25.1.14 Global potentials and positivity lociF9B38BEB0BCF

For elliptic operators on compact manifolds and the Laplacian on forms compare [Chapter VI, Section 2, pp. 289–290, and Sections 3.1–3.2, pp. 291–294, DemBook]. The results on Sobolev spaces, compactness and regularity used here are proved directly in lemma 10.2.1, lemma 10.2.5, by Fourier truncation, difference quotients and duality. The proof of the $\mathrm{dd}^{\mathrm{c}}$-lemma, theorem 10.2.15, uses the scalar trace equation and the identity for the Hodge star of primitive $(1,1)$-forms; the factor $-1/(4\pi )$ in the trace identity (10.2.13) comes from our normalization of $\mathrm{dd}^{\mathrm{c}}$ and from the sign convention that makes the Riemannian Laplacian nonnegative.

The method of primitive representatives for the mixed Hodge–Riemann relations is developed in [Propositions 2.3–2.4 and proof of Theorem A, pp. 5–8, DN2005mixedv2]. For real $(1,1)$-classes, lemma 10.2.9, theorem 10.4.3 make a representative primitive at every point by solving a scalar equation in divergence form. The operator $\mathrm{dd^ c}=\mathrm{i}\partial \bar\partial /\pi $ of that source is twice ours; doubling its correcting function gives the same representative. Corollary 10.4.5 derives the lower bound of mixed intersection numbers of nef classes by the geometric mean of their volumes from the concavity of the logarithms of mixed intersection numbers, first for Kähler classes and then, by perturbation, for nef classes.

For positive cones, potentials with minimal singularities and the divisorial Zariski decomposition compare [Sections 2.1–2.3, pp. 47–49, Sections 2.7–2.8, pp. 51–53, and Section 3.1, pp. 54–55, Bou04]. These locators refer to the published article, whose title and numbering differ from those of the arXiv version of 2002. Siu's decomposition, theorem 10.4.8, including the uniqueness of the decomposition, is proved by subtracting divisors with the help of the uniform logarithmic growth of local potentials along a divisor (proposition 6.9.8). The proof of proposition 10.5.2 is the argument of Proposition 3.6 of the published article, by convex combination with a Kähler current. As in that article, minimal multiplicities are defined by means of currents $T\geq -\varepsilon \omega $; for big classes they coincide with the Lelong numbers of a potential with minimal singularities, and in general they are bounded above by these Lelong numbers.

That the non-Kähler locus of a big class is the pole set of a single Kähler current with analytic singularities is [Definition 3.16 and Theorem 3.17, pp. 59–60, Bou04]. Our proof of theorem 10.5.4 follows Boucksom's argument, based on the stationarity of increasing sequences of coherent ideals on a compact manifold and on taking maxima, with the approximation results of section 6.13. The theorem also contains the identity (10.5.5), with conventions that include classes that are not big or not pseudo-effective. The last part of Boucksom's Theorem 3.17, a criterion in terms of restrictions to arbitrary analytic subsets, is not treated here.

The conventions for psh metrics, written as quadratic forms, and for the Fubini–Study metrics defined by sections are those of [Chapter on plurisubharmonic functions, Definitions 1.8.3, 1.8.6 and 1.8.7, XiaSampleBook]. In proposition 10.6.9 the convex combinations have rational coefficients, and closure under these operations is proved by clearing denominators of the degrees and multiplying sections. The local weight of a metric in [Section 2.4, pp. 49–50, Bou04] is $\mathrm e^{-2\varphi }$; doubling the potential gives our weight $\mathrm e^{-\varphi }$ for quadratic metrics, and the curvature current is the same.

For the $L^2$ estimate for line bundles on Kähler manifolds, lemma 10.7.6, compare [Chapter VI, Section 6.1, pp. 304–306, and Chapter VII, Theorem 1.1, Theorem 1.2 and Corollary 1.3, pp. 329–330, DemBook]. We prove the commutator identity, the density of smooth forms for the graph norm and the solvability in Hilbert space, and we show that for forms of top degree the curvature term in the estimate does not depend on the auxiliary complete metric.

