25.3 Plurisubharmonic functions and regularization9D79D5D188C6

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Compactness, integrability and regularization are local properties, and they survive the passage to a nonclosed background because a quasi-plurisubharmonic function becomes plurisubharmonic after a smooth local correction. The $L^1$ compactness of normalized potentials in theorem 5.25.2, also stated by Dinew [Proposition 6.2, Dinew16], follows by propagating local compactness over a connected coordinate cover. Local exponential integrability, as recalled by Zeriahi [Section 1, Zeriahi01], becomes uniform over normalized potentials in lemma 5.26.2, through a contradiction argument with a weighted sum of potentials.

The comparison of convolutions in different coordinate charts in lemma 5.30.1, corollary 5.30.3 is the bounded part of the argument of Błocki and Kołodziej [§2, Lemmas 4–5, BK07]. Their full regularization theorem [Theorem 1, BK07] allows unbounded potentials and a positive continuous background. The proof of theorem 5.32.1 treats only the bounded case, which is all that the construction of products requires; it includes the extraction of a decreasing diagonal sequence and the rescaling that keeps the regularized forms strictly positive.

Demailly's regularization by the flow of the Chern connection [Theorem 4.1, Dem94] applies to $\mathrm{dd}^{\mathrm{c}}u$ for every $u\in \mathrm{PSH}(X,\theta )$, where $\theta $ is a smooth real $(1,1)$-form: the theorem concerns a closed current with an arbitrary smooth lower bound, here $-\theta $, and this lower bound need not be closed. The monotonicity and logarithmic convexity of the regularization enter the estimate (5.42.11), whose linear error rate follows from the curvature inequality [Lemma 4.4, Dem94] and the expansion in [Proposition 3.8, equations (3.11), (4.3), and (4.5), Dem94]. In the present situation the curvature inequality needs neither a small $\epsilon $ term nor an $M_\epsilon $ correction, and local trace bounds make the constants uniform in the center and the radius.

The same expansion underlies the truncated Kiselman transform of lemma 5.42.1; see [Theorem 4.1 and Remark 4.7, Dem94] and Kołodziej–Nguyen [Lemma 4.1, KN19]. In lemma 5.42.4 the full base–fiber error is kept as a modulus tending to zero, and, as in [Remark 4.7, Dem94], a term linear in the radius absorbs the mixed and vertical errors. This formulation accommodates the unbounded potentials with finite $L^1$ norm that occur in theorem 15.8.1, where localizing the averaging radius makes the minimum-principle argument available away from the singular barrier.

For a continuous potential, the logarithmic derivative of the Chern averages tends to zero. Together with Dini's theorem and uniform continuity, this gives the uniform smooth approximation with vanishing curvature loss in lemma 5.42.5. For a Hölder reference, lemma 5.42.7 retains a common Hölder norm and rescales to genuine potentials. Dinh, Ma and Nguyên [Theorem 2.1, pp. 551–555, DMNFekete17] prove quantitative smooth regularization in a Kähler background using its local potentials. The proof here uses Chern averages and the continuous-potential curvature-loss estimate on the Hermitian background. Their quantitative Kähler estimate is not used.

Demailly's regularization in the form of theorem 5.36.5, whose approximants become divisorial with a smooth remainder after finitely many blowups, is the statement of Demailly and Păun [Theorem 3.2, DP04]. It allows an arbitrary smooth lower bound on a Hermitian manifold, and the smooth remainder after resolution is essential for its applications here. This sequence is in general different from the multiplier-ideal sequence of theorem 5.36.1, and we do not assume that a single sequence has the additional properties of both.

The analyticity of the sets where the Lelong number is at least $c>0$, recalled in 5.34.2 and used in lemma 12.19.4, theorem 11.8.1, is Siu's theorem [Main Theorem, p. 53, Siu74]; a smooth local correction makes it applicable to quasi-plurisubharmonic functions. The dimension bound for the Lelong level set and the existence of global irreducible components are stated in Demailly's book [Chapter III, Corollary 8.5, p. 174; Chapter II, Theorem 5.3, pp. 102–103, DemCADG]. The non-strict inequality is essential. In example 5.34.4, a summable series of line currents on the projective plane has the property that the set where the Lelong number exceeds one is a sequence of points accumulating at a point outside that set.

