Positivity of the CM line bundle for families of K-stable klt Fano varieties

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.

Valuative stability of polarised varieties

“…Definition 3. 18 We say that a dreamy prime divisor F over (X , L) is of product type if its associated test configuration (X F , L F ) is a product test configuration.…”

“…This is true both abstractly, in the sense that one can now construct moduli spaces of K-stable Fano varieties (though properness remains open 1 ), and concretely, in the sense that one can now give a very thorough understanding of which Fano varieties are actually K-stable. There are many results along these lines, such as [2,8,18,29] to name only a few. The valuative approach to K-stability of Fano varieties has been essential to all of these developments.…”

Openness of uniformly valuative stability on the Kähler cone of projective manifolds

“…where we used F ∈ D ud L ′ , i.e. (15) and Lemma 3.4 for the second inequality. Since L ′ is ample, by (17), we have…”

“…From this powerful theory, one can construct a satisfactory moduli space of K-stable Fano varieties, the so-called K-moduli space which is proper as proved recently in [32]. There are many works along these lines, see [7], [15], [6], [41], [1], [4], [42], etc. The valuative criterion for K-stability of Fano varieties has played an essential role in all of these developments.…”

A note on families of K-semistable log-Fano pairs

“…The Chow-Mumford (CM) line bundle is an ample line bundle on this moduli space. We refer to the introductions of [CP21] and [XZ20], to the survey [?] and to the recent groundbreaking paper [?]…”