/n)n 1 2 +a 1 1 u/n) N 1 N 1 1 z) 2 +a 1 + N 1 z) N ue u/2 e u 1 = 1 u 2 sinh u. 2 Lemma 3 Let b, α be fixed non-negative integers and z a real number. Then the limit α z + p lim N α N α 1) α p N p=0 b p equals the coefficient of x α in α! x+z b. 10

11 Proof. Let us write f b z) = α p=0 z + p N b ) α 1) α p. p The recursion f b+1 z + 1) = f b+1 z) + f b z) implies by induction that We evaluate N α f j 0) = N α α p=0 p N j f b z) = b j=0 ) α 1) α p = N α p z f j 0). b j where cj, i) are the coefficients defined by the expansion x = j j cj, i) x i. i=0 We use the Euler identities ) α p i α 1) α p 0 if i < α = p=0 p α! if i = α. Making N we obtain Therefore, lim N α f j 0) = α! cj, α). N lim f bz) = α! N b j=0 j α i p α cj, i) 1) α p, i=0 p=0 N p z cj, α). b j This expression can be computed as the coefficient of x α in b z x x + z α! = α!. j=0 b j j b References [1] A. Bertram, G. Daskalopoulos and R. Wentworth, Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians, J. Amer. Math. Soc. 11

12 9 1996), no 2, [2] S. K. Donaldson, Gluing techniques in the cohomology of the moduli spaces, Topological methods in modern mathematics Stony Brook, NY, 1991), , Publish or Perish, Houston, TX, [3] L. Jeffrey, F. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. 2) ), no. 1, [4] A. Marian, Intersection theory on the moduli spaces of stable bundles via morphism spaces, Harvard University Thesis [5] A. Marian, D. Oprea, Virtual intersections on the Quot scheme and the Vafa- Intriligator formula, preprint, math.ag/ [6] P. E. Newstead, Characteristic classes of stable bundles over an algebraic curve, Trans. Amer. Math. Soc ), [7] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom ), no. 1, [8] M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math ), no. 2, [9] E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys ), no. 4, [10] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, and physics, , Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, [11] D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, The moduli space of curves Texel Island, 1994), , Progr. Math., 129, Birkhäuser Boston, Boston, MA,

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