The characterization of K-stable varieties is well-studied when $K_X$ is ample or X is a Calabi-Yau or Fano variety. However, K-stability of Calabi-Yau fibrations (i.e., $K_X$ is relatively trivial) is not known much in algebraic geometry. We introduce uniform adiabatic K-stability (if $f\colon (X,H)\to (B,L)$ is a fibration of polarized varieties, which means that K-stability of $(X,aH+L)$ for sufficiently small $a>0$).In this talk, I would like to explain that uniform adiabatic K-stability of a Calabi-Yau fibration over a curve is equivalent to K-stability of the base curve in some sense. Furthermore, we construct separated moduli spaces of polarized uniformly adiabatically K-stable Calabi-Yau fibrations over curves. This talk is based on a joint work with Kenta Hashizume.
