Finite generation for valuations computing stability thresholds and applications to K-stability
aa r X i v : . [ m a t h . AG ] F e b FINITE GENERATION FOR VALUATIONS COMPUTING STABILITYTHRESHOLDS AND APPLICATIONS TO K-STABILITY
YUCHEN LIU, CHENYANG XU, AND ZIQUAN ZHUANG
Abstract.
We prove that on any log Fano pair of dimension n whose stability thresholdis less than n +1 n , any valuation computing the stability threshold has a finitely generatedassociated graded ring. Together with earlier works, this implies: (a) a log Fano pairis uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable(resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combiningwith the previously known equivalence between the existence of K¨ahler-Einstein metricand reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs. Introduction
In recent years, the algebro-geometric study of the K-stability of Fano varieties hasmade remarkable progress. See [Xu20] for a comprehensive up-to-date survey.The theory has naturally driven people’s attention to valuations which are not neces-sarily divisorial. In fact, to further advance the theory, one main question is to show thefinite generation property of the associated graded rings for quasi-monomial valuationsof higher (rational) rank which minimize functions on the space of valuations arisen fromK-stability theory.While the finite generation property for divisorial valuations follows from [BCHM10],the higher rank case posts a completely new problem. In fact, there were very few stud-ies on higher rank quasi-monomial valuations from the viewpoint of the minimal modelprogram (MMP), which is our fundamental tool to study K-stability.In this paper, we prove that quasi-monomial valuations that compute the stabilitythresholds (or δ -invariants) of log Fano pairs satisfy the finite generation property (see[Xu20c, Conjecture 1.2]). Theorem 1.1 (=Theorem 4.1, Higher Rank Finite Generation Conjecture) . Let ( X, ∆) be a log Fano pair of dimension n and let r > be an integer such that r ( K X + ∆) isCartier. Assume that δ ( X, ∆) < n +1 n . Then for any valuation v that computes δ ( X, ∆) ,the associated graded ring gr v R , where R = L m ∈ Z ≥ H ( X, − mr ( K X + ∆)) , is finitelygenerated. The assumption δ ( X, ∆) < n +1 n might look a bit surprising since the original conjectureonly assumes δ ( X, ∆) ≤
1. However, the improvement becomes quite natural using thetrick of compatible divisors invented in [AZ20].1.1.
Corollaries of the main theorem.
Together with many earlier works in recentyears, Theorem 1.1 solves some central questions in the field of K-stability theory. Com-bining with [BLZ19], the first consequence we have is the following, which says that any log Fano pair that is not uniformly K-stable has an optimal destabilizing degeneration (interms of preserving the stability threshold).
Theorem 1.2 (=Theorem 5.1, Optimal Destabilization Conjecture) . Let ( X, ∆) be a logFano pair of dimension n such that δ ( X, ∆) < n +1 n , then δ ( X, ∆) ∈ Q and there exists adivisorial valuation E over X such that δ ( X, ∆) = A X, ∆ ( E ) S X, ∆ ( E ) .In particular, if δ ( X, ∆) ≤ , then there exists a non-trivial special test configuration ( X , ∆ X ) with a central fiber ( X , ∆ ) such that δ ( X, ∆) = δ ( X , ∆ ) , and δ ( X , ∆ ) iscomputed by the G m -action induced by the test configuration structure. The second main application of Theorem 1.1 is on the general construction of the K-moduli space. Indeed, it is proved in [BHLLX20] that Theorem 1.2 implies that there is aΘ-stratification on the stack M Fano n,V,C of Q -Gorenstein families of n -dimensional log Fanopairs ( X, ∆) → B with a fixed volume V and C · ∆ being integral. By the general theoryof Θ-stratification [AHLH18], this yields the properness of the K-moduli space. We alsoconclude the projectivity following [XZ20]. Theorem 1.3 (=Theorem 5.3, Properness and Projectivity of K-moduli spaces) . TheK-moduli space M Kps n,V,C is proper, and the CM line bundle on M Kps n,V,C is ample.Remark . Theorem 1.3 is the last step in the general construction of the K-modulispace. We briefly review the previously known steps here:A notion of a family of log pairs over a general base was introduced in [Kol19]. [Jia20](which heavily relies on [Bir19, Bir16]) proved the boundedness of K-semistable Fanovarieties with a fix volume. A different proof, which only uses the solution to Batyrev’sconjecture [HMX14], is given in [XZ20b]. Then using boundedness of complements [Bir19],[BLX19] and [Xu20b] gave two different proofs for the openness of K-semistability, andas a consequence the open subfunctor M Kss n,V,C ⊆ M
Fano n,V,C parametrizing K-semistable logFano pairs yields an Artin stack of finite type, which is called the K-moduli stack .By [LWX18], [BX19] and [ABHLX20], we know that M Kss n,V,C admits a separated goodmoduli space M Kps n,V,C which parametrizes K-polystable log Fano pairs and is called theK-moduli space .The remaining part is the properness and projectivity of the K-moduli space. In[BHLLX20], it is proved that Theorem 1.2 implies the properness of M Kps n,V,C ; and in [XZ20],it is shown that Theorem 1.1 implies the ampleness of the CM line bundle (introduced in[Tia97, PT09]) on M Kps n,V,C .The next major consequence of Theorem 1.1 is the complete solution of the Yau-Tian-Donaldson conjecture for log Fano pairs, including singular ones.
Theorem 1.5 (=Theorem 5.2, Yau-Tian-Donaldson Conjecture) . A log Fano pair ( X, ∆) is uniformly K-stable ( resp. reduced uniformly K-stable ) if and only if it is K-stable ( resp.K-polystable ) . In particular, ( X, ∆) admits a weak K¨ahler-Einstein ( KE ) metric if andonly if it is K-polystable.Remark . In this generality, the direction which says that the existence of KE metricsimplies K-polystability was settled in [Ber16].For the converse direction, [BBJ18] initiated a variational approach to the Yau-Tian-Donaldson conjecture, and the analytic side of this approach was completed in [LTW19,
INITE GENERATION FOR VALUATIONS AND K-STABILITY 3
Li19], which shows that a log Fano pair ( X, ∆) admits a weak KE metric if and only ifit is reduced uniformly K-stable . Therefore, what remains to show is the purely algebro-geometric statement that K-polystability is equivalent to reduced uniform K-stability.When | Aut( X, ∆) | < ∞ , this means for any log Fano pair ( X, ∆) which is not uni-formly K-stable, i.e. δ ( X, ∆) ≤
1, we need to show it is also not K-stable by producinga degeneration that destabilizes the log Fano pair. In [BLX19] (see also [BLZ19]), a steptoward constructing such a degeneration was made. More precisely, it was shown that thevaluation which computes δ ( X, ∆)( ≤
1) is quasi-monomial, and the degeneration shouldbe the Proj of the graded ring associated to this valuation, provided it is finitely gen-erated. Theorem 1.1 addresses the finite generation and then it follows that the soughtdegeneration as in Theorem 1.2 exists. As a consequence, it establishes the equivalencebetween K-stability and uniform K-stability. In the more general case when the auto-morphism group is positive dimensional, it is shown in [XZ20] that Theorem 1.1 impliesthe equivalence between K-polystability and reduced uniform
K-stability, by refining theargument from [BLX19] and finding a quasi-monomial valuation that is not induced by aone parameter subgroup of Aut( X, ∆), which yields a non-product type degeneration.We note that when X is smooth and ∆ = 0, the above theorem was first proved in[CDS15, Tia15] using Cheeger-Colding-Tian theory, which seems difficult to generalize tothe case of general (possibly singular) log Fano pairs.We also prove the following statement, which is a (necessarily) modified version ofa conjecture first raised by Donaldson in [Don12, Conjecture 1] (see also [Sz´e13] and[BL18, Section 7] for some further discussions of the problem). Theorem 1.7 (=Theorem 5.5) . Let ( X, ∆) be a log Fano pair such that δ := δ ( X, ∆) < .For any sufficiently divisible integer m > and any general member D of the linearsystem | − m ( K X + ∆) | , if we take D = m D , then the pair ( X, ∆ + (1 − δ ) D ) is K-semistable. In particular, ( X, ∆ + (1 − δ ′ ) D ) is uniformly K-stable for any ≤ δ ′ < δ . For a smooth Fano manifold X , the above theorem was essentially implied by a com-bination of [Zhu20b] with analytic results in [CDS15, Tia15]. We also note that, as onecan easily see from our proof of Theorem 5.5, the integer m can be chosen uniformly forany bounded family ( X, ∆) → S of log Fano pairs, e.g. the family of all smooth Fanomanifolds with a fixed dimension.1.2. Outline of the proof of Theorem 1.1.
