24.2 Threshold equations and singular limitsA1CD1701B975

Solving a Monge–Ampère equation with singular data involves two separate steps: one constructs a subsolution at the optimal coefficient, and then passes to a limit of approximate solutions without losing Monge–Ampère mass. Throughout this section, $h$ is a Hermitian form on $X$ and $\mathrm{d}V$ a smooth positive probability volume form. We write $v_-(h)$ and $v_+(h)$ for the lower and upper masses of $h$ (definition 13.14.14), $\mathrm{MA}^{\mathrm{ext}}_ h(u)$ for the extended Monge–Ampère measure of $u\in \mathrm{PSH}(X,h)$ (theorem 13.14.3), which may have infinite mass, and $\mathcal E_{\mathrm{env}}(X,h,0)$ for the relative envelope class with reference $0$. For a nonzero nonnegative density $f\in L^1(\mathrm{d}V)$, the density threshold $b_ h(f)$ is the largest $a\geq 0$ for which some $u\in \mathrm{PSH}(X,h)$ satisfies $\mathrm{MA}^{\mathrm{ext}}_ h(u)\geq af\, \mathrm{d}V$ (theorem 15.28.4); such a $u$ with $a=b_ h(f)$ is called a threshold subsolution. Finally, $\mathcal D_ h$ is the space of density roots $p\in L^ n(\mathrm{d}V)$ whose root norm $\rho _ h(p)=b_ h(|p|^ n)^{-1/n}$ is finite (proposition 15.32.5).

In two regimes the equations are solved for arbitrary finite non-pluripolar measures. For a semipositive big background $\theta $ with positive lower mass and a model reference satisfying (16.28.67), theorem 15.37.4 solves both the equation with a positive exponent and the normalized equation. For an arbitrary smooth big background $\theta $, theorem 10.31.44, theorem 10.31.45 solve both equations under bounded mass and positive lower volume of the model reference. The existence theorem for $L^ p$ densities in the minimal ordinary singularity type, in section 15.16, requires neither mass property; in the second regime, corollary 10.31.56 also places the solutions for $L^ p$ densities in the prescribed ordinary model type.

Under bounded mass the reference may also vary. Suppose the ordinary model references converge for the volume comparison $d_\theta $ of section 13.22 and have a common positive lower bound for their lower volumes, and the densities converge in $L^1$ of a fixed finite non-pluripolar measure. Then, by theorem 10.31.30, theorem 10.31.47, for a fixed positive exponent the solutions converge in capacity and their Monge–Ampère measures converge in total variation. At exponent zero the scalars and the measures converge, and every $L^1$-convergent subsequence of normalized solutions converges to a solution of the limiting equation; singling out one limit potential requires a further uniqueness or selection argument.

To remove these mass hypotheses, one must replace each place where they enter the compactness arguments. The proof in the semipositive regime uses finite auxiliary contact measures to construct a common lower envelope, and then obtains the limit for a moving measure from convergence in capacity and control of deep sublevel tails. The proof for an arbitrary big background uses finite upper relative volumes to construct a rooftop between classes and to maximize the weighted measure. In either case one must still prove that the measure of the solution is finite; the formal existence of the envelopes does not give this.

24.2.1 Individual density thresholds33D327BAD4E0

One must distinguish solvability for all measures from solvability for one fixed density. By corollary 15.27.5, the normalized equation $\mathrm{MA}^{\mathrm{ext}}_ h(u)=c\mu $ has a solution in $\mathcal E_{\mathrm{env}}(X,h,0)$ for every nonzero finite non-pluripolar measure $\mu $ exactly when $v_-(h)>0$. When $v_-(h)=0$, theorem 15.27.3 produces a probability density in $L^1(\mathrm{d}V)$, bounded below by a positive constant, that admits no positive scaled subsolution. Such obstructing densities are dense in $L^1$, although every smooth strictly positive density is solvable. The compact threefold of theorem 12.19.1 has $v_-(h)=0$ for every Hermitian form $h$, so the obstruction does occur.

