The $J$-Null Locus and Local Regularity of Weak Solutions of the Semistable $J$-Equation

Abstract: For the semistable $J$-equation, we prove that the numerical $J$-null locus coincides with the ambient $C<sup>2$-singular locus of Murakami's weak solution. Consequently, this singular locus is a proper analytic subset with finitely many irreducible components, and the weak solution is locally smooth outside the $J$-null locus. The proof relies on two analytic ingredients. First, we establish a regularization theorem for singular $J$-subsolutions, showing that Demailly's global regularization preserves quantitative strict cone conditions. Thus, singular strict subsolutions with prescribed analytic poles can be replaced by smooth strict subsolutions away from their pole sets. Second, we derive relative a priori estimates adapted to these subsolutions: a relative $L<sup>\infty$-estimate from determinant control and a weighted second order estimate yielding uniform $C<sup>2$-bounds on compact subsets of the regular locus.