K-stability was introduced to detect a constant scalar curvature K\"{a}hler metric that is a generalization of a Ricci flat K\"{a}hler metric.
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
adiabatic K-stability (If $f : (X, H) \to(B, L)$ 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 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.
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