Anti-holomorphic involutive isometry of hyper-K\"ahler manifolds and branes

TL;DR

This paper investigates the properties of complex Lagrangian submanifolds in hyper-Kähler manifolds, revealing fixed point loci of certain involutions and the existence of Lagrangian submanifolds not arising from such involutions.

Contribution

It proves that antiholomorphic involutions with respect to one complex structure have fixed point loci that are complex Lagrangian in another structure, and shows the existence of Lagrangians not fixed by any such involution.

Findings

Involutions with specific antiholomorphic properties have fixed point loci that are complex Lagrangian.

Existence of Lagrangian submanifolds not fixed by any antiholomorphic involution.

Results deepen understanding of symmetries and submanifold structures in hyper-Kähler geometry.

Abstract

We study complex Lagrangian submanifolds of a compact hyper-K\"ahler manifold and prove two results: (a) that an involution of a hyper-K\"ahler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is complex Lagrangian with respect to one of the other complex structures, and (b) there exist Lagrangian submanifolds which are complex with respect to one complex structure and are not the fixed point locus of any involution which is anti-holomorphic with respect to one of the other complex structures.