On a hyper-Kähler manifold X, a real structure - or equivalently, an anti-holomorphic involution -
is referred to as a brane involution. Up to hyper-Kähler rotation, an anti-holomorphic involution
can become an anti-symplectic involution. When the Smith-Thom inequality is realized as an
equality, the brane involution is said to be maximal. While examples of non-compact hyper-
Kähler manifolds admitting maximal branes are known, the compact case presents a more
intriguing picture. In particular, although some K3 surfaces admit maximal brane involutions,
the main result that I will show is the non-existence of maximal branes on compact hyper-
Kähler manifolds of K3[n]-type when n = 2 or n is odd. This talk is based on a joint work in
progress with Simone Billi and Lie Fu.