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Uniqueness of \(\mathrm{K}\)-polystable degenerations of Fano varieties. (English) Zbl 1427.14084

An essential difficulty in the construction of moduli spaces of Fano varieties is non-uniqueness of limits of families. In recent years the notion of \(K\)-stability, originating in complex geometry, has assumed a prominent role in algebraic geometry as a plausible solution to this problem.
This paper makes an important step towards proving the existence of a projective moduli space for \(K\)-polystable \(\mathbb Q\)-Fano varieties of fixed dimension and volume. The main result, Theorem 1.1, shows that
1. \(K\)-semistable limits of \(\mathbb Q\)-Gorenstein families of log Fano pairs are unique up to S-equivalence;
2. \(K\)-polystable limits of \(\mathbb Q\)-Gorenstein families of log Fano pairs are unique up to isomorphism;
3. if a family as above has a \(K\)-stable limit, then any \(K\)-semistable limit is isomorphic to the \(K\)-stable limit.
In combination with other recent progress in the area due to Jiang and Blum-Yiu, this yields the result (Corollary 1.4) that the functor of uniformly \(K\)-stable Fano varieties of fixed dimension and volume is a separated Deligne-Mumford stack of finite type, which has a coarse moduli space that is a separated algebraic space.
The methods of the paper are purely algebraic.

MSC:

14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
14D20 Algebraic moduli problems, moduli of vector bundles
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