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.

23 pages. Comments are welcome

References in corpus (6)

• K-stability and Kähler-Einstein metrics
• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• CM Stability and the Generalized Futaki Invariant I
• Compact moduli space of Kahler-Einstein Fano varieties
• K-stability implies CM-stability
• CM Stability of Projective varieties

Cited by in corpus (2)

• Compact moduli space of Kahler-Einstein Fano varieties
• Generalized Kähler-Einstein metric along $\mathbb{Q}$-Fano fibration