24.4.4 Positive lower mass and concentrationE1772510D012

Positive lower mass is an integrability property of the Gauduchon tests at an endpoint exponent. It is independent of finite upper mass: the threefold of theorem 12.19.1 has $v_-(h)=0$ for every Hermitian metric and carries a balanced metric of finite upper mass, while the normalized Hopf metric has $v_-=6$ and $v_+=\infty $.

Let $n\geq 2$, let $h$ be a Hermitian form and $\mathrm{d}V$ a smooth positive volume form. Recall the set $\mathscr G_ h$ of normalized Gauduchon currents (section 11.7) and the determinant measure $\Gamma _0$ of (12.7.1), and write

By theorem 12.9.2,

and the exact value of the supremum is $v_-(h)^{-1/n}$, including the value $+\infty $. Uniform estimates for every smaller exponent $1<q<n/(n-1)$ hold without either mass property. In the example of zero lower mass, the endpoint estimate fails already for smooth strictly positive tests. Positivity, the condition $\mathrm{dd}^{\mathrm{c}}T=0$ and the trace normalization alone therefore do not control concentration at the endpoint exponent.

Corollary 12.18.1 proves the endpoint bound when some Hermitian metric $g$ satisfies $\mathrm{dd}^{\mathrm{c}}g=\mathrm{dd}^{\mathrm{c}}(g^2)=0$, which includes all compact Kähler manifolds and all compact complex surfaces. For a fixed smooth strictly positive density, theorem 12.7.3 gives a smooth maximizing current with bounded determinant density. However, proposition 12.18.2 produces maximizing sequences in fixed Kähler classes whose moments diverge at every exponent above the endpoint. Exact maximizers and maximizing sequences therefore require different estimates.

For principal elliptic threefolds over Kähler surfaces, theorem 12.17.7 gives a geometric sufficient condition: the sum of the squares of the two real curvature forms is nonnegative and not identically zero. A bounded cubic identity then gives positive lower mass, and the determinant bound transfers to every Hermitian background. These manifolds admit no pluriclosed metric; proposition 12.17.11, proposition 12.17.14 give explicit families. A primitive pair of curvature forms has the opposite sign, by proposition 12.17.9, so this criterion leaves the general problem open.

The balanced condition and bounded upper mass do not suffice, even together, as the threefold of theorem 12.19.1 shows. On this elliptic fibration, the identity (12.19.19) relates the failure of the lower bound to a vertical Dirichlet integral, and the horizontal curvature forms $\alpha ,\beta $ of section 12.19 have a positive exact product $\alpha \wedge \beta $, which excludes pluriclosed metrics and closed Kähler currents. These calculations give geometric information about this example, but not a classification of all degenerations of the lower mass.

For equations the consequence is exact: in a Hermitian background, corollary 15.27.5 characterizes positive lower mass by solvability of the normalized equation for all finite non-pluripolar measures. When the lower mass vanishes, even a positive density in $L^1$ bounded away from zero can admit no positive scaled subsolution. The remaining questions about equations concern specified measures or additional geometric hypotheses, and, separately, the effect of the exponential weight.