Recall that by [BLX19], any valuationcomputing δ ( X, ∆) ≤ Q -complement Γ, and every divisorial lcplace w of the complement (parametrized by the rational points of the dual complex DMR ( X, ∆ + Γ)) induces a weakly special degeneration of the log Fano pair. In addition,if finite generation holds for the minimizer, it is observed in [LX18] as w gets sufficientlyclose to the minimizer, the central fibers of the induced degenerations would be isomorphicto each other and in particular are bounded. Therefore, in order to prove Theorem 1.1, ourmain technical goal is to construct a specific Q -complement Γ with the given minimizer v as an lc place, and show that in a neighborhood of v in the dual complex DMR ( X, ∆+Γ),the rational points correspond to degenerations with bounded central fibers. It turns outthat the existence of such a complement is in fact equivalent to the finite generationproperty (Theorem 4.4). YUCHEN LIU, CHENYANG XU, AND ZIQUAN ZHUANG
The choice of the complement Γ is indeed tricky, and the argument takes several steps.In the first step, using compatible basis type divisors as first introduced in [AZ20], weprove a crucial improvement of [BLX19] that the complement Γ can be chosen to containsome fixed multiple of any effective divisor D ∼ Q − ( K X + ∆). The same argument alsoallows us to realize valuations computing δ ( X, ∆) < n +1 n as lc places of complements.Next, starting with some complement Γ ′ that has the minimizer as an lc place and alog resolution π : ( Y, E ) → ( X, ∆ + Γ ′ ), we run the construction in the previous step toget another Q -complement Γ ∼ Q − ( K X + ∆) containing a multiple of the pushforwardof some generally positioned ample Q -divisor G on Y . The key idea is to look at theintersection DMR ( X, ∆ + Γ ′ ) ∩ DMR ( X, ∆ + Γ) (which is just the set of lc places for thecomplement (Γ+Γ ′ )). See Definition 3.3 for the precise definition. The advantages of thevaluations in the above intersections are twofold: firstly, rational points in this new dualcomplex corresponds simply to weighted blowups on the fixed log smooth model Y , whichare somewhat easier to analyze; secondly and more importantly, they all correspond tospecial divisors, i.e. they induce klt degenerations of the log Fano pair (this follows from atie-breaking argument, see Lemma 3.5). For general complements the second property ishard to come by, which suggests that we are in the right direction. In particular, we caninvoke results from [Jia20, XZ20b] to prove the boundedness once we establish a uniformpositive lower bound on the α -invariants of the central fibers.For this purpose, we combine an analysis of the construction (especially the tie-breakingargument) in the previous steps with the Koll´ar-Shokurov connectedness lemma and showthat, roughly speaking, α -invariants of the central fibers can be uniformly bounded frombelow if there are uniform positive lower bounds for: firstly, α -invariants of the exceptionaldivisors F of the corresponding weighted blowups on Y ; secondly, nef thresholds of theample divisor G with respect to F . Since the divisors F have rather explicit descriptions,we are able to verify that both desired uniform lower bounds exist. Putting these factstogether, we get the sought estimate on the α -invariants and hence the boundedness. Acknowledgement : We would like to thank Xiaowei Wang for helpful discussions. YLis partially supported by NSF Grant DMS-2001317. CX is partially supported by NSFGrant DMS-1901849. 2.
Preliminaries
Throughout this paper, we work over an algebraically closed field k of characteristic 0.We follow the standard terminology from [KM98, Kol13]. Definition 2.1. A pair ( X, ∆) is a normal variety X together with an effective Q -divisor∆ such that K X + ∆ is Q -Cartier. A log Fano pair ( X, ∆) is a pair such that X is proper, − K X − ∆ is ample, and ( X, ∆) is klt.A log smooth model ( Y, E ) over a pair ( X, ∆) consists of a log resolution π : Y → ( X, ∆)and a reduced divisor E on Y , such that E + Ex( π ) + π − ∗ ∆ has simple normal crossing(SNC) support.2.1. Valuations.
In this subsection, we assume that X is a normal variety. Definition 2.2. A valuation v on X is a R -valued valuation v : K ( X ) × → R such that v has a center on X and v | k × = 0. By convention, we set v (0) = + ∞ . We denote by Val X INITE GENERATION FOR VALUATIONS AND K-STABILITY 5 the set of all valuations on X . Recall that the center of v on X , denoted by c X ( v ), is ascheme-theoretic point ζ ∈ X such that v ≥ O X,ζ , and v > m X,ζ . Denote by C X ( v ) := c X ( v ). Since X is separated, a center of v on X is unique ifexists. If, in addition, X is proper, then every valuation v has a center on X . The trivialvaluation v triv is defined as v triv ( f ) = 0 for any f ∈ K ( X ) × .For a valuation v ∈ Val X , we define its valuation ideal sheaf a p ( v ) for p ∈ R ≥ as a p ( v ) := { f ∈ O X | v ( f ) ≥ p } . We also define the valuation ideal sequence of v as a • ( v ) := ( a m ( v )) m ∈ Z ≥ .For a valuation v ∈ Val X , we define its valuation semigroup Φ + v := { v ( f ) | f ∈ O X,c X ( v ) \{ }} and its valuation group Φ v := { v ( f ) | f ∈ K ( X ) × } . The rational rank of v is definedas rat . rk( v ) := rank Z Φ v . Definition 2.3.
Let π : Y → X be a proper birational morphism where Y is normal.A prime divisor E on Y is called a prime divisor over X . It induces a valuation ord E : K ( X ) × → Z by taking the vanishing order along E . A valuation v ∈ Val X is called divisorial if v = c · ord E for some prime divisor E over X and some c ∈ R > . Definition 2.4.
Let π : Y → X be a birational morphism where Y is normal. Let η ∈ Y be a scheme-theoretic point such that Y is regular at η . For a regular system ofparameters ( y , · · · , y r ) of O Y,η and α ∈ R r ≥ , we define a valuation v α as follows. For f ∈ O Y,η \ { } , we may write f in d O Y,η ∼ = κ ( η ) J y , · · · , y r K as f = P β ∈ Z r ≥ c β y β , where c β ∈ κ ( η ) and y β = y β · · · y β r r with β = ( β , · · · , β r ). We set v α ( f ) := min {h α, β i | c β = 0 } . A valuation v ∈ Val X is called quasi-monomial if v = v α as above for some π : Y → X , η , ( y , · · · , y r ) and α . It is proven in [ELS03] that a valuation v is quasi-monomial if andonly if it is an Abhyankar valuation, i.e. v satisfies tr . deg( v ) + rat . rk( v ) = dim X wheretr . deg( v ) is the transcendental degree of v .If, in addition, π : ( Y, E = P li =1 E i ) → X is a log smooth model where ( y i = 0) = E i for 1 ≤ i ≤ r as an irreducible component of E , then we denote the set { v α | α ∈ R r ≥ } by QM η ( Y, E ). We also set QM(
Y, E ) := ∪ η QM η ( Y, E ) where η runs through all genericpoints of ∩ i ∈ J E i for some non-empty subset J ⊆ { , · · · , l } .2.2. Stability thresholds.
We first define the log discrepancy function for valuations.
Definition 2.5 ([JM12, BdFFU15]) . For a pair ( X, ∆), we define the log discrepancy function A X, ∆ : Val X → R ∪ { + ∞} as follows.(1) If v = c · ord E is divisorial, then A X, ∆ ( v ) := c · A X, ∆ ( E ) = c (1 + coeff E ( K Y − π ∗ ( K X + ∆))) . (2) If v = v α is quasi-monomial for a log smooth model ( Y, E ) over ( X, ∆), then A X, ∆ ( v ) := r X i =1 α i · A X, ∆ ( E i ) . It is clear that A X, ∆ is linear on QM η ( Y, E ). YUCHEN LIU, CHENYANG XU, AND ZIQUAN ZHUANG (3) According to [JM12], there is a retraction map r Y,E : Val X → QM(
Y, E ) for anylog smooth model π : ( Y, E ) → ( X, ∆) satisfying Supp(Ex( π ) + π − ∗ ∆) ⊆ E . Forany v ∈ Val X , we define A X, ∆ ( v ) := sup { A X, ∆ ( r Y,E ( v )) | ( Y, E ) is a log smooth model over ( X, ∆) } . From the definition, we know that A X, ∆ ( λv ) = λ · A X, ∆ ( v ) for any λ ∈ R ≥ . Note thata pair ( X, ∆) is lc (resp. klt) if and only if A X, ∆ ( v ) ≥ >
0) for all valuations v ∈ Val X \ { v triv } . We also setVal ◦ X := { v ∈ Val X | v = v triv and A X, ∆ ( v ) < + ∞} . Then it is clear that Val ◦ X contains all non-trivial quasi-monomial valuations on X . If( X, ∆) is lc, then v ∈ Val X is an lc place of ( X, ∆) if A X, ∆ ( v ) = 0. If ( Y, E ) is alog smooth model over an lc pair ( X, ∆) satisfying Supp(Ex( π ) + π − ∗ ∆) ⊆ E , thenby [JM12, Corollary 5.4] we know that the set of all lc places of ( X, ∆) coincides withQM( Y, E ′ ) where E ′ is the sum of irreducible components E i of E satisfying A X, ∆ ( E i ) = 0.In particular, any lc place of ( X, ∆) is a quasi-monomial valuation in QM( Y, E ).In the rest of this subsection, we assume that ( X, ∆) is a log Fano pair. Let r be apositive integer such that L := − r ( K X + ∆) is Cartier. Then the section ring of ( X, L )is given by R ( X, L ) := R = M m ∈ Z ≥ R m = M m ∈ Z ≥ H ( X, O X ( mL )) . Definition 2.6 ([FO18]) . Let m be a positive integer such that N m := h ( X, O X ( mL )) >
0. An m -basis type divisor D on X is a divisor of the following form D = 1 mrN m N m X i =1 ( s i = 0) , where ( s , · · · , s N m ) is a basis of the vector space R m = H ( X, O X ( mL )). It is clear that D ∼ Q − ( K X + ∆). Definition 2.7 ([BJ20]) . Let v ∈ Val ◦ X be a valuation. Let m be a positive integer. Wedefine the invariants T m ( v ) := max { mr v ( s ) | s ∈ R m \ { }} ,S m ( v ) := max { v ( D ) | D is of m -basis type } . We define the T -invariant and S -invariant of v as T X, ∆ ( v ) := sup m ∈ Z > T m ( v ) and S X, ∆ ( v ) := lim m →∞ S m ( v ) . Note that the above limit exists as finite real numbers by [BJ20, Corollary 3.6].
Definition 2.8 ([FO18, BJ20]) . Let m be a positive integer. Then we define δ m ( X, ∆) := inf { lct( X, ∆; D ) | D is of m -basis type } . The above infimum is indeed a minimum since m -basis type divisors are bounded, and lcttakes finitely many values on a bounded Q -Gorenstein family [Amb16, Corollary 2.10]. Inparticular, there exists some divisor E over X such that δ m ( X, ∆) = A X, ∆ ( E ) S m ( E ) . INITE GENERATION FOR VALUATIONS AND K-STABILITY 7
The stability threshold (also called δ -invariant ) of a log Fano pair ( X, ∆) is defined as δ ( X, ∆) := lim m →∞ δ m ( X, ∆) . Equivalently, we have δ ( X, ∆) = inf v ∈ Val ◦ X A X, ∆ ( v ) S X, ∆ ( v ) . We say that δ ( X, ∆) is computed by a valuation v ∈ Val ◦ X if δ ( X, ∆) = A X, ∆ ( v ) S X, ∆ ( v ) . Theorem 2.9 (Fujita-Li valuative criterion, see [Fuj19b, Li17]) . Let ( X, ∆) be a log Fanopair. Then ( X, ∆) is K-semistable ( resp. uniformly K-stable ) if and only if δ ( X, ∆) ≥ resp. δ ( X, ∆) > . When Aut( X, ∆) is positive dimensional and T ⊆ Aut( X, ∆) is a torus, there is also areduced version δ T ( X, ∆). See [XZ20, Appendix A] for more discussions. Definition 2.10 ([Tia87, CS08, BJ20]) . The α -invariant of a log Fano pair ( X, ∆) isdefined as α ( X, ∆) := inf { lct( X, ∆; D ) | D ∈ | − K X − ∆ | Q } . Equivalently, we have α ( X, ∆) = inf v ∈ Val ◦ X A X, ∆ ( v ) T X, ∆ ( v ) . Definition 2.11.
