PomeLemma 19.9.12.976C06708A77 If a compact complex manifold $X$ admits a modification $\pi :Z\to X$ from a compact Kähler manifold, then $X$ has the bounded mass property.
19.9.11 Moishezon manifolds and bounded massF8F1527B70F2
A compact complex manifold of dimension $n$ is Moishezon if it has $n$ algebraically independent meromorphic functions. By Moishezon's theorem, a compact irreducible complex space of dimension $n$ with $n$ algebraically independent meromorphic functions admits a modification from a smooth projective manifold; the source of this form of the theorem is given in section 25.16. Such a modification bounds the masses of Hermitian potentials on $X$: the pulled-back Hermitian form is dominated by a fixed Kähler form, whose Monge–Ampère mass does not change when a smooth potential is added. Recall that $X$ has the bounded mass property if the masses $\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}v)^ n$ of bounded $\omega $-plurisubharmonic functions $v$ are uniformly bounded for one, equivalently every, Hermitian form $\omega $ (definition 13.14.14, proposition 13.14.15).
Proof. Fix a Hermitian form $\omega $ on $X$ and a Kähler form $\kappa $ on $Z$, and choose $C>0$ with $\pi ^*\omega \leq C\kappa $. By proposition 13.14.15, it suffices to bound the masses of smooth functions $u\in \mathrm{PSH}(X,\omega )$. Their pullbacks satisfy
\[ 0\leq \pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ) \leq C\kappa +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ). \]
Taking top powers and integrating gives
\[ \begin{split} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n & =\int _ Z(\pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ))^ n\\ & \leq \int _ Z(C\kappa +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ))^ n =C^ n\int _ Z\kappa ^ n. \end{split} \]
The first equality follows by change of variables on the set where $\pi $ is biholomorphic, since smooth top-degree forms give the exceptional analytic sets zero mass. The inequality follows by expanding in the smooth semipositive difference $C\kappa -\pi ^*\omega $, and the last equality follows from Stokes' theorem because $\kappa $ is closed. The bound does not depend on $u$, which proves the bounded mass property. $\square$
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. $\square$
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