14.20 Uniqueness for degenerate equations6B1310DDF7B5

Let $X$ be a connected compact complex manifold of dimension $n\geq 1$, let $\omega $ be a Hermitian form, and let $\mathrm dV$ be a smooth positive volume form. For a nonzero nonnegative density $f\in L^ p(X,\mathrm dV)$, $p>1$, the preceding chapter determines the constant $c$ in $(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n=cf\, \mathrm dV$. We now ask whether the potential is determined up to a constant.

For real potentials $u,v$, with the regularity stated in each result, set

Then

For $A,B\geq 0$ with equal Monge–Ampère measures, $w$ solves $\mathrm{dd}^{\mathrm{c}}w\wedge T=0$, whose cofactor may be degenerate or nonclosed. Coefficient products for Sobolev potentials are interpreted almost everywhere. For $C^{1,1}$ solutions with a common continuous density, perturbing $w$ locates its extrema in elliptic regions; the maximum principle propagates their values along the corresponding components.

For $C^2$ potentials, a continuous weight $h>0$ with $\mathrm{dd}^{\mathrm{c}}(hT)=0$ in distributions gives a global energy identity forcing $\mathrm dw$ to vanish in the positive directions of the cofactor.

For bounded solutions, an energy measure replaces the pointwise gradient calculation. A continuous density whose positive set is connected and dense then gives uniqueness. When the background has invariant mass, ordinary comparison proves uniqueness for nonzero nonnegative $L^ p$ densities, $p>1$, even if the density vanishes on an open set.

The examples on $\mathbb P^3$ have bounded mass and a smooth nonclosed Hermitian background, yet distinct smooth normalized solutions share one density. Its zero set separates the elliptic regions: it can be an open band or two hypersurfaces with quadratic vanishing, even for a real analytic density. Finite upper mass therefore does not ensure uniqueness.