13.1 Non-pluripolar productsE478B9E17C75

In section 7.1 we multiplied the currents $\mathrm{dd}^{\mathrm{c}}u_1,\ldots ,\mathrm{dd}^{\mathrm{c}}u_ p$ of locally bounded plurisubharmonic functions by the Bedford–Taylor recursion $\mathrm{dd}^{\mathrm{c}}u_1\wedge T=\mathrm{dd}^{\mathrm{c}}(u_1T)$. The potentials that arise in global geometry are often unbounded: by corollary 10.2.19, a closed positive $(1,1)$-current on a compact Kähler manifold is $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ with $\theta $ a smooth form and $\varphi $ quasi-plurisubharmonic, and $\varphi =-\infty $ wherever the current has a positive Lelong number, for instance along a hypersurface carrying a current of integration. Indeed, at a point where $\varphi $ is finite, the spherical means of a local potential are bounded below by its value at that point, so the Lelong number vanishes there by (6.7.8). For such potentials the recursion breaks down, since it multiplies the measure coefficients of a current by an unbounded function. In this chapter we construct a product for arbitrary plurisubharmonic potentials which keeps positivity, closedness, multilinearity and locality, at the price of ignoring everything that happens on pluripolar sets.

The construction truncates the potentials and discards their poles. We replace each $u_ j$ by $\max \{ u_ j,-k\} $, take the Bedford–Taylor product, and keep only its restriction to the set where all $u_ j>-k$, on which the truncated potentials coincide with the original ones. Since bounded products are local in the plurifine topology (theorem 5.47.22), the restrictions obtained for different $k$ are compatible and increase with $k$. Their limit is the non-pluripolar product $\langle \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle $, which by construction gives no mass to pluripolar sets. Two phenomena without counterpart in the bounded theory appear. The limit may have infinite mass on compact sets (example 13.2.9). And the mass carried by the poles is lost: in degree one, $\langle \mathrm{dd}^{\mathrm{c}}u\rangle $ is the restriction of $\mathrm{dd}^{\mathrm{c}}u$ to $\{ u>-\infty \} $ (proposition 13.4.7), so the current of integration at a point of $\mathbb P^1$ has zero non-pluripolar product, and even a potential with zero Lelong numbers and trivial multiplier ideals can have zero non-pluripolar mass (example 13.4.11). When the local masses are finite, the product is a closed positive current (theorem 13.3.1).

On a compact Kähler manifold $X$ the local masses are always finite: every tuple of closed positive $(1,1)$-currents has a non-pluripolar product, and its total mass is bounded by a cohomological quantity (theorem 13.5.4). The total mass is, however, no longer determined by the cohomology classes of the factors; it depends on the singularities of the potentials. The central result of the second half of the chapter is that it is monotone in the singularities: if $\theta $ is a smooth closed real $(1,1)$-form, $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$, and $\psi $ is more singular than $\varphi $ in the sense of definition 10.3.7, then the mass of $\varphi $ is at least that of $\psi $ (theorem 13.8.7). When all potentials are locally bounded outside the closed pole set of a fixed quasi-plurisubharmonic function, this follows from Stokes' formula; arbitrary singularities are reduced to this case by an auxiliary potential on $X\times \mathbb P^ N$, whose total mass is an average of the masses along a segment of potentials. Along the way we prove that non-pluripolar products are lower semicontinuous under convergence in capacity, and continuous when no mass is lost in the limit (theorem 13.7.5).

Monotonicity has several consequences that are used in the following constructions. Mixed masses do not change under bounded perturbations of the potentials, and in a big class the potentials with minimal singularities have the largest masses; these maximal masses define the positive intersection numbers and the volume of pseudo-effective classes. The full-mass classes and the comparison and domination principles at the end of the chapter are basic tools for the envelopes of section 16.10, and the fiber formula of section 13.6 is used for relative energy in section 17.1.

Historical remarks and references for this chapter are collected in subsection 25.1.15.

Let $X$ be a complex manifold and let $\varphi _1,\ldots ,\varphi _ p\in \mathrm{PSH}(X)$. When the functions $\varphi _ i$ are smooth, the differential form

is defined by the usual calculus. A central problem in pluripotential theory is to extend this product to singular plurisubharmonic functions in a reasonable way.

We use the non-pluripolar product. This construction, originating in the work of Bedford–Taylor and developed in the global setting by Guedj–Zeriahi and Boucksom–Eyssidieux–Guedj–Zeriahi, has two features that are essential for us: it is defined for arbitrary quasi-plurisubharmonic singularities, and it satisfies the monotonicity theorem for non-pluripolar masses. These two properties make it possible to compare singularity types through Monge–Ampère mass.

The chapter is organized as follows. We first recall the Bedford–Taylor product for locally bounded plurisubharmonic functions. We then introduce the non-pluripolar product and record its basic functorial and locality properties. After a short discussion of quasi-continuous functions and capacity, we prove the convergence and concentration results used later in the theory of envelopes. The key tools extracted from this chapter are the monotonicity theorem, semicontinuity theorem, the concentration theorem for envelopes and the domination principle.