POLARIZATIONS OF PRYM VARIETIES FOR WEYL GROUPS VIA ABELIANIZATION

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POLARIZATIONS OF PRYM VARIETIES FOR WEYL GROUPS VIA ABELIANIZATION HERBERT LANGE AND CHRISTIAN PAULY Abstract. Let pi : Z ? X be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group G. For any dominant weight ? consider the curve Y = Z/Stab(?). The Kanev correspondence defines an abelian subvariety P? of the Jacobian of Y . We compute the type of the polarization of the restriction of the canonical principal polarization of Jac(Y ) to P? in some cases. In particular, in the case of the group E8 we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety to the moduli stack of principal G-bundles on the curve X. 1. Introduction 1.1. Verlinde spaces. Let X be a smooth complex projective curve of genus g and let G be a simple, simply-connected complex Lie group. We denote by MX(G) the moduli stack of principal G-bundles and by L the ample generator of its Picard group. The celebrated Verlinde formula ([Fa1], [So1], [So2]) computes the dimension Ng,l(G) of the space of global sections H0(MX(G),L?l) for any level l.

• map ??


• polarized abelian


• map ∆?


• ?a ?l ?


• prym


• connected complex


• abelian subvarieties


• galois group


• weyl group

Sujets

• Galois group
• Weyl group

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

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POLARIZATIONS OF PRYM VARIETIES FOR WEYL ABELIANIZATION

HERBERT LANGE AND CHRISTIAN PAULY

GROUPS VIA

Abstract.Letπ:Z→XGalois covering of smooth projective curves with Galois groupbe a the Weyl group of a simple and simply connected Lie groupG. For any dominant weightλ consider the curveY=Z/Stab(λThe Kanev correspondence defines an abelian subvariety). P λ of the Jacobian ofYthe polarization of the restriction of the. We compute the type of canonical principal polarization of Jac(Y) toP λ in some cases. In particular, in the case of the groupE 8 we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety to the moduli stack of principalG-bundles on the curveX.

1.Introduction

1.1.Verlinde spaces.LetXbe a smooth complex projective curve of genusgand letG be a simple, simply-connected complex Lie group. We denote byM X (G) the moduli stack of principalG-bundles and byLthe ample generator of its Picard group. The celebrated Verlinde formula ([Fa1], [So1], [So2]) computes the dimensionN g,l (G) of the space of global sections H 0 (M X (G)L ⊗l ) for any levell. The Verlinde numbers at levell= 1 for the groups of type ADE are given in the following table.

GSL(m) Spin(2m)E 6 E 7 E 8 N g,1 (G)m g 4 g 3 g 2 g 1

The numberm g for SL(m) coincides with the number of level-mtheta functions on the Jacobian ofX(see [BNR]). For the even Spin group the Verlinde number equals the number of theta characteristics ofX(see [O]). The striking simplicity of the Verlinde numbers forE 6 , E 7 andE 8 main motivation for us to try to relate these Verlinde spaces to spaces ofwas the theta functions on polarized abelian varieties (the Prym varieties) and compute the induced polarizations — see Main Theorem and Remark 8.4.

1.2.Abelianization of principalG-bundles.The abelianization program of principalG- bundles, or more preciselyG-Higgs bundles, takes its origin in Hitchin’s papers [Hi1] and [Hi2]. For the caseG= SL(m) it is shown in [BNR] that for a sufficiently ramified spectral cover ψ:Y→Xthe direct image map

Prym(Y /X)−→M X (SL(m)) induces by pull-back an isomorphism between the SL(mspace at level 1 and the)-Verlinde space of abelian theta functionsH 0 (Prym(Y /X) L Y ). For general structure groupsGthe abelianization theory has been worked out by Faltings [Fa2], by Donagi [Don1], [Don2] and Donagi-Gaitsgory [DG]. 1

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1.3.Correspondences on spectral and cameral covers.For general structure groupsG Prym varieties can be constructed via correspondences on covers of the curveX:

In [K1] Kanev constructs from the data of a rational mapf:C→g= Lie(G) and an irreducible representationρ λ :G→GL(V) a spectral coverψ:Y→P 1 equipped with a correspondence. He shows that ifGis of type ADE, the weightλminuscule andfsufficiently general, then the Prym varietyP λ ⊂Jac(Y) associated with Kanev’s correspondence is a Prym- Tyurin variety (i.e. the polarization onP λ induced from the principal polarization of Jac(Y) is a multiple of a principal polarization). Note that Kanev’s construction is carried out in the caseX=P 1 .

