Separatedness of moduli of K-stable varieties

Abstract

Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM line bundle, and some applications to moduli. Our methods work for arbitrary projective schemes in any characteristic.

Withdrawn for now due to some extra components in a blow up that we had missed. We still hope to get an appropriate birational model of the total space using MMP, to recover the result under appropriate conditions in the future. Comments would be appreciated!

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