Theorem 11.42.1.1DCF668EBCCF For every such pair $(\theta ,\varphi )$ and every bounded measurable $F$, the equation in the ordinary type of $\varphi $ has a unique constant and a solution unique modulo constants. Its normalization scalar is
\[ B_{\theta ,\varphi }(F)=\log m_\varphi -\log Z_ F. \]
For continuous real $F$, its supporting set is
\[ \mathcal A_{\theta ,\varphi }(F)=\{ \mathrm{e}^ F\mathrm{d}V/Z_ F\} . \]
For two such pairs on the same $X$, with the same $\mathrm{d}V$ and possibly in different classes,
B169C3B4AD17
11.42.2
\begin{equation} \sup _{F\in L^\infty }|B_{\theta ,\varphi }(F)-B_{\theta ',\psi }(F)| =|\log m_\varphi -\log m_\psi |. \end{equation}
Thus an invariant of these pairs is determined by the whole functional if and only if it is constant on every fiber of the map $(\theta ,\varphi )\mapsto m_\varphi $.
Proof.
Existence follows from corollary 10.31.56. Indeed, a Kähler manifold has the bounded mass property by lemma 19.9.12, and since in a closed background all potentials of the ordinary type of $\varphi $ have the same mass $m_\varphi >0$, the reference satisfies the positive-volume model condition (13.20.5). Uniqueness modulo constants is the classical theorem on equations with prescribed singularities in closed big classes, which applies under the model and small-unbounded-locus hypotheses; see the notes in section 25.13.
Every potential of the ordinary type of $\varphi $ has mass $m_\varphi $, so integrating the equation forces $cZ_ F=m_\varphi $, which is the scalar formula. For continuous $F$, Jensen's inequality shows, as for (11.37.2), that the normalized exponential measure is a supporting probability, and differentiating at zero in both signs of every continuous direction shows that it is the only one. Subtracting the scalar formulas gives (11.42.2). Finally, $m_\varphi =\mathrm{d}V(X)\exp B_{\theta ,\varphi }(0)$, so the functional and the mass determine each other, and an invariant factors through either one exactly when it is constant on the fibers of the mass map.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.