On the fixed locus of the antisymplectic involution of an EPW cube

Abstract

EPW cubes are polarized hyper-Kähler varieties of K$3^{[3]}$-type that carry an anti-symplectic involution. We study the geometry of the fixed locus $\sW_A$ of this involution and prove that it is a \emph{rigid} atomic Lagrangian submanifold. Our proof is based on a detailed description of certain singular degenerations of EPW cubes and the degeneration methods of Flappan--Macrì--O'Grady--Saccà.

Remark 2.7 added. Comments are welcome!