Hitchin Fibration on Moduli of Symplectic and Orthogonal Parabolic Higgs Bundles
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Hitchin Fibration on Moduli of Symplectic and Orthogonal Parabolic Higgs Bundles Sumit Roy1 Received: 11 March 2020 / Accepted: 22 October 2020 / © Springer Nature B.V. 2020
Abstract Let X be a compact Riemann surface of genus g 2, and let D ⊂ X be a fixed finite subset. Let M(r, d, α) denote the moduli space of stable parabolic G-bundles (where G is a complex orthogonal or symplectic group) of rank r, degree d and weight type α over X. Hitchin, in his paper Hitchin (Duke Math. J. 54(1), 91–114, 1987) 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. In this paper we study the Hitchin fibers for M(r, d, α). Keywords Integrable system · Moduli space · Parabolic bundle
1 Introduction Let X be a compact Riemann surface, and let D ⊂ X be a fixed finite subset. The notion of parabolic bundles over a curve and their moduli space was described in [10]. In [4], Bhosle and Ramanathan extended this notion to parabolic G-bundles where G is a connected reductive group. The moduli space of parabolic Higgs bundles was constructed by Yokogawa [11]. The notion of symplectic and orthogonal parabolic bundles were described in [1]. Those bundles are parabolic vector bundles with a nondegenerate (in a suitable sense) symmetric or anti-symmetric form taking values in a parabolic line bundle. When all parabolic weights are rational, the notion of parabolic symplectic or orthogonal bundles coincides with the notion of parabolic principal G-bundles where G is an orthogonal or symplectic group.
Sumit Roy
[email protected] 1
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India
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Math Phys Anal Geom
(2020) 23:41
Hitchin in his paper [5] showed that the moduli space of stable G-Higgs bundles (where G = GL(m, C), Sp(2m, C), SO(2m, C) or SO(2m + 1, C)) on an algebraic curve forms an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. Later, in [6], Markman proved the same for the moduli space of stable L-twisted Higgs bundles where L is a positive line bundle on X satisfying L KX . In this paper, we will prove 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 our context, Higgs fields are strongly parabolic, meaning that the Higgs field is nilpotent with respect to the flag. Here is a brief outline of this paper: In Section 2 we give the necessary details regarding parabolic symplectic or orthogonal Higgs bundles and their moduli. In Section 3 we give a description of the Hitchin fibration and the spectral data. In Section 4 we prove the main result for three different cases, i.e. for G = Sp(2m, C), SO(2m, C)
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