16.10 Envelopes and model potentialsCC5E34310EEC

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Throughout this chapter $X$ is a connected compact Kähler manifold of dimension $n\geq 1$ with a fixed Kähler form $\omega $, and $\theta $ is a smooth closed real $(1,1)$-form on $X$, as in the two preceding chapters. Many potentials are defined as envelopes: given an obstacle $f$, one takes the largest $\theta $-plurisubharmonic function that lies below $f$ outside a pluripolar set. For instance, the extremal potential $V_\theta $ of definition 10.3.7 is the envelope of the obstacle $0$. Two features of the global setting require care. The obstacle is often only a Borel function, so we must keep track of which inequalities hold everywhere and which only quasi-everywhere. And there may be no $\theta $-plurisubharmonic function below the obstacle at all: two $\theta $-plurisubharmonic functions need not have a common $\theta $-plurisubharmonic minorant (example 16.11.7).

The basic property of an envelope is that its non-pluripolar Monge–Ampère measure is carried by the contact set, where the envelope touches the obstacle; for quasi-continuous obstacles in a big class this is theorem 16.13.4. Away from the contact set, the envelope of a lower semicontinuous obstacle can be replaced locally by a maximal plurisubharmonic function, so its measure vanishes there, and quasi-continuity lets us approximate a general obstacle by lower semicontinuous ones. Regularity of the obstacles also passes to the envelope. If finitely many obstacles have bounded Laplacian with respect to the Riemannian metric of $\omega $, then so does the envelope of their minimum. For a Kähler background we prove this with an exponentially penalized Monge–Ampère equation (theorem 16.15.24), which also shows that envelopes of finite minima of $C^{1,1}$ obstacles are $C^{1,1}$ (theorem 16.16.7); in a big class the bound holds locally on the ample locus, the complement of the non-Kähler locus (theorem 16.19.1). On the contact set of such an obstacle with an arbitrary potential lying below it, the non-pluripolar measure of the potential is then computed from the second derivatives of the obstacle (theorem 16.20.2).

The second half of the chapter studies envelopes that prescribe singularities rather than values. The singularity envelope $P_\theta [u]$ of a $\theta $-plurisubharmonic function $u$ is the upper envelope of all nonpositive $\theta $-plurisubharmonic functions $v$ with $v\leq u+C$ for some constant $C$, which may depend on $v$. It has the same non-pluripolar masses as $u$ but may be less singular, and its fixed points of positive mass are called model potentials. The main tool is extrapolation (theorem 16.22.4): if $u$ is more singular than $v$ and has almost the same mass, then for some $a>1$ the function $v+a(u-v)$, which continues the segment from $v$ to $u$ beyond $u$, still has a $\theta $-plurisubharmonic minorant of positive mass. From this we derive the domination principle for potentials of full mass relative to a reference potential (theorem 16.22.10) and the characterization of $P_\theta [u]$, for $u$ of positive mass, as the largest nonpositive potential that is less singular than $u$ and has the same mass (theorem 16.22.11). Rooftop envelopes satisfy a mass inequality (theorem 16.23.5), and upper envelopes preserve the comparison of singularities through multiplier ideals (section 16.24).

Finally we introduce the Monge–Ampère capacity $\operatorname {Cap}_{\theta ,\phi }$ relative to a reference potential $\phi $, prove that it is comparable with the capacity of a Kähler form when $\phi $ has positive mass (theorem 16.27.17), and use it to measure the deep sublevel sets $\{ u<\phi -s\} $ of potentials of full mass relative to $\phi $. These estimates are what the next chapter needs in order to integrate by parts with singular potentials. Singularity envelopes of potentials of zero mass, and contact concentration in arbitrary pseudo-effective classes, depend on the Monge–Ampère equations solved there (section 17.6, subsection 17.5.27). Historical remarks and references for this chapter are collected in subsection 25.1.16.