A simple remark on a flat projective morphism with a Calabi-Yau fiber

TL;DR

The paper proves that the property of being a K3 surface in fibers of a flat projective morphism does not extend to higher-dimensional Calabi-Yau fibers, providing a counterexample to a posed question.

Contribution

It offers a counterexample showing higher-dimensional Calabi-Yau fibers do not necessarily share the same properties as K3 surfaces in flat projective morphisms.

Findings

Smooth connected fibers are K3 surfaces in the K3 fiber case

Counterexample for higher-dimensional Calabi-Yau fibers

Initial observation confirmed for K3 surfaces

Abstract

If a K3 surface is a fiber of a flat projective morphisms over a connected noetherian scheme over the complex number field, then any smooth connected fiber is also a K3 surface. Observing this, Professor Nam-Hoon Lee asked if the same is true for higher dimensional Calabi-Yau fibers. We shall give an explicit negative answer to his question as well as a proof of his initial observation.