24.7 Weak Gauduchon adjointsC528B286A8D7

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Let $\theta $ be a background with a model reference $\varphi $ and $\mathrm{d}V$ a smooth positive volume form, as in section 11.32. For a bounded logarithmic density $F$, the normalized equation $\mathrm{MA}^{\mathrm{ext}}_\theta (u)=c_\varphi (F)\mathrm{e}^ F\, \mathrm{d}V$ in the ordinary type of $\varphi $ determines a unique scalar, and $B_\varphi (F)=\log c_\varphi (F)$ is a concave functional of $F$ (section 11.33). For smooth data, differentiating the smooth equation produces a Gauduchon adjoint probability. At a continuous logarithmic density $F$, the supporting probabilities of $B_\varphi $, which form the set $\mathcal A_\varphi (F)$ of (11.34.1), provide a weak replacement. In the regimes of background and reference considered in section 11.32, $\mathcal A_\varphi (F)$ is nonempty, convex and weakly compact, and its members are absolutely continuous (theorem 11.34.2). If this set consisted of a single probability, the scalar functional would be differentiable at $F$ even where no classical linearized equation is available.

Conjecture 24.7.1.7C0A1CE5EC77 Let $\omega $ be a Hermitian form on $X$ and $\mathrm{d}V$ a smooth positive volume form, and take the nonsingular reference $\varphi =0$. For every real $F\in C(X)$, the set $\mathcal A_0(F)$ of supporting probabilities of the normalization functional $B_0$ at $F$ consists of a single probability measure.

Equivalently, by proposition 11.34.5, the concave functional $B_0$ is Gâteaux differentiable at every continuous logarithmic density. The conjecture holds at smooth $F$, by the linearized equation, and in the closed Kähler setting, where the normalization is explicit. By proposition 11.34.3, it would imply that for any smooth approximations of $F$ the Gauduchon adjoint probabilities have a unique weak limit.

A further sufficient condition is a bounded solution with continuous, everywhere positive complex Hessian. Then theorem 11.34.11 identifies a unique stationary supporting probability and the Gâteaux derivative. Proposition 11.34.13 gives its formula on $\mathbb P^1$ times any compact Hermitian factor for every positive continuous density of the stated product form; example 11.34.14 includes a non-Dini density on a non-Kähler threefold. The criterion does not prove that an arbitrary continuous density has a solution with such a Hessian.

For bounded potentials there is a direct result. Suppose that $\omega +\mathrm{dd}^{\mathrm{c}}v$ is Gauduchon for every smooth $v$ with $\omega +\mathrm{dd}^{\mathrm{c}}v>0$; this holds for every Gauduchon form on a compact surface. Then, by proposition 11.25.1, the normalized Monge–Ampère measures of bounded $\omega $-plurisubharmonic functions extend the smooth adjoint probabilities and are continuous under convergence in capacity of uniformly bounded potentials. This defines no weak linearized operator, and it does not settle approximation by unbounded potentials in a general background.

A Baire category argument gives a partial result. In every regime of background and reference considered in section 11.32, theorem 11.34.7 shows that $\mathcal A_\varphi (F)$ is a single probability for $F$ in a dense $G_\delta $ subset of $C(X)$, and that at such $F$ all choices of supporting probabilities along uniform approximations of $F$ have the same weak limit. The assertion is generic for the uniform topology and does not exclude exceptional continuous densities.

For $n\geq 2$, the compact formulation with currents gives a more precise target. By theorem 12.8.2, the supporting probabilities at $F$ are exactly the normalized determinant-weighted measures of the currents $T\in \mathscr G_\omega $ maximizing the functional $\mathfrak D_ F$ of (12.7.1). The conjecture thus asks whether all maximizers have the same weighted measure. Corollary 12.9.3 bounds the densities of these measures in $L^ q$, for every $1<q<n/(n-1)$, uniformly as $F$ ranges over a bounded subset of $C(X)$; the endpoint $q=n/(n-1)$ is included when $v_-(\omega )>0$. For $n\geq 3$, theorem 12.9.7 gives uniform $L^\infty $ bounds for both the adjoint density and the determinant density for smooth logarithmic densities in a small $C^{2,\alpha }$ ball around the logarithmic density of $\omega ^ n$ with respect to $\mathrm{d}V$, even when the lower mass vanishes, and by corollary 12.9.10 this ball contains distinct data with distinct exact maximizing currents. The controlled-Hessian family of theorem 12.9.18 gives a larger sufficient condition: if the smooth solution forms are uniformly comparable to the background, both densities have uniform two-sided bounds depending only on that comparison and fixed geometry. The divergence of the cofactor uses only the fixed torsion, so derivatives of the varying Hessian do not enter the constants. Example 12.9.21 gives such a smooth family whose densities have unbounded $C^1$ norms and no common modulus of continuity. Arbitrary bounded families of continuous data remain outside these hypotheses. These estimates alone give compactness but not uniqueness at continuous data. For a fixed smooth density the maximizing current is unique, smooth and strictly positive; nevertheless, proposition 12.18.2 shows that a sequence approaching the maximum can have unbounded determinant moments above the critical exponent. Another approach to the conjecture would be to show directly that the upper and lower directional derivatives of $B_0$ agree.

More is known about the functional itself. Relative inequalities between positive residual forms give uniform convergence of the whole scalar functional, by the explicit bound (11.40.2) of theorem 11.40.1. Under $L^1$ convergence of tests with a common $L^\infty $ bound, corollary 11.40.3 shows that the supporting probabilities have uniformly integrable densities whose weak $L^1$ cluster limits lie in the limiting supporting set. The adjoint probabilities at smooth tests, together with the value of the functional at zero, determine the whole functional by (11.40.5). None of these statements decides the conjectured uniqueness at continuous tests.

For a singular reference or a degenerate background, the same uniqueness question makes sense under the hypotheses of section 11.32. A geometric interpretation of the supporting probabilities, however, requires a well-defined weak linearization. For a smooth background $\theta $ and $u\in \mathrm{PSH}(X,\theta )$, write $\theta _ u=\theta +\mathrm{dd}^{\mathrm{c}}u$, and let $\rho $ be a measurable weight. The expression $\mathrm{dd}^{\mathrm{c}}(\rho \theta _ u^{n-1})$ is not a definition until multiplication of the product $\theta _ u^{n-1}$ by $\rho $ has been justified.

An explicit divisorial model shows what can go wrong. For the admissible reference of lemma 11.34.16, the supporting set is a single probability and the residual linearization is well defined (proposition 11.34.18). Nevertheless, theorem 11.34.21 gives decreasing smooth approximations, with positive forms, whose classical adjoint probabilities have different weak limits, each carrying mass one half on the divisor. These potentials converge in capacity and their logarithmic densities converge in every finite $L^ p$, but their exponential densities concentrate and their smooth mass exceeds the relative mass. Within this fixed divisorial type, corollary 11.34.24 proves recovery, with an explicit estimate, under convergence in capacity and a uniform bound on the residual potentials. On the non-Gauduchon product surface of 11.45.14, theorem 11.45.16 constructs a weak linearization for a logarithmic solution with continuous positive residual and identifies its full supporting set as a singleton. Uniform convergence of the positive residual forms, with the divisor fixed, gives convergence of both the weighted currents and their probabilities in proposition 11.45.18; pole-filling smoothings still retain unwanted divisor mass (example 11.45.19). For general singular references, adjoints that do not depend on the approximation still require control of the singularity type, of the mass and of the conformal multiplier.