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

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 ${\text{sW}}_A$ 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.