Donagi–markman Cubic for the Generalized Hitchin System

Abstract

have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi-Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system.

Lectures on Higgs moduli and abelianisation

“…It is only natural, then, to try to compute c for the Hitchin integrable system and its various generalisations. For the Hitchin system per se this was done by T.Pantev (for SL n , unpublished, but see [DDP07] for SL 2 ), and by D.Balduzzi ([Bal06]) for arbitrary reductive G. For meromorphic Higgs bundles this was done by U.Bruzzo and the author ( [BD14]). We shall recall the statement of the main theorem and sketch the key steps of the proof.…”

“…We shall recall the statement of the main theorem and sketch the key steps of the proof. For more details one can refer to [BD14] or [Dal16].…”

Meromorphic Higgs bundles and related geometries

“…Related results. We should note that several instances of Theorem A (and its reformulation, Theorem B in [BD14]) have already been established in the literature. First of all, for the usual Hitchin system (D = 0) and G = SL n (C) the formula appears in unpublished work of T.Pantev, while for the case G = SL 2 (C) the formula can be found in [DDD + 06], (47).…”

“…We direct the reader to the beautiful expositions in [DM93], §1 and [DM96], §7 for a different version of this argument, discussion and applications. For the case of the generalised Hitchin system, our Theorem B in [BD14] contains a formula for the infinitesimal period map which makes it evident that c is a section of Sym 3 T ∨ B .…”

“…One of our goals in this note is to outline a calculation of the infinitesimal period map of the generalised (ramified) Hitchin system. This is done in §5, and we refer to [BD14] for more details. In short, our main result in §5 is that the Balduzzi-Pantev formula ( [Bal06], [DDP07]) holds along the maximal rank symplectic leaves of the generalised Hitchin system.…”