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.
12.19.14 Consequences for dimension and equations8E3D46779F5F
The example occurs in the smallest possible dimension: on a surface, comparison with a Gauduchon metric gives positive lower mass. Products with projective spaces carry the threefold degeneration to every higher dimension, and a product with the Hopf threefold produces, in addition, unbounded upper mass.
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 $v_-(h)\leq A^ n v_-(h_0)=0$. $\square$
Corollary 12.19.16.940F7BB9270C In every dimension $n\geq 6$ there is a connected compact complex manifold with
\[ v_-(h)=0,\qquad v_+(h)=+\infty \]
for every Hermitian form $h$.
Proof. Take the product of the threefold above with the Hopf threefold of section 23.1, with a split metric. Potentials pulled back from the first factor make the masses tend to zero, while the smooth potentials of theorem 23.1.1, pulled back from the second factor, make them tend to infinity. In each case the product integral is $\binom 63$ times the product of the two factor masses. The metric comparison in the preceding proof handles the lower mass for every Hermitian form, and the metric independence in proposition 13.14.15 handles the upper mass. Products with projective spaces give every higher dimension by the same expansion of wedge powers. $\square$
Finally, fix a smooth positive probability measure $\mu $ on the threefold $X$. The smooth equation still gives $\mathcal V_\mu (h)>0$ for each Hermitian form $h$. With $\mathscr M$ and $\lambda _*$ as in subsection 12.15.3, theorem 12.15.4 gives
\[ \inf _{\nu \in \mathscr M}\mathcal V_\nu (h)=0, \qquad \lambda _*(h)=0. \]
Thus even a smooth strictly positive background can have vanishing infimum over densities. For the metrics $\eta _ j$ and numbers $R_ j,P_ j$ in the proof of theorem 12.19.1, the measures $\mu _ j=\eta _ j^3/R_ j$ satisfy $\mathcal V_{\mu _ j}(h)\leq P_ j^3/R_ j^2\to 0$. Hence the densities $\mu _ j/\mu $ cannot be uniformly bounded above: a bound $\mu _ j\leq M\mu $ would give $\mathcal V_{\mu _ j}(h)\geq M^{-1}\mathcal V_\mu (h)>0$, by the comparison of denominators in the proof of proposition 12.15.1.
The example also shows that the endpoint estimate in theorem 12.9.2 fails for the full family of normalized Gauduchon currents: the supremum of the $L^{3/2}$ norms of their determinant densities is infinite, although every subcritical $L^ q$ norm, $1<q<3/2$, remains uniformly bounded by theorem 12.9.1. For the same fixed background, theorem 12.9.7, corollary 12.9.10 give uniform $L^\infty $ bounds for the exact maximizers and their adjoint probabilities in a small $C^{2,\alpha }$ family of smooth densities. A larger sufficient family is given by theorem 12.9.18: uniformly comparable smooth solution forms yield uniform bounds for both densities, without derivative bounds on the varying Hessian. Endpoint estimates for arbitrary bounded families of continuous densities remain open.
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