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Let $f:X\rightarrow Y$ be a flat surjective morphism between reduced complex analytic spaces. Assume that $Y$ is locally irreducible (i.e. unibranch). We assume that $X$ (resp. $Y$) is pure-dimensional of dimension $m$ (resp. $n$). Let $\alpha$ be a smooth $(m-n+r,m-n+r)$-form ($r\geq 0$) with compact support on $X$. Is it true that the push-forward current $f_*[\alpha]$ is always represented by a form with continuous coefficients?

As far as I know, the following cases are known:

  1. When $X$, $Y$ are both smooth. 2. When $r=0$ by King's fibering theorem.

If the general case fails, what if we assume that $f$ is smooth?

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    $\begingroup$ Perhaps this is worth reading: Chia-Chi Tung, "The first main theorem of value distribution on complex spaces. (English)", Atti della Accademia Nazionale dei Lincei. Memorie. Classe di Scienze Fisiche, Matematiche e Naturali. Serie VIII. Sezione I (Matematica, Meccanica, Astronomia, Geodesia e Geofisica), 15, 93-261 (1979), MR0563153, Zbl 0496.32018. In order to prove his main result, Tung proves a general Stokes theorem on general analytic spaces with singularities. $\endgroup$ Nov 3, 2021 at 8:07
  • $\begingroup$ @DanieleTampieri Thanks! But unfortunately I am unable to find Tung's book online or in our local library. Do you happen to have a copy of this book? $\endgroup$ Nov 3, 2021 at 8:31
  • $\begingroup$ I have a paper copy: however, since it is not a book but a (very long) paper in one of the main journals of the Accademia dei Lincei, I am trying to see if there are copies of this journal in a library near you (I know that there are copies at the Mittag -Leffler Institute). Let me see if I'm able to locate one. $\endgroup$ Nov 4, 2021 at 8:24
  • $\begingroup$ I did a research on WorldCat: I was not able to find what copies of the Journal are located in Sweden, but I hope you'll succeed in finding a library near you. $\endgroup$ Nov 4, 2021 at 10:26
  • $\begingroup$ Thanks again. But could you let me know if my problem is proved or disproved in that paper? $\endgroup$ Nov 4, 2021 at 14:17

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