The slope of surface fibrations and applications

The slope is one of the most fundamental invariants for a locally non-trivial surface fibration. The slope inequality proved by Xiao (and by Cornalba-Harris in the semi-stable case) provides a nice lower bound for the slope, which turns to be strict unless the fibration is regular and hyperelliptic. It arises thus a natural problem to find improved bounds for irregular or non-hyperelliptic fibrations. In this talk, I will report the joint works with Professor Kang Zuo on these questions and also some applications.