25.21 Regularity and singular spaces2200D43D6A26

2200D43D6A26

Regularity in a big class is first a local question on the locus where strict barriers are smooth. For a closed class on a compact Kähler manifold, existence and uniqueness for (10.20.2) are due to Boucksom, Eyssidieux, Guedj and Zeriahi [Theorem A, equivalently Theorem 3.1, BEGZ10], and the bounds (10.20.3) in terms of minimal singularities are [Theorem B, quantitatively Theorem 4.1, BEGZ10]. The same uniqueness theorem for full mass identifies the solutions in closed classes in section 15.22, section 15.23, section 10.22, section 10.23. Smoothness on the ample locus in the nef case is [Theorem C, BEGZ10]; theorem 10.20.1 gives this smoothness in every closed big class, for a smooth positive volume form.

For a big background that need not be closed, the local Hölder theorem theorem 15.17.1 is Dang's [v1, Theorem A and Theorem 3.1, Dang26]. It gives the local input for the argument on the ample locus, while allowing the global potential to be unbounded and the lower curvature bound to be nonclosed. Formulating the equation with a positive scalar and only on the regular locus is equivalent, since the complementary analytic set carries neither smooth volume nor non-pluripolar mass.

The proof of local regularity also needs a quantitative stability estimate. The estimate with a strict barrier in section 15.15 is [v1, Theorem 3.3 and Remark 3.4, Dang26] in the case of minimal singularities, with the comparison potential normalized explicitly; the construction with auxiliary subsolutions in section 15.4 first treats a semipositive big background.

The interior regularity theorem theorem 15.18.1 combines this Hölder control with the assumption that the density is smooth and positive on the whole regular locus of one fixed strict analytic barrier. This domain is part of the hypothesis: smoothness of the density on a smaller open set does not give the same conclusion by restriction. The compact Pogorelov example example 10.21.10 has a smooth positive density on an inner chart, but its density loses regularity along the divisor in the gluing region, so it does not satisfy this hypothesis.

Local smoothing uses the theorem of Guan and Li on the Dirichlet problem [Theorem 1.1, p. 1186, GL10]. In (15.21.3) the background is zero and, in local coordinates $z$, the metric is $(\mathrm i/2)\partial \bar\partial |z|^2$; for a smooth positive density $G$, a large quadratic correction gives a strict subsolution. The same theorem constructs the approximants in subsection 15.18.7 and starts the passage to continuous boundary values and to zero density in section 15.7; in dimension one it is replaced by the Dirichlet problem for the Poisson equation. Once (15.18.6) is uniformly elliptic, the Evans–Krylov theorem for variable coefficients [Definition 3.2, Theorem 3.4, and the paragraph following it, TWWY15], followed by difference quotients and Schauder estimates, gives smoothness.

The local identification in section 15.21 uses Bedford and Taylor's uniqueness theorem for the bounded Dirichlet problem [Corollary 4.4, p. 17, BT82]. Degeneration of the density also matters for global stability: Kołodziej and Nguyen [Theorem 4.1, KN21] assume that the density satisfies $f\geq c_0$ for a constant $c_0>0$, in addition to their hypotheses on capacity. The densities giving nonuniqueness in section 14.20 have no such lower bound, even when they are positive almost everywhere.

For compact sets in a Hermitian manifold, Ahn [Theorems 1.1–1.2, Ahn26] proves that Hölder continuity of the extremal function is a pointwise local property. The weighted version assumes that both the unweighted extremal function and the weight are Hölder continuous, and that the set is uniformly dense in capacity; the same density condition enters the equivalence with the local and weakly local Hölder properties. These are the regularity settings referred to in section 24.1.