The closedness of polar curvature currents, proposition 5.46.7, is proved by the complete pluripolar restriction argument of Boucksom, Eyssidieux, Guedj and Zeriahi [author preprint v2, Theorem 1.8, Lemma 1.9 and Remark 1.10, BEGZ10]. The chain rule for bounded functions allows cutoffs that converge only pointwise, so the complete pluripolar set need not be closed, and a smooth local correction extends the argument to quasi-plurisubharmonic potentials with an arbitrary smooth background. Combined with the recovery of closed currents carried by a Lebesgue-null set from their traces, this removes the hypothesis that the pole set be closed from corollary 18.26.11, proposition 18.26.12. The simultaneous realization of the limiting currents is a separate question.

The Hausdorff estimate of corollary 5.34.3 follows from the Lelong mass formulas in Demailly's book [Chapter III, formulas (5.5)–(5.8), pp. 158–159, DemCADG] and holds in every positive bidimension. When a closed polar current vanishes outside an exceptional locus, the support theorem for currents carried by analytic sets [Chapter III, Corollaries 2.11 and 2.14, pp. 141–143, DemCADG] identifies the remaining divisorial part. Full local Monge–Ampère products at an unbounded locus of Hausdorff dimension zero rely instead on [Chapter III, Theorem 4.5 and Proposition 4.9, pp. 152–156, DemCADG], as stated in theorem 5.27.18. These products keep their top-degree mass on the pole set, whereas every lower-degree coefficient measure vanishes there. The local maximum argument of lemma 5.27.19 compares the residual measures and shows that they are invariant under bounded perturbations. Their global consequences for masses and models are discussed in section 25.4, section 25.11.

The local reference-integral estimate lemma 5.29.7 is related to the Hölder estimates for measure functionals of Dinh and Nguyên [arXiv v1, Lemma 3.5 and Proposition 3.6, pp. 9–10, DNHolder14]. Here the tests are bounded plurisubharmonic functions on nested balls. Keeping the cutoff-gradient terms gives a bound uniform in the Hölder norm of the reference; a finite cover yields corollary 5.29.9 for bounded tests with a common curvature lower bound.

The local moderation theorem of Dinh, Nguyên and Sibony [Theorem 1.1 and Corollary 1.2, pp. 466–467; proof pp. 468–471, DNSModerate10] gives exponential integrability with respect to locally moderate currents. The top-degree estimate lemma 5.29.10 follows here by applying the reference-integral estimate to consecutive truncations and absorbing their polynomial growth into the exponential tail. Guedj and Zeriahi's extremal-capacity comparison [Theorem 5.2, pp. 622–623, and Proposition 7.1, p. 632, GZIntrinsic05] motivates lemma 5.29.12. Its proof uses conservation of mass only for the closed Fubini–Study form on an auxiliary projective space. Local capacity comparison gives the Hermitian estimate proposition 5.29.13 with constants uniform for a common Hölder norm.

The convergence against a fixed non-pluripolar measure in lemma 5.44.2 rests on Cegrell's classical fixed-measure argument, which Kołodziej and Nguyen also state in a form allowing extraction of subsequences [Lemma 2.1 and Corollary 2.2, KN22]. Applying that statement to every subsequence gives convergence of the whole bounded sequence.

Let $f$ be a holomorphic function and $\chi $ a convex nondecreasing function. The sharp Sobolev criteria for the potentials $\chi (\log |f|)$ in 5.27.20 are due to Chen, Shao and Wang [v1, Theorems 1.1–1.2, pp. 2–3 and 19–22, CSW26]; their estimate in terms of the smallest multiplicity and their energy bound on small sublevel sets are [v1, Lemma 4.1 and Proposition 3.8, pp. 16–20, CSW26]. The proofs given here use the regularizations $\chi (\frac12\log (|f|^2+\varepsilon ^2))$, $\varepsilon >0$, to identify the weak gradient and to keep track of the residual divisor term in the distributional Laplacian, a term that the classical Laplacian omits. The power examples and their endpoint obstruction in example 5.27.24 are the calculation in normal coordinates and the table of [v1, Lemma 2.1 and Section 6.1, pp. 6–7 and 24–25, CSW26]. These local criteria for derivatives should not be confused with the theorem on products at a small unbounded locus discussed above.

The concentration example of proposition 5.27.31 is the point-mass case of Qin's [v1, Corollary 3.6, p. 14, QinLimits26]. Here the mass is computed directly from the transverse identity (5.27.30), which gives a smooth variant of the logarithmic truncation in [Lemma 2.4, pp. 9–10, QinLimits26]. The conclusion of lemma 5.27.28 for sequences with a common local bound agrees with [Proposition 7.2, p. 26, QinLimits26]; in the book it follows from the local range estimate proved earlier.

