# $m$-th root of holomorphic section of direct image of relative line bundle

Question edited after the answer of Sándor Kovács:

Let $f:X\to B$ be a holomorphic fibre space of smooth projective varieties which $f$ is relatively semi-ample and take $\mu$ as $m$-th root of holomorphic section of direct image of relative line bundle $f_*(K_{X/B}^{\otimes m})$ then why $\mu$ must be a family of canonical forms on a family of cyclic $m$ covers of fibres $X_b$ , where $b\in B$.

• What does it mean for a morphism to be relatively semi-ample? Edit: OK, I see that this addition is a response to S\'andor's comment about $K_{X/B}$ in his answer. Again, inaccuracy makes things confusing. Commented Aug 9, 2016 at 14:36
• That reference has the (standard) definition of what it means for a divisor to be relatively semi-ample. Your question talks about a morphism being relatively semi-ample. I think I will withdraw from this question now. Commented Aug 9, 2016 at 14:54
• fibres are semi-ample. Ineeded it when $X_b$ is polarized CY
– user21574
Commented Aug 9, 2016 at 14:57
• My previous comment refers to a now-deleted comment of the OP. Commented Aug 9, 2016 at 15:01
• It was my mistake, sorry
– user21574
Commented Aug 9, 2016 at 15:02

For any line bundle $L$ on $Z$, and a global section of $L^{\otimes m}$ there is an associated cyclic cover of $Z$ of degree $m$ branched over exactly the zero locus of the given section. Furthermore, the pull-back of the given section to this cover is the $m^\text{th}$ power of a section of the pull-back of $L$. (For more on this, see for example page 67 of this book.)
A section of $f_*(K_{X/B}^{\otimes m})$ could be considered a family of global sections of $K_{X_b}^{\otimes m}$ and looking at the construction of the cyclic cover it is relatively easy to see that these covers form a family.
From what you are saying I assume that there might be an additional assumption on $K_{X/B}$, maybe something like it is relatively semi-ample or something like that.
The point is that if you $K_{X/B}^{\otimes m}$ is relatively generated by global sections, then choosing a general section will give you something smooth, at least globally, so the zero locus on the fiber is smooth at least for a general $b\in B$. Then performing the cyclic cover gives another smooth variety (and you need to have some control of the singularities if you want to talk about canonical forms on the cover). Now if the cover is indeed smooth, then by Hurwitz's theorem the pull-back of $K_{X/B}$ to the family of the covers embeds into the relative canonical of that family. By the above general result about the section pulling back to an $m^\text{th}$ power on the family of covers gives you the section of the relative canonical of the family of covers which then can be considered as a family of canonical forms on these covers.