A duality for Spin Verlinde spaces and Prym theta functions C Pauly and S Ramanan

Description

Niveau: Supérieur, Master, Bac+4
A duality for Spin Verlinde spaces and Prym theta functions C. Pauly and S. Ramanan August 29, 2000 Abstract We prove canonical isomorphisms between Spin Verlinde spaces,i.e., spaces of global sec- tions of a determinant line bundle over the moduli space of semistable Spinn-bundles over a smooth projective curve C, and the dual spaces of theta functions over Prym varieties of unramified double covers of C. 1 Introduction To any smooth, projective curve C, one classically associates a collection of principally polarized abelian varieties: the Jacobian JC, parametrizing degree zero line bundles, as well as, for any unramified double cover C˜? ? C, depending on a non-zero 2-torsion point ? ? JC[2], a Prym variety P?. The projective geometry of the configuration JC ? ? ? P? has been much studied [M1], [vGP], [B1] and encodes e.g. the Schottky-Jung identities among theta-constants [M1]. Less classically, one can consider the moduli space M(G) of semistable principal G-bundles over the curve C, where G is a simple and simply-connected algebraic group. For some ample line bundle L over M(G), the vector space of global sections H0(M(G),L) has been identified to a space of conformal blocks arising in conformal field theory (see e.

• theta


• line bundle


• divisors associated


• associated half-spinor


• any smooth


• spinor vector


• spin group


• bundles over


• prym varieties

Sujets

• Line bundle
• Spin group
• Prym variety

Informations

• Publié par: larec
• Nombre de lectures: 150
• Langue: English

Extrait

A duality for Spin Verlinde spaces and Prym theta functions C. Pauly and S. Ramanan August 29, 2000

Abstract We prove canonical isomorphisms between Spin Verlinde spaces,i.e., spaces of global sec- tions of a determinant line bundle over the moduli space of semistable Spin n -bundles over a smooth projective curveC, and the dual spaces of theta functions over Prym varieties of unramied double covers ofC.

1 Introduction To any smooth, projective curveC, one classically associates a collection of principally polarized abelian varieties: the JacobianJC, parametrizing degree zero line bundles, as well as, for any ˜ unramied double coverC  →C, depending on a non-zero 2-torsion point∈JC[2], a Prym varietyP  . The projective geometry of the congurationJC∪ S  P  has been much studied [M1], [vGP], [B1] and encodes e.g. the Schottky-Jung identities among theta-constants [M1]. Less classically, one can consider the moduli spaceM(G) of semistable principalG-bundles over the curveC, whereGis a simple and simply-connected algebraic group. For some ample line bundleLoverM(G), the vector space of global sectionsH 0 (M(G)L) has been identied to a space of conformal blocks arising in conformal eld theory (see e.g.[S] for a survey or [LS] for a proof), which made the computation of its dimension possible. This is the celebrated Verlinde formula. In this paper we are interested in dualities between Verlinde spacesH 0 (M(G)L) and spaces of abelian theta functions ,i.e.,sections of some multiple of a principal polarization overJCor, more generally,JC∪ S  P  . Such dualities were rst proved for the structure groupG= SL 2 and the line bundlesL l forl= 124 [B1], [B2], [OP] or, more generally,G= SL n and the line bundle L[BNR], whereLis the ample generator of the Picard group ofM(SL n ). In the articles [O1] and [O2], W.M. Oxbury constructs linear maps between a Verlinde space for the complex spin group,i.e.,G= Spin n , and a space of abelian theta functions overJC∪ S  P  . Our main theorem states that these linear maps are actually isomorphisms. More precisely we show 1.1. Theorem.For any curveCand any integerm1, we have canonical isomorphisms (1) X s   : X H 0 (P  ev (2m+ 1)  )  ∨   →H 0 (M  (Spin 2m+1 )(C 2m+1 )) (1.1) ∈JC[2] (2) X s   : X £ H 0 (P  ev (2m)  )  ∨ H 0 (P  odd (2m)  )  ∨ ¤   →H 0 (M  (Spin 2m )(C 2m )) ∈JC[2] (1.2)

1

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We refer to section 2 and 3 for the rather technical details encoded in the notation. We note that theorem 1.1 for the group Spin 3 coincides with the above mentioned duality for Verlinde spaceH 0 (M(SL )L 4 ) under the exceptional isomorphism Spin = SL . In this case

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