11.31 Gauduchon traces and the topology of potentials15943A2C3613

Proof. The set $\mathcal K^0_{\theta ,\mathrm{d}V}$ is convex. It is the image of the compact set $\mathcal K_\theta $ of theorem 5.25.2 under the continuous map $v\mapsto v-\int _ Xv\, \mathrm{d}V$, and is therefore compact.

If $u_ j\to u$ in $L^1$, then $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}u$ as currents, so the trace measures converge in the sense of distributions. Since they are positive measures of the same finite mass, and every continuous function is a uniform limit of smooth functions, they also converge weakly as measures. Thus $\mathcal T_\gamma $ is continuous.

To prove injectivity, suppose that $\mathcal T_\gamma (u)=\mathcal T_\gamma (v)$. Then $A(u-v)=0$ in the sense of distributions. Elliptic regularity for $A$ shows that $u-v$ agrees almost everywhere with a smooth function, and the maximum principle shows that this function is constant. Its mean is zero, so $u=v$ almost everywhere, and hence everywhere by lemma 5.24.3.

A continuous injection from a compact space into a Hausdorff space is a homeomorphism onto its image; here the target is the space of positive measures of mass $M_{\theta ,\gamma }$ with the weak topology. Since $\mathcal T_\gamma $ is affine, its image is convex. If $M_{\theta ,\gamma }=0$, every trace measure is zero, so injectivity shows that $\mathcal K^0_{\theta ,\mathrm{d}V}$ has at most one point; it has exactly one when $\mathrm{PSH}(X,\theta )$ is nonempty, since the mean of any $\theta $-plurisubharmonic function can be subtracted. In this case lemma 11.6.3 also gives $\theta +\mathrm{dd}^{\mathrm{c}}u=0$ for the unique normalized potential $u$, which is therefore smooth. $\square$

Proof. If $\nu =\mathcal T_\gamma (u)$ with $u\in \mathcal K^0_{\theta ,\mathrm{d}V}$, then $u$ is a mean-zero solution of (11.31.3) with $\sigma =\nu -\theta \wedge \gamma ^{n-1}$, so uniqueness gives $w=u$ as distributions. Conversely, if $\theta +\mathrm{dd}^{\mathrm{c}}w\geq 0$, then proposition 5.22.2 gives a $\theta $-plurisubharmonic representative $u$ of $w$. It has mean zero by construction, and (11.31.3) gives $\mathcal T_\gamma (u)=\nu $. $\square$

Proof. Symmetry and the triangle inequality follow directly from the definition, and $d_{\gamma ,d}\leq 2M_{\theta ,\gamma }$ because both trace measures have mass $M_{\theta ,\gamma }$. Let $\mathcal F$ denote the family of test functions in the supremum. After rescaling, $\mathcal F$ contains a positive multiple of every Lipschitz function, and Lipschitz functions are uniformly dense in $C(X)$. Hence $\mathcal F$ separates finite measures, and together with the injectivity of $\mathcal T_\gamma $ this shows that $d_{\gamma ,d}$ is a metric. The same density, together with the common mass bound, shows that convergence in $d_{\gamma ,d}$ implies weak convergence of the trace measures. Conversely, $\mathcal F$ is uniformly bounded and equicontinuous, so the Arzelà–Ascoli theorem gives a finite uniform $\epsilon $-net in $\mathcal F$ for every $\epsilon >0$. Weak convergence on the finitely many functions of the net, and the bound $2M_{\theta ,\gamma }\epsilon $ for the remaining error, give convergence in $d_{\gamma ,d}$.

Thus $d_{\gamma ,d}$ is the pullback by the homeomorphism of theorem 11.31.2 of a metric for the weak topology. The resulting metric space is compact, hence complete. For two choices of the auxiliary data, both metrics induce the same compact $L^1$ topology, so the identity map is uniformly continuous in both directions. $\square$

Proof. The lower bound follows by duality from the inverse of $A^*$, and the upper bound by smoothing the test functions. Put $w=u-v$, $\sigma =\mathcal T_\gamma (u)-\mathcal T_\gamma (v)$ and $M=M_{\theta ,\gamma }$, and write $\| \sigma \| _{\mathrm{TV}}$ for the total variation of $\sigma $. Then $\int _ Xw\, \mathrm{d}V=0$, $\sigma =\mathrm{dd}^{\mathrm{c}}w\wedge \gamma ^{n-1}$ and $\| \sigma \| _{\mathrm{TV}}\leq 2M$. If $M=0$, theorem 11.31.2 gives $u=v$, and both estimates hold trivially.

