Suppose I have a complex n-fold $X$. Let $\tau$ be a holomorphic multi-valued function from $X$ to upper half plane. $\tau$ is allowed to be singular at a codimension one locus in $X$. And let the multi-valuedness of $\tau$ be valued in $PSL(2,\mathbb{Z})$.
Is it necessarily the case that such a $\tau$ can be interpreted as a complex structure of an elliptic fibration over $X$?
P.S. It is something that is assumed in string theory literature. See, for instance, F-theory.