More generally, for any projective klt pair ( X, ∆) and any effective Q -Cartier Q -divisor G on X we definelct( X, ∆; | G | Q ) := inf { lct( X, ∆; D ) | D ∈ | G | Q } . For any valuation v ∈ Val ◦ X , if we let T ( G ; v ) = sup { m v ( s ) | m ∈ N sufficiently divisible, s ∈ H ( X, O X ( mG )) } , then as above we have lct( X, ∆; | G | Q ) = inf v ∈ Val ◦ X A X, ∆ ( v ) T ( G ; v ) . In our argument later, we need the following theorem.
Theorem 2.12.
Fix positive integers n, C and three positive numbers
V, α , δ . If weconsider the set P of all n -dimensional log Fano pairs { ( X, ∆) } such that C · ∆ is integral, ( − K X − ∆) n = V and α ( X, ∆) ≥ α ( resp. δ ( X, ∆) ≥ δ ) . Then P is bounded.Proof. When ∆ = 0, this is first proved in [Jia20] which heavily relies on [Bir19, Bir16].See also [Che20]. Later a proof which only uses the boundedness result from [HMX14]was given in [XZ20b]. (cid:3)
YUCHEN LIU, CHENYANG XU, AND ZIQUAN ZHUANG
Filtrations and compatible basis type divisors.
In this subsection, we assumethat ( X, ∆) is a log Fano pair, and L = − r ( K X + ∆) is an ample Cartier divisor for some r ∈ Z > . Let R = ⊕ m ∈ Z ≥ H ( X, O X ( mL )) be the section ring of ( X, L ). Definition 2.13. A filtration F on R is a collection of vector subspaces F λ R m ⊆ R m forany m ∈ Z ≥ and λ ∈ R ≥ satisfying the following properties.(1) F λ R m ⊆ F λ ′ R m if λ ≥ λ ′ ;(2) F λ R m = ∩ λ ′ <λ F λ ′ R m if λ > F R m = R m and F λ R m = 0 for λ ≫ F λ R m · F λ ′ R m ′ ⊆ F λ + λ ′ R m + m ′ .A filtration F induces a function ord F : R m → R ≥ as ord F ( s ) := max { λ | s ∈ F λ R m } .By convention, we set ord F (0) = + ∞ .In this paper, we are mainly interested in the following two types of filtrations comingfrom valuations or divisors. Example 2.14.
Any valuation v ∈ Val X induces a filtration F v on R as F λv R m := { s ∈ R m | v ( s ) ≥ λ } . Any effective Q -divisor G on X induces a filtration F G on R as F λG R m := { s ∈ R m | ( s = 0) ≥ λG } . Definition 2.15.
Let F be a filtration on R . The associated graded ring gr F R of F isdefined asgr F R := M m ∈ Z ≥ M λ ∈ R ≥ gr λ F R m , where gr λ F R m := F λ R m / ∪ λ ′ >λ F λ ′ R m . We say that F is finitely generated if gr F R is a finitely generated k -algebra. For avaluation v ∈ Val X , we define the associated graded ring of v by gr v R := gr F v R . Notethat the grading of gr v R can be chosen as ( m, λ ) ∈ Z ≥ × Φ + v where Φ + v is the valuationsemigroup of v . Definition 2.16.
Let F be a filtration on R . A basis ( s , · · · , s N m ) of R m is said to be compatible with F if F λ R m is spanned by some of the s i ’s for every λ ∈ R ≥ . An m -basistype divisor D = mrN m P N m i =1 ( s i = 0) is said to be compatible with F if ( s , · · · , s N m ) iscompatible with F . By abuse of notation, we will say that an m -basis type divisor D iscompatible with a valuation v (resp. an effective Q -divisor G ) if D is compatible with thefiltration induced by v (resp. G ).From the definition, it is easy to see that for any v ∈ Val ◦ X , we have v ( D ) = S m ( v )for any m -basis type divisor D that is compatible with v . Another useful fact aboutcompatible divisors is the following. Lemma 2.17 ([AZ20, Lemma 3.1]) . Let F and G be two filtrations of R . Then for any m ∈ Z > there exists an m -basis type divisor that is compatible with both F and G . Definition 2.18.
Let F be a filtration of R . We define the T -invariant of F as T X, ∆ ( F ) := sup m ∈ Z > T m ( F ) ∈ [0 , + ∞ ] , where T m ( F ) := max { λmr | F λ R m = 0 } . INITE GENERATION FOR VALUATIONS AND K-STABILITY 9
By Fekete’s lemma, we know that T X, ∆ ( F ) = lim m →∞ T m ( F ). We say that F is linearlybounded if T X, ∆ ( F ) < + ∞ . Definition 2.19.
Let F be a linearly bounded filtration. Let ( s , · · · , s N m ) be a basis of R m that is compatible with F . We define the invariant S m ( F ) := 1 mrN m N m X i =1 ord F ( s i ) . The S -invariant of F is defined as S X, ∆ ( F ) := lim m →∞ S m ( F ) . Note that the above limit exists as finite real numbers by [BJ20, Lemma 2.9]. In fact, by loc. cit. , we have(2.1) S X, ∆ ( F ) = 1( − K X − ∆) n Z ∞ vol( F ( t ) R )d t where vol( F ( t ) R ) := lim m →∞ dim F mrt R m ( mr ) n /n ! .By [BJ20, Lemma 3.1], any valuation v ∈ Val ◦ X induces a linearly bounded filtration F v . From our definitions, it is easy to see that each invariant from T m , S m , T X, ∆ , and S X, ∆ has the same value for v and F v . For an effective Q -divisor G on X , we define S m ( G ) := S m ( F G ) and S X, ∆ ( G ) := S X, ∆ ( F G ). As before, we note that D ≥ S m ( G ) · G forany m -basis type divisor D that is compatible with G . The following calculation is alsovery useful for us. Lemma 2.20.
Let λ ∈ Q > and let G ∼ Q − λ ( K X + ∆) be an effective Q -divisor. Then S X, ∆ ( G ) = λ ( n +1) where n = dim X .Proof. We have vol( F ( t ) G R ) = vol( − K X − ∆ − tG ) = (1 − λt ) n vol( − K X − ∆), thus theresult follows from (2.1). (cid:3) Special divisors and complements.
Let ( X, ∆) be a log Fano pair. We first recallthe concepts of (weakly) special test configurations. Note that we omit the polarizationin the following definition, because (weakly) special test configurations are naturally anti-canonically polarized. Definition 2.21. A weakly special test configuration ( X , ∆ X ) of ( X, ∆) consists of thefollowing data: • a normal variety X together with a flat proper morphism π : X → A ; • a G m -action on X such that π is G m -equivariant with respect to the standard G m -action on A by multiplication; • X \ X is G m -equivariantly isomorphic to X × ( A \ { } ) where the G m -action on X is trivial; • an effective Q -divisor ∆ X on X such that ∆ X is the componentwise closure of∆ × ( A \ { } ) under the identification between X \ X and X × ( A \ { } ). • − ( K X + ∆ X ) is Q -Cartier and π -ample; • ( X , X + ∆ X ) is log canonical. A weakly special test configuration ( X , ∆ X ) of ( X, ∆) is special if ( X , X + ∆ X ) is plt.The central fiber ( X , ∆ ) of a special (resp. weakly special) test configuration ( X , ∆ X )is called a special (resp. weakly special ) degeneration of ( X, ∆). A weakly special testconfiguration is trivial if X is G m -equivariantly isomorphic to X × A where the G m -actionon X is trivial. Definition 2.22.
Let E be a prime divisor over X . We say that E is weakly special over( X, ∆) if there exists a weakly special test configuration ( X , ∆ X ) with integral centralfiber X , such that ord X | K ( X ) = b · ord E for some b ∈ Z > . A weakly special divisor E over ( X, ∆) is called special if ( X , ∆ X ) is a special test configuration. By abuse ofnotation, we will say ord E or any valuation v proportional to ord E is weakly special (resp. special ) if E is weakly special (or special).If E is a weakly special divisor over ( X, ∆) such that ord X | K ( X ) = b · ord E , then thecentral fiber ( X , ∆ ) is uniquely determined by E up to isomorphism, and X ∼ = Proj gr E R (see e.g. [Xu20, Section 3.6 and Lemma 3.7]). Definition 2.23. A Q -complement of ( X, ∆) is an effective Q -Cartier Q -divisor D ∼ Q − K X − ∆ such that ( X, ∆ + D ) is log canonical. A Q -complement D is called an N -complement for N ∈ Z > if N ( K X + ∆ + D ) ∼
0, and N (∆ + D ) ≥ N ⌊ ∆ ⌋ + ⌊ ( N + 1) { ∆ }⌋ where { ∆ } := ∆ − ⌊ ∆ ⌋ .For any Q -complement D of ( X, ∆), we define the dual complex of ( X, ∆ + D ) as DMR ( X, ∆ + D ) := { v ∈ Val ◦ X | A X, ∆+ D ( v ) = 0 and A X, ∆ ( v ) = 1 } . In particular, the space of all lc places of ( X, ∆ + D ) is a cone over DMR ( X, ∆ + D ). Theorem 2.24 ([BLX19, Theorems 3.5 and A.2]) . There exists N ∈ Z > depending onlyon dim( X ) and Coeff(∆) such that the following statements are equivalent for a primedivisor E over X . (1) E is a weakly special divisor over ( X, ∆) ; (2) E is an lc place of a Q -complement of ( X, ∆) ; (3) E is an lc place of an N -complement of ( X, ∆) . The following observation is made by the third named author (see [Xu20, Theorem4.12]).
