6.6 Lelong numbers and multiplier ideals8494F0D4F79D

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A plurisubharmonic function can have a complicated pole set, but near each point its growth is governed by a single logarithmic rate. The Lelong number $\nu (u,a)$ of a plurisubharmonic function $u$ at a point $a$ is the largest $c\geq 0$ such that $u(z)\leq c\log |z-a|^2+O(1)$ near $a$; with this normalization $\nu (\log |z|^2,0)=1$. By the Jensen–Lelong formula, $\nu (u,a)$ is also the limiting density at $a$ of the positive current $\mathrm{dd}^{\mathrm{c}}u$, which links the pointwise growth of $u$ with the measures studied in the preceding chapters. Lelong numbers do not, however, determine the integrability properties of a singular weight $\varphi $ (example 6.9.4). These are recorded by the multiplier ideal $\mathcal I(\varphi )$, which consists of the holomorphic germs $f$ for which $|f|^2\mathrm e^{-\varphi }$ is locally integrable.

The $L^2$ estimates and extension theorems of section 6.1 make multiplier ideals accessible. Solving a $\bar\partial $-equation with a weight that has a logarithmic pole at a point, we approximate every germ of $\mathcal I(\varphi )$, to arbitrarily high order, by holomorphic functions on a polydisc that are square integrable against $\mathrm e^{-\varphi }$; with the Krull intersection theorem and the strong Noetherian property, this shows that $\mathcal I(\varphi )$ is a coherent sheaf (theorem 6.8.9). Comparing the vanishing orders of the generators of $\mathcal I(m\varphi )$ with the growth of $\varphi $ (lemma 6.9.1), we prove Siu's theorem that the sets where the Lelong number is at least $c>0$ are analytic. The same comparison shows that $\mathrm e^{-t\varphi }$ is integrable near a point $x$ for every $t>0$ if $\nu (\varphi ,x)=0$, a fact used in section 13.1 for an example on the projective line.

The Ohsawa–Takegoshi theorem gives the restriction and subadditivity properties of multiplier ideals. Since its constant does not depend on the weight, it can be applied along a moving family of hyperplanes; comparing the resulting extensions on one fixed holomorphic curve, we prove that multiplier ideals are stable under increasing limits of weights and deduce the strong openness theorem $\mathcal I(\varphi )=\bigcup _{\varepsilon >0}\mathcal I((1+\varepsilon )\varphi )$ (corollary 6.11.6).

The last section studies integrability on modifications. For a weight with analytic singularities, the change of variables formula on a log resolution reduces membership in the multiplier ideal to explicit one-variable integrals (theorem 6.12.8). For a general weight, the ideals $\mathcal I(m\varphi )$ recover the generic Lelong numbers of $\varphi $ along all prime divisors over the ambient manifold, and strong openness gives a valuative criterion for membership in $\mathcal I(\varphi )$ in which one positive parameter serves for all divisors (theorem 6.12.16). These results are the local foundation for comparing singularities through their multiplier ideals in later chapters.

Historical remarks and references for this chapter are collected in subsection 25.1.12.