For one nonnegative density $f\in L^1(\mathrm{d}V)$, the threshold $b_ h(f)$ is the decreasing limit of the normalization constants of the bounded truncations $\min (f,k)$, see (15.28.1). An argument with upper envelopes and damped continuity shows that a subsolution attains this coefficient, but it proves neither an exact equation nor finiteness of the mass. Smooth comparison tests alone do not detect the threshold, by theorem 15.28.9. On the other hand, the Gauduchon formula of theorem 15.29.1 expresses the root norm as a supremum of linear tests, which makes $b_ h$ upper semicontinuous in $L^1$. The densities with positive threshold are those whose roots $f^{1/n}$ lie in $\mathcal D_ h$, and this space does not depend on the Hermitian form (proposition 15.29.5). With the norm $\| p\| _{L^ n}+\rho _ h(p)$ it is a Banach space; it equals $L^ n$ when $v_-(h)>0$, and it is complete for $\rho _ h$ alone exactly when $v_+(h)<\infty $, by proposition 15.32.5, proposition 15.32.6.

The hypothesis $v_-(h)=0$ is what makes the problem open. For $\lambda >0$, write $u_\lambda $ for the greatest solution of the exponential equation $\mathrm{MA}^{\mathrm{ext}}_ h(u)=\mathrm{e}^{\lambda u}f\, \mathrm{d}V$, when it exists (theorem 15.35.2). When $v_-(h)>0$, corollary 15.37.32 solves the threshold equation in $\mathcal E_{\mathrm{env}}(X,h,0)$ for every nonzero nonnegative $L^1$ density: comparison of finite measures identifies the scalar, and dominated continuity shows that every $L^1$-convergent subsequence of the normalized greatest exponential solutions $u_\lambda -\sup _ Xu_\lambda $ converges in capacity. This covers the Hopf example of section 23.1, which has positive lower mass but not bounded mass. For $L^ p$ densities with $p>1$, the continuous solution of the normalized equation and proposition 15.28.7 answer the question without any lower mass assumption.

At zero lower mass, theorem 15.30.3 solves the threshold equation with finite mass for explicit densities on a split product with projective space; these densities are integrable but belong to no $L^ p$ with $p>1$. There the background $h$ is a nonclosed Hermitian form, and the canonical solutions $u$ belong to $\mathcal E_{\mathrm{env}}(X,h,0)$; moreover, for every $0<\delta <1$, the envelope $P_{\delta h}(u)$ of section 5.40, the largest $\delta h$-plurisubharmonic function below $u$, is not identically $-\infty $. More generally, theorem 15.30.6 solves the exact finite-mass equation for every nonzero integrable density from a closed Kähler factor, with the appropriate change of reference volume. The threshold is the split background's mass divided by the integral of the factor density. On the product of the elliptic threefold with $\mathbb P^2$, this gives threshold $240\pi $ for every probability density from that factor. An arbitrary density on the product need not have this form, and the general problem remains open.

For product measures from a projective-line factor, proposition 15.31.13 gives an exact finite-mass equation for every non-polar factor probability. In the separated sector, boundedness and continuity are exactly those of its logarithmic potential. When that potential is finite and continuous, theorem 15.31.15 identifies the attained coefficient over all genuine subsolutions, constructs the greatest exponential solution for every positive exponent, and gives whole-family uniform selection with error $O(\lambda )$. The Sobolev measure trace supplies existence, and bounded comparison extends greatestness to singular competitors. Weighted circle probabilities in proposition 15.31.16 give measures singular with respect to smooth volume. The zero-lower-mass specialization retains the factor hypothesis. These product criteria do not classify arbitrary measures on a fixed Hermitian manifold.

A different geometric condition applies directly to singular measures on a fixed Hermitian manifold. The probability $\rho $ of (15.26.1) is chosen from a coordinate ball before the datum. If $\nu $ is finite, nonzero and has bounded density with respect to $\rho $, proposition 15.26.2, theorem 15.26.4 give a strict Hölder scaled subsolution and a unique bounded exponential solution for every positive exponent. In addition, corollary 15.26.3 gives a normalized Hölder solution of $\mathrm{MA}_ h(w)=c\nu $ with $c>0$ obtained as part of the solution. This includes bounded densities on one of the spheres and needs neither a product decomposition nor positive lower mass. It is a sufficient condition; it does not classify arbitrary individual measures or identify a homogeneous threshold coefficient.

