Prym-Tjurin varieties and the Hitchin map (Q1907179)

Let \({\mathcal M}\) be the moduli space of stable principal \(G\)-bundle over a compact Riemann surface \(C\), with \(G\) a reductive algebraic complex group and denote by \(K\) the canonical bundle over \(C\). In Duke Math. J. 54, 90- 114 (1987; Zbl 0627.14024), \textit{N. Hitchin} defined a map \({\mathcal H}\) from the cotangent bundle \(T^* {\mathcal M}\) to the ``characteristic space'' \({\mathcal K}\) by associating to each \(G\)-bundle \(P\) and section \(s \in H^0 (C,\text{ad}P \otimes K)\) the spectral invariants of \(s\). Next he described in the classical cases \((G = Gl(n), SO(n), Sp(n))\) the generic fibre of \({\mathcal H}\) in terms of a suitable abelian variety inside the Jacobian of a spectral curve covering \(C\). Let \(T \subset G\) be some fixed maximal torus with Weyl group \(W\) and denote by \(X(T)\) the group of characters on \(T\). One may associate to each generic \(\varphi \in {\mathcal K}\) a \(W\)-Galois covering \(\widetilde C\) of \(C\) and consider the ``generalized Prym variety'' \({\mathcal P} = \Hom_W (X(T), J(\widetilde C))\). In this paper the author explicitly defines a map \({\mathcal F} : {\mathcal H}^{-1} (\varphi) \to {\mathcal P}_0\), \({\mathcal P}_0\) being the connected component in \({\mathcal P}\) containing the identity element. Such \({\mathcal F}\) depends in fact on the choice of one basis of roots \(\Delta \subset R(G,T)\). The author proved [``An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups'' (preprint 1994)] that the restriction of \({\mathcal F}\) to each connected component of \({\mathcal H}^{-1} (\varphi)\) has finite fibres. In the present paper, for each dominant weight \(\lambda \in \Lambda (T)^+\) (equivalently for each irreducible representation of \(G)\) the author defines by suitable divisorial correspondences a Prym-Tjurin variety \(P_\lambda \subset J (\widetilde C)\) and shows that there is an isogeny: \({\mathcal P} \to P_\lambda\). Her construction of \(P_\lambda\) is a natural generalization of the one given by \textit{V. Kanev} [in: Algebraic geometry, Proc. Conf., Sitges 1983, Lect. Notes Math. 1124, 166-215 (1985; Zbl 0575.14037)]. Next the author considers an irreducible spectral covering \(C_\lambda\) of \(C\) and, in analogy with the above, defines a Prym-Tjurin variety \({\mathcal A}_\lambda \subset J (C_\lambda)\). In the classical cases, when \(\lambda\) corresponds to the natural representation \({\mathcal A}_\lambda\) is exactly the abelian variety found by Hitchin. Following Hitchin's procedure, the author defines a map \(h_\lambda : {\mathcal H}^{-1} (\varphi) \to {\mathcal A}_\lambda\) and using her previous results and the fact that \(P_\lambda\) is isogenous to \({\mathcal A}_\lambda\) proves that the restriction of \(h_\lambda\) to each connected component of \({\mathcal H}^{-1} (\varphi)\) has finite fibres. Similar results concerning isogenies between a generalized Prym variety not depending on the representation of \(G\) and spectral Prym-Tjurins are also given by \textit{R. Donagi} [in: Journées de géométrie algébrique, Orsay 1992, Astérisque 218, 145-175 (1993; Zbl 0820.14031)].