Degeneration of natural Lagrangians and Prymian integrable systems

TL;DR

This paper explores how natural Lagrangians and Prymian integrable systems associated with K3 surfaces degenerate into fixed loci of involutions, extending known degenerations from K3 surfaces to more general surfaces.

Contribution

It demonstrates that these integrable systems and Lagrangians degenerate into fixed loci of involutions, generalizes the Donagi--Ein--Lazarsfeld degeneration to arbitrary smooth projective surfaces, and analyzes involution behavior under degeneration.

Findings

Degeneration of integrable systems into fixed loci of involutions.

Generalization of symplectic degeneration from K3 surfaces to all smooth projective surfaces.

Behavior of involutions under degeneration.

Abstract

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