17.1 Relative energy and Perron segmentsBDB8E5100372

BDB8E5100372

Let $(X,\omega )$ be a connected compact Kähler manifold of dimension $n\geq 1$, and let $\theta $ be a smooth closed real $(1,1)$-form on $X$. In this chapter we study the Monge–Ampère energy of $\theta $-plurisubharmonic potentials relative to a model potential $\phi $ (definition 16.21.1); recall that model potentials have positive mass. The energy is a concave functional whose derivative is the non-pluripolar Monge–Ampère measure. We use it for two purposes: to solve Monge–Ampère equations with prescribed non-pluripolar measures by maximizing suitable functionals, and to show that the energy is affine along weak geodesics, which we construct as Perron envelopes of subgeodesics.

Everything rests on integration by parts for non-pluripolar products of potentials with bounded difference. Such potentials may be unbounded, so Stokes' formula for bounded potentials (lemma 13.5.2) does not apply to them directly. In section 17.2 we truncate them and control the truncation errors with the relative capacities of section 16.10. This control requires a strictly positive background, a restriction that a polynomial expansion in an auxiliary Kähler form removes at the end. The resulting formula (theorem 17.2.5) makes no assumption on the masses of the potentials.

The energy $E_\theta ^\phi $ is first defined for potentials with bounded difference from $\phi $, where integration by parts computes its first and second variations, and is then extended by decreasing truncation to all potentials $u$ with $u\leq \phi +O(1)$. It is finite exactly on the relative finite-energy class $\mathcal E^1(X,\theta ,\phi )$ (theorem 17.3.14), continuous along monotone sequences, and upper semicontinuous for $L^1$ convergence, so that sets of normalized potentials with energy bounded below are compact.

In section 17.5 these properties turn maximizers of suitable functionals into solutions of Monge–Ampère equations, first for bounded densities and for the exponential equations $\mu _ u=\mathrm e^{\lambda u}\nu $, and then, letting $\lambda \to 0$, for $\mu _ u=\nu $ with an arbitrary non-pluripolar measure $\nu $ of the same mass as $\phi $ (theorem 17.5.15). The solution is unique up to an additive constant (theorem 17.5.17), and it has bounded difference from $\phi $ when $\nu $ has an $L^ p$ density with $p>1$ (corollary 17.5.26). Two further consequences are that a pseudo-effective class of positive volume is big (theorem 17.5.37) and that mixed non-pluripolar masses dominate the geometric mean of the individual masses (theorem 17.5.41).

With these equations we show in section 17.6 that, in a big class, the singularity ceiling of a potential $u$ (definition 17.6.4) coincides with its singularity envelope $P_\theta [u]$, and that $P_\theta [P_\theta [u]]=P_\theta [u]$ (theorem 17.6.5). This includes the case of zero mass, $\int _ X\langle \theta _ u^ n\rangle =0$, to which the characterization of $P_\theta [u]$ in theorem 16.22.11 does not apply.

In the last three sections we return to a fixed model potential $\phi $ and join two potentials of full mass relative to $\phi $ by a Perron segment, the upper envelope of all subgeodesics lying below them at the endpoints. Rooftop envelopes show that the segment attains the prescribed endpoints. Its complexification is locally maximal on the product of $X$ with a strip, so its top-degree non-pluripolar measure vanishes, and integration by parts on $X\times \mathbb P^1$ then shows that the energy is affine along the segment (theorem 17.8.3). We conclude with estimates for the extremal slopes of a segment and for long segments; the latter produce profiles that depend only on the singularity type of the endpoint (proposition 17.9.2).

Historical remarks and references for this chapter are collected in subsection 25.1.17, subsection 25.1.18, subsection 25.1.19.