Siu's semicontinuity theorem

Remarks

That $E_c(T)$ is closed is elementary, since $x \mapsto \nu(T, x)$ is upper semicontinuous. The content is that $E_c(T)$ is analytic. Siu's original proof was long; a shorter proof by Demailly writes $E_c(T)$ locally as a common zero set of holomorphic functions built with Hörmander's L2 estimates.

Example: let $A \subseteq X$ be an analytic subset of pure dimension $p$ and $T = [A]$ its current of integration. By Thie's theorem, $\nu([A], x)$ is the multiplicity of $A$ at $x$, so $E_c([A])$ is the set of points of multiplicity at least $c$, an analytic set.

For a plurisubharmonic function $u$ on an open set of $\C^n$, apply the theorem to $T = dd^c u$ with $p = n - 1$: the sets $\set{x : \nu(u, x) \ge c}$ are analytic of dimension at most $n - 1$.

Going deeper

For a student

The theorem is the starting point of Siu's decomposition of a closed positive current. It writes the current as a countable sum of currents of integration over analytic sets plus a residual part. For the residual part, the upper level sets of Lelong numbers have smaller dimension. It is used with multiplier ideals and with regularization of currents in the study of positivity of line bundles.