Slope inequalities for KSB-stable and K-stable families

TL;DR

This paper extends slope inequalities to higher dimensions for KSB-stable and K-stable families, providing new tools for understanding moduli spaces and stability conditions in algebraic geometry.

Contribution

It introduces higher-dimensional slope inequalities for KSB-stable and K-stable families, utilizing Harder-Narasimhan filtrations and classical inequalities.

Findings

Generalized slope inequalities for higher dimensions

Applications to moduli space of KSB-stable varieties

Insights into the structure of stable families

Abstract

We prove some higher dimensional generalizations of the slope inequality originally due to G. Xiao, and to M. Cornalba and J. Harris. We give applications to families of KSB-stable and K-stable pairs, as well as to the study of the ample cone of the moduli space of KSB-stable varieties. Our proofs relies on the study of the Harder-Narasimhan filtration, and some generalizations of Castelnuovo's and Noether's inequalities.