# Regularity of fiber integration between complex analytic spaces

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?

• 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. Nov 3, 2021 at 8:07
• @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? Nov 3, 2021 at 8:31
• 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. Nov 4, 2021 at 8:24
• 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. Nov 4, 2021 at 10:26
• Thanks again. But could you let me know if my problem is proved or disproved in that paper? Nov 4, 2021 at 14:17