13.16 Volumes under bounded massEEFEDC3AE6FE

EEFEDC3AE6FE

In a Hermitian background, Monge–Ampère masses need not be bounded above or away from zero. On the standard Hopf threefold the masses of smooth admissible potentials of a Gauduchon form are unbounded (theorem 23.1.1), while on the threefold of theorem 12.19.1 every Hermitian form $h$ admits smooth admissible potentials whose masses tend to zero. In this part such bounds are imposed explicitly. The upper bound is the bounded mass property of definition 13.14.14: $v_+(\omega )<+\infty $ for one, and hence for every, Hermitian form $\omega $ (proposition 13.14.15), where $v_+(\omega )$ and $v_-(\omega )$ denote the supremum and the infimum of the masses of bounded $\omega $-plurisubharmonic functions. A lower bound takes one of two forms. In section 15.37, which does not assume bounded mass, it is positive lower mass $v_-(\theta )>0$ of a semipositive background $\theta $. In this chapter and in section 10.31, which assume bounded mass, it is usually a positive lower singularity volume of the particular potentials involved, rather than a bound for all bounded potentials.

In a closed background, the non-pluripolar mass of a potential depends only on its ordinary singularity type. In a nonclosed background, potentials of the same type can have different masses, so a singularity type carries two numbers, the infimum and the supremum of the masses of its members. These lower and upper singularity volumes keep the order information needed for comparison. Bounded mass makes the upper volume finite, and a positive lower volume prevents the auxiliary envelopes used below from collapsing. Together these bounds turn a small upper-volume deficit into a large admissible stretch, and this relates equality of volumes to the relative full-mass class and to ordinary envelopes.

The upper volume of the maximum of two potentials measures how far apart their ordinary envelopes are. On models with a uniform positive lower-volume bound, the stretching estimate controls three-point comparisons, monotone limits and convergence in capacity. The comparison itself need not satisfy the triangle inequality, even on the projective plane, and at zero volume it can identify distinct models (example 13.22.3). Finite chains nevertheless give a complete metric once the reciprocal of the lower volume is included to prevent escape to zero volume. Analytic regularization identifies the multiplier-ideal models among the resulting limits, and strict barriers allow these comparisons to persist under small changes of background.

On a curve, the polar part of the curvature current determines the ordinary model, and the volume comparison is the total variation of polar measures. This description extends to the zero-volume boundary. Taking the atomic part gives the nearest ideal model, while diffuse polar curvature measures the failure of analytic approximation. Area rescaling and smooth gauges then describe changes of background, including the collapse of all normalized potentials when the backgrounds converge uniformly to a form of zero area.

Throughout the chapter, $X$ is a connected compact complex manifold of dimension $n\geq 1$ with the bounded mass property, and $\theta $ is a big smooth real $(1,1)$-form on $X$, not necessarily closed; bigness means that $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \varepsilon \omega $ for some $\rho \in \mathrm{PSH}(X,\theta )$, some $\varepsilon >0$ and a Hermitian form $\omega $ (section 5.40). From section 13.28 on, we work instead on a compact Riemann surface with the backgrounds specified there.

We recall the notation used from earlier chapters. For quasi-plurisubharmonic functions, $u\preceq v$ means $u\leq v+C$ for a constant $C$, and $u\sim v$ means $u\preceq v\preceq u$; the classes of $\sim $ are the ordinary singularity types (definition 16.7.1). Similarly, $u\preceq _{\mathcal I}v$ means $\mathcal I(tu)\subseteq \mathcal I(tv)$ for every $t>0$, and $u\sim _{\mathcal I}v$ means that these multiplier ideals are equal (section 5.33). For $u\in \mathrm{PSH}(X,\theta )$, $\mathrm{MA}_\theta (u)$ is the non-pluripolar Monge–Ampère measure of theorem 13.14.3, which is finite under bounded mass by proposition 13.14.16. The lower and upper singularity volumes $\underline{\operatorname {vol}}_\theta (u)$ and $\overline{\operatorname {vol}}_\theta (u)$ of definition 16.28.58 are the infimum and the supremum of $\int _ X\mathrm{MA}_\theta (v)$ over all $v\in \mathrm{PSH}(X,\theta )$ with $v\sim u$. A statement about $\operatorname {vol}_\theta $, without a bar, is asserted for both volumes.

Given an obstacle $f$, we denote by $P_\theta (f)$ the regularized upper envelope of the $\theta $-plurisubharmonic functions lying below $f$, and we write $P_\theta (u,v)=P_\theta (\min (u,v))$ for the rooftop envelope and $V_\theta =P_\theta (0)$ (section 5.40). The ordinary envelope $P_\theta [u]$ is the regularized supremum of all $h\in \mathrm{PSH}(X,\theta )$ with $h\preceq u$ and $h\leq 0$ (definition 5.40.4), and $P_\theta [u]_{\mathcal I}$ is the multiplier-ideal envelope of definition 18.4.1, defined in the same way with $h\sim _{\mathcal I}u$ in place of $h\preceq u$. We call $u$ an ordinary model if $P_\theta [u]=u$, and an ideal model if $P_\theta [u]_{\mathcal I}=u$. Both kinds have supremum zero, and every ideal model is an ordinary model by corollary 18.4.4. Finally, $\nu (u,x)$ denotes the Lelong number at $x$, and a potential is called analytic if it has analytic singularities in the rational-coefficient convention of section 5.34.