Are bounded solutions of complex Monge-Ampère equations smooth on Hermitian manifolds?

The theorem claims smoothness only on the regular locus

The paper studies the complex Monge-Ampère equation (θ + dd^c u)^n = e^{f(·,u)+ψ+−ψ−} ω_X^n on a compact Hermitian manifold X of dimension n, where θ is a smooth semipositive (1,1)-form that is big, f is smooth on X×R, and ψ± are quasi-psh functions smooth on a fixed open set Ω ⊂ X. Theorem 1.1 states that if φ is a bounded θ-psh function solving this equation, then φ is smooth in Ω\D, where D is the analytic subset along which the big form θ degenerates.

Two boundaries matter here. First, Ω need not be Zariski open; it can be a small open set in X. Second, the theorem is conditional: it says that if a bounded solution exists, it is smooth on Ω\D. The paper states explicitly that existence of a bounded θ-psh solution is not guaranteed. So the result is a regularity statement about solutions, not an existence statement about them.

The existence of a bounded solution implicitly forces the function e^{ψ+−ψ−} to be integrable over X. When f is non-decreasing in t, the paper notes that this integrability condition is also sufficient for solvability, and in the Kähler case with Ω Zariski open the unique solution is smooth in Ω\D.

The right-hand side may decrease in the unknown

Most classical treatments assume f is non-decreasing in t, which makes the equation monotone and yields uniqueness of solutions. This paper allows f to depend on u in a way that may decrease. The authors state that when f is not increasing in t, the situation becomes more delicate: solutions, when they exist, are generally not unique, except in certain special cases.

That loss of uniqueness is not a technicality. It changes what a regularity theorem can say. The paper cannot rely on identifying a single canonical solution; instead it must show that any bounded solution, whatever its provenance, is smooth on the regular locus. The proof is therefore built to work without uniqueness.

The authors also note that the domination principle reveals that, when f is non-decreasing in t, the constant c(t) appearing in the associated eigenvalue problem is non-increasing in t. They do not assume f is increasing, so c(t) may be constant in t.

Domination principle replaces Kähler-Ricci flow arguments

Earlier results in this direction used the regularizing property of the Kähler-Ricci flow. Székelyhidi and Tosatti proved the Kähler case with ψ± = 0 and θ Kähler by using the smoothing property of the weak Kähler-Ricci flow. Nie adapted that idea to compact Hermitian manifolds under a mass-preservation condition, which Chiose showed is equivalent to a curvature assumption of Guan and Li. Kołodziej and Nguyen later supplied the stability result that made the Hermitian case work.

This paper takes a different route. Its proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. The authors describe their approach as a simpler alternative proof when ψ± = 0, and they emphasize that their assumptions on the data are flexible enough to extend the result to singular varieties.

The key tool is the comparison principle for bounded θ-psh functions. In the form used here, if (θ + dd^c u)^n ≤ e^{u−v}(θ + dd^c v)^n then u ≥ v; and if (θ + dd^c u)^n ≤ c(θ + dd^c v)^n for some c ≥ 0 then c ≥ 1. The last conclusion is the domination principle. It lets the authors compare solutions without invoking a flow.

Bounded solutions are approximated by smooth ones

The proof strategy is easiest to see in the model case where θ = ω is Kähler and ψ± = 0, so Ω = X and D = ∅. The authors approximate the bounded solution φ by a decreasing sequence (v_j) of smooth ω-psh functions using Demailly's approximation theorem, with −C_0 ≤ v_j ≤ C_0 for all j.

For each j they solve (ω + dd^c u_j)^n = e^{u_j − v_j + f(·,v_j)} ω_X^n, where u_j is smooth. Proposition 3.1 gives a uniform bound sup_X |u_j| + sup_X |Δ_ω u_j| ≤ C_1, where C_1 depends on C_0 and on an upper bound for the C^2 norm of the relevant nonlinearity on X×[−C_0,C_0]. The Laplacian bound is obtained by maximizing H = log Tr_ω(ω_φ) + A_1(v+C_0+1)^2 − A_2 φ and choosing A_1, A_2 appropriately.

Uniform estimates let the authors extract a subsequence converging uniformly to a function u ∈ PSH(X, ω) solving (ω + dd^c u)^n = e^{u+g(x,φ)} ω_X^n with g(x,t) = −t + f(x,t). Since φ solves the same equation, the comparison principle forces u = φ. A separate argument shows the sequence (Δ_ω v_j) is also uniformly bounded, and then C^{2,α} bounds and Schauder estimates upgrade φ to smooth.

