Divisorial Rigidity and Regularity of Boundary Cases for the Supercritical LYZ Equation

We study the supercritical LYZ equation on compact Kähler manifolds at the boundary of stability. Under numerical semistability, we prove a quantitative positivity estimate on the modified nef cone. It implies that the destabilizing prime divisors form a finite exceptional family. Assuming the existence of a smooth semisubsolution, we prove that the associated intersection form is negative definite on the span of classes of these divisors. Using this rigidity, we then construct a logarithmic singular subsolution along their union and obtain a bounded Bedford–Taylor solution which is smooth on its complement.