On singular spaces, regularity on a resolution must be combined with descent. Dang [v1, Theorem B, Dang26] proves continuity results for bounded solutions on resolutions under additional geometric and integrability hypotheses, notably for log terminal singularities; this result does not cover the arbitrary unibranch singularities in the problem posed in section 24.1. Pan [Theorems A and A', Pan25] constructs a Gauduchon metric on the regular locus of a compact reduced irreducible Hermitian space of dimension at least two; it is conformal and uniformly comparable to the given metric, and unique up to scale among such metrics. Control of the conformal factors on a resolution is central to the construction. This result motivates the questions about singular traces in section 24.1.

The boundary-weighted quadric surface in 9.11.10 is Seo's [v1, Example 6.2, pp. 29–30, SeoToricIdeals26]. The paper uses the weight $\mathrm{e}^{-2\varphi }$, so the potential here is $u=2\varphi $. The adapted volume comes from a generator of $\omega _ X^{[4]}(4D)$ and pulls back to a constant times $|s|^{-1}$ times Euclidean volume on the finite quotient cover. Splitting the logarithmic quadrant gives the strict monomial test; angular orthogonality and division of convergent series give the full ideal. This proves the explicit example directly, without using the paper's general Newton-body description. The boundary-weighted ideal and the resolution-ideal relation in proposition 9.11.9 use different volume conventions.

The affine toric monomial test in theorem 9.11.20 is the holomorphic-character part of Seo's [v1, Theorem 6.1, pp. 28–29, SeoToricIdeals26], here restricted to full-dimensional cones and effective rational boundary. Its convex-cone criterion is [v1, Lemma 4.4 and Theorem 4.6, pp. 18–19, SeoToricIdeals26]; the proof here obtains coercivity by a compactness argument and sandwiches the actual logarithmic neighborhoods between translated cones. The subtraction test of corollary 9.11.22 is the integral calculation behind [v1, Proposition 5.10 and Corollary 6.3, pp. 27 and 30, SeoToricIdeals26]. For this holomorphic-character integrability assertion the exponent can be any positive real number, without assigning a rational boundary to the subtracted weight. The quadric threefold and its whole-germ ideal in proposition 9.11.23 are computed directly from the same criterion and the homogeneous coordinate ring. No general fractional-sheaf generation or ideal-envelope statement is used.

In the construction on unibranch curves in subsection 9.11.50, the Green function, the exact mass of the pole, uniqueness in the full singularity type, descent through disks and the projective cusp are all obtained directly. $L^ p$ data on the normalization, $p>1$, give a continuous residual even when the background coming from a local embedding pulls back to a degenerate form. The example of the cusp also computes the exact threshold for densities on the singular curve. This is a result in dimension one; it neither extends the theorem on log terminal singularities cited above nor gives a general equation on nonclosed singular spaces.

The calculation at the endpoint in theorem 9.11.60 is another direct construction from the Green function of the curve and the radial primitive of theorem 10.23.1. For one logarithmic family it distinguishes three properties: finite mass of the density, a bounded residual in the ordinary singularity type, and equivalence of multiplier ideals at all weights. It does not characterize arbitrary integrable data.

The equations with nonclosed backgrounds on products with cuspidal curves in subsection 9.11.63 combine a correction of the reference to a smooth positive form, quantitative Fourier inversion, and a comparison of bounded energies in a background all of whose powers are $\mathrm{dd}^{\mathrm{c}}$-closed. The weighted limit lemma 5.27.2 is proved by adding one complex variable $w$ and passing to the maxima with $\log |w|^2$, which reduces it to the ordinary convergence of mixed products along decreasing sequences, including top-degree products in one higher dimension. The corresponding classical input is Bedford and Taylor's [Theorem 2.4 and its proof, pp. 7–9, BT82]; their operator is $4\pi $ times ours, so products of degree $r$ scale by $(4\pi )^{-r}$. The resulting family with small data in all variables has Hölder continuous solutions on the ambient space, unique among all bounded competitors. The existence of bounded solutions for arbitrary nonnegative $L^ p$ data on the normalization, $p>1$, follows separately from theorem 10.31.3. These results complement the construction with Green functions on curves, without giving a general equation on singular spaces or a sharp criterion for densities on the singular space.