Classification of Lagrangian planes in Kummer-type Hyperkähler Manifolds

We generalise a result from Bakker on K3 type hyperkähler manifolds proving that a line in a Lagrangian plane on an hyperkähler manifold X of Kummer type has Beauville-Bogomolov-Fujiki square −(n+1)/ 2 and order 2 in the discriminant group of H^2(X, Z). Viceversa, an extremal primitive ray of the Mori cone verifying these conditions is in fact the class of a line in some Lagrangian plane. In doing so, we show, on moduli spaces of Bridgeland stable objects on an abelian surface, that Lagrangian planes on the fiber of the Albanese map correspond to sublattices of the Mukai lattice verifying some numerical condition.