Theorem 2.25.
The following statements are equivalent for a prime divisor E over X . (1) E is a special divisor over ( X, ∆) ; (2) A X, ∆ ( E ) < T X, ∆ ( E ) and there exists a Q -complement D ′ of ( X, ∆) such that, upto rescaling, E is the only lc place of ( X, ∆ + D ′ ) ; (3) there exists an effective Q -divisor D ∼ − K X − ∆ and t ∈ (0 , such that ( X, ∆ + tD ) is lc with E as the only lc place (up to rescaling). Log Fano pairs with δ ( X, ∆) < n +1 n In this section, for a valuation v which computes δ ( X, ∆), we carefully construct a Q -complement Γ such that v is an lc place of ( X, ∆ + Γ). In fact, Γ satisfies a numberof other technical properties (we call it a special complement , see Definition 3.3 for itsdefinition), which are indispensable for our proof of Theorem 1.1. INITE GENERATION FOR VALUATIONS AND K-STABILITY 11
As a by-product, in Section 3.2, we show that the various results in [BLX19] can beimproved using the construction of compatible basis type divisors introduced in [AZ20](see Definition 2.16).3.1.
Complements for higher rank valuations.
Recall that when δ ( X, ∆) ≤
1, anyvaluation computing δ ( X, ∆) is an lc place of a Q -complement [BLX19, Theorem A.7].Using compatible divisors, we first generalize this result to log Fano pairs with δ ( X, ∆) < n +1 n and investigate the degree of freedom when choosing such complements. Lemma 3.1.
Let ( X, ∆) be a log Fano pair of dimension n such that δ ( X, ∆) = δ < n +1 n ,and let v be a valuation that computes δ ( X, ∆) . Let α ∈ (0 , min { δn +1 , − nδn +1 } ) ∩ Q . Thenfor any effective divisor D ∼ Q − ( K X + ∆) , there exists some Q -complement Γ of ( X, ∆) such that Γ ≥ αD and v is an lc place of ( X, ∆ + Γ) .Proof. Up to rescaling, we may assume that A X, ∆ ( v ) = 1. By [BJ20, Proposition 4.8(ii)]and [Xu20b, Theorem 1.1], the valuation v is quasi-monomial. Let r be the rational rankof v . Let π : Y → X be a log resolution such that v ∈ QM(
Y, E ) for some simple normalcrossing divisor E = E + · · · + E r on Y . By [LX18, Lemma 2.7], for any ε > v , · · · , v r ∈ QM(
Y, E ) and positive integers q , · · · , q r such that • v is in the convex cone generated by v i , • for all i = 1 , · · · , r , the valuation q i v i is Z -valued and has the form ord F i for somedivisor F i over X , and • | v i − v | < εq i for all i = 1 , · · · , r .We claim that when ε is sufficiently small, there exists a Q -complement Γ ≥ αD of ( X, ∆)that has all v i as lc places. Since v is contained in their convex hull, the statement of thelemma then follows.To prove the claim, we first argue as in [LX18, Lemma 2.51] . Let a • be the gradedsequence of valuation ideals of v , i.e. a m = a m ( v ). The log discrepancy function w A X, ∆+ a • ( w ) is convex on QM( Y, E ). In particular, it is Lipschitz, hence there exists someconstant
C > | A X, ∆+ a • ( w ) − A X, ∆+ a • ( v ) | ≤ C | w − v | for any w ∈ QM(
Y, E ). Applying this to the divisorial valuations v i above, we find A X, ∆+ a • ( F i ) = q i A X, ∆+ a • ( v i ) ≤ Cq i | v i − v | ≤ Cε.
Therefore, for some 0 < ε ≪ ε ) we have(3.1) A X, ∆+ a − ε • ( F i ) < Cε for all 1 ≤ i ≤ r . Let 0 ≤ D ′ ∼ Q − ( K X + ∆) be general (so that it does not contain thecenter of v in its support). Now for any m ∈ N such that − m ( K X + ∆) is very ample,let G = βD ′ + (1 − β ) D where β = max { , ( n +1)( δ − δ } , and let D m be an m -basis type Q -divisor that is compatible with both G and v . Then we have D m ≥ S m ( G ) · G and v ( D m ) = S m ( v ). There is a small error in the proof of [LX18, Lemma 2.51], where the log discrepancy function A X, ∆+ a c • ( · ) was treated as a linear function on QM( Y, E ); nonetheless it is Lipschitz (since it is convex)and that is enough for the argument there.
Denote by D ′ m := D m − S m ( G ) · βD ′ ∼ Q − (1 − βS m ( G ))( K X + ∆) . Note that G ∼ Q − ( K X + ∆), thus lim m S m ( G ) = S X, ∆ ( G ) = n +1 (see Lemma 2.20) andlim m →∞ (1 − βS m ( G )) = min { , δ } by a direct calculation. It follows that we can choose asequence of rational numbers δ m > m ∈ N ) such that δ m < δ m ( X, ∆), lim m →∞ δ m = δ and δ m (1 − βS m ( G )) < m . In particular, ( X, ∆ + δ m D ′ m ) is log Fano and by ourassumption on α , we have δ m D ′ m ≥ (1 − β ) δ m S m ( G ) · D ≥ αD as m ≫ v computes δ ( X, ∆) and D ′ is general, we also see that as m ≫ δ m v ( D ′ m ) = δ m v ( D m ) ≥ (1 − ε ) δ ( X, ∆) S X, ∆ ( v ) = (1 − ε ) A X, ∆ ( v ) = 1 − ε . Combined with (3.1) we obtain A X, ∆+ δ m D ′ m ( F i ) ≤ A X, ∆+ a − ε • ( F i ) < Cε < ε < C . By [BCHM10, Corollary 1.4.3] we know that there exists a Q -factorial birationalmodel p : e X → X that extracts exactly the divisors F i . Let e D denote the strict transformof a divisor D on X . Let K e X + e ∆ + δ m e D ′ m + r X i =1 (1 − a i ) F i = p ∗ ( K X + ∆ + δ m D ′ m )be the crepant pullback, then a i ∈ (0 , Cε ) and as ( X, ∆ + δ m D ′ m ) is log Fano, we seethat ( e X, e ∆ + α e D + P ri =1 (1 − a i ) F i ) has a Q -complement.By the following Lemma 3.2, if ε is sufficiently small (depending only on C and thecoefficients of ∆ and αD ), then ( e X, e ∆+ α e D + P ri =1 F i ) also have a Q -complement. Pushingit forward to X , we obtain a Q -complement Γ ≥ αD of ( X, ∆) that realizes all F i as lcplaces. The proof is now complete. (cid:3) We have used the following well-known consequence of [HMX14] in the proof above.
Lemma 3.2.
Let ( X, ∆) be a projective pair and let G be an effective Q -Cartier Q -divisoron X . Assume that X is of Fano type. Then there exists some ε > depending only onthe coefficients of ∆ and G such that: if ( X, ∆ + (1 − ε ) G ) has a Q -complement, then thesame is true for ( X, ∆ + G ) .Proof. This should be well-known to experts (see e.g. [Bir16, Proof of Proposition 3.4]),but we provide a proof for readers’ convenience.Replacing X by a small Q -factorial modification, we may assume that X itself is Q -factorial. Let n = dim X and let I ⊆ Q + be the coefficient set of ∆ and G . By the ACCof log canonical thresholds and global ACC of log Calabi-Yau pairs [HMX14, Theorems1.1 and 1.5], we know that there exists some rational constant ε > n, I which satisfies the following property: for any pair ( X, ∆) of dimension at most n and any Q -Cartier divisor G on X such that the coefficients of ∆ and G belong to I , wehave ( X, ∆ + G ) is lc as long as ( X, ∆ + (1 − ε ) G ) is lc; if in addition there exists somedivisor D with (1 − ε ) G ≤ D ≤ G such that K X + ∆ + D ∼ Q
0, then D = G . INITE GENERATION FOR VALUATIONS AND K-STABILITY 13
Now let ( X, ∆ + (1 − ε ) G ) be a pair with a Q -complement Γ. As X is of Fano type, wemay run the − ( K X + ∆ + G )-MMP f : X X ′ . Let ∆ ′ , G ′ , Γ ′ be the strict transformsof ∆ , G, Γ. Note that by construction K X + ∆ + G ≤ f ∗ ( K X ′ + ∆ ′ + G ′ ) , hence ( X, ∆ + G ) has a Q -complement if and only if ( X ′ , ∆ ′ + G ′ ) has one. Since K X + ∆ + (1 − ε ) G + Γ ∼ Q , the MMP is crepant for the lc pair ( X, ∆ + (1 − ε ) G + Γ), hence ( X ′ , ∆ ′ + (1 − ε ) G ′ + Γ ′ ) isalso lc. It follows that ( X ′ , ∆ ′ + (1 − ε ) G ′ ) is lc, thus by our choice of ε , ( X ′ , ∆ ′ + G ′ ) is lcas well. Suppose that X ′ is Mori fiber space g : X ′ → S . Then K X ′ + ∆ ′ + G ′ is g -ample.Since K X ′ + ∆ ′ + (1 − ε ) G ′ ∼ Q − Γ ′ ≤ ρ ( X ′ ) = ρ ( S ) + 1, there exists some ε ′ ∈ (0 , ε ]such that K X ′ + ∆ ′ + (1 − ε ′ ) G ′ ∼ g. Q
0. But if we restrict the pair to the general fiberof X ′ → S we would get a contradiction to our choice of ε . Thus X ′ is a minimal modeland − ( K X ′ + ∆ ′ + G ′ ) is nef. As X ′ is also of Fano type, we see that − ( K X ′ + ∆ ′ + G ′ )is semiample, hence ( X ′ , ∆ ′ + G ′ ) has a Q -complement. By the previous discussion, thisimplies that ( X, ∆ + G ) has a Q -complement as well. (cid:3) To proceed, we make the following definition. Recall that a log smooth model (
Y, E )over ( X, ∆) consists of a log resolution π : Y → ( X, ∆) and a reduced divisor E on Y such that E + Ex( π ) + π − ∗ ∆ has SNC support (see Definition 2.1). Definition 3.3. A Q -complement Γ of ( X, ∆) will be called special with respect to alog smooth model π : ( Y, E ) → ( X, ∆) if Γ Y = π − ∗ Γ ≥ G for some effective ample Q -divisor G whose support does not contain any stratum of ( Y, E ). Any valuation v ∈ QM(
Y, E ) ∩ DMR ( X, ∆ + Γ) is called a monomial lc place of the special Q -complementΓ with respect to ( Y, E ).The following immediate consequence of Lemma 3.1 says a valuation computing δ ( X, ∆)when δ ( X, ∆) < n +1 n is a monomial lc place of a special complement. Later we will showfor a (possibly higher rank) valuation, if it is a monomial lc place of a special complement,its associated graded ring is finitely generated (see Theorem 4.2). Corollary 3.4.
