24.2.3 Dual attainment and exact equations23806B61F487

For the current formulation of the dual problem, suppose $n\geq 2$. By Gauduchon duality, the threshold is determined by a maximization problem over test currents. Let $\mathscr G_ h$ be the compact convex set of normalized Gauduchon currents, the positive $(n-1,n-1)$-currents $T$ with $\mathrm{dd}^{\mathrm{c}}T=0$ and $\int _ Xh\wedge T=1$ (section 11.7), and let $k_ T$ be the determinant density of $T$ (theorem 12.9.1). For $0\leq p\in \mathcal D_ h$, the root norm $\rho _ h(p)$ is the supremum of $D_ p(T)=\int _ Xp\, k_ T\, \mathrm{d}V$ over $T\in \mathscr G_ h$. A maximizing current can turn the threshold inequality into an equality, provided its regularity, or control of its weighted moments, prevents loss of mass in the limit. For roots in the closed order ideal $\mathcal D_ h^0$, theorem 12.10.1, proposition 12.10.9 show that the supremum is attained and represent all supporting probabilities of the threshold functional by maximizing currents. These probabilities are absolutely continuous, and by proposition 12.10.11 the trace errors along smooth maximizing sequences tend to zero in total variation. Uniform integrability along a single maximizing sequence already gives attainment and convergence of the determinant-weighted measures of that sequence (proposition 12.10.6), but it does not give the representation of all supporting probabilities.

Regularity of a maximizer has strong consequences. Let $n\geq 2$ and let $f$ be a nonzero density with $b_ h(f)>0$. If $D_{f^{1/n}}$ has a smooth strictly positive maximizer, then $f^{1/n}$ is smooth and nonnegative, and every threshold subsolution is smooth and solves the equation exactly, uniquely up to constants (corollary 12.10.12). For a Kähler background, corollary 12.10.14 shows that such a maximizer exists exactly when $f$ is smooth and strictly positive. The strict positivity fails for nonclosed backgrounds: on every compact complex manifold of dimension at least two, example 12.10.15 gives a smooth strictly positive maximizer for a density vanishing on an open set.

Theorem 12.10.17 weakens the hypothesis to a smooth semipositive maximizer that is strictly positive outside a proper analytic subset $Z\subset X$. It still gives an exact equation on all of $X$, smoothness of the solutions off $Z$, and uniqueness of threshold subsolutions up to constants. The proof takes the maximum of two subsolutions and uses a Gauduchon integral to eliminate the divisorial part of their difference; no singularity type is prescribed in advance. The prime divisors along which the threshold solution has positive Lelong number have linearly independent Bott–Chern classes, by corollary 12.10.39. In this regime write $b=b_ h(f)$ and $c_\lambda =\exp (\lambda \sup _ Xu_\lambda )$ for the greatest exponential solutions $u_\lambda $, $\lambda >0$. By proposition 12.10.20, the whole family of normalized solutions converges in $L^1$ as $\lambda \downarrow 0$, it has a fixed lower barrier, and $b\leq c_\lambda \leq b\mathrm{e}^{C\lambda }$ for a finite constant $C$ independent of $\lambda $. If the envelope $P_{\delta h}(u_0)$ of the normalized threshold solution $u_0$ is not identically $-\infty $ for some $0<\delta <1$, the convergence also holds in capacity (corollary 12.10.21); for $L^ p$ data with $p>1$ the threshold solution is continuous and the convergence is uniform (corollary 12.10.31). In example 12.10.41 the solution is smooth, but the root $f^{1/n}$ is not differentiable on $Z$. For general $L^1$ data we do not know whether $P_{\delta h}(u_0)\not\equiv -\infty $ holds automatically, nor how regular the solution is across $Z$.

The moment criterion of theorem 12.10.43 requires no regularity of a maximizing current. Suppose that the supremum-normalized $h$-plurisubharmonic functions have uniformly bounded first moments against the determinant-weighted probabilities of one smooth maximizing sequence. Then the exponential solutions have a fixed lower barrier, and their normalized cluster limits lie below the greatest normalized threshold subsolution $\Phi _ f$ of proposition 15.28.6. If, in addition, $P_{\delta h}(\Phi _ f)\not\equiv -\infty $ for some $0<\delta <1$, the lower-barrier criterion gives convergence in capacity along subsequences and an exact threshold equation, without either mass property. An entropy bound on these test probabilities implies both the bound on first moments and the uniform integrability along one sequence used above. By proposition 11.38.1, such an entropy bound is equivalent to a uniform bound for $H_0$, the convex dual of the normalization functional with zero reference (section 11.32). An entropy bound does not, however, give the condition on $P_{\delta h}(\Phi _ f)$.

Attainment by a singular current therefore does not settle the equation by itself. The roots from a closed factor form a closed subspace of $\mathcal D_ h^0$ isomorphic to $L^ n$ on that factor, by proposition 15.32.11; some of these roots belong to no higher $L^ r$ space, and their tails decay only as a power of $\log K$. At zero lower mass we do not know in general whether the root-tail condition $f^{1/n}\in \mathcal D_ h^0$ suffices for an exact threshold equation, or even whether $\mathcal D_ h^0=\mathcal D_ h$. By proposition 12.10.5, the tail condition is equivalent to continuity of the threshold under logarithmic perturbations $\mathrm{e}^{F_ j}f$ with $F_ j$ uniformly bounded and tending to zero in measure. This is continuity on order intervals, not continuity of the threshold under arbitrary $L^1$ perturbations.

Convergence in capacity cannot be replaced by $L^1$ convergence in these arguments. On a curve, the ideal envelopes with finitely many logarithmic poles in theorem 18.6.4 have zero non-pluripolar mass, yet they converge in $L^1$ to a smooth potential of positive mass. The existence theorems obtain the stronger convergence from their measure bounds; representing the measure by a carrier, even within a prescribed singularity type, does not provide it.