The Hermitian case and singular varieties follow

The Kähler argument is then extended to the Hermitian setting, where θ is only semipositive and big, and where ψ± may be singular along D. The paper devotes Section 4 to the a priori estimates and the proof of Theorem 1.1 in this general case. The same domination-principle machinery is used, but the estimates must be carried out without the Kähler condition.

A corollary extends the result to singular varieties. If V is a compact complex variety with klt singularities equipped with a Hermitian form ω_V, an adapted measure μ_V, and a smooth f : V×R → R, then any bounded solution to (ω_V + dd^c u)^n = e^{f(·,u)} μ_V with ω_V + dd^c u ≥ 0 is smooth in the regular locus V_reg. The proof pulls back along a desingularization π : X → V, where u∘π is bounded and π^*ω_V-psh and solves the equation for appropriate ψ±.

In the special case where V is a Q-Fano compact Kähler variety with klt singularities, the corollary applies and shows that any singular Kähler-Einstein metric is smooth in the regular locus of V. The authors note that this corollary was announced in another work as provable via the smoothing property of the Kähler-Ricci flow; here it follows from Theorem 1.1.

Sub- and supersolutions do not guarantee existence

Theorem 1.3 addresses existence directly. If the equation admits a bounded subsolution u ∈ PSH(X, θ) and a bounded supersolution ū ∈ PSH(X, θ) with u ≤ ū, then it admits a bounded solution u ∈ PSH(X, θ) that is smooth in Ω and satisfies u ≤ u ≤ ū on X. The ordering u ≤ ū is essential to the statement.

The paper is explicit that ordering cannot be dropped in general. Example 5.8 takes f(x,t) = −t + h(x) on a Fano manifold without Kähler-Einstein metrics, where the equation has no bounded solution; the constant function u = C is a subsolution for large C, while −C is a supersolution. So the existence of sub- and supersolutions without the right ordering does not ensure existence.

Remark 5.5 makes a related point: a subsolution alone is not enough. If f is strictly increasing but f ≤ −1, ψ± = 0, and θ = ω_X, then a bounded solution would satisfy (ω_X + dd^c u)^n ≤ e^{−1} ω_X, and the domination principle would force u ≥ C for all C > 0, which is impossible.

When f is non-decreasing, existence becomes a threshold condition

For non-decreasing f the paper proves an equivalence. Theorem 5.4 states that the following three conditions are equivalent: there exists t_0 ∈ R such that c(t_0) = 0; there exists a bounded subsolution and a bounded supersolution of the equation; and there exists a bounded solution. Here c(t) is the unique constant for which (θ + dd^c φ_t)^n = e^{c(t)+f(·,t)} g ω_X^n has a solution φ_t ∈ PSH(X, θ).

The ordering between the subsolution and supersolution is not assumed in condition (2); the paper notes that it holds automatically after adding constants. The proof uses the domination principle to show c(t_1) ≥ 0 and c(t_2) ≤ 0 for t_1 = inf_X u and t_2 = sup_X ū, and then uses continuity of t ↦ c(t) to find t_0 with c(t_0) = 0.

This generalizes an earlier Kähler result where condition (1) could be reformulated as ∫_X e^{f(·,t_0)} ω_X^n = ∫_X θ^n. The authors point out that in the general Hermitian context the argument via computing Monge-Ampère mass fails, so their proof instead relies on the comparison principle and on the resolution of the equation given sub- and supersolutions in Theorem 1.3.

The decisive distinction is the regular locus, not the whole manifold

The most useful way to read this paper is as a statement about where smoothness holds, not about whether solutions exist. Theorem 1.1 gives smoothness on Ω\D, the open set where the data are smooth, and it does so for any bounded solution, including solutions of equations whose right-hand side decreases in the unknown and which therefore need not be unique.

The practical threshold is the pair (Ω, D). If Ω is a small open set in X, the conclusion is correspondingly local. If D is nonempty, smoothness is not claimed across D. And if no bounded solution exists, the theorem has nothing to regularize. The paper's own examples show that sub- and supersolutions without the correct ordering can coexist with nonexistence, so the presence of a subsolution is not by itself a reason to expect a smooth solution.

For readers comparing this with earlier work, the relevant contrast is the method rather than the conclusion in the smooth positive case. When the right-hand side is strictly positive and smooth, the paper recovers known results of Nie and Kołodziej-Nguyen in the Hermitian case and Székelyhidi-Tosatti in the Kähler case, which relied on the regularizing property of the Kähler-Ricci flow. Here the same smoothness is reached through the domination principle and a priori estimates, and the same machinery extends to singular varieties with klt singularities.

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