Degeneration of natural Lagrangians and Prymian integrable systems

Abstract Starting from an anti-symplectic involution on a K3 surface, one can consider natural Lagrangian subvariety inside the moduli space of sheaves over K3. One also construct Prymian integrable system following construction Markushevich–Tikhomirov, extended by Arbarello–Saccà–Ferretti, Matteini and Sawon–Shen. In this article we address question Sawon, showing that these systems their associated Lagrangians degenerate, respectively, into fix loci involutions considered Heller–Schaposnik, García-Prada–Wilkin Basu–García-Prada. Along way find interesting results such as proof Donagi–Ein–Lazarsfeld degeneration is symplectic varieties, generalization degeneration, originally described for surfaces, to case arbitrary smooth projective description behaviour certain under degeneration.