The J-equation at the birational minimal slope

Abstract

We prove the existence, uniqueness, and partial regularity outside a proper analytic subset for the Kähler current solving the $J$-equation at the birational minimal slope. This confirms Datar--Mete--Song's conjecture 1.5 in \cite{DMS26}. We introduce an analytic threshold, and prove its equivalence to the birational threshold introduced by Datar--Mete--Song. One of the key tools is the approximation of subsolution by Bergman's kernels, which is motivated by the work of Demailly on the approximation of plurisubharmonic functions with analytic singularities. As an application of the Bergman kernel approximation, we combine the results of Fang--Ma \cite{FM26} to give an analytic characterization of the $J$-null locus of a semistable pair $(α,β)$. This removes the technical assumptions in \cite{L26b}.