On boundedness and moduli spaces of K-stableCalabi–Yau fibrations over curves

We show boundedness of polarized Calabi-Yau fibrations over curves only with fixed volumes of general fibers and Iitaka volumes.As its application, we construct a separated coarse moduli space of K-stable Calabi-Yau fibrations over curves in an adiabatic sense (Hattori 2022) and show that all members (resp.smooth members) of the moduli are simultaneously uniformly K-stable (resp.have cscK metrics) for a certain choice of polarizations.