Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.arXiv:1706.04548v1 [math.AG] 14 Jun 2017for all positive integers p, q. This, together with the Noetherianity of X, implies that {J (X, (c/p) · a p )} p∈N has a unique maximal element that is called the c-th asymptotic multiplier ideal and denoted by J (X, c · a • ). Note that J (X, c · a • ) = J (X, (c/p) · a p ) for all p divisible enough.Asymptotic multiplier ideals also satisfy a subadditivity property. See [Laz04, Theorem 11.2.3] for the case when X is smooth.Corollary 5.6. Let a • be a graded sequence of ideals on X. If m ∈ N * and c ∈ Q * + , thenNext we give a containment relation for the multiplier ideal associated to the graded sequence of valuation ideals. The result appears in [ELS03] in the case when v is divisorial.Proof. It is an immediate consequence of the valuative criterion for membership in the multiplier ideal [BdFFU15, Theorem1.2] that J (X, c · a • (v)) ⊂ a cv(a•(v))−A(v) (v).Since v(a • (v)) = 1 (see [Blu16b, Lemma 3.5]), the proof is complete.

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In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and Θ-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space. Contents 1. Introduction 1 2. Preliminaries 4 3. S-completeness 11 4. Reductivity of the automorphism group 19 5. Θ-reductivity 23 References 29Throughout, we work over an algebraically closed field k of characteristic 0.

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.