Several complex variables and analytic spaces Pluripotential theory Theorem

Demailly's regularization of currents

Builds on Plurisubharmonic function Lelong number Pseudoconvexity Positive current Multiplier ideal Ohsawa–Takegoshi extension theorem

Statement

Let $\Omega \subseteq \C^n$ be a bounded pseudoconvex domain and let $\varphi$ be a plurisubharmonic function on $\Omega$ , not identically $-\infty$ . For each integer $m \ge 1$ , let $\mathcal{H}_m$ be the Hilbert space of holomorphic functions $f$ on $\Omega$ with $\int_\Omega \abs{f}^2 e^{-2m\varphi} \, dV < \infty$ . Let $(\sigma_{m,l})_{l \ge 1}$ be an orthonormal basis of $\mathcal{H}_m$ , and let $\varphi_m = \frac{1}{2m} \log \sum_{l} \abs{\sigma_{m,l}}^2 .$ Then there are constants $C_1, C_2 > 0$ , independent of $m$ , such that for all $z \in \Omega$ and all $r$ with $0 < r < \operatorname{dist}(z, \partial\Omega)$ :

$\varphi(z) - \dfrac{C_1}{m} \le \varphi_m(z) \le \sup_{\abs{\zeta - z} < r} \varphi(\zeta) + \dfrac{1}{m} \log \dfrac{C_2}{r^n}$ ;
$\nu(\varphi, z) - \dfrac{n}{m} \le \nu(\varphi_m, z) \le \nu(\varphi, z)$ , where $\nu$ denotes the Lelong number.

In particular $\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\mathrm{loc}}(\Omega)$ as $m \to \infty$ .

Remarks

Each $\varphi_m$ is plurisubharmonic and has analytic singularities: locally it is $\frac{1}{2m} \log \sum \abs{g_j}^2$ plus a bounded function, for finitely many holomorphic $g_j$ , which generate the multiplier ideal $\mathcal{I}(m\varphi)$ . Proof idea: $\varphi_m(z) = \frac{1}{m} \sup\set{\log\abs{f(z)} : f \in \mathcal{H}_m, \norm{f} \le 1}$ , so it is the logarithm of a weighted Bergman kernel on the diagonal. The upper bound follows from the mean value inequality for $\abs{f}^2$ on the ball of radius $r$ around $z$ . The lower bound follows from the Ohsawa–Takegoshi extension theorem, which gives $f \in \mathcal{H}_m$ with $f(z) = e^{m\varphi(z)}$ and $\norm{f}$ bounded independently of $z$ and $m$ . The Lelong number inequalities follow from these bounds. Example: for $\varphi = \log\abs{z}$ on the unit disc of $\C$ , $\nu(\varphi, 0) = 1$ , and the bounds force $1 - \frac{1}{m} \le \nu(\varphi_m, 0) \le 1$ . The method can be globalised with a partition of unity. On a compact complex manifold, it approximates a closed positive current of bidegree $(1, 1)$ by closed currents in the same cohomology class. The approximating currents have analytic singularities, lose only a small amount of positivity, and have Lelong numbers converging to those of the current.

Going deeper

For a student

This is the main regularisation tool of analytic methods in algebraic geometry. Functions with analytic singularities can be resolved by blowing up, so statements about them reduce to divisors with normal crossings. The global version for closed positive $(1,1)$ -currents is used to study pseudo-effective and nef line bundles. It also gives a proof of Siu's theorem that the upper level sets of Lelong numbers are analytic. It relies on the $L^2$ theory of $\bar\partial$ , through the Ohsawa–Takegoshi theorem.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Holomorphic function of several variables concept
Lelong number concept
Multiplier ideal concept
Plurisubharmonic function concept
Positive current concept
Pseudoconvexity concept
Ohsawa–Takegoshi extension theorem statement
Demailly's regularization of currents statement

Tag	`31CV5K5SAJ`
URI	`m2://31CV5K5SAJ`
Kind	statement, theorem
Writing	W0 stub
MSC	32U40, 32U05, 32U25
Status	This copy of the site has no build data. Run `m2 status 31CV5K5SAJ`.
About	Positive current, Plurisubharmonic function
Uses	Plurisubharmonic function, Lelong number, Pseudoconvexity, Positive current, Multiplier ideal, Ohsawa–Takegoshi extension theorem