For the lower bound, let $g$ be a smooth real function. Since $\ker A=\ker A^*=\mathbb R$, the equation

has a unique smooth solution. For $p>2n$, elliptic $W^{2,p}$ estimates on functions of mean zero and the Sobolev embedding theorem give $\| \phi \| _\infty +\operatorname {Lip}_ d(\phi )\leq C\| g\| _\infty $. Pairing the distributional equation $Aw\, \mathrm{d}V=\sigma $ with $\phi $, and using $\int _ Xw\, \mathrm{d}V=0$, we obtain

Taking the supremum over smooth real $g$ with $\| g\| _\infty \leq 1$ gives $\| w\| _{L^1}\leq C d_{\gamma ,d}(u,v)$.

For the upper bound, let $f$ be any test function in the definition of $d_{\gamma ,d}$. Using a fixed finite coordinate cover, a smooth partition of unity and convolution at scale $0<\epsilon <\epsilon _0$, we obtain smooth functions $f_\epsilon $ satisfying

with constants uniform over this family of test functions. To see the second bound, localize $f$ by the fixed partition of unity. The localized functions have uniformly bounded values and first weak derivatives. In each second derivative of their convolutions, move one derivative onto the mollifier, whose derivative has $L^1$ norm $O(\epsilon ^{-1})$. The lower-order terms of $A^*$ have fixed smooth coefficients and satisfy the same bound. Since $\int _ Xf_\epsilon \, \mathrm{d}\sigma =\int _ XwA^*f_\epsilon \, \mathrm{d}V$, it follows that

If $0<\| w\| _{L^1}<\epsilon _0^2$, choose $\epsilon =\| w\| _{L^1}^{1/2}$. For larger norms use $d_{\gamma ,d}(u,v)\leq 2M$, and for $w=0$ use injectivity. Taking the supremum over $f$ proves the upper bound. $\square$

Proof. Set $v=a-b$. The identity $\mathrm{dd}^{\mathrm{c}}(v^2)=2\mathrm{d}v\wedge \mathrm{d}^ cv+2v\mathrm{dd}^{\mathrm{c}}v$, integrated against $G$, and the equation $\mathrm{dd}^{\mathrm{c}}G=0$ give

The measures $(\mathrm{dd}^{\mathrm{c}}a+C\omega )\wedge G$ and $(\mathrm{dd}^{\mathrm{c}}b+C\omega )\wedge G$ are positive, and each has mass $CM$, because the terms $\mathrm{dd}^{\mathrm{c}}a\wedge G$ and $\mathrm{dd}^{\mathrm{c}}b\wedge G$ integrate to zero. Their difference $\mathrm{dd}^{\mathrm{c}}v\wedge G$ therefore has total variation at most $2CM$, which proves the upper bound. The lower bound holds because $\mathrm{d}v\wedge \mathrm{d}^ cv=(\mathrm{i}/2\pi )\partial v\wedge \bar\partial v$ is a positive $(1,1)$-form.

For the second estimate, work in a local unitary frame for $\omega $. Positivity of $G$ means that its coefficients form a positive semidefinite matrix of measures. These entries are absolutely continuous with respect to the trace measure $\mu $: the diagonal entries are dominated by the trace, and the off-diagonal entries by the Cauchy–Schwarz inequality for positive semidefinite matrices. The resulting matrix of densities has trace bounded by a constant depending only on the normalization. Pairing $\mathrm{d}v\wedge G$ or $\mathrm{d}^ cv\wedge G$ with a test one-form $\eta $, and applying the matrix Cauchy–Schwarz inequality and then the integral Cauchy–Schwarz inequality, we obtain

A finite cover by such frames and a partition of unity give the global estimate. Taking the supremum over smooth test forms of norm at most one yields the total variation bound. When $M=0$, positivity gives $G=0$, and both estimates hold trivially. $\square$

Proof. We first define the gradient currents $\mathrm{d}u\wedge G$ and $\mathrm{d}^ cu\wedge G$ as limits, then define the product by an integration-by-parts formula, and finally prove joint continuity.

