Let $X$ be a compact (Hausdorff reduced) complex space. It is asserted (and used in an essential way) in a famous paper by Demailly and Păun ("Numerical characterization of the Kähler cone of a compact Kähler manifold") Proposition 3.3(iii), that a Kähler current $T$ on $X$ can be regularised by currents with analytic singularities. This result in turn is used in the proof of Theorem 0.1.

This seems rather surprising to me. In some special cases, for example when $T$ represents a Kähler class and $X$ is normal, this follows simply from a resolution argument. In general, is there a reference for this?

  • 1
    $\begingroup$ This statement in that paper is indeed problematic, or at the very least it needs more details. Here are two comments: 1) arxiv.org/abs/1304.5216 gives another proof of the main theorems of Demailly-Paun without using regularisation of currents on singular analytic spaces; 2) the recent arxiv.org/abs/2205.12205 has a relevant discussion in section 2.4 $\endgroup$
    – YangMills
    May 19, 2023 at 20:59
  • $\begingroup$ @YangMills Thanks! The proof of Collins--Tosatti seems to work. I also realised that the statement of Prop 3.3(iii) itself is a non-sense. After all, Lelong numbers do not make sense on non-normal spaces. $\endgroup$ May 20, 2023 at 5:56


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.