  The $J$-equation proposed by Donaldson is a complex Hessian quotient equation
on K\"ahler manifolds. The solvability of the $J$-equation is proved by
Song-Weinkove to be equivalent to the existence of a subsolution. It is also
conjectured by Lejmi-Szekelyhidi to be equivalent to a stability condition in
terms of holomorphic intersection numbers as an analogue of the Nakai-Moishezon
criterion in algebraic geometry. The conjecture is recently proved by Chen
under a stronger uniform stability condition. In this paper, we establish a
Nakai-Moishezon type criterion for pairs of K\"ahler classes on analytic
K\"ahler varieties. As a consequence, we prove Lejmi-Szekelyhidi's original
conjecture for the $J$-equation. We also apply such a criterion to obtain a
family of constant scalar curvature K\"ahler metrics on smooth minimal models.