Step 1: gradient currents. Choose smooth approximants $a_ k\to u$ as in lemma 5.42.5. There is a common constant $C>0$ such that $\mathrm{dd}^{\mathrm{c}}a_ k\geq -C\omega $. Write $M=\int _ X\omega \wedge G$. Since $a_ k$ converges uniformly, lemma 11.31.10 shows that $\mathrm{d}a_ k\wedge G$ and $\mathrm{d}^ ca_ k\wedge G$ are Cauchy sequences in total variation. Denote their limits, which are currents of order zero, by $\mathrm{d}u\wedge G$ and $\mathrm{d}^ cu\wedge G$. Applying the same lemma to two uniformly convergent smooth sequences with a common lower Hessian bound shows that the limits do not depend on the sequence. Uniform convergence also gives $a_ kG\to uG$ in total variation.

Step 2: the product. Define

in the sense of distributions. For smooth $u$, the Leibniz rule and $\mathrm{dd}^{\mathrm{c}}G=0$ show that this agrees with the usual product. Consequently $(\alpha +\mathrm{dd}^{\mathrm{c}}a_ k)\wedge G$ converges in distributions to the right-hand side of (11.31.12). With $\varepsilon _ k\to 0$ as in lemma 5.42.5, the measures $(\alpha +\mathrm{dd}^{\mathrm{c}}a_ k+\varepsilon _ k\omega )\wedge G$ are positive, with masses $\int _ X\alpha \wedge G+\varepsilon _ kM$. Their distributional limit is therefore a positive measure, and testing against the constant function one gives its mass. The uniform mass bound upgrades distributional convergence to weak convergence of measures. If $G$ is smooth, the uniform convergence of $a_ k$ also identifies the limit with the product of the current $\alpha +\mathrm{dd}^{\mathrm{c}}u$ and the smooth form $G$.

Step 3: independence of the decomposition. Adding a smooth function $h$ to $u$ adds $\mathrm{d}h\wedge G$ and $\mathrm{d}^ ch\wedge G$ to the gradient currents, so replacing $(\alpha ,u)$ by $(\alpha -\mathrm{dd}^{\mathrm{c}}h,u+h)$ leaves the product unchanged. If two continuous decompositions define the same $(1,1)$-current, the difference of their potentials has smooth $\mathrm{dd}^{\mathrm{c}}$; taking its trace in a coordinate chart and applying elliptic regularity shows that this difference is smooth. Thus the product depends only on the current $\alpha +\mathrm{dd}^{\mathrm{c}}u$ and on $G$.

Step 4: joint continuity. For $u$ and for each $u_ j$, keep only those smooth approximants from lemma 5.42.5 whose curvature errors are at most one. Choosing $C$ with $\alpha \leq (C-1)\omega $, all these approximants satisfy the same lower Hessian bound $-C\omega $. The numbers $M_ j=\int _ X\omega \wedge G_ j$ are bounded, by weak convergence. Fix a smooth approximant $a_ k$ of $u$, and put $\delta _{j,k}=\| u_ j-a_ k\| _\infty $. Applying lemma 11.31.10 to $a_ k$ and to smooth approximants of $u_ j$, and passing to the gradient limits, we find that the total variations of the differences $(\mathrm{d}u_ j-\mathrm{d}a_ k)\wedge G_ j$ and $(\mathrm{d}^ cu_ j-\mathrm{d}^ ca_ k)\wedge G_ j$ are at most $C'M_ j\sqrt{\delta _{j,k}}$, where $C'$ is independent of $j$ and $k$.

For a smooth real test function $\phi $, formula (11.31.12) now gives

where $C_\phi $ depends on the first two derivatives of $\phi $, on $\omega $ and on $C$. The same estimate holds with $u_ j,G_ j$ replaced by $u,G$. For fixed $k$, the products $(\alpha +\mathrm{dd}^{\mathrm{c}}a_ k)\wedge G_ j$ of a fixed smooth form with $G_ j$ converge weakly to $(\alpha +\mathrm{dd}^{\mathrm{c}}a_ k)\wedge G$. Let first $j\to \infty $ and then $k\to \infty $; since $u_ j\to u$ uniformly, the error tends to zero. This proves convergence in distributions, and positivity together with the bounded masses $\int _ X\alpha \wedge G_ j$ upgrades it to weak convergence of measures. $\square$