Abstract
Let \(f:\,X \rightarrow \mathbb {P}^1\) be a non-isotrivial semi-stable family of varieties of dimension m over \(\mathbb {P}^1\) with s singular fibers. Assume that the smooth fibers F are minimal, i.e., their canonical line bundles are semiample. Then \(\kappa (X)\le \kappa (F)+1\). If \(\kappa (X)=\kappa (F)+1\), then \(s>\frac{4}{m}+2\). If \(\kappa (X)\ge 0\), then \(s\ge \frac{4}{m}+2\). In particular, if \(m=1\), \(s=6\) and \(\kappa (X)=0\), then the family f is Teichmüller.
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The authors would like to thank the referees for many useful suggestions for the correction of the original manuscript.
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This work is supported by SFB/Transregio 45 Periods, Moduli Spaces and Arithmetic of Algebraic Varieties of DFG, by NSF of China and by the Science Foundation of Shanghai (No. 13DZ2260400).
