Monogamous subvarieties of the nilpotent cone

Let G be a reductive algebraic group over an algebraically closed field k of prime characteristic not 2, whose Lie algebra is denoted g. We call a subvariety X of the nilpotent cone N \subset g$ _monogamous_ if for every e\in X, the \sl_2-triples (e,h,f) with f\in X are conjugate under the centraliser C_G(e). Building on work by the first two authors, we show there is a unique maximal closed G-stable monogamous subvariety V\subset N and that it is an orbit closure, hence irreducible. We show that V can also be characterised in terms of Serre's G-complete reducibility.