25.1.15 Non-pluripolar products0948B3754F28

The definition of non-pluripolar products by truncation at finite levels, locally and on compact Kähler manifolds, is developed in [Section 1.2, Proposition 1.4, Lemma 1.5 and Proposition 1.6, pp. 203–207, BEGZ10]. Multilinearity is proved in lemma 13.2.2, proposition 13.2.6 by truncating all factors at a common level. In lemma 13.2.2 the convergence of the restricted products is stated for Borel sets $E_ k$ such that $\mathbf1_{E_ k}\to 1$ almost everywhere for the trace measure of the product; this holds if the $E_ k$ increase and their union has pluripolar complement, whereas the condition on the union alone does not suffice. In example 13.2.9 the computation of the Hessian shows that the function is plurisubharmonic with the sign opposite to the one displayed in [Example 1.3, pp. 204–205, BEGZ10].

That non-pluripolar products are closed, theorem 13.3.1, is [Theorem 1.8 and Lemma 1.9, pp. 207–210, BEGZ10]. The gradient $\mathrm dv\wedge \Theta $ of the product of a bounded plurisubharmonic function $v$ and a closed positive current $\Theta $, and the corresponding chain rule, are established in lemma 7.14.3 by decreasing approximation; the proof of theorem 13.3.1 then estimates the error made by the cutoff functions. The gradient energy $\mathrm dv\wedge \mathrm d^{\mathrm c}v\wedge \Theta $, with $\mathrm d^{\mathrm c}$ as in (5.2.10), is used only as a positive measure, and no closedness property of it is needed. On a compact Kähler manifold, Stokes' formula for bounded potentials (proposition 13.5.1) and the mass bound with shifted backgrounds (theorem 13.5.4) are proved before global truncations are used.

The lower bound for mixed products in terms of a common measure is [Proposition 1.11 and its proof, p. 211, BEGZ10]; it applies Dinew's inequality for bounded functions before passing to the restrictions at finite levels. Proposition 13.5.8 is the statement for $n$ currents, deduced from theorem 7.12.1; the case of two currents, corollary 13.5.10, follows by repeating factors. Since non-pluripolar products put no mass on pluripolar sets, the domination hypotheses force the densities to vanish almost everywhere for the common measure on the union of the local pole sets; this allows us to remove the restrictions to finite levels for an arbitrary measure.

The compatibility of non-pluripolar products with Bedford–Taylor products of bounded factors, proposition 13.4.3, is [Proposition 2.2.1(iii), XiaSampleBook]. Our proof uses the fact that Bedford–Taylor products with a current that does not charge pluripolar sets again do not charge them, lemma 13.4.2, which follows from the Chern–Levine–Nirenberg estimate relative to a current, lemma 13.4.1; the products are then identified with the help of bounded plurisubharmonic cutoff functions.

For the identity in degree one, proposition 13.4.7, compare [Proposition 3.3, pp. 3–4, ElKadiri2023v3] (third arXiv version). That proposition concerns the domain of definition of the Monge–Ampère operator and refers to an earlier theorem for the equality. We prove the identity first in one variable, by approximating the Riesz measure from below by measures with continuous potentials, and then in several variables by integration over the fibers of coordinate projections. Strong positivity (proposition 13.2.5) passes from products of smooth forms (proposition 5.3.2) to Bedford–Taylor products and then, by lemma 13.2.1, to their restrictions to Borel sets.

The fiber integration formula and its application to mixed products on compact manifolds follow [Chapter on non-pluripolar products, Lemma 2.1.1 and Corollary 2.2.1, XiaSampleBook]. The measurability of the fiber measures that is needed to integrate indicator functions of level sets is proved in lemma 13.4.4, the formula for Bedford–Taylor products of bounded functions is lemma 13.4.5, and corollary 13.6.1 passes to the limit along the increasing restricted products.

The comparison of masses in lemma 13.8.1, for potentials that are locally bounded outside the closed pole set of a quasi-plurisubharmonic function, follows the argument with compact supports in [Theorem 1.16 and its proof, pp. 217–218, BEGZ10]. The auxiliary projective construction is due to Witt Nyström [arXiv version, Sections 3–4, pp. 4–9, WN19]. The projection onto the simplex and the integration over angles are carried out in lemma 13.8.2. With the exterior powers used in this book, which are not divided by factorials, the formula of that lemma contains the factor $\binom {n+N}{n}$, and the Hessian contains $2\sum _ i\mathrm dx_ i\wedge \mathrm d^{\mathrm c}x_ i$. Both factors follow by direct differentiation and expansion, and they are missing from the calculation displayed in the preprint; the common positive binomial factor cancels in the proof of monotonicity. Theorem 13.8.7 handles classes that need not be big by adding $\varepsilon \omega $, so it does not depend on the criterion for bigness by positive mass, theorem 17.5.37.

The invariance of mixed masses under bounded perturbations, the lower bound for limits in capacity and the monotonicity of mixed masses are proved in [Proposition 2.1, Theorem 2.3 and Theorem 2.4, pp. 2054–2057, DDNL18mono]. Our theorem 13.7.5 uses the local continuity of weighted products under convergence in capacity and truncations at fixed levels, and corollary 13.8.8 obtains invariance from theorem 13.8.7 by polarization and then monotonicity by decreasing maxima. Positive mixed intersection numbers and full mass are introduced in [Definition 1.17, Proposition 1.20 and Definition 1.21, pp. 218–222, BEGZ10]; proposition 13.9.2 gives the bound for these masses that is used later. The comparison and domination principles, proposition 13.9.5, corollary 13.9.6, follow [arXiv version, Corollaries 4.1–4.2, pp. 8–9, WN19]; they keep the comparison hypotheses on singularities, and pointwise domination requires a potential with minimal singularities, as stated there.

The version with varying weights, corollary 13.7.9, is [arXiv version, Theorem 2.6 and its proof, pp. 10–12, DDNLsurv], the reference given in [Theorem 2.4.2, XiaSampleBook]. We keep the argument with a cutoff at a finite level and complementary weights, but work locally in every mixed degree before gluing on the compact manifold; the limit is taken with the theorem for bounded weights proved earlier. The survey uses $\mathrm d^{\mathrm c}=\mathrm i(\bar\partial -\partial )/(2\pi )$, so its operator $\mathrm d\mathrm d^{\mathrm c}$ is twice ours. At fixed potentials and with doubled backgrounds, each product of $p$ factors is $2^ p$ times ours, and this common factor cancels in the inequalities for lower limits and for masses. Proposition 13.7.8 establishes the statement of [Proposition 2.3.1, XiaSampleBook] for arbitrary monotone nets; the equivalence of the two notions of convergence in capacity is asserted for plurisubharmonic functions, not for arbitrary Borel functions with values in $[-\infty ,\infty ]$.