Polarizations of Prym varieties for Weyl groups via abelianization

Herbert Lange (1) , Christian Pauly (2)

1 Mathematisches Institut
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Let $\pi: Z \ra X$ 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 $\lambda$ consider the curve $Y = Z/\Stab(\lambda)$. The Kanev correspondence defines an abelian subvariety $P_\lambda$ of the Jacobian of $Y$. We compute the type of the polarization of the restriction of the canonical principal polarization of $\Jac(Y)$ to $P_\lambda$ in some cases. In particular, in the case of the group $E_8$ 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 principal $G$-bundles on the curve $X$.

Domaines: Géométrie algébrique [math.AG]

Herbert Lange, Christian Pauly. Polarizations of Prym varieties for Weyl groups via abelianization. 2007. ⟨hal-00128775v2⟩