Classification of Lagrangian planes in generalised Kummer manifolds

Abstract

Abstract We prove that the class of a line contained in a Lagrangian plane on a dimension 2 n hyperkähler manifold X of Kummer type has Beauville-Bogomolov-Fujiki square $$-\frac{n+1}{2}$$ - n + 1 2 in "Equation missing" and order 2 in the discriminant group of $$H^2(X,\mathbb {Z})$$ H 2 ( X , 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.