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

Abstract

Let $X$ be a compact K\"ahler manifold and $(\alpha,\beta)$ be a pair of K\"ahler 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. This confirms the finiteness and uniform-gap expectation of Khalid-Sj\"ostr\"om Dyrefelt \cite[Section 1.2, Theorem 1.5; Remark 1.8]{KSD26}. As a corollary, we show that the collection of optimal destabilizers is finite when the stability threshold is nonpositive.