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12.7 A compact dual problem over Gauduchon currents54ADA5D399DA

54ADA5D399DA

We now fix a Hermitian background and vary the density. For a smooth positive density, theorem 12.2.3 provides a minimizing Gauduchon metric; for a density that is only continuous or integrable, there is no smooth equation to produce one. The extremal problem becomes compact when positive $\mathrm{dd}^{\mathrm{c}}$-closed $(n-1,n-1)$-currents are allowed as tests, provided the denominator $D_\mu $ is extended to currents. For this we take the determinant of the matrix density of a current relative to its trace measure. The integral of its $n$th root is an infimum of continuous linear functionals, hence upper semicontinuous under weak convergence, and its maximum over normalized currents is attained. The smooth equation identifies this maximum with the supremum over smooth tests.

From here until section 12.11, we assume $n\geq 2$, and we fix a Hermitian form $\omega $ and a smooth positive probability volume form $\mathrm{d}V$ on $X$; integrability and convergence in measure refer to $\mathrm{d}V$. As in section 13.14, $\mathrm{MA}_\omega (u)=(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n$ for bounded $u\in \mathrm{PSH}(X,\omega )$, and $\mathrm{MA}^{\mathrm{ext}}_\omega (u)$ is the extended non-pluripolar Monge–Ampère measure of theorem 13.14.3. Let $\mathscr G_\omega $ be the compact convex set of normalized Gauduchon currents of section 11.7, that is, of positive $(n-1,n-1)$-currents $T$ with $\mathrm{dd}^{\mathrm{c}}T=0$ and $\int _ X\omega \wedge T=1$. Finally, let $F\in C(X,\mathbb R)$.

For $T\in \mathscr G_\omega $, the trace measure $\sigma _ T=\omega \wedge T$ is a probability measure, and all coefficient measures of $T$ are absolutely continuous with respect to it. In a unitary frame for $\omega $, let $B$ be the matrix of a smooth real $(1,1)$-form $\beta $ relative to $\omega $. The coefficient densities of $T$ form a measurable positive semidefinite Hermitian matrix $P_ T$, determined $\sigma _ T$-almost everywhere by

\[ \beta \wedge T=\frac1n\operatorname {tr}(B P_ T)\, \mathrm{d}\sigma _ T, \qquad \operatorname {tr}P_ T=n\quad \sigma _ T\text{-almost everywhere}. \]

The matrix $P_ T$ transforms by unitary conjugation when the frame changes, so its determinant is well defined. Define

A99038CD10B4 12.7.1
\begin{equation} f_ F=\left(\frac{\mathrm{e}^ F\mathrm{d}V}{\omega ^ n}\right)^{1/n},\qquad \Gamma _ F(T)=f_ F(\det P_ T)^{1/n}\sigma _ T,\qquad \mathfrak D_ F(T)=\Gamma _ F(T)(X). \end{equation}

These quantities are finite, since the arithmetic–geometric mean inequality gives $(\det P_ T)^{1/n}\leq 1$; the measure $\Gamma _ F(T)$ may vanish for some currents. The same definitions apply to every positive $(n-1,n-1)$-current $T$ of finite mass, with the finite measure $\sigma _ T=\omega \wedge T$ and $\operatorname {tr}P_ T=n$, and $\Gamma _ F$ is positively homogeneous of degree one in $T$. If the coefficient measures of $T$ are written instead as a matrix density $P'$ relative to another positive measure $\sigma '$ with respect to which they are absolutely continuous, then $(\det P_ T)^{1/n}\sigma _ T=(\det P')^{1/n}\sigma '$, by homogeneity of the determinant.

