24.2.4 Regularity of singular solutions228EC8A6CA8D

Regularity estimates provide the compactness needed to single out a limit as the exponential parameter tends to zero. In this subsection $\theta $ is a smooth big real $(1,1)$-form, and $\rho $ is the strict barrier with analytic singularities fixed in section 15.13, with pole set $Z$ and regular locus $\Omega =X\setminus Z$. For bounded solutions with $L^ p$ density, $p>1$, theorem 15.6.1 gives continuity wherever $\rho $ is locally bounded. A locally Hölder continuous solution with minimal singularities is smooth on $\Omega $ when the density is smooth and positive there (theorem 15.18.1), and global $L^ p$ control of the density gives the Hölder hypothesis (theorem 15.17.1, corollary 15.18.50).

For a fixed reference $\eta $ with $\rho \leq \eta \leq 0$, corollary 15.15.3 shows that the potentials $u$ with $\eta -C_0\leq u\leq \eta $ and uniformly bounded $L^ p$ densities form a compact family for uniform convergence of differences on $\Omega $. Since the reference need not have minimal singularities, this compactness does not by itself make the potentials regular. In the minimal type, the convergence of scalars and densities in proposition 15.17.3 identifies the equation satisfied by every cluster limit of the normalized exponential solutions.

Smoothness of the density on an open set is not enough. On a compact Kähler manifold with a smooth positive volume form, theorem 10.20.1 proves that the full-mass solution is smooth on the ample locus of every big class, nef or not. Smoothness of the density only on the ample locus, however, does not imply global $L^ p$ integrability. On an arbitrary open set where the density is smooth, it does not even imply continuity of a bounded solution with minimal singularities: theorem 15.21.12 gives a compact Kähler counterexample with constant density on a ball. When the background has smooth local potentials, theorem 15.21.2 gives a sufficient condition in terms of smooth boundary traces on small balls. Among prescribed singularity types, the resolved logarithmic types of theorem 15.22.2 admit Hölder remainders when their residual form is positive, while for exceptional minimal poles theorem 15.23.5, theorem 15.23.10 give a Hölder remainder on the target and quantify the loss of integrability caused by the Jacobian. Neither result covers arbitrary prescribed types, and neither derives a global Hölder bound from information on the density confined to the regular locus. At a zero of the density, even smooth data need not give a smooth residual: proposition 10.21.9, example 10.21.11 give a necessary vanishing order and show that it is not sufficient. Beyond these resolved and explicit geometric cases, the regularity questions are open.

For the nonlogarithmic projective types of section 11.46, the central density criterion gives residual regularity across the pole and smoothly through density zeros. Perturbing the same singular reference as an everywhere radial obstacle instead gives a globally $C^{1,1}$ residual, zero near the pole, with open free regions of zero measure (theorem 11.47.5, corollary 11.47.8). Fixed-support bounded $C^2$ obstacle families satisfy the derivative stability of theorem 11.47.11, and proposition 11.47.14 transfers the full envelope to any fixed positive Hermitian product factor. These explicit ordinary-type regimes do not cover general singular references or ideal weighted envelopes.