Application of slope inequality for semi-stable families

Abstract: In this talk I shall first discuss the relation of Miyaoka-Yau inequality and sharp slope inequality for \({\cal F} \subseteq f_*\omega_{X/Y}^{\otimes\nu}\) of semi-stable family of curves in char 0. Then I’ll talk about the case of semi-stable family of curves in char p. It includes szpiro’s inequality and slope inequality for \({\cal F} \subseteq f_*\omega_{X/Y}^{\otimes\nu}\) (joint work with Sheng Mao and Zuo). Finally I’ll mention the volume inequality for semi-stable family of high-dimension manifolds. This is a joint work with with Tan, Viehweg and Zuo.