Theorem 14.10.1 (Székelyhidi–Tosatti–Weinkove).76AF759A4024 Fix a Hermitian form $\gamma _0$, a Gauduchon form $\gamma $, and a smooth real function $F$. There is a unique pair $(u,b)\in C^\infty (X,\mathbb R)\times \mathbb R$ with $\sup _ Xu=0$ such that the $(n-1,n-1)$-form
\[ \Gamma _ u=\gamma _0^{n-1}+\mathrm{dd}^{\mathrm{c}}u\wedge \gamma ^{n-2} +\operatorname {Re}\left( \frac{\mathrm i}{2\pi }\partial u\wedge \bar\partial (\gamma ^{n-2}) \right) \]
is strictly positive, and the Hermitian form $\eta _ u$ with $\eta _ u^{n-1}=\Gamma _ u$ satisfies
\[ \eta _ u^ n=\mathrm{e}^{F+b}\gamma ^ n. \]
of theorem 14.10.1.
Let
\[ F_0=\log \frac{\gamma _0^ n}{\gamma ^ n},\qquad F_ t=(1-t)F_0+tF,\quad 0\leq t\leq 1. \]
We use the continuity method along the path $F_ t$. At $t=0$, the pair $u=0$, $b=0$ solves the equation, since $\eta _0=\gamma _0$. During the continuity argument we normalize potentials by $\int _ Xu\, \gamma ^ n=0$; adding a constant does not affect the equation, and at the end we impose $\sup _ Xu=0$ instead.
For openness, the derivative of the equation with respect to the mean-zero potential and the scalar is $(w,a)\mapsto \mathcal L_ uw-a$. Given a right-hand side $q$, integration against the adjoint probability measure of proposition 14.12.1 forces $a=-\int _ Xq\, \mathrm{d}\mu _ u^\gamma $. The Fredholm alternative then gives a unique mean-zero solution $w$, and Schauder estimates bound this inverse between Hölder spaces. Since positivity of $\Gamma _ u$ is an open condition, the implicit function theorem gives solutions for all nearby parameters.
For closedness, note that the functions $F_ t$ are uniformly bounded in every $C^ k$ norm. After supremum normalization, proposition 14.11.1 bounds $u$ and $b$, the second-order and rescaling estimates bound the gradient and the complex Hessian, and lemma 14.17.1 gives all higher bounds. Subtracting the mean preserves all derivative bounds and changes the potentials by uniformly bounded constants. A subsequence of mean-zero solutions along a convergent sequence of parameters therefore converges smoothly to a solution at the limiting parameter, and the bounds $aI\leq Q\leq b_0I$ preserve strict positivity in the limit. Thus the set of solvable parameters is nonempty, open and closed, hence equal to $[0,1]$.
Finally, consider two solutions $(u,b)$ and $(w,c)$ for the same $F$. At a maximum of $u-w$ their gradients agree and their complex Hessians are ordered, so the first-order terms are equal there and, since $P_ n$ preserves the order, $Q_ u\leq Q_ w$. The equation then gives $b\leq c$, and at a minimum the reverse inequality gives $b\geq c$. Hence $b=c$. Along the segment between $u$ and $w$, the matrices $Q$ remain positive because the ansatz is affine in the potential. Integrating the derivative of the logarithmic determinant along that segment produces a uniformly elliptic scalar operator with first-order terms and no zeroth-order term that annihilates $u-w$. The maximum principle shows that $u-w$ is constant, and the supremum normalization makes the constant zero.
$\square$
Pome
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