Uniqueness of bounded potentials in the nonclosed Hermitian setting

Establish uniqueness of the bounded normalized potential solving the nonclosed non-pluripolar Monge–Ampère equation \(\langle(\theta+\sqrt{-1}\partial\bar\partial\varphi)^n\rangle=c_0\Omega\) on a compact Fujiki-class \(\mathcal C\) manifold when \(\theta\) is smooth, semipositive, and generally nonclosed.

Background

The paper studies degenerations of Chern-Ricci-flat Hermitian metrics on a smooth small crepant resolution $\pi:X\to Y$, where $X$ belongs to Fujiki class $\mathcal C$. The degenerating metrics converge, after subsequence extraction, to a bounded $\theta$-plurisubharmonic potential $\varphi_0$ solving a nonclosed non-pluripolar Monge–Ampère equation.

In the Kähler case, the limiting equation descends to the singular projective variety $Y$, where established pluripotential theory gives uniqueness and therefore convergence of the full family. For a nonclosed background form $\theta$, the paper proves only subsequential convergence because uniqueness of the bounded solution is not established.