Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds

Abstract

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.

arXiv comment23 pages. Comments are welcome

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Cited by in corpus (8)

• Kähler-Einstein metrics and volume minimization
• Nakamaye's theorem on complex manifolds
• Compact moduli space of Kahler-Einstein Fano varieties
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• Ampleness of the CM line bundle on the moduli space of canonically polarized varieties
• Restricted volumes on Kähler manifolds
• Fano-Ricci limit spaces and spectral convergence
• Generalized Kähler-Einstein metric along $\mathbb{Q}$-Fano fibration