For a smooth positive $T=\gamma ^{n-1}$, we may therefore use the reference measure $\omega ^ n$. The coefficient matrix of $T$ relative to $\omega ^ n$ is the cofactor matrix of the matrix of $\gamma $, and its determinant is $(\gamma ^ n/\omega ^ n)^{n-1}$. The definition therefore gives

\[ \mathfrak D_ F(\gamma ^{n-1}) =\int _ X\left(\frac{\mathrm{e}^ F\mathrm{d}V}{\gamma ^ n}\right)^{1/n}\gamma ^ n. \]

Thus $\mathfrak D_ F$ extends the denominator $D_\mu $ of definition 12.2.1 to currents, with $\mu $ replaced by the possibly unnormalized density $\mathrm{e}^ F\mathrm{d}V$.

Lemma 12.7.2.FB6C2382E066 For every positive $(n-1,n-1)$-current $T$ of finite mass on $X$,

\[ \mathfrak D_ F(T) =\inf _{\substack {\beta >0\ \mathrm{continuous}\\ \beta ^ n=\mathrm{e}^ F\mathrm{d}V}} \int _ X\beta \wedge T. \]

In particular, $\mathfrak D_ F$ is concave, positively homogeneous and weakly upper semicontinuous on $\mathscr G_\omega $, and its value does not depend on the auxiliary frame or on the metric used to express the matrix density.

Proof. Let $I$ be the identity matrix of size $n$. For any positive semidefinite Hermitian matrix $P$ of that size,

\[ (\det P)^{1/n} =\inf _{Q>0,\ \det Q=1}\frac1n\operatorname {tr}(QP). \]

The inequality $\leq $ is the arithmetic–geometric mean inequality. For the reverse inequality, take, for real $\varepsilon >0$,

\[ Q_\varepsilon = \det (P+\varepsilon I)^{1/n}(P+\varepsilon I)^{-1}. \]

Its determinant is one. If $P$ is positive definite, the trace expression $\frac1n\operatorname {tr}(Q_\varepsilon P)$ converges to $(\det P)^{1/n}$ as $\varepsilon \downarrow 0$. If $P$ is singular, it converges to zero: each ratio of an eigenvalue of $P$ to that eigenvalue plus $\varepsilon $ is bounded, while the prefactor tends to zero. When $\operatorname {tr}P=n$, the trace expression is at most $1+\varepsilon $.

We apply this formula to $P_ T$ and multiply by $f_ F$. Every continuous positive $\beta $ in the asserted infimum has matrix $f_ FQ$ with $\det Q=1$, so it gives an upper bound for $\mathfrak D_ F(T)$. Conversely, for fixed $\varepsilon >0$ the measurable field $Q_\varepsilon $ formed with $P=P_ T$ is bounded above and below by positive constants. It can be approximated in $L^1(\sigma _ T)$ by continuous positive definite Hermitian fields with bounds of the same type. One way to construct them is to apply Lusin's theorem in finitely many bundle charts, extend the continuous fields from the resulting compact sets, and glue with a partition of unity; spectral truncation to a fixed positive interval preserves the bounds. Finally we divide each approximating field by the $n$th root of its determinant. This normalization is uniformly continuous on fields with spectrum in a fixed compact positive interval, so the normalized fields still converge in $L^1(\sigma _ T)$ and have determinant one. Their trace integrals converge to that of $Q_\varepsilon $.

Now let $\varepsilon \downarrow 0$. The bound $1+\varepsilon $ permits dominated convergence, and we obtain equality of the infimum and the integral. For a current of arbitrary positive finite trace mass we use homogeneity, and the zero current is immediate. The infimum is taken over functionals $T\mapsto \int _ X\beta \wedge T$ that are linear and weakly continuous, which proves concavity and upper semicontinuity. It also expresses $\mathfrak D_ F(T)$ without a matrix frame or an auxiliary metric, which proves the final assertion. $\square$

Let $B_0(F)$ be the normalization scalar of section 11.33 for the background $\omega $ and the reference potential zero. Thus the equation $\mathrm{MA}_\omega (u)=\mathrm{e}^{B_0(F)+F}\mathrm{d}V$ has a bounded solution $u\in \mathrm{PSH}(X,\omega )$, and $B_0(F)$ is the only constant with this property.

