Chern numbers of a singular fiber, modular invariants and isotrivial families of curves (Q977143)

Let \(f:X\rightarrow C\) be a family of curves of genus \(g\) with smooth general fiber. When \(g=1\), \textit{K. Kodaira} proves a fundamental formula in [Ann. Math. (2) 78, 1--40 (1963; Zbl 0171.19601)] which represents the global invariants of the surface \(X\) by the local invariants of the singular fibers together with the modular invariant. The purpose of this paper is to generalize Kadaira's formula to the higher genus cases. In the author's previous paper [Math. Z. 222, No. 4, 655--676 (1996; Zbl 0864.14016)], he defines the Chern numbers \(c_{1}^{2}(F),c_{2}(F),\chi_{F}\) of a singular fiber \(F\), and shows that, if \(g=1\), then \(c_{1}^{2}(F)=0\) and \(c_{2}(F)\) is exactly the coefficient in Kodaira's formula according to the type of the singular fiber. His generalization is formulated with the help of these local invariants together with some natural modular invariants constructed from a holomorphic map from \(C\) to the moduli space of semistable curves of genus \(g\).