Special Lagrangian Geometry in irreducible symplectic 4-folds

TL;DR

This paper studies special Lagrangian submanifolds in irreducible symplectic 4-folds, showing they are bi-Lagrangian, obtained via hyperkähler rotation, and are real analytic, revealing their geometric rigidity.

Contribution

It demonstrates that all special Lagrangian submanifolds in these 4-folds are bi-Lagrangian and can be derived from complex submanifolds through hyperkähler rotation, highlighting their rigidity.

Findings

All special Lagrangian submanifolds are bi-Lagrangian.

They are obtained by hyperkähler rotation from complex submanifolds.

All such submanifolds are real analytic.

Abstract

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are bi-Lagrangian and that they are obtained by complex submanifolds via a sort of "hyperkaehler rotation trick"; thus they retain part of the rigidity of complex submanifolds: indeed all special Lagrangian submanifolds of X turn out to be real analytic.