Let ( X, ∆) be a log Fano pair of dimension n such that δ ( X, ∆) = δ < n +1 n , and let v be a valuation that computes δ ( X, ∆) . Then there exists a log smooth model π : ( Y, E ) → ( X, ∆) and a special Q -complement ≤ Γ ∼ Q − ( K X + ∆) with respect to ( Y, E ) , such that v ∈ QM(
Y, E ) ∩ DMR ( X, ∆ + Γ) .Proof. Since v is quasi-monomial, we may find a log smooth model π : ( Y, E ) → ( X, ∆)whose exceptional locus supports a π -ample divisor F such that v ∈ QM(
Y, E ). Choosesome 0 < ε ≪ L := − π ∗ ( K X + ∆) + εF is ample and let G be a generaldivisor in the Q -linear system | L | Q whose support does not contain any stratum of ( Y, E ).Let D = π ∗ G ∼ − ( K X + ∆) and let α < min { δn +1 , − nδn +1 } be a fixed rational positivenumber. By Lemma 3.1, there exists some complement Γ of ( X, ∆) such that Γ ≥ αD and v is an lc place of ( X, ∆ + Γ). Replace G by αG . By construction, the strict transformof Γ is larger or equal to G , so Γ is a special Q -complement with respect to ( Y, E ). (cid:3) To help understand the importance of special complements, we prove the followingstatement. In Section 4, we will need a stronger version of it (see Theorem 4.7).
Lemma 3.5.
Let Γ be a special Q -complement of ( X, ∆) with respect to a log smoothmodel ( Y, E ) . Denote by Π := QM(
Y, E ) ∩ DMR ( X, ∆ + Γ) the set of monomial lcplaces. Then every divisorial valuation v ∈ Π is special (see Definition 2.22).Proof. Fix v ∈ Π( Q ). By Theorem 2.25, it suffices to find an effective divisor D ∼ Q − ( K X + ∆) such that λ = lct( X, ∆; D ) ∈ (0 ,
1) and that (up to rescaling) v is the uniquelc place of ( X, ∆ + λD ). To see this, let W = C Y ( v ), and let ρ : Z → Y be the weightedblowup corresponding to v , so that v = c · ord F for some c >
0, where F is the exceptionaldivisor of ρ .By assumption, there exists an effective ample Q -divisor G on Y such that Γ Y ≥ G and C Y ( v ) Supp( G ). Since ( Y, E ) is log smooth, it is clear that F is a weighted projectivespace bundle over the smooth center W and − F is ρ -ample. In particular, there existssome ε > ρ ∗ G − εF is ample on Z . Let G be a general divisor in the Q -linearsystem | ρ ∗ G − εF | Q and consider D := Γ − π ∗ G + π ∗ ρ ∗ G . It is not hard to see that π ∗ D = π ∗ Γ − G + ρ ∗ G and D ∼ Q − ( K X + ∆). We claim that this divisor D satisfies thedesired conditions.Let K Y + ∆ Y = π ∗ ( K X + ∆) and K Z + ∆ Z = ρ ∗ ( K Y + ∆ Y ) be the crepant pullbacks.We first show that the above claim is a consequence of the following two properties:(1) ( Y, ∆ Y + π ∗ Γ − G ) is lc and v is an lc place of this pair;(2) F is the only divisor that computes lct( Y, ∆ Y ; ρ ∗ G ).This is because, (1) implies that A X, ∆ ( w ) = A Y, ∆ Y ( w ) ≥ w ( π ∗ Γ − G )for all valuations w on X , and equality holds when w is proportional to v ; on the otherhand, if we let µ = lct( Y, ∆ Y ; ρ ∗ G ) >
0, then (2) implies that A X, ∆ ( w ) = A Y, ∆ Y ( w ) ≥ µ · w ( ρ ∗ G )for all valuations w on X , and equality holds if and only if w is proportional to v . Com-bining the two inequalities we have w ( D ) = w ( π ∗ D ) = w ( π ∗ Γ − G + ρ ∗ G ) ≤ (1 + µ − ) A X, ∆ ( w )for all valuations w on X , and equality holds if and only if w is proportional to v . Inparticular, lct( X, ∆; D ) = µ − ∈ (0 ,
1) and up to rescaling, v is the unique lc place thatcomputes this lct, which is exactly what we want.It remains to prove the two properties above. Point (1) is quite straightforward sinceby assumption v is an lc place of the lc pair ( Y, ∆ Y + π ∗ Γ) and G does not contain C Y ( v ). To see point (2), we note that by assumption, ∆ Y has simple normal crossingsupport, ⌊ ∆ Y ⌋ ≤ X, ∆) is klt), and therefore one can easily check that the sub-pair( Z, ∆ Z ∨ F ) is plt. Here we denote by D ∨ D the smallest divisor D such that D ≥ D i for i = 1 ,
2. Let t = A Y, ∆ Y ( F ) ε . Then ρ ∗ ( K Y + ∆ Y + tρ ∗ G ) = K Z + ∆ Z ∨ F + tG by construction. Since G is general, the pair ( Z, ∆ Z ∨ F + tG ) is also plt. This proves(2). The proof is now complete. (cid:3) INITE GENERATION FOR VALUATIONS AND K-STABILITY 15
Minimizers and constructibility.
The existence of a valuation computing δ ( X, ∆)is proved to exist in [BJ20] if the ground field k is uncountable, and in [BLX19] when δ ( X, ∆) ≤ Q -complement. Here we extend these results to the case when δ ( X, ∆) is bounded by n +1 n . Theorem 3.6.
Let ( X, ∆) be a log Fano pair of dimension n such that δ ( X, ∆) < n +1 n .Then (1) there exists a valuation computing δ ( X, ∆) ; and (2) there exists a positive integer N depending only on dim( X ) and the coefficients of ∆ such that for any valuation v computing δ ( X, ∆) , there exists an N -complement D of ( X, ∆) which satisfies that v is an lc place of ( X, ∆ + D ) .Proof. First we prove (1). For any sufficiently divisible m ∈ N , let δ m := δ m ( X, ∆), andlet E m be a divisor over X such that A X, ∆ ( E m ) S m ( E m ) = δ m .Let H be a general divisor in | − m ( K X + ∆) | for some sufficiently divisible m whichdoes not contain the center of any E m . For any sufficiently divisible m , we can find an m -basis type divisor D m which is compatible with both E m and H by Lemma 2.17. Wewrite D m = Γ m + a m H where Supp(Γ) does not contain H . We know thatlct( X, ∆; D m ) ≤ A X, ∆ ( E m )ord E m ( D m ) = A X, ∆ ( E m ) S m ( D m ) = δ m , where the equality ord E m ( D m ) = S m ( D m ) follows from the fact that D m is chosen to becompatible with E m . However, we have lct( X, ∆; D m ) ≥ δ m by the definition of δ m . Thuslct( X, ∆; D m ) = δ m and the lct is computed by E m . Since H does not contain the centerof E m , it follows that ( X, ∆ + δ m Γ m ) is lc and E m is an lc place of this pair.Note that lim m →∞ δ m = δ ( X, ∆) < n +1 n . By Lemma 2.20, we also have lim m →∞ a m = m ( n +1) . So for sufficient large m , we get δ m Γ m = δ m ( D m − a m H ) ∼ Q − λ m ( K X + ∆)where λ m = δ m (1 − m a m ) ∈ (0 , E m is an lc place of a Q -complement. The restof the proof is the same as in [BLX19, Theorems 4.6 and A.7]: we know that E m is indeedan lc place of an N -complement for some N that only depends on dim( X ) and Coeff(∆).Therefore, after passing to a subsequence, we can find a fix N -complement D , togetherwith lc places F m of ( X, ∆ + D ), such that A X, ∆ ( E m ) S X, ∆ ( E m ) = A X, ∆ ( F m ) S X, ∆ ( F m ) for all m ∈ N + . If wetake v to be the limit of F m in DMR ( X, ∆ + D ), then v computes δ ( X, ∆), as A X, ∆ ( v ) S X, ∆ ( v ) = lim m →∞ A X, ∆ ( F m ) S X, ∆ ( F m ) = lim m →∞ A X, ∆ ( E m ) S X, ∆ ( E m ) = lim m →∞ δ m = δ ( X, ∆) . For (2), it follows immediately from Lemma 3.1 that v is an lc place of a Q -complementΓ. Let µ : Y → X be a dlt modification of ( X, ∆ + Γ). In particular, Y is a Fano typevariety. Then we can argue as in the proof of [BLX19, Theorem 3.5]: ( Y, Ex( µ ) + µ − ∗ ∆)has an N -complement, whose pushforward on X gives the N -complement D of ( X, ∆)that we seek for. (cid:3) Combining Theorem 3.6 with the argument in [BLX19, Section 4], we get the followinggeneralization of [BLX19, Theorem 1.1].
Corollary 3.7.
For a Q -Gorenstein family of log Fano pairs ( X, ∆) → S over a normalbase, the function t ∈ S min (cid:26) n + 1 n , δ ( X ¯ t , ∆ ¯ t ) (cid:27) is lower semi-continuous and constructible, where ( X ¯ t , ∆ ¯ t ) is the base change to the alge-braic closure of k ( t ) .Remark . It is proved in [Zhu20b] thatmin { δ ( X ¯ t , ∆ ¯ t ) , } = min { δ ( X t , ∆ t ) , } . However, in general we may have δ ( X ¯ t , ∆ ¯ t ) < δ ( X t , ∆ t ) (see [CP21, Remark 4.16]).4. Finite generation
In this section, we prove the following finite generation result.
