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Let $X$ be a complex manifold. Any dimension $p$ subvariety(or reduced analytic subscheme) will determine a real closed positive $(p,p)$-current. But the converse is not true.

My question:

  1. Is there a typical example of a closed positive real $(p,p)$-form which is not a subvariety?

  2. Intuitively, what is the essential difference between closed positive real $(p,p)$-form and subvariety?

I realize there are some trivial examples. For example, let $A\subseteq X$ be a subvariety and $[A]$ be the current associated to it. Then $\lambda[A]$ is not a current from subvariety. But is there any essential examples?

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    $\begingroup$ any smooth $(p,p)$ form, seen as a current, will do the job $\endgroup$
    – Henri
    Commented Sep 30, 2022 at 17:33
  • $\begingroup$ @Henri Could we roughly say any positive closed real $(p,p)$-current is a smooth subvariety(though this concept is not well defined)? Is there any other possible subtly here? $\endgroup$
    – Hydrogen
    Commented Sep 30, 2022 at 21:52

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