Theorem 12.7.3.0537569F9F52 For every $F\in C(X,\mathbb R)$,

053F006B114F 12.7.4
\begin{equation} M(F):=\max _{T\in \mathscr G_\omega }\mathfrak D_ F(T) =\mathrm{e}^{-B_0(F)/n}>0. \end{equation}

The maximum equals the supremum over the smooth strictly positive members of $\mathscr G_\omega $. The set $\mathscr M(F)$ of maximizing currents is nonempty, compact and convex.

If $F$ is smooth, there is exactly one maximizing current, namely the normalized $(n-1)$st power of the Gauduchon metric conformal to the smooth solution form $\omega +\mathrm{dd}^{\mathrm{c}}u$ of $\mathrm{MA}_\omega (u)=\mathrm{e}^{B_0(F)+F}\mathrm{d}V$.

Proof. Upper semicontinuity of $\mathfrak D_ F$ on the compact set $\mathscr G_\omega $ gives attainment, and testing a smooth positive member gives a positive value. Concavity makes the set of maximizers convex, and upper semicontinuity makes it closed.

Suppose first that $F$ is smooth, and let $\Omega =\omega +\mathrm{dd}^{\mathrm{c}}u>0$ be the smooth solution form with $\Omega ^ n=\mathrm{e}^{B_0(F)+F}\mathrm{d}V$. The form $\beta =\mathrm{e}^{-B_0(F)/n}\Omega $ is an admissible test in the preceding lemma, so Stokes' theorem for currents gives, for every $T\in \mathscr G_\omega $,

\[ \mathfrak D_ F(T)\leq \int _ X\beta \wedge T =\mathrm{e}^{-B_0(F)/n}\int _ X\omega \wedge T=\mathrm{e}^{-B_0(F)/n}. \]

If $\gamma $ is the Gauduchon metric conformal to $\Omega $, equality holds in the pointwise matrix inequality for $T$ proportional to $\gamma ^{n-1}$; normalizing $T$ so that $\int _ X\omega \wedge T=1$, we obtain the formula and its attainment by a smooth test.

For uniqueness, let $T$ be a maximizer. Equality in the integrated inequalities forces equality in the matrix arithmetic–geometric mean inequality $\sigma _ T$-almost everywhere. Since the matrix of $\beta $ is positive definite and $\operatorname {tr}P_ T=n$, this means that the coefficient measures of $T$ are a nonnegative scalar measure times the smooth coefficients of $\gamma ^{n-1}$. Thus $T=a\gamma ^{n-1}$, where $a$ is a positive measure, regarded as a distribution. The equation $\mathrm{dd}^{\mathrm{c}}(a\gamma ^{n-1})=0$ is the adjoint of a scalar elliptic equation, so elliptic regularity makes $a$ smooth. Since $\gamma $ is Gauduchon, proposition 11.23.2 identifies the kernel of this adjoint operator with the constants: the trace operator $v\mapsto n\mathrm{dd}^{\mathrm{c}}v\wedge \gamma ^{n-1}/\gamma ^ n$ has the constants as kernel and index zero, and integration against $\gamma ^ n$ annihilates its range. Thus $a$ is constant, and the normalization fixes the constant. This proves uniqueness.

For continuous $F,G$, the pointwise definition gives

\[ \mathrm{e}^{-\| F-G\| _\infty /n}\mathfrak D_ G(T) \leq \mathfrak D_ F(T) \leq \mathrm{e}^{\| F-G\| _\infty /n}\mathfrak D_ G(T). \]

The same bounds hold for the maxima, and for the suprema over smooth positive members. We approximate $F$ uniformly by smooth functions. Since $B_0$ is $1$-Lipschitz for the uniform norm by proposition 11.33.2, the formula just proved for smooth data gives (12.7.4) in the limit, including the equality with the smooth supremum. Note that only $F$ is approximated here: some normalized Gauduchon currents are not limits of smooth positive ones, as example 10.18.3 shows. $\square$

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