15.1 Equations with prescribed singularitiesF9AE3BF05078

Throughout this chapter we fix a Hermitian metric $\omega $ on $X$ and a smooth positive probability volume form $\mathrm{d}V$. Unless another measure is named, $L^ p$ norms, volumes and the phrase “almost everywhere” refer to $\mathrm{d}V$, while capacity and convergence in capacity refer to $\operatorname {Cap}_\omega $ from section 5.29. The background $\theta $ is a smooth real $(1,1)$-form whose positivity is specified in each section. For a smooth real $(1,1)$-form $\alpha $ and $u\in \mathrm{PSH}(X,\alpha )$ we write $\alpha _ u=\alpha +\mathrm{dd}^{\mathrm{c}}u$, and we call $\{ u=-\infty \} $ the pole set of $u$; it is pluripolar and has zero volume. We write $\mathrm{MA}_\theta (u)$ for the extended non-pluripolar Monge–Ampère measure $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ of theorem 13.14.3. It equals $(\theta +\mathrm{dd}^{\mathrm{c}}u)^ n$ when $u$ is bounded, charges no pluripolar set, and may have infinite total mass; we keep the superscript where this possibility matters. Finally, $s_+=\max (s,0)$ for real $s$.

We use the following notation from earlier chapters. As in definition 16.7.1, $u\preceq v$ means that $u\leq v+C$ for some constant $C$, and $u\sim v$ means that $u\preceq v$ and $v\preceq u$; the equivalence classes of $\sim $ are the ordinary singularity types. For an obstacle $f$, $P_\theta (f)$ is the largest $\theta $-plurisubharmonic function lying below $f$ quasi-everywhere, $P_\theta (u,v)=P_\theta (\min (u,v))$ is the rooftop envelope, and $V_\theta =P_\theta (0)$; a potential has minimal singularities if it is equivalent to $V_\theta $ (section 5.40). The ordinary envelope $P_\theta [u]$ and the envelope fixed points, those $u$ with $P_\theta [u]=u$, are defined in definition 5.40.4. Recall that $\theta $ is big if some $\rho \in \mathrm{PSH}(X,\theta )$ satisfies $\theta _\rho \geq \delta \omega $ with $\delta >0$. We call such a $\rho $ a strict barrier; by theorem 5.36.5 it can be chosen with analytic singularities.

We seek potentials $u$ in the ordinary singularity type of a given reference $\varphi \in \mathrm{PSH}(X,\theta )$ solving

where $\mu $ is a given measure, typically $f\, \mathrm{d}V$ with $f\in L^ p$ and $p>1$, and the constant $c>0$ is part of the unknown. When $\varphi $ is unbounded, the comparison principles for bounded potentials no longer apply, and a limit of approximate solutions may leave the prescribed type. For a semipositive big background, both difficulties are overcome by a smaller background. If $P_{\delta \theta }(\varphi )\not\equiv -\infty $ for some $0<\delta <1$, then the part $\delta \theta $ of the background can carry the singularity of $\varphi $, while bounded envelopes in the remaining big background $(1-\delta )\theta $ have Monge–Ampère measures of positive mass. This gives a domination principle for singular potentials (theorem 15.2.2), and hence uniqueness. Existence starts from bounded equations with $L^ p$ densities, solved with a subsolution built from a strict barrier. A compactness criterion with a common lower barrier identifies limits of Monge–Ampère measures, and an addition theorem for weighted measures in a fixed type removes a possible deficit of mass in a limit. Together with a uniform estimate relative to the reference (theorem 15.12.3), these results solve both equations in the type of an envelope fixed point that admits such a smaller-background envelope (theorem 15.12.1). At exponent zero, comparison determines the constant $c$ but, in general, not the potential.

If $\theta $ is big but not semipositive, $V_\theta $ may be unbounded and the smaller-background construction is not available. A strict analytic barrier $\rho $ takes over its role. For two potentials bounded below by $\rho $ minus a constant, a small perturbation by $\rho $ forces a negative infimum of their difference to be attained away from the poles of $\rho $, where a local comparison applies. Resolving the singularities of analytic approximants of $V_\theta $ reduces the equations to semipositive big backgrounds. Stability above the barrier then produces solutions with minimal singularities which are locally Hölder continuous off the poles of $\rho $ (theorem 15.16.2, theorem 15.16.7, theorem 15.17.1). They are smooth off these poles when the density is smooth and strictly positive on the whole complement (theorem 15.18.1).

The same barrier controls smooth equations as a semipositive background degenerates (theorem 15.19.7). For small resolutions of threefold nodes with a closed background, the potentials converge smoothly away from the exceptional curves, and comparing the limiting and approximating metrics reduces to controlling the diameters of neighborhoods of these curves. In the projective example of theorem 14.26.1, where the normalized potential is not unique, symmetry selects the zero-exponent limit, while local examples show that a positive density on a small open set does not by itself give regularity. Resolved logarithmic poles and minimal types on blowups give explicit equations whose data are determined by the residual form and the Jacobian of the modification.

The last part of the chapter concerns a Hermitian background $h$ and general right-hand sides. Solvability of $\mathrm{MA}_ h(u)=c\mu $ for every nonzero finite non-pluripolar measure $\mu $ is equivalent to a positive lower bound for the masses of bounded $h$-plurisubharmonic functions. For singular data supported on shrinking coordinate spheres, a bounded-density test against a fixed probability supplies a strict Hölder reference, a normalized Hölder solution of the homogeneous equation and bounded exponential solutions (proposition 15.26.2, corollary 15.26.3, theorem 15.26.4), even when the lower mass vanishes. For an individual nonnegative density $f\in L^1$, bounded truncations and Gauduchon tests determine the largest coefficient $a$ for which $\mathrm{MA}_ h(u)\geq af\, \mathrm{d}V$ has a solution. Passing from this threshold to an exact equation at exponent zero requires control of the potentials as well as of their measures. At a positive exponent, exponential decay controls the mass on deep sublevels (theorem 15.34.2); this control is lost as the exponent tends to zero.