PomeCorollary 12.19.15.435BD74C27A9 Every Hermitian form on a compact complex manifold of dimension at most two has positive lower mass. In every dimension $n\geq 3$ there is a connected compact complex manifold on which every Hermitian form has zero lower mass.
Corollary 12.19.15435BD74C27A9
Proof. In dimension one the mass is $\int _ Xh>0$, and the surface case is corollary 12.18.1. In higher dimensions we use the monotonicity and the homogeneity of degree $n$ of $v_-$, which follow directly from (12.15.5), to pass from a split metric to every Hermitian metric.
For $n=3$ this is theorem 12.19.1. For $n>3$, take the product of the threefold $X$ with $\mathbb P^{n-3}$, let $\kappa $ be a Fubini–Study form on the projective factor, and consider the split metric $h_0=\omega +\kappa $, with pullbacks understood. For the potentials $u_ j$ given by the theorem for the background $\omega $,
\[ \int _{X\times \mathbb P^{n-3}}(h_0+\mathrm{dd}^{\mathrm{c}}u_ j)^ n =\binom n3\left(\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u_ j)^3\right) \left(\int _{\mathbb P^{n-3}}\kappa ^{n-3}\right) \longrightarrow 0. \]
For another Hermitian form $h$, choose $A$ with $h\leq Ah_0$. Monotonicity and homogeneity give $v_-(h)\leq A^ n v_-(h_0)=0$. $\square$
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