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

In 1963, Kodaira found a local-global formula to compute the Chern numbers of an elliptic surface by using the local invariants of singular fibers and the J-invariants of smooth fibers. The main purpose of this talk is to generalize Kodaira’s formula to surfaces fibered by a family of curves of genus g>1.

We shall define the first and the second local Chern numbers of a singular fiber, which measure its stability and can be computed locally. The natural generalizations of the J-invariant are the modular invariants of generic fibers defined from the moduli space of stable curves of genus g. Then we obtain the generalized local-global formulas.

There are some interesting inequalities between the local Chern numbers of singular fibers. For example, the local Miyaoka-Yau Inequality and the local Canonical Class Inequality. We shall present also some applications of these inequalities in the classification of singular fibers.