Theorem 5.27.36, which adds an arbitrary positive Radon measure to the Monge–Ampère measure of a fixed bounded potential in the limit, is Qin's [v1, Theorem B, p. 3, QinLimits26]. The addition along finitely many test functions and the passage to a determining family correspond to [Proposition 3.2 and Theorem 3.4, pp. 11–14, QinLimits26]. In the proof given here, the smooth ridge gives the limits of the lower-degree currents directly, finite iteration over atoms controls every mixed term, and the one-sided logarithmic bound also gives convergence in local capacity. The smoothness of the approximants is asserted for a smooth background, whereas Qin also treats continuous backgrounds.

The classification of vague cluster sets in theorem 5.27.39, according to which every vaguely closed set of positive measures is the full cluster set of a sequence of potentials tending to zero, is the case of the zero potential in [v1, Theorem A, pp. 2–3, QinLimits26]. The metric for the vague topology and the concatenation of dense sequences are [Lemma 3.3, p. 12 and Lemma 4.1, p. 14, QinLimits26], and the comparison example with moving measures is [Example B.1, p. 28, QinLimits26]. In the book, finite addition of atoms gives smooth approximants that converge in local capacity directly. The construction with moving measures over a fixed bounded background in theorem 5.27.42 also corresponds to [Theorem 3.4, pp. 13–14, QinLimits26]. The cutoff argument of lemma 5.27.41 shows that, under convergence in local capacity, every limit measure dominates the Monge–Ampère measure of the background; this gives a complete classification of the cluster sets for bounded backgrounds approximated from below. Convergence in local capacity is a stronger requirement than the $L^1$ convergence in Qin's general classification of reachable sets, whose argument for reaching the zero measure needs additional hypotheses on maximal approximation and on the domain; that argument is not used here.

Let $\phi _1,\ldots ,\phi _ r$ be plurisubharmonic weights with exponents $p_1,\ldots ,p_ r>1$, as in theorem 5.37.15. The division estimate with two summands, lemma 5.37.10, is the case of two weights in Takakura's [v1, Theorem 1.3, pp. 2 and 5–6, TakDivision26]. Since the relevant kernel is a line, the proof can use the Hessian formula for a scalar function instead of the general curvature theorem for subbundles. The constant in the a priori inequality (5.37.13) follows from the twisted Bochner identity in Demailly's notes [Section 13.A, pp. 123–124, DemAM11] by a weighted Cauchy inequality. Proposition 13.4 there states the weaker constant $2$; keeping the full factor $\eta +\lambda $ gives the constant $1$ used here. For a single weight, proposition 5.37.14 is the case $r=1$ of [Theorem 1.2, pp. 1–2 and 6–7, TakDivision26]. Applying it successively gives the product constant of theorem 5.37.15, with no condition on $\sum _ i1/p_ i$. The coordinate example in its proof extends [Example 3.2, p. 7, TakDivision26] and shows that this constant is optimal when $r\leq n$. For $2\leq r\leq n$ the product is strictly smaller than $1-r^{-1}\sum _ i1/p_ i$, the constant stated in Takakura's theorem. For $r>n$ the product need not improve Takakura's dimension-dependent estimate, and by itself it does not determine the optimal constant.

In dimension one, theorem 5.37.22 gives the optimal constant for every number $r$ of weights, including the range $\sum _ i1/p_ i\geq 1$ in which it vanishes. When $\sum _ i1/p_ i<1$, it equals the dimension-one constant of [Theorem 1.2, pp. 1–2, TakDivision26], and a common logarithmic pole shows that it is sharp. The local argument in lemma 5.37.18 uses the logarithmic fundamental solution and the characterization of positive distributions in [Chapter I, Sections 4.A–4.C, pp. 30–36, DemCADG]; in our normalization, $\mathrm{dd}^{\mathrm{c}}\log |z|^2=\delta _0$. The global lifting in lemma 5.37.20 is the locally free planar case of vanishing for coherent sheaves. It is proved here with the vector-valued $L^2$ estimate for complete metrics in [Theorem 5.1 and its proof, pp. 33–34, DemAM11], whose Bochner identity is [Section 4.C, p. 30, DemAM11]. We do not determine the optimal constant when $n\geq 2$ and $r>n$.