Several complex variables and analytic spaces Local analytic geometry Theorem

Thie's theorem on multiplicity

Builds on Lelong number Complex analytic variety Dimension of an analytic set Regular and singular points of an analytic set Current

Statement

Let $\Omega \subseteq \C^n$ be open, let $A \subseteq \Omega$ be an analytic subset of pure dimension $p \ge 1$ and let $x \in A$ . Let $\pi \colon \C^n \to \C^p$ be a generic complex linear map. Then the restriction of $\pi$ to $A$ near $x$ is a branched covering of a neighbourhood of $\pi(x)$ . Let $m_x(A)$ be its number of sheets, the multiplicity of $A$ at $x$ . Let $\operatorname{vol}_{2p}$ denote the $2p$ -dimensional volume of the set of regular points of $A$ . Then $\lim_{r \to 0^+} \frac{\operatorname{vol}_{2p}\bigl(A \cap B(x, r)\bigr)}{\pi^p r^{2p} / p!} = m_x(A).$ In other words, the Lelong number at $x$ of the current of integration $[A]$ equals $m_x(A)$ .

Remarks

The denominator $\pi^p r^{2p} / p!$ is the volume of a ball of radius $r$ in $\C^p = \R^{2p}$ . The proof uses a generic linear projection $\pi$ , under which $A$ covers a neighbourhood of $\pi(x)$ with $m_x(A)$ sheets. It then compares, as $r \to 0$ , the volume of the part of each sheet near $x$ with the volume of its image in $\C^p$ .

Example: $A = \set{z_1 z_2 = 0} \subseteq \C^2$ is the union of two lines through $0$ . The set $A \cap B(0, r)$ is two discs of area $\pi r^2$ , so the ratio is $2$ , and indeed a generic projection to $\C$ has two sheets. At a regular point the multiplicity is $1$ , and the ratio tends to $1$ .

Going deeper

For a student

The theorem links an algebraic invariant, the multiplicity of the local ring of $A$ at $x$ , with a metric invariant, the density of $[A]$ . It is the reason why the points of multiplicity at least $c$ appear as upper level sets of Lelong numbers. It is also the model case for Lelong numbers of general closed positive currents.

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Complex analytic variety concept
Current concept
Dimension of an analytic set concept
Lelong number concept
Regular and singular points of an analytic set concept
Thie's theorem on multiplicity statement

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Siu's semicontinuity theorem

Tag	`G5JKJEAKZ9`
URI	`m2://G5JKJEAKZ9`
Kind	statement, theorem
Writing	W0 stub
MSC	32U25, 32C30
Status	This copy of the site has no build data. Run `m2 status G5JKJEAKZ9`.
About	Lelong number, Complex analytic variety
Uses	Lelong number, Complex analytic variety, Dimension of an analytic set, Regular and singular points of an analytic set, Current

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Siu's semicontinuity theorem