Finiteness of null subvarieties and optimal destabilizers for the J-equation

Let $X$ be a compact Kähler manifold and $(\alpha ,\beta )$ be a pair of Kähler classes. We show that when $(\alpha ,\beta )$ is $J$-semistable, the collection of positive-dimensional irreducible $J$-null subvarieties is finite. We also obtain a uniform positive lower bound for the normalized $J$-slope of every irreducible non-$J$-null subvariety. As a corollary, we show that the collection of optimal destabilizers is finite when the stability threshold is nonpositive.