25.1.12 Lelong numbers and multiplier ideals2BFBF0A17586

For the Jensen–Lelong formula compare [Chapter III, (6.1)–(6.11), pp. 162–166, DemBook]. The book uses only its form on Euclidean balls, theorem 6.7.6. For smooth functions the formula follows from the derivative of spherical means, proposition 3.1.5, and the monotonicity of the density from the convexity of circle means on complex lines through the center; regularization, with lower bounds for spherical means given by the Poisson kernel, gives the general case. Demailly's operator there is twice ours, and he measures growth against $\log |z|$; the density in (6.7.5) is written with our $\mathrm{dd}^{\mathrm{c}}$ and $\log |z|^2$, so that $\nu (\log |z|^2,0)=1$.

The coherence of multiplier ideals, theorem 6.8.9, and its proof by generation from a weighted Hilbert space of holomorphic functions are presented in [Proposition 5.7, pp. 36–37, DemAnalyticOnline]; page numbers refer to the manuscript on Demailly's web page. The weight there is $\mathrm e^{-2\varphi }$, which becomes our $\mathrm e^{-\varphi }$ after doubling the potential. For the strong Noetherian property and the Artin–Rees lemma used in the proof compare [Chapter II, (3.22), p. 90, and Chapter IX, (1.1)–(1.3), pp. 403–404, DemBook]. Hilbert's basis theorem, the Artin–Rees lemma and Krull's intersection theorem are proved in lemma 6.8.1. We prove the strong Noetherian property in lemma 6.8.4, and the uniform generation by elements of the Hilbert space, with bounded coefficients, in lemma 6.8.10, using the completeness of the Hilbert space.

Siu's theorem on the analyticity of the upper level sets of Lelong numbers, theorem 6.9.5, goes back to [Siu74]. The proof through the orders of vanishing of multiplier ideals is Demailly's method; compare [Corollary 14.3, p. 138, DemAnalyticOnline] and [Theorem 10.2 and Lemma 10.3, p. 42, Bou23Notes], the version of Boucksom's notes dated 11 September 2023. The proofs of the restriction formula and of subadditivity by products and the diagonal (theorem 6.10.1), and the proof of the pullback formula by the graph (corollary 6.10.5), are taken from [(15.1)–(15.3), pp. 158–159, DemAnalyticOnline]. In (6.10.2), the restriction of $\mathcal I(\varphi )$ to $Z$ means its image ideal in $\mathcal O_ Z$.

Strong openness, corollary 6.11.6, is due to Guan and Zhou [GZ15Strong]. Our proof of theorem 6.11.4, corollary 6.11.6 follows their argument by analytic curves and extension in Sections 3.1–3.2, pp. 609–611, of the published article, and its monotone version in Section 3.3, pp. 611–612. The curve is constructed in lemma 6.11.1 by blowing up an ideal, and the lower bound for the norms of extensions is lemma 6.11.2. In the induction the slices are taken in a fixed polydisc, so that the constant of the extension theorem does not depend on the slice or on the weight.

That a weight with zero Lelong number at a point has trivial multiplier ideals there, corollary 6.9.3, is a special case of Skoda's integrability theorem; compare [Lemma 5.6(a), pp. 35–36, DemAnalyticOnline]. Here it is deduced from the bounds for the orders of multiplier ideals in lemma 6.9.1, which rely on the extension with prescribed point value of corollary 6.5.8. In that source the potential $\varphi =u/2$ corresponds to our $u$, by the conversion described at the beginning of these notes.

For discrepancies, the change of variables formula and the valuative criterion for multiplier ideals compare [Proposition 10.12, Theorem 10.13 and Corollary 10.18, pp. 44–46, Bou23Notes]; Boucksom's conventions $\mathrm e^{-2\varphi }$ and $\log |z|$ are converted as above. We state the criterion, theorem 6.12.16, for a holomorphic germ, with one positive parameter $\varepsilon $ that works for all divisors over a neighborhood of the point. For weights of the form $c\log \sum _ j|g_ j|^2+O(1)$, the strict threshold on a log resolution in theorem 6.12.8 involves every component of the resolved divisor that meets the fiber over the point, including the strict transforms of hypersurfaces in $X$, not only the exceptional ones.