24.11 Equations on singular spacesC6BAF3541DDF

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On a resolution of a singular space, a pulled-back Hermitian form degenerates, and the pulled-back density can vanish or become singular. Existence on the resolution must therefore be combined with an estimate that allows the solution to descend. 9.11 develops potentials, envelopes, ideal equivalence and extended Monge–Ampère measures on reduced irreducible compact unibranch spaces, keeping the original nonclosed background. On projective cones, theorem 9.11.34, theorem 9.11.43 solve explicit equations, for prescribed angular and radial data, in full ordinary logarithmic classes and in the bounded class of the closed background. The logarithmic construction includes nonclosed deformations that preserve the determinant, and its full solution descends even when the smooth residual is nonconstant on the exceptional fiber. The continuity results discussed in section 25.21 use additional geometric and integrability hypotheses, notably log terminal singularities.

On the regular locus of a compact reduced irreducible Hermitian space of dimension at least two there is a conformal Gauduchon metric uniformly comparable to the original metric; see section 25.21. This existence theorem does not by itself give the nonlinear equations asked for below.

In dimension one, theorem 9.11.54 gives a different sufficient condition: the normalization is a torus, the prescribed ordinary type has finitely many logarithmic poles of total weight smaller than the mass of the background, and the measure has a nonnegative $L^ p$ density, $p>1$, with respect to the flat area on the normalization. Convolution with a Green function compensates for the degeneracy of the pulled-back background. The atoms at the poles determine the mass of every potential in the prescribed type and give uniqueness even among solutions that are a priori rough, and the residual descends continuously. The projective cusp of example 9.11.57 has an unbounded density on the curve itself although the data on the normalization are smooth. Every background on a curve is closed, so this construction gives no equation with a nonclosed background in higher dimension.

The family of logarithmic densities in theorem 9.11.60 makes the integrability issue precise. Densities on the normalization with local form $C_0|z|^{-2}(\log (1/|z|))^{-\gamma }$ are integrable for $\gamma >1$, but they give a solution in the prescribed ordinary type exactly when $\gamma >2$. For $1<\gamma \leq 2$, the constructed solution keeps all multiplier ideals and the same non-pluripolar mass, but loses the lower bound defining the type. Multiplier-ideal equivalence alone therefore does not give a bounded residual. A criterion for general integrable data remains to be found.

Problem 24.11.1.A63319B4B931 Consider reduced irreducible compact unibranch complex spaces with a Hermitian form, as in 9.11. Beyond the projective cones of subsection 9.11.27, determine geometric conditions on such spaces and conditions on the measure that give existence and uniqueness of solutions in specified singularity classes. Identify estimates ensuring finite mass, descent from resolutions and continuity, and describe their behavior under common refinements of resolutions.

In the cone examples, the pulled-back density vanishes on the exceptional divisor, and the logarithmic remainder solves an equation in a strictly positive background; the bounded equation for the closed background uses instead an energy that is coercive off that divisor. More general spaces of higher dimension require a replacement for these two controls. The curve construction uses a Green operator on the normalization; a linear construction by traces on a singular space needs further estimates across the singular locus.