# Herbert Lange

## Polarizations of Prym varieties via abelianization

Abstract: This is a report on a recent joint paper with Christian
Pauly:
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 $X$.