Abstract
In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space 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 M and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifoldsM. As a consequence, we show that the CM line bundle is nef and big on M and its restriction on M is ample.
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