The $J$-equation proposed by Donaldson is a complex Hessian quotient equation on Kahler 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 Kahler classes on analytic Kahler varieties. As a consequence, we prove Lejmi-Szekelyhidis original conjecture for the $J$-equation. We also apply such a criterion to obtain a family of constant scalar curvature Kahler metrics on smooth minimal models.
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