On positivity of the CM line bundle on K-moduli spaces

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.

52 pages; v3: final version, to appear in Ann. of Math

Cited by in corpus (9)

• Optimal destabilizing centers and equivariant K-stability
• The existence of the Kähler-Ricci soliton degeneration
• On K-stability of some del Pezzo surfaces of Fano index 2
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• Weakly special test configurations of log canonical Fano varieties
• On semi-ampleness of the moduli part
• Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability
• On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves
• On local delta invariant of del Pezzo surfaces