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Positivity of the CM line bundle for families of \(K\)-stable klt Fano varieties. (English) Zbl 1462.14044

The moduli part of the classification theory expects a projective, compactified moduli space for the different kind of fibers of the Minimal Model Program. In particular, for Fano varieties, one should construct (i) the stack of \(K\)-semistable Fano varieties of fixed dimension and anticanonical volume, and (ii) its projective good moduli space. These constructions are known (see the Introduction of the paper under review and references therein) except for the the properness and projectivity of (ii). This is the subject of the main result of this paper (see Theorem 1.1): (a) the Chow-Mumford line bundle \(\lambda\) on (i) and its descent \(L\) to (ii) are nef; (b) on any proper closed subspace \(V\) of (ii) meeting the uniformly \(K\)-stable locus, the restriction of \(L\) is also big; and (c) the normalization of the intersection of \(V\) with the uniformly \(K\)-stable locus is a quasi-projective scheme.

MSC:

14J45 Fano varieties
14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)
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