For the data of 24.2.2, corollary 15.35.3 shows that the greatest exponential solutions $u_\lambda $ exist for every $\lambda >0$, that the scalars $c_\lambda =\mathrm{e}^{\lambda \sup _ Xu_\lambda }$ converge to $b_ h(f)$ as $\lambda \downarrow 0$, and that the measures $\mathrm{e}^{\lambda u_\lambda }f\, \mathrm{d}V$ converge in total variation. What is missing in general is convergence in capacity of the normalized potentials. If $f^{1/n}$ lies in the subspace $\mathcal D_ h^0$ of density roots with uniformly vanishing tails (proposition 15.32.9), proposition 15.35.5 gives a modulus for the errors of the scalars and of the measures; a rate linear in the exponent would require further control.

For $n\geq 3$, a power capacity bound (15.33.2) supplies both root tails and the compactness needed for an exact equation. By theorem 15.33.1, the threshold is positive and finite, the root belongs to $\mathcal D_ h^0$, and a subsequence of the normalized clipped solutions converges uniformly to a continuous solution at $b_ h(f)$. The logarithmic density of theorem 15.33.6 satisfies this condition on every fixed Hermitian background, while belonging to no $L^ p$ with $p>1$. Its reference potential is continuous but not Hölder, and its root tails have the bound (15.33.7). These conclusions use the capacity assumption on the fixed datum. Whether all positive-threshold roots have vanishing tails, and whether tails alone imply an exact equation at zero lower mass, remain open.

Positivity and integrability of $f$ do not make the threshold positive, and exponential equations can be solvable even when it vanishes. For zero threshold, theorem 15.35.9 gives a sufficient growth condition in the radial variable, and theorem 15.35.10 constructs a single strictly positive probability density with $b_ h(f)=0$ whose exponential equation has, for every positive exponent, a greatest solution of finite mass; all its solutions are unbounded. In that construction the projective factors are chosen after the collapsing sequence on the base.

For this density, proposition 15.35.13 shows that any choice of exact exponential solutions has total mass tending to zero as the exponent decreases to zero; the argument says nothing about convergence of the normalized potentials. In general, the set $\Lambda _ h(f)$ of positive exponents for which the exponential equation is solvable is an upper interval, by proposition 15.35.12. A necessary criterion on the density for solvability at zero threshold, solvability for an arbitrary prescribed concentration family, and an extension to arbitrary non-pluripolar measures are open.

Suppose that $a=\inf \Lambda _ h(f)$ is positive. By proposition 15.35.16, $a$ belongs to $\Lambda _ h(f)$ if and only if there are exact solutions at exponents decreasing to $a$ whose suprema are bounded below; if $a\notin \Lambda _ h(f)$, the total masses of every such sequence of solutions tend to zero. The common upper barrier of lemma 15.35.15 also shows that the greatest solutions become monotone in the exponent after a logarithmic correction. Consequently, by proposition 15.35.17, they are continuous from the right in $L^1$ and in capacity at every exponent in $\Lambda _ h(f)$, and their Monge–Ampère measures converge in total variation. At every interior exponent, proposition 15.35.18 gives a left limit in $L^1$ and in capacity that solves the equation at that exponent and lies below the greatest solution; the equation measures converge in total variation. By proposition 15.35.19, these potentials can differ at only countably many exponents. Whether a positive endpoint can fail to belong to $\Lambda _ h(f)$, or an actual jump from the left can occur, remains open.

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.

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.

24.2.5 Uniqueness and selectionDFEF338E1977

Where the density vanishes, the linearized equation degenerates, and the zero set can separate regions where it is elliptic. The question is whether the difference of two solutions can vary from one such region to another. Throughout this subsection $\omega $ is a Hermitian form on $X$, $f$ is smooth and nonnegative, and potentials are normalized by $\sup _ Xu=0$.

