Theorem 19.9.13.0F8B7237A1BA If $X$ fails the bounded mass property, then for every complete flag on $X$, every holomorphic line bundle $L$ and every graded subalgebra $W_\bullet $ of its section ring as above,
\[ \operatorname {vol}_{\mathbb R^ n}\Delta (W_\bullet )=0, \qquad \dim W_ m=o(m^ n). \]
Proof.
If $\Delta (W_\bullet )$ had nonempty interior, $X$ would have $n$ algebraically independent meromorphic functions by proposition 19.9.10. Moishezon's theorem would then give a modification of $X$ from a smooth projective, hence Kähler, manifold, contradicting lemma 19.9.12. The body therefore has empty interior, and proposition 19.9.6 gives both conclusions.
$\square$
Pome
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