Abstract
In this paper, we consider the CM line bundle on the $\mathrm {K}$-moduli space, i.e., the moduli space parametrizing $\mathrm {K}$-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly $\mathrm {K}$-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable $\mathrm {K}$-polystable Fano varieties is projective.
During the course of proof, we develop a new invariant for filtrations that can be used to test various $\mathrm {K}$-stability notions of Fano varieties.
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mrnumber = {4088355},
doi = {10.4007/annals.2020.191.3.6},
url = {https://doi.org/10.4007/annals.2020.191.3.6},
zblnumber = {07190308},
} -
@ARTICLE{Zhu19,
author = {Zhuang, Ziquan},
title = {Product theorem for {K}-stability},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {371},
year = {2020},
pages = {107250, 18},
issn = {0001-8708},
mrclass = {14E30 (14J45)},
mrnumber = {4108221},
doi = {10.1016/j.aim.2020.107250},
url = {https://doi.org/10.1016/j.aim.2020.107250},
zblnumber = {07219698},
}