On $\mathbb P^3$, theorem 14.26.1 gives a nonclosed Hermitian form and a smooth nonnegative density with two distinct smooth normalized solutions; there the density vanishes on an open set. The construction of theorem 14.27.1 has uncountably many smooth normalized solutions for a density that is positive off two disjoint real hypersurfaces, and after a modification of the background, theorem 14.28.1 achieves exactly quadratic vanishing on those hypersurfaces. Thus neither a zero set of zero volume nor vanishing of finite order guarantees uniqueness. In all these examples the background is a Hermitian form with bounded mass; the degeneracy responsible for nonuniqueness lies in the density and in the solution forms.

These examples restrict any geometric use of the equation. A construction assigning a canonical potential to arbitrary nonnegative data must specify a selection procedure and prove that it is well defined; existence in the minimal ordinary singularity type does not provide one. The scalar, which is unique, and for strictly positive data the corresponding variational volume, remain meaningful without uniqueness of the potential. In low dimension uniqueness holds under additional conditions. On a compact Hermitian surface, 14.23.7 proves uniqueness for $C^2$ potentials when the common density is continuous and positive on a dense set, with no differential condition on the metric; in dimension three, 14.23.1 uses instead a sign condition on $\mathrm{dd}^{\mathrm{c}}\omega $. The two arguments rest on different integration identities: the cofactor identity (14.22.14) and a signed estimate for the quadratic difference.

When $\{ f>0\} $ is connected and dense, theorem 14.24.1 shows that two bounded normalized solutions with the same positive scalar coincide; the theorem applies to the measure $cf\, \mathrm{d}V$ after the smooth positive ratio $\mathrm{d}V/\omega ^ n$ is absorbed into the density. In all the counterexamples above whose zero set consists of hypersurfaces, including the real-analytic example of 14.29.1, the positivity set has three connected components. None of them contradicts 24.7.1, which concerns strictly positive densities $\mathrm{e}^ F$ with $F$ continuous.

For a fixed nonzero density $f\in L^ p(\mathrm{d}V)$, $p>1$, the exponential equation does select a solution. By theorem 11.44.2, a weakly compact set of norming probabilities determines the limit: as the exponent tends to zero, the suitably shifted exponential solutions converge to the greatest subsolution of the critical equation that satisfies the integral normalizations given by these probabilities. In a Hermitian background the convergence is uniform; in an arbitrary smooth big background it holds in $L^1(X)$ and locally uniformly off the pole set of the strict barrier. The selected limit inherits the symmetries of the data (proposition 15.20.1), and in the projective example theorem 15.20.4 identifies it with the explicit symmetric $C^{1,1}$ solution of proposition 14.26.14. Selection thus coexists with nonuniqueness of the normalized critical equation. For the merely integrable radial densities on zero-lower-mass products, theorem 15.36.5 determines the whole supremum-normalized limit, even with joint variation of the density parameter and exponent. Scalar centering has a finite limit exactly below the moment boundary of proposition 15.36.8; at the boundary and above it diverges. For logarithmic product models with positive residual, theorem 11.45.9 also gives selection under independent variation of the reference coefficients, background, density and exponent; centering follows each varying critical scalar. Selection for arbitrary integrable data and general varying singular reference types remains open.

Dual attainment gives a different criterion. In dimension at least two, suppose that $f\neq 0$, that $b_\omega (f)>0$, and that $D_{f^{1/n}}$ has a smooth strictly positive maximizing Gauduchon current. Then the normalized threshold solution is unique, and the whole family of normalized solutions with positive exponent converges uniformly to it (corollary 12.10.12, corollary 12.10.13). The density of example 12.10.15 satisfies this condition although it vanishes on an open set. The size of the zero set alone therefore does not decide uniqueness; the dual condition also depends on the background.

For prescribed radial nonlogarithmic types, theorem 11.46.8 gives a different kind of regularity: a relative density error $O(r^\delta )$ controls the residual after the singularity is subtracted. The critical $C^1$ and $C^2$ thresholds are attained and are sharp by proposition 11.46.17. Positive cumulative central mass allows smooth crossing of density zeros, and proposition 11.46.20 gives an open zero annulus with a globally smooth residual. The product construction corollary 11.46.22 transfers this regularity under a zero-lower-mass factor hypothesis. These are radial criteria; they do not imply regularity or uniqueness for general nonsymmetric data.