Theorem 4.1.
Let ( X, ∆) be a log Fano pair of dimension n and let r > be an integersuch that r ( K X + ∆) is Cartier. Let R = L m ∈ Z ≥ H ( X, − mr ( K X + ∆)) . Assume that δ ( X, ∆) < n +1 n . Then for any valuation v that computes δ ( X, ∆) , the associated gradedring gr v R is finitely generated. To tackle Theorem 4.1, we need some finite generation criterion for lc places of com-plements. As shown by the examples in [AZ20, Theorem 1.4] and Section 6, one needsextra assumptions on the valuation and the complement. It turns out that the specialcomplements as defined in Definition 3.3 will be the correct one for proving Theorem4.1. In other words, monomial lc places of special complements have finitely generatedassociated graded rings (see Theorem 4.2).To prove this, we will show that in a neighborhood of a monomial lc place of a spe-cial complement, the divisorial ones induce degenerations to log Fano pairs whose alphainvariants are bounded from below by a positive constant (see Theorem 4.7), and this issufficient for the finite generation (see Theorem 4.4).4.1.
Finite generation criterion.
The next statement gives us the finite generationcriterion. Clearly, it implies Theorem 4.1 by Corollary 3.4.
Theorem 4.2.
Let ( X, ∆) be a log Fano pair, and let v be a quasi-monomial valuationon X . Let R = L m ∈ N H ( X, − mr ( K X + ∆)) for some integer r > such that r ( K X + ∆) is Cartier. Then the following are equivalent. (1) The associated graded ring gr v R is finitely generated and the central fiber ( X v , ∆ v ) of the induced degeneration is klt. (2) The valuation v is a monomial lc place of a special Q -complement Γ with respectto some log smooth model ( Y, E ) (see Definition 3.3). The remaining part of this section will be devoted to the proof of this theorem. In thissubsection, we reduce the proof to showing the boundedness of the degenerations inducedby divisorial valuations that are sufficiently closed to v .We first prove the easier direction in Theorem 4.2. Lemma 4.3.
Assume that gr v R is finitely generated and ( X v , ∆ v ) is klt. Then v is amonomial lc place of a special Q -complement. INITE GENERATION FOR VALUATIONS AND K-STABILITY 17
Proof.
Let π : ( Y, E ) → ( X, ∆) be a log smooth model whose exceptional locus supportsa π -ample divisor F such that v ∈ QM(
Y, E ). Let D ∼ Q − ( K X + ∆) be the divisorconstructed in the proof of Corollary 3.4 so that the strict transform G = π − ∗ D is ample.We have D = r { f = 0 } for some r ∈ N and some f ∈ H ( X, − r ( K X + ∆)). By the proofof [LX18, Lemma 2.10], we know that there exists some f := f, f , · · · , f m ∈ R whoserestrictions form a (finite) set of generators ¯ f , · · · , ¯ f m of gr v R (in particular, f , · · · , f m generates R ). Moreover, for all valuations w ∈ QM(
Y, E ) that are contained in theminimal rational affine subspace Σ of QM(
Y, E ) containing v and sufficiently close to v ,we have an isomorphism gr w R ∼ = gr v R and the restriction of f , · · · , f m ∈ R also generatesgr w R .By assumption, ( X v , ∆ v + εD v ) is klt for some constant 0 < ε ≪
1, thus the sameholds for divisorial valuations w in a sufficiently small neighbourhood U ⊆ Σ of v . Fora fixed divisorial valuation w ∈ U , by Theorem 2.24, there exists a Q -complement 0 ≤ Γ ∼ Q − ( K X + ∆ + εD ) such that ( X, ∆ + εD + Γ ) is lc and has w as an lc place. LetΓ = εD + Γ . Since the isomorphism gr w R ∼ = gr v R sends ¯ f i to ¯ f i for any w ∈ U , the factthat w (Γ) = A X, ∆ ( w ) implies v (Γ) = A X, ∆ ( v ), i.e., v is also an lc place of ( X, ∆ + Γ).Since π − ∗ Γ ≥ εG and G is ample, it is a special Q -complement with respect to ( Y, E ) byconstruction. (cid:3)
The reverse direction of Theorem 4.2 is much harder. To this end, we drop the assump-tions on the complements and prove a weaker finite generation criterion.
Theorem 4.4.
Let ( X, ∆) be a log Fano pair and let ≤ Γ ∼ Q − ( K X + ∆) be a Q -complement. Let v be an lc place of ( X, ∆ + Γ) and let Σ ⊆ DMR ( X, ∆ + Γ) be theminimal rational PL subspace containing v . Then the following are equivalent: (1) The associated graded ring gr v R is finitely generated. (2) There exists an open neighbourhood v ∈ U ⊆ Σ such that the set { ( X v , ∆ v ) | v ∈ U ( Q ) := U ∩ Σ( Q ) } is bounded. (3) The S -invariant function v S X, ∆ ( v ) is linear on a neighbourhood of v in Σ . We first handle the implication (3) ⇒ (1). This is done in the next two lemmas bystudying the concavity of the S -invariant function. Lemma 4.5.
Let ≤ Γ ∼ Q − ( K X + ∆) be a Q -complement of the log Fano pair ( X, ∆) .Let v , v ∈ DMR ( X, ∆ + Γ) be divisorial valuations in the same simplex determined bya fixed log smooth model ( Y, E ) . There exists a natural linear map [0 , → QM(
Y, E ) sending v and v . For t ∈ (0 , we then denote v t to be the valuationcorresponding to t . Then (1) S X, ∆ ( v t ) ≥ (1 − t ) S X, ∆ ( v ) + tS X, ∆ ( v ) . (2) When equality holds, we have F λv t R m = Span { s ∈ R m | (1 − t ) v ( s ) + tv ( s ) ≥ λ } for all λ ∈ R and all m ∈ N . In particular, the filtration F v t is finitely generated. Proof.
Let F = F v , F = F v and let F t be the filtration given by F λt R m = Span { s ∈ R m | (1 − t ) v ( s ) + tv ( s ) ≥ λ } . We claim that S ( F t ) = (1 − t ) S ( v ) + tS ( v ). To see this, we pick a basis s , · · · , s N m of R m that is compatible with both F and F by Lemma 2.17. It is straightforward tocheck that this basis is also compatible with F t , thus S m ( F t ) = 1 mrN m N m X i =1 ((1 − t ) v ( s i ) + tv ( s i )) = (1 − t ) S m ( v ) + tS m ( v ) . The claim then follows by letting m → ∞ . Clearly F λt R m ⊆ F λv t R m , hence S ( v t ) ≥ S ( F t ) = (1 − t ) S ( v ) + tS ( v ), which proves (1).Next assume that equality holds for some t ∈ (0 , t ∈ (0 , F ( λ ) v t R ) = vol( F ( λ ) t R ) for all λ ∈ R . Assumefor the moment that t ∈ Q . Let q > qv t is Z -valued; it is also the smallest integer such that the set of jumping numbers F t lies in q Z . By [Xu20c, Section 3], F t is finitely generated and induces a weakly specialdegeneration of ( X, ∆), thus by (the proof of) [BLX19, Theorem A.2], there exists some Z -valued divisorial valuations w i ( i = 1 , · · · , r ) and some a i ∈ Z such that F λt R m = { s ∈ R m | w i ( s ) ≥ λq + ma i for all 1 ≤ i ≤ r } . Suppose that F t = F v t . Then there exists some f ∈ R m such that v t ( f ) = µ > f ∈ F λt R m \ F >λt R m for some λ < µ . In particular, λ ∈ q Z , and for at least one of thevaluations w i , say w , we have w ( f ) = λq + ma . Let ε ∈ Q + be sufficiently small sothat η := ( µ − λ ) q + εma >
0. Then for sufficiently divisible integer k ∈ N , the kernel ofthe map R εmk · f k −→ F µkv t R (1+ ε ) mk / F µkt R (1+ ε ) mk is contained in F ηkw R εmk . It follows thatdim( F µkv t R (1+ ε ) mk / F µkt R (1+ ε ) mk ) ≥ dim( R εmk / F ηkw R εmk )and thus dividing out by k n /n ! and letting k → ∞ we obtainvol( F ( λ ′ ) v t R ) − vol( F ( λ ′ ) t R ) > λ ′ = µ (1 + ε ) m , a contradiction. Hence the two filtrations F t and F v t coincides when t ∈ Q .In general, for any fixed m ∈ N and any irrational t ∈ (0 , F (1 − t ) a + tbt R m = F (1 − t ′ ) a + t ′ bt ′ R m and F (1 − t ) a + tbv t R m = F (1 − t ′ ) a + t ′ bv t ′ R m for some t ′ ∈ Q that is sufficiently close to t . Thus by the previous discussion, it followsthat F t = F v t for irrational t as well. Since gr F t R ∼ = gr v (gr v R ) is finitely generated by[Xu20c, Section 3], the same holds for F v t . The proof is now complete. (cid:3) Lemma 4.6.
Let ≤ Γ ∼ Q − ( K X + ∆) be a Q -complement of the log Fano pair ( X, ∆) ,and let P be a convex subset in some simplex of DMR ( X, ∆ + Γ) . Assume that the linearspan of P is rational, and S X, ∆ ( · ) is linear on P . Then for any v ∈ P , the associatedfiltration F v is finitely generated. INITE GENERATION FOR VALUATIONS AND K-STABILITY 19
Proof.
We first claim that for any valuations v , v ∈ P and any t ∈ (0 , F λv t R m = Span { s ∈ R m | (1 − t ) v ( s ) + tv ( s ) ≥ λ } where v t = (1 − t ) v + tv . Indeed, by Lemma 4.5 we know that this holds when v , v aredivisorial; on the other hand, by assumption we know that the set of divisorial valuationsare dense in P , thus (4.1) holds for all v i ∈ P and t ∈ (0 ,
1) as in the proof of Lemma 4.5.Now for any v ∈ P , we may find some t ∈ (0 ,
1) and some valuation v ∈ P such that v t = (1 − t ) v + tv is divisorial. By (4.1), we see that gr v (gr v R ) ∼ = gr v t R as in [Xu20c] andsince v t is divisorial, the associated graded ring gr v t R is finitely generated by [BCHM10].It follows that gr v (gr v R ) is finitely generated, hence the same holds for gr v R by liftingthe generators of gr v (gr v R ). Thus F v is finitely generated as desired. (cid:3) We now present the proof of Theorem 4.4.