Complete metrics and weights with logarithmic poles are also used in [Theorem 5.1 and its proof, pp. 18–22, Hisamoto2012v2] (second arXiv version), where Theorem 5.1 assumes a smooth metric and Theorem 1.4 assumes projectivity. Theorem 10.7.9 concerns extension from a point of a compact Kähler manifold. It is proved by applying the gentle approximations of section 6.13 to the integrable error introduced by a fixed cutoff function. The weighted spaces of sections in lemma 10.7.11 provide the finite-dimensional Hilbert spaces and the curvature of the background used in [Proposition 1.8.2, XiaSampleBook].

For the twisted $L^2$ estimate and the treatment of the error from the initial gluing compare [Section 2, pp. 5–9, and Appendix, pp. 14–15, Cao2017v2]; page numbers refer to the second arXiv version, posted on 3 October 2017 and dated 4 October in the manuscript. Theorem 1.3 there treats a submanifold defined by a global section of a vector bundle, under assumptions on curvature and approximation. Theorem 10.8.18 applies to an arbitrary closed connected submanifold $Y$. The pole along $Y$ is given by the function $R_ Y$ of lemma 10.8.8, a sum of local squared normal distances weighted by squares of cutoff functions.

The estimate (10.8.19) is first proved for weights with gentle analytic singularities. The local gain from the logarithmic factor is lemma 10.8.6, the error of the initial gluing is controlled by strong openness and Hölder's inequality in lemma 10.8.11, and lemma 10.8.15 computes the limit over thin annular neighborhoods of $Y$ for gentle weights. Approximation by decreasing gentle weights then gives the estimate for the original metric, with the same constant. Corollary 10.8.22 recovers [Theorem 1.8.1, XiaSampleBook]; the threshold for the tensor power and the conversion between the volume forms, which involves factorials, are stated explicitly.

The extension of potentials from a submanifold, theorem 10.8.26, follows [arXiv version, Theorem 6.1 and its proof, Lemma 6.2, pp. 19–22, DRWNXZ]. That source uses the same operator $\mathrm{dd}^{\mathrm{c}}=\mathrm i\partial \bar\partial /(2\pi )$ (p. 5) and the same notion of analytic singularities, with rational coefficients and bounded remainder (Definition 2.2, p. 6). The lemma on poles along the submanifold quoted there is replaced by lemma 10.8.8. Strict positivity near the submanifold follows from an estimate for block matrices, and the local extensions are formed as logarithms of sums of exponentials (lemma 10.8.25); the exponent of the normal pole is taken small enough to absorb the error coming from the Hölder continuity of the projection. Since the comparison of the local extensions on overlaps holds also at common poles, the glued maximum involves a fixed set of indices near each point of the submanifold.

Josefson's theorem on compact complex manifolds, theorem 10.9.19, is due to Vu [arXiv version, Theorem 1.1 and its proof, pp. 8–9, Vu19]. The operator $\mathrm i\partial \bar\partial /\pi $ used there is twice ours. We follow Vu's argument, with a weighted sum of global poles and two extremal limits, but replace his comparison of capacities in exponential form by the estimate (10.9.17). Positivity of the mass is needed only for the one bounded limit function, not as a uniform lower bound over all bounded potentials.

For products with a background form that is not closed and the modified comparison principle compare [Section 1, pp. 3–4, and Theorem 2.3 and its proof, pp. 5–6, KN2015v2], the second arXiv version of the article by Kołodziej and Nguyen, whose operator is again twice ours. The one-factor inequality on which the modified comparison principle rests is lemma 10.9.4; we prove it for smooth approximations with a convex cutoff and a partition of unity, and pass to the limit by the continuity of Bedford–Taylor products with polynomial weights (corollary 7.3.8) and uniform approximation of the weights by polynomials. The modified comparison principle is proved here only with the fixed margin $\omega _ v\geq \frac12\omega $, proposition 10.9.10.

In proposition 10.9.16 the background form is frozen on small balls, so that the balayage of ordinary plurisubharmonic functions applies. In this way, the proof that the extremal function puts no Monge–Ampère mass outside the compact set avoids solving the homogeneous Dirichlet problem or smooth Monge–Ampère equations on Hermitian manifolds.