On the Datar-Mete-Song minimal slope conjecture

We prove a conjecture of Datar-Mete-Song \cite{DMS} characterizing $J$-slope semi-stability by the minimal $J$-slope. More precisely, for a semi-stable pair of K\"ahler classes $(\alpha,\beta)$ on a compact K\"ahler manifold $X$, every big and nef birational test class has slope at least the topological $J$-slope, whereas an unstable pair admits a test class with strictly smaller slope. We also introduce the $J$-null locus of a semi-stable pair and prove that it is an analytic subset of $X$ if $X$ is a compact K\"ahler surface or a compact toric K\"ahler manifold. In the toric invariant case, we show that Murakami's \cite{Murakami} weak solution to the $J$-equation is smooth and K\"ahler on the dense big torus $(\mathbb{C}^*)^n$ of $X$.