Proof of Theorem 4.4. (1) ⇒ (2) is well-known (in fact in a suitable neighbourhood U all ( X v , ∆ v ) are isomorphic), see [LX18, Lemma 2.10]. (3) ⇒ (1) by Lemma 4.6. So itremains to prove (2) ⇒ (3). The key point is that if a concave function takes rationalvalues with linearly bounded denominators on rational points, then it is linear.By Lemma 4.5, the S -invariant function is concave on the simplex containing v . Inparticular, it is Lipschitz and we may find some constant C such that(4.2) | S X, ∆ ( v ) − S X, ∆ ( v ) | ≤ C | v − v | in a neighbourhood of v . Let S = S X, ∆ ( v ). By [LX18, Lemma 2.7], for any ε > v , · · · , v r ∈ Σ, rational numbers S , · · · , S r and positiveintegers q , · · · , q r such that • ( v , S ) is in the convex cone generated by ( v i , S i ), i.e., there exists some λ i > v = P ri =1 λ i v i and S = P ri =1 λ i S i . • ( q i v i , q i S i ) is an integer vector for all i = 1 , · · · , r . • | v i − v | + | S i − S | < εq i for all i = 1 , · · · , r .In particular, by the last condition, we may assume that v i ∈ U , where U ⊆ Σ is the openneighbourhood of v in the condition (2).Since the set { ( X v , ∆ v ) | v ∈ U ( Q ) } is bounded, there exists some integer M > M ( K X + ∆) is Cartier and that M · Fut X v , ∆ v ( ξ ) ∈ Z for any v ∈ U ( Q ) and anyone parameter subgroup ξ : G m → Aut( X v , ∆ v ). If we let q be an integer such that qv is integral, then M q · A X, ∆ ( v ) ∈ Z . Note that qv induces a one parameter subgroup ξ v : G m → Aut( X v , ∆ v ) with Fut X v , ∆ v ( ξ v ) = q · β X, ∆ ( v ). It follows that M q · S X, ∆ ( v ) ∈ Z for any v ∈ U ( Q ) and any integer q such that qv is integral. In particular, we have M q i · S X, ∆ ( v i ) ∈ Z .On the other hand, by (4.2) we have | S X, ∆ ( v i ) − S | ≤ Cεq i and hence | M q i · S X, ∆ ( v i ) − M q i S i | ≤ M q i · | S X, ∆ ( v i ) − S | + M q i · | S i − S | ≤ ( C + 1) M ε.
Note that the constants C and M are independent of the choice of ε . Thus if we take ε = C +1) M , then as M q · S X, ∆ ( v ) and q i S i are both integers we deduce that S X, ∆ ( v i ) = S i .But as S X, ∆ ( · ) is concave on U and v = P ri =1 λ i v i , we also have r X i =1 λ i S X, ∆ ( v i ) ≤ S X, ∆ ( v ) = S = r X i =1 λ i S i = r X i =1 λ i S X, ∆ ( v i ) . Hence the first inequality is an equality, which forces S X, ∆ ( · ) to be linear on the conegenerated by v , · · · , v r . In particular, it is linear in a neighbourhood of v in Σ. (cid:3) Estimate of alpha invariants.
We next proceed to check the condition of Theorem4.4(2) when the complement is special. In order to control the boundedness of ( X v , ∆ v )for v ∈ Σ( Q ), we wish to apply the boundedness result as in Theorem 2.12. In lightof Lemma 3.5, we already know ( X v , ∆ v ) is klt. Then we need to further analyze the α -invariant of ( X v , ∆ v ).The following theorem is our main result in this section. Theorem 4.7.
Let ( X, ∆) be a log Fano pair and Γ a special complement with respect toa log smooth model ( Y, E ) (see Definition 3.3). Let K ⊂ DMR ( X, ∆ + Γ) be a compactsubset that is contained in the interior of a simplicial cone in QM(
Y, E ) . Then thereexists some constant α > such that for all rational points v ∈ K , the alpha invariants α ( X v , ∆ v ) of the induced degenerations ( X v , ∆ v ) are bounded from below by α . Our main tool is the following characterization of α -invariants. Lemma 4.8.
Let v be a divisorial valuation such that gr v R is finitely generated and let α ∈ (0 , . Then α ( X v , ∆ v ) ≥ α if and only if for all ≤ D ∼ Q − ( K X + ∆) , there existssome ≤ D ′ ∼ Q − ( K X + ∆) such that ( X, ∆ + αD + (1 − α ) D ′ ) is lc and has v as an lcplace. For ease of notation, we call such D ′ an ( α, v ) -complement of D . Proof.
Note that ( X v , ∆ v ) has an induced G m -action. By taking the limit under the G m -action, we see that any effective Q -divisor G ∼ Q − ( K X v + ∆ v ) degenerates to some G m -invariant divisor G . By the semi-continuity of log canonical thresholds, we havelct( X v , ∆ v ; G ) ≥ lct( X v , ∆ v ; G ), hence α ( X v , ∆ v ) ≥ α if and only if lct( X v , ∆ v ; G ) ≥ α for all G m -invariant divisors G ∼ Q − ( K X v + ∆ v ). Any such G is also the specializationof some divisor 0 ≤ D ∼ Q − ( K X + ∆) on X , and lct( X v , ∆ v ; G ) ≥ α means that v induces a weakly special degeneration of ( X, ∆ + αD ). By Theorem 2.24, this is the caseif and only if v is an lc place of a complement of ( X, ∆ + αD ); in other words, there existssome 0 ≤ D ′ ∼ Q − ( K X + ∆) such that ( X, ∆ + αD + (1 − α ) D ′ ) is lc and has v as an lcplace. (cid:3) Corollary 4.9.
Let v be a divisorial lc place of a complement of ( X, ∆) . Then α ( X v , ∆ v ) ≤ − A X, ∆ ( v ) T X, ∆ ( v ) . Proof.
We may assume that α ( X v , ∆ v ) >
0, otherwise since v is an lc place of somecomplement we have T X, ∆ ( v ) ≥ A X, ∆ ( v ) and the result is clear. Let α = α ( X v , ∆ v ). Notethat α ( X v , ∆ v ) < G m -action. Choose some effective divisor D ∼ Q − ( K X + ∆) whose supportdoes not contain C X ( v ). By Lemma 4.8, there exists some 0 ≤ D ′ ∼ Q − ( K X + ∆)such that v is an lc place of ( X, ∆ + αD + (1 − α ) D ′ ). In particular, (1 − α ) v ( D ′ ) = v ( αD + (1 − α ) D ′ ) = A X, ∆ ( v ), which implies A X, ∆ ( v ) ≤ (1 − α ) T X, ∆ ( v ). In other words, α ( X v , ∆ v ) ≤ − A X, ∆ ( v ) T X, ∆ ( v ) . (cid:3) INITE GENERATION FOR VALUATIONS AND K-STABILITY 21
In order to construct ( α, v )-complements for some uniform constant α (and thereforeproduce a uniform lower bound for α ( X v , ∆ v ) by Lemma 4.8), our strategy is to refine theproof of Lemma 3.5 using the alpha invariants and nef thresholds of the correspondingexceptional divisors F . This is done in the next three lemmas.To this end, we introduce some notation. Under the notation and assumptions ofTheorem 4.7, we fix an effective ample Q -divisor G on Y that does not contain any stratumof E such that Γ Y ≥ G . For any divisorial valuation v ∈ DMR ( X, ∆ + Γ) ∩ QM(
Y, E ),let ρ : Z → Y be the corresponding weighted blowup, F the exceptional divisor (i.e. v = c · ord F ), and ( Z, ∆ Z ), ( Y, ∆ Y ) the crepant pullbacks as in the proof of Lemma3.5. Let ∆ + := ∆ Z ∨ ∨ F . Note that ( Z, ∆ + ) is plt. By adjunction we may write K F + Φ = ( K Z + ∆ + ) | F . Let L := − ρ ∗ π ∗ ( K X + ∆) − A X, ∆ ( F ) · F. Since v = c · ord F is an lc place of ( X, ∆ + Γ), F is not contained in the support of ρ ∗ π ∗ Γ − A X, ∆ ( F ) · F ∼ Q L , thus the Q -linear system | L | F | Q is non-empty and we maydefine α v := lct( F, Φ; | L | F | Q ) . We also let ε v := sup { t ≥ | ρ ∗ G − tA X, ∆ ( F ) · F is nef } . Note that as − F is ρ -ample, we have ε v > t ∈ (0 , ε v ), the divisor ρ ∗ G − tA X, ∆ ( F ) · F is ample. Lemma 4.10.
