PomeTheorem 14.2.1 (Tosatti–Weinkove).F33DA8C9DF9F For every smooth real $F$, (14.1.1) has a unique pair $(u,b)\in C^\infty (X,\mathbb R)\times \mathbb R$ with $\sup _ Xu=0$.
Theorem 14.2.1 (Tosatti–Weinkove)F33DA8C9DF9F
of theorem 14.2.1. The case $n=1$ was treated after the statement of the theorem. For $n\geq 2$ put $G=F+\log (\mathrm{d}V/\omega ^ n)$ and consider
\[ (\omega +\mathrm{dd}^{\mathrm{c}}u_ t)^ n=\mathrm{e}^{tG+b_ t}\omega ^ n, \qquad \sup _ Xu_ t=0,\qquad 0\leq t\leq 1. \]
At $t=0$, take $u_0=0$ and $b_0=0$. The local implicit-function argument in proposition 14.4.1, with the additive constant temporarily fixed by a linear mean, shows that the set of solvable parameters is open.
At every solvable parameter, proposition 14.2.2 gives
\[ -\sup _ X(tG)\leq b_ t\leq -\inf _ X(tG), \qquad \mathrm{e}^{tG+b_ t}\leq \mathrm{e}^{\operatorname {osc}_ XG}. \]
The constants $b_ t$ are therefore uniformly bounded, and theorem 14.3.5 bounds $u_ t$ in $C^0$. By corollary 14.5.8, the solutions are bounded in every $C^ k$ norm. Along any sequence of solvable parameters converging in $[0,1]$, a subsequence of $(u_ t,b_ t)$ therefore converges smoothly to a solution; uniform equivalence of the metrics preserves strict positivity, and the supremum normalization passes to the limit. Thus the solvable set is closed. Since it is nonempty, open and closed, connectedness of $[0,1]$ gives a solution at $t=1$, and proposition 14.2.4 gives uniqueness. $\square$
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