6.13 Analytic singularities and Demailly approximation1A5FE90C1245

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Let $u$ be a plurisubharmonic function on a domain in $\mathbb C^ n$. Convolution with a smoothing kernel approximates $u$ by smooth plurisubharmonic functions, but these approximants have lost all singularities: their Lelong numbers vanish and their multiplier ideals are trivial. Many problems require approximants whose singularities are still visible but as simple as possible, namely functions locally of the form $c\log \sum _ i|g_ i|^2+O(1)$ with holomorphic $g_ i$ and rational $c>0$; such functions are said to have analytic singularities. On a compact Kähler manifold a second difficulty appears: convolution is defined only in charts, and gluing local approximations costs positivity. This chapter presents Demailly's answer to both problems, in which convolution is replaced by weighted Bergman kernels.

For $u$ plurisubharmonic on a bounded pseudoconvex domain $\Omega $ and an integer $m\geq 1$, the holomorphic functions $F$ on $\Omega $ with $\int _\Omega |F|^2\mathrm e^{-mu}<\infty $ form a Hilbert space, which generates the multiplier ideal $\mathcal I(mu)$ of section 6.6. If $K_ m(z)$ denotes the supremum of $|F(z)|^2$ over its unit ball, then $u_ m=m^{-1}\log K_ m$ is plurisubharmonic and has analytic singularities described by $\mathcal I(mu)$. The Ohsawa–Takegoshi estimate of section 6.1 bounds $u_ m$ below by $u-O(1/m)$, and the mean-value inequality bounds it above by the supremum of $u$ over a small ball, up to an error that tends to zero as $m\to \infty $. Hence $u_ m\to u$, and the Lelong numbers of $u_ m$ differ from those of $u$ by at most $n/m$ (theorem 6.15.4).

On a compact Kähler manifold $(X,\omega )$ we apply this construction in finitely many coordinate balls to a function $\varphi $ with $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq 0$, where $\theta $ is a smooth closed real $(1,1)$-form. A $\bar\partial $-estimate with a logarithmic pole shows that the kernels of two overlapping balls are comparable up to a factor that depends only on the charts (lemma 6.16.2). Gluing the local potentials by a maximum gives the quantitative regularization of theorem 6.16.3, with a curvature loss of order $1/m$ and a lower bound by the generators of $\mathcal I(m\varphi )$. Gluing them instead by a sum with smooth cutoffs gives approximants $G_ m$ with gentle analytic singularities (lemma 6.17.1): $\exp (mG_ m)$ is smooth, and $G_ m$ has a smooth remainder on every log resolution of $\mathcal I(m\varphi )$. Comparing the kernels at the scales $m$ and $2m$ by extension from the diagonal, we show that the dyadic sequence $G_{2^ j}$, corrected by small constants, decreases to $\varphi $ and that its multiplier ideals approximate those of $\varphi $, so that it is a quasi-equisingular approximation of $\varphi $ in the sense of definition 6.17.5 (theorem 6.17.6). Any two quasi-equisingular approximations of $\varphi $ dominate each other up to an arbitrarily small loss in the coefficient (proposition 6.18.2).

The first section collects the elementary properties of quasi-plurisubharmonic functions, introduces $\theta $-plurisubharmonic functions and analytic singularities, and shows, using metrics on line bundles, that blowups with smooth centers preserve the Kähler property. The last section studies the ideal of holomorphic germs $f$ with $|f|^2\leq C\mathrm e^ v$: for $v$ with analytic singularities it is coherent, and on a compact manifold the ideal attached to $kv$ contains $\mathcal I(k(1+\varepsilon )v)$ for all large $k$.

Historical remarks and references for this chapter are collected in subsection 25.1.13.