A different but related construction of Prym varieties is given by Donagi in [Don1], [Don2]: letT⊂Gbe a maximal torus,Wthe Weyl group ofGandS ω = Hom(T C ∗ ) the weight lattice. For any cameral cover, i.e. a Galois cover with Galois groupW

π:Z−→X

satisfying some conditions on the ramification, Donagi introduces the Prym variety Prym(πS ω ) := Hom W (S ω Jac(Z)) parametrizingW-equivariant homomorphisms fromS ω to Jac(Z). In this paper we generalize Kanev’s construction to an arbitrary base curveX. Given a Galois coverπ:Z→Xand a dominant weightλ∈S ω we consider the cover of curves ψ:Y−→XwithY=Z/Stab(λ).

Kanev’s construction generalizes (see section 3) to give a correspondenceK λ on the curveY defining an abelian subvariety P λ ⊂Jac(Y) which is isogenous to the Donagi-Prym variety Prym(πS ω ) (Proposition 6.13). 1.4.The polarization on the Prym varietyP λ .LetL Y denote a line bundle defining the canonical principal polarization on the Jacobian Jac(Y). The aim of this paper is to compute the induced polarizationL Y |P λ under certain assumptions. In fact, ifq λ denotes the exponent of the correspondenceK λ andd λ the Dynkin index ofλ, our main result is the following theorem (we prove a slightly more precise version, see Theorem 8.1) . We use the notation of [Bo] for the weights.

Main Theorem.We suppose that theW-Galois coverπ:Z→Xista´e.Tleehintnehacess given in the table below the induced polarizationL Y |P λ is divisible byq λ ,i.e.L Y |P λ =M ⊗q λ and the polarizationMonP λ is of typeK(M) = (Z/mZ) 2g :

Weyl group of type weightλ q λ =d λ A n ;n >1$ i ; (i n+ 1) = 1  i n − − 1 1  −3 D n ;nodd$ n−1  $ n 2 n E 6 $ 1  $ 6 6 E 7 $ 7 12 E 8 $ 8 60

K(M) (Z/(n+ 1)Z) 2g (Z/4Z) 2g (Z/3Z) 2g (Z/2Z) 2g 0

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Using the table in section 1.1 we observe that in all the cases of the Main Theorem we have an equality of dimensions dimH 0 (P λ  M) =N g,1 (G). Moreover there exists a morphismγ:P λ → M X (G) (see Remark 8.3) constructed via the abelianization map Δ θ which induces by pull-back a linear map between— see below — and spaces of global sections having the same dimension γ ∗ :H 0 (M X (G)L)−→H 0 (P λ  M). We discuss in Remark 8.4 the natural question whetherγ ∗ is an isomorphism.

It is well known that the Weyl group of typeE k is closely related to the del Pezzo surface of degree 9−kfor 5≤k≤8. In fact, a slightly modified lattice of the weight lattice is isomorphic to the Picard lattice of the corresponding del Pezzo surface (see [K1, section 8.7]). Moreover, for 4≤k≤is given essentially by the incidence correspondence of7 the Kanev correspondence lines of the corresponding del Pezzo surface. Fork= 8 there are multiplicities due to the fact that the weight$ 8 is only quasi-minuscule (see [K1]). Notice that in these cases the polarization MonP λ is of type (Z/dZ) 2g wheredof the correponding del Pezzo surface.is the degree In particular in the case ofW=W(E 8 family of Prym-Tyurin varieties, i.e. the) we obtain a pairs (P λ  M) are principally polarized abelian varieties. It is easy to see that any curveXof genusg≥a4mdoisgrouplaGhtiwgnirevocsoialeGaletn´saitW(E 8 ) and we get a family of Prym-Tyurin varieties of dimension 8(g−1) of exponent 60. We plan to study this family in a subsequent paper.

Probably there is an analogous result in the case that the Galois coveringπ:Z→Xadmits simple ramification. Certainly the paper [DG] will be essential for this. We plan to come back to it subsequently.

Note that our results are disjoint from the results in [K1]. First of all, Kanev considers only Galois coverings overP 1 which are necessarily ramified. Moreover, his correspondence satisfies a quadratic equation and he uses his criterion ([K2]) to show that the associated abelian subvarieties are principally polarized. In our case the corresponding correspondence satisfies a cubic equation (see Theorem 3.9) and a quadratic equation only on the Prym variety Prym(Y /X) of the coveringψ:Y→X. However, the abelian variety Prym(Y /X) is not principally polarized and thus we cannot apply Kanev’s criterion [K2] in order to compute the polarization ofP λ . Instead we proceed as follows: We work out a general result on restrictions of polarizations to abelian subvarieties (Proposition 2.10) roughly saying that, if the restricted polarization equals theq-fold of a polarization, whereqis the exponent of the abelian subvariety, then its type can be computed. The main idea of the proof is then the use of an abelianization map

Δ θ : Prym(πS ω ) n →M− X (G) (see Section 7). Here Prym(πS ω ) n denotes a certain connected component of the Donagi-Prym variety Prym(πS ω that the restricted polarization is the). The fact, q λ -fold of a polarization is