Donagi-Markman cubic Hitchin systems (Q862069)

Let \(G\) be a complex connected semisimple Lie group \(G\). We write \({\mathcal M}_G\) for the smooth locus of the moduli space of semistable principal \(G\)-bundles over a smooth complex projective curve \(M\). Write \({\mathcal X}\) for its cotangent bundle, a symplectic manifold. Points of \(\mathcal X\) correspond to \textit{Higgs bundles}, pairs \((P, \Phi)\) where \(P\) is a principal \(G\)-bundle and \(\Phi\) a section of the twisted adjoint bundle \(K_{M} \otimes \mathfrak{g}_P\). \textit{N. Hitchin} [Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)] showed that there is a Lagrangian fibration of \({\mathcal X}\) over a space \(B\) of the form \(\bigoplus_{i} H^{0}(M, K_{M}^{d_i})\), whose generic fibre is an open subset of an abelian variety. Almost all fibres \({\mathcal X}_b\) are diffeomorphic to some fixed fibre \(X\), so one can consider the period map \(B \dashrightarrow \mathrm{Grass}(\dim(H^{0}({\mathcal X}_{b},\Omega^{1})), H^{1}(X,\mathbb{C}))\) as studied by \textit{P. A. Griffiths} [Am. J. Math. 90, 805--865 (1968; Zbl 0183.25501)]. The differential of this period map gives rise to a symmetric cubic tensor on \(B\) called the \textit{Donagi--Markman cubic}. In unpublished notes, Pantev gave a formula for this cubic in the case \(G = \mathrm{SL}_n\). The author generalises this formula to the case of arbitrary \(G\). At a point \(\phi\) of \(B\), the cubic is expressed in terms of quadratic residues of differential forms on a cameral cover \(\tilde{M_{\phi}}\) of \(M\). Furthermore, the author derives a second, more symmetric formula for the cubic, in terms of simple residues on \(\tilde{M_{\phi}}\) and roots of the Lie algebra of \(G\).