Abstract
The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.
Highlights
The interest in the moduli space of singular K -polystable Fano varieties stems from the classification theory of algebraic varieties
Unwinding definition (1.7.a), we obtain that, in the case of one dimensional base, Theorem 1.9 states that (−K X/T )n+1 is at most zero/smaller than 0. Using this in conjunction with the base-change property of the CM line bundle proved in Proposition 3.8 we obtain that Theorem 1.9, especially the last 3 points, prove strong negativity properties of −K X/T for families of klt Fano varieties
We note that the Yau-Tian-Donaldson conjecture asserts that a klt Fano variety admits a singular Kähler-Einstein metric if and only if it is K-polystable
Summary
The interest in the moduli space of singular K -polystable Fano varieties stems from the classification theory of algebraic varieties. The projective good moduli space MnK,-vps of MKn,-vss parametrizing K polystable Fano varieties of dimension n and anti-canonical volume v. Remark 1.5 An equivalent way of stating point (a) and (b) of Theorem 1.1 is the following: λ and L are nef, and for every proper closed subspace V ⊆ MnK,-vps the augmented base locus B+(L|V ) is contained in V \ Mnu-,Kv -s. This follows immediately from [86, Thm 0.3]. Question 1.7 Is the deformation space of uniformly K -stable Fano varieties unobstructed?
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