25.1.13 Analytic singularities and regularizationC1D97719C08E

Approximation by Bergman kernels is Demailly's construction; compare [Proposition 3.1 and its proof, pp. 13–14, DemRegOnline] and [Theorem 14.2, pp. 136–138, DemAnalyticOnline]. Page numbers of the former refer to the manuscript on Demailly's web page. These sources use the weight $\mathrm e^{-2m\varphi }$ and the factor $(2m)^{-1}$, whereas theorem 6.15.4 uses $\mathrm e^{-m\varphi }$ and $m^{-1}$, in accordance with our normalization of Lelong numbers. The comparison of the Bergman kernel with finitely many generators and its form on a log resolution (lemma 6.15.3) are derived from the bounded generation of lemma 6.15.1(3), a consequence of lemma 6.8.10.

The computation for the logarithm of a sum of local Bergman kernels weighted by cutoff functions (lemma 6.17.1) is [Lemma 3.5, p. 15, DemRegOnline], and the gluing over overlapping balls in Lemma 3.6, pp. 16–17, of that paper is explained by the more general Lemma 4.6 and its proof, pp. 19–22. The comparison of the local kernels on overlaps and the gluing by maxima (lemma 6.16.2, lemma 6.16.1) follow the argument of Vu and Xia in the unpublished manuscript [Lemmas patching and regularization, VuXiaNakamaye]. Theorem 6.16.3 has exactly the two conclusions of the regularization lemma there, and its curvature constant depends only on the fixed geometry.

Our definition 6.14.12 asks for the local form with smooth remainder on every log resolution of the ideal, that is, on every composite of blowups along closed submanifolds that transforms it into the ideal of a divisor with simple normal crossings support (section 6.14). The definition in [Chapter on plurisubharmonic functions, definition of gentle analytic singularities, XiaSampleBook] asks for such a form on one Kähler modification. Since log resolutions exist and are Kähler (proposition 6.14.11), our requirement is the stronger one, and the approximations of theorem 6.17.6 satisfy it.

For the comparison by extension from the diagonal, the dyadic correction and the passage from the integrability of a gap to inclusions of ideals compare [arXiv version, Section 2.2, Steps 2–3, pp. 12–14, DPS01]. Theorem 2.2.1 there, proved on pp. 9–14 of the same version, concerns equisingular approximation, and Remark 2.2.2 distinguishes the case of logarithmic poles. Theorem 6.17.6 is a different statement: the approximations have the prescribed gentle analytic types $(2^{-j},\mathcal I(2^ j\varphi ))$, and their monotonicity and the inclusions of ideals between consecutive dyadic scales are derived from the kernels with cutoffs, the comparison on the diagonal and lemma 6.17.4. To compare the approximations, the growth estimate lemma 6.18.1, proved on finitely many local log resolutions, suffices, so that no division theorem is needed.

The ideal $\mathcal I_\infty $ of germs of bounded growth (definition 6.19.1) is used in [Chapter on plurisubharmonic functions, the lemma following the definition of $\mathcal I_\infty $ and Lemma 1.6.3, XiaSampleBook]. Its coherence for potentials with analytic singularities, theorem 6.19.2, is proved with a cone of fractional powers, finite normalization and finitely many homogeneous generators; the direct image under the finite map is handled by division by monic polynomials, so the general theorem on direct images under proper maps is not needed. Corollary 6.19.3 derives from the discrepancies on log resolutions the inclusion for large multiples that is needed to compare approximations; the sharper estimate in the source, with a loss depending on the dimension, is not proved here.