Theorem 12.7.30537569F9F52

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 $,

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

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. $\square$