The cone, ampleness, and contraction conjecture for four-dimensional generalized Kummer varieties

Let $(X,g)$ be a polarized variety deformation equivalent to a four-dimensional generalized Kummer variety. Let $(R,R)$ denote the Beauville–Bogomolov form on homology. The generalized Kummer cone and contraction conjecture.

$$\mathrm{NE}_1(X)=\left\langle R\in \mathrm{N}_1(X,\mathbb Z):(R,R)\geq -3/2,\ R\cdot g>0\right\rangle.$$

A divisor class $h$ on $X$ is ample if and only if $h\cdot R>0$ for every $R\in \mathrm{N}_1(X,\mathbb Z)$ satisfying $g\cdot R>0$ and $(R,R)\geq -3/2$. Moreover, every extremal ray $R$ with $(R,R)<0$ has one of these interpretations: if $(R,R)=-1/6$, there is a divisor $E\subset X$ with $[E]=2e$, $e=6R$, ruled over an abelian surface with fibers of class $R$; if $(R,R)=-2/3$, there is a divisor $F\subset X$ with $F=3R$, ruled over an abelian surface with fibers of class $R$; and if $(R,R)=-3/2$, there is a plane $P\subset X$ whose lines have class $R$. The conjecture gives a predicted numerical Mori and ample cone together with geometric descriptions of all negative extremal rays in this four-dimensional generalized Kummer case; the supplied source gives no resolution, so it remains open.