On positivity of the CM line bundle on K-moduli spaces
Algebraic Geometry
2021-02-22 v3 Differential Geometry
Abstract
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.
Cite
@article{arxiv.1912.12961,
title = {On positivity of the CM line bundle on K-moduli spaces},
author = {Chenyang Xu and Ziquan Zhuang},
journal= {arXiv preprint arXiv:1912.12961},
year = {2021}
}
Comments
52 pages; v3: final version, to appear in Ann. of Math