Hitchin fibration on moduli of symplectic and orthogonal parabolic Higgs bundles (Q2220905)

Let \(G\) be a complex orthogonal or symplectic group and \(\mathcal{M}(r, d, \alpha)\) the moduli space of stable parabolic \(G\)-bundles of rank \(r\), degree \(d\) and weight type \(\alpha\) over a compact Riemann surface \(X\) of genus \(g\geq2\). \textit{N. Hitchin} [Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)] discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic curve is an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. The aim of this paper is to study the Hitchin fibers for \(\mathcal{M}(r, d, \alpha)\). The author proves that the Hitchin fibers for the moduli space of stable parabolic symplectic or orthogonal Higgs bundles on an algebraic curve are Prym varieties of the spectral curve with respect to an involution. In this context, Higgs fields are strongly parabolic, meaning that the Higgs field is nilpotent with respect to the flag. This paper is organized as follows: In Section 2, the author gives the necessary details regarding parabolic symplectic or orthogonal Higgs bundles and their moduli. Section 3 deals with a description of the Hitchin fibration and the spectral data. In Section 4, the author proves the main result for three different cases, i.e., for \(G=\mathrm{Sp}(2m, \mathbb{C}), \mathrm{SO}(2m, \mathbb{C})\) and \(\mathrm{SO}(2m +1, \mathbb{C})\). For \(G=\mathrm{Sp}(2m, \mathbb{C})\) or \(\mathrm{SO}(2m+1, C)\), fibers are Prym variety of the spectral curve with respect to an involution wih fixed points. For \(G=\mathrm{SO}(2m, \mathbb{C})\), the spectral curve is singular but the fibers are Prym variety of the desingularised spectral curve with respect to an involution without fixed points.