Classification of Lagrangian planes in generalised Kummer manifolds

Abstract

We prove that the class of a line contained in a Lagrangian plane on a dimension $2n$ hyperkähler manifold $X$ of Kummer type has Beauville-Bogomolov-Fujiki square $-\frac{n+1}{2}$ in $H_2(X,\mathbb{Z})\cong (H^2(X,\mathbb{Z}))^\vee$ and order 2 in the discriminant group of $H^2(X,\mathbb{Z}).$ Vice versa, an extremal primitive ray of the Mori cone verifying these conditions is in fact the class of a line in some Lagrangian plane.