On Lagrangian fibrations by Jacobians, II

Abstract

We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties. THE GL-HITCHIN SYSTEMThe degeneration of Donagi, Ein, and Lazarsfeld can be generalized to some Lagrangian fibrations by Prym varieties, giving a very concrete connection between certain compact and noncompact examples. In other cases, we can detect some analogies between compact and non-compact examples without there being a clear connection. Still, there remain many non-compact Lagrangian fibrations which do not have obvious compact counterparts (and conversely).Here is a summary of the paper. In Section 2 we describe the GL-Hitchin system, a noncompact Lagrangian fibration by Jacobians of curves, and how it is related to its dual fibration. In Section 3 we consider compact Lagrangian fibrations, particularly the Beauville-Mukai system. Again we describe how it is related to its dual fibration, we describe the Donagi-Ein-Lazarsfeld degeneration to a compactification of the GL-Hitchin system, and we summarize what is known about the classification of Lagrangian fibrations by Jacobians of curves. In Section 4 we describe the other Hitchin systems, which are fibred by Prym varieties, and the duality relations between them. In Section 5 we turn to compact Lagrangian fibrations by Prym varieties. We describe the Markushevich-Tikhomirov system, its dual fibration, the Matteini system, and some other examples. In Section 6 we mention a few original results of the author (some obtained jointly with Chen Shen): we extend the Donagi-Ein-Lazarsfeld degeneration result to Lagrangian fibrations by Prym varieties, we speculate on the dual fibration of the Matteini system, and we describe the dual fibration of the Debarre system. Finally, in Section 7 we summarize all of the known examples of Lagrangian fibrations by Jacobians and Prym varieties in Table 1; this suggests where we should perhaps search for new examples.This work was presented at the satellite meeting of the ICM 2018 "Moduli spaces in algebraic geometry and applications" held in Campinas, 26-31 July, 2018. The author is grateful to the organizers for the invitation to speak at this meeting, and also grateful for support from the National Science Foundation, award DMS-1555206.Theorem 1 (Hitchin [20]) For coprime n and d, M GL is a holomorphic symplectic manifold, i.e., it admits a natural holomorphic symplectic form σ. In fact, M GL admits a hyperkähler metric. The Hitchin system3 Remark Denote by Bun GL the moduli space of stable bundles on Σ of rank n and degree d. If E ∈ Bun GL , then the inequality above is satisfied for all subbundles F . Therefore we could choose any Higgs field Φ, and the pair (E, Φ) would be a stable Higgs bundle. Consider the cotangent space to Bun GL at E. By Serre dualitySo in this case (E, Φ) is really a point of the cotangent bundle T * Bun GL . This shows that T * Bun GL ⊂ M GL , and in fact it is a dense open subset. The holomorphic symplectic form σ on M GL is an extension of the canoni...

Birational geometry of Beauville-Mukai systems I: the rank three and genus two case

“…In fact, Markushevich [Marku02] conjectured that any Lagrangian fibration whose fibres are Jacobians of curves must be of this type, i.e., the family of curves must be a complete linear system on a K3 surface. He proved this for genus two curves [Marku96], and the second author proved this for genus three, four, and five [Saw15b], and for all genera under an additional assumption [Saw15a].…”

Fibrations on four-folds with trivial canonical bundles

“…but this is Lemma 5 in [26]. More precisely, Lemma 5 is for Lagrangian fibrations, but the same argument works provided that the generic singular curves of C/P 2 have only single nodes, which is guaranteed by the mild degenerations hypothesis.…”

“…We include a slight simplification of his argument here for completeness. Markushevich's theorem has been generalized to fibrations on higher-dimensional holomorphic symplectic manifolds by the author [26]. In [23,25] the author described twisted Fourier-Mukai transforms between Lagrangian fibrations X → P n on holomorphic symplectic manifolds and their dual fibrations X → P n .…”

On Lagrangian fibrations by Jacobians I

“…In a sequel [35] to this paper we will prove Conjecture 1 in a number of low dimensional cases: when n = 3, 4, and 5 and all of the curves in the family Y /P n are non-hyperelliptic, and when n = 3 and all of the curves are hyperelliptic.…”