Let a, b > be constants. Then there exists some constant α > dependingonly on a, b, ( X, ∆) and Γ such that α ( X v , ∆ v ) ≥ α as long as α v > a and ε v > b .Proof. We may assume that a <
1. By Lemma 4.8, it is enough to find some constant α > α, v )-complement exists for any 0 ≤ D ∼ Q − ( K X + ∆).As a reduction step, we first show that it suffices to check the existence of ( α, v )-complement for those D such that v ( D ) = A X, ∆ ( v ). Indeed, as G is ample, we may findsome constant 0 < λ ≪ G + λπ ∗ ( K X + ∆) remains ample. It follows that T ( G ; v ) ≥ λ · T X, ∆ ( v ), thus T X, ∆ ( v ) = T ( π ∗ Γ; v ) ≥ v ( π ∗ Γ − G ) + T ( G ; v ) ≥ v (Γ) + λ · T X, ∆ ( v ) = A X, ∆ ( v ) + λ · T X, ∆ ( v ) , or (1 − λ ) T X, ∆ ( v ) ≥ A X, ∆ ( v ). On the other hand, α ( X, ∆) T X, ∆ ( v ) ≤ A X, ∆ ( v ) by thedefinition of alpha invariants. Thus there exists some constant µ ∈ (0 ,
1) depending onlyon λ and α ( X, ∆) such that for any 0 ≤ D ∼ Q − ( K X + ∆), we can always find some0 ≤ D ∼ Q − ( K X + ∆) and ν ≥ µ such that νv ( D ) + (1 − ν ) v ( D ) = A X, ∆ ( v ). Clearly if( α, v )-complement exists for νD + (1 − ν ) D , then ( αµ, v )-complement exists for D . Thisproves the reduction.Next we fix a sufficiently small t ∈ (0 ,
1) such that s := (1 − a ) t − t < b . By assumption, ρ ∗ G − sA X, ∆ ( F ) · F is ample. Fix any 0 ≤ D ∼ Q − ( K X + ∆) with v ( D ) = A X, ∆ ( v ), let H ′ be a general member of the Q -linear system | ρ ∗ G − sA X, ∆ ( F ) · F | Q , and let H = ρ ∗ H ′ .We claim that along ρ ( F ) the pair( Y, ∆ Y + a · π ∗ D + 1 − tt H ) is lc and has F as its unique lc place. To see this, first we have A Y, ∆ Y ( F ) − ord F ( a · π ∗ D + 1 − tt H ) = A X, ∆ ( F ) − aA X, ∆ ( F ) − (1 − a ) A X, ∆ ( F ) = 0 . Then let D ′ = ρ ∗ π ∗ D − ord F ( D ) · F = ρ ∗ π ∗ D − A X, ∆ ( F ) · F ∼ Q L. By assumption, ( F, Φ+ aD ′ | F ) is klt, hence since H ′ is general, we see that ( F, Φ+ aD ′ | F + − tt H ′ | F ) is also klt. By inversion of adjunction, ( Z, ∆ + + aD ′ + − tt H ′ ) is plt along F .Since ∆ + ≥ ∆ Z ∨ F , we deduce that ( Z, ∆ Z ∨ F + aD ′ + − tt H ′ ) is also plt along F . Byconstruction and the above calculation, we can check that K Z + ∆ Z ∨ F + aD ′ + 1 − tt H ′ = ρ ∗ ( K Y + ∆ Y + a · π ∗ D + 1 − tt H ) , thus ( Y, ∆ Y + a · π ∗ D + − tt H ) is lc along ρ ( F ) and F is the only lc place there, provingthe previous claim.We also know that ( Y, ∆ Y + π ∗ Γ − G ) is lc and F is an lc place of the pair. Takingconvex linear combination as in the proof of Lemma 3.5, it follows that (cid:18) Y, ∆ Y + t (cid:18) a · π ∗ D + 1 − tt H (cid:19) + (1 − t )( π ∗ Γ − G ) (cid:19) = (cid:0) Y, ∆ Y + at · π ∗ D + (1 − t )( π ∗ Γ − G + H ) (cid:1) is lc along ρ ( F ) and F is the only lc place of the pair in a neighbourhood of ρ ( F ). Inparticular, ρ ( F ) is a connected component of the non-klt locus of the pair. Note that K Y + ∆ Y + at · π ∗ D + (1 − t )( π ∗ Γ − G + H ) = π ∗ ( K X + ∆ + atD + (1 − t )(Γ − π ∗ G + π ∗ H )) , thus ( X, ∆ + atD + (1 − t )(Γ − π ∗ G + π ∗ H )) is lc along C X ( v ) = π ( ρ ( F )) and C X ( v )is a connected component of its non-klt locus, since otherwise in some neighbourhood of π − C X ( v ) there would be another non-klt center of ( Y, ∆ Y + at · π ∗ D +(1 − t )( π ∗ Γ − G + H ))that is disjoint from ρ ( F ), contradicting the Koll´ar-Shokurov connectedness lemma.Similarly, as − ( K X + ∆ + atD + (1 − t )(Γ − π ∗ G + π ∗ H )) ∼ Q − (1 − a ) t ( K X + ∆)is ample, we deduce that ( X, ∆ + atD + (1 − t )(Γ − π ∗ G + π ∗ H )) is indeed lc everywhere,as otherwise there would be some non-klt center of the pair that is disjoint from C X ( v ),contradicting Koll´ar-Shokurov connectedness. Since by construction v = ord F is an lcplace of ( X, ∆ + atD + (1 − t )(Γ − π ∗ G + π ∗ H )), we may add some general divisor0 ≤ D ∼ Q − (1 − a ) t ( K X +∆) to the pair and conclude that D has an ( at, v )-complement.Since D is arbitrary and t only depends on a, b , this completes the proof. (cid:3) The argument for the following lemma is similar to the one in [Zhu20].
Lemma 4.11.
Notation as above. Let K ⊆ DMR ( X, ∆ + Γ) be a compact subset thatis contained in the interior of some simplicial cone in QM(
Y, E ) . Then there exist someconstants a > such that α v ≥ a for all divisorial valuations v ∈ K .Proof. Let E i ( i = 1 , · · · , r ) be the irreducible components of E so that W = E ∩ · · · ∩ E r is the common center of valuations in K on Y . Any divisorial valuation v ∈ K correspondsto a weighted blowup at W with weights wt( E i ) = a i for some integers a i > INITE GENERATION FOR VALUATIONS AND K-STABILITY 23 gcd( a , · · · , a r ) = 1. Since K is compact, there exists some constant C > a i a j < C for all 1 ≤ i, j ≤ r . Let c i = A X, ∆ ( E i ) > b i = max { , ord E i (∆ Y ) } < q i = gcd( a , · · · , ˆ a i , · · · , a r ), q = q · · · q r , and let a ′ i = a i q i q . Then it is not hard to checkthat (with notation as in the paragraph before Lemma 4.10):(1) the exceptional divisor F is a weighted projective space bundle over W with fiber F ∼ = P ( a ′ , · · · , a ′ r ),(2) A X, ∆ ( F ) = P ri =1 a i A X, ∆ ( E i ) = a c + · · · + a r c r ,(3) L F := L | F ∼ Q A X, ∆ ( F ) q L where L is the class of O (1) on P ( a ′ , · · · , a ′ r ),(4) Φ F := Φ | F = P ri =1 q i − b i q i { x i = 0 } where x , · · · , x r are the weighted homoge-neous coordinates on P ( a ′ , · · · , a ′ r ),(5) b m := ρ ∗ O Z ( − mF ) /ρ ∗ O Z ( − ( m + 1) F ) ∼ = L O W ( − ( m E + · · · + m r E r )), wherethe direct sum runs over all ( m , · · · , m r ) ∈ N r such that a m + · · · a r m r = m ,(6) ρ ∗ O F ( mL ) ∼ = O Y ( − mπ ∗ ( K X + ∆)) ⊗ b mA X, ∆ ( F ) for all sufficiently divisible m ∈ N .Combining (2), (5), (6) and the fact that a i a j < C , we see that there exists an ampleline bundle H (independent of a i ) on W such that ρ ∗ O F ( mL ) ֒ → O W ( mH ) ⊕ N m for someinteger N m (the rank of b mA X, ∆ ( F ) ). Since F is toric, by [BJ20, Theorem F] we know thatlct( F , Φ F ; | L F | Q ) is computed by one of torus invariant divisors { x i = 0 } , thus by (2),(3) and (4) we get lct( F , Φ F ; | L F | Q ) = min ≤ i ≤ r a i (1 − b i ) a c + · · · + a r c r ≥ a for some constants a that only depend on b i , c i and C . Replacing a by a smaller number,we may further assume that lct( W ; | H | Q ) ≥ a . We claim that α v ≥ a .To see this, let t ∈ (0 , a ) and let Φ ′ ∼ Q L | F be an effective divisor. Suppose that( F, Φ + t Φ ′ ) is not lc. Then since ( F, Φ + t Φ ′ ) is lc along the general fiber of F → W by our choice of a , we know that there exists a divisorial valuation v over F such that A F, Φ+ t Φ ′ ( v ) < v does not dominate W . By [Zhu20, Lemma 2.1], v restricts to a divisorial valuation w on W .Let g : W → W be a birational morphism such that the center of w is a divisor Q on W , let F = F × W W , Φ = g ∗ Φ (we also denote the projection F → F by g ),and let P be the preimage of Q in F . Since t < a , ρ ∗ O F ( mL | F ) ֒ → O W ( mH ) ⊕ N m andlct( W ; | H | Q ) ≥ a , we haveord P ( t Φ ′ ) < a · T ( H ; ord Q ) ≤ A W ( Q ) = A F, Φ ( P ) . Thus if we write g ∗ ( K F + Φ + t Φ ′ ) = K F + Φ + λP + D where P Supp( D ), then λ ≤ P is vertical, over a general fiber of P → Q (whichwe still denote by F ), we have D | F ∼ Q tg ∗ Φ ′ | F ∼ Q tL | F , thus by our choice of a ,( P, (Φ + D ) | P ) is lc along the general fibers of P → Q , hence by inversion of adjunctionwe see that ( F , Φ + λP + D ) is also lc along the general fibers of P → Q . In particular,it is lc at the center of v , a contradiction. Thus ( F, Φ + t Φ ′ ) is always lc and α v ≥ a asdesired. (cid:3) Lemma 4.12.
Notation as above. Let K ⊆ DMR ( X, ∆ + Γ) be a compact subset thatis contained in the interior of some simplicial cone in QM(
Y, E ) . Then there exist someconstants b > such that ε v ≥ b for all divisorial valuations v ∈ K .Proof. We continue to use the notation from the proof of Lemma 4.11. Let a m := ρ ∗ O Z ( − mA X, ∆ ( F ) · F ) . Since a i a j < C for all 1 ≤ i, j ≤ r , there exists some constant M ∈ N such that1 A X, ∆ ( F ) ord F ( f ) ≥ M mult W ( f )for all regular function f around the generic point of W . In particular, I MmW ⊆ a m for all m ∈ N . As in the proof of Lemma 4.11, we can find a sequence of ideals(4.3) O Y ⊇ I W ⊇ · · · ⊇ a m ⊇ · · · ⊇ I MmW on Y such that the quotients of consecutive terms are all isomorphic to O W ( − n E −· · · − n r E r ) for some ( n , · · · , n r ) ∈ N with P ri =1 n i ≤ M m . Thus we can choose somesufficiently large and divisible integer p > pG and ( pG − E i ) | W are globally generated for all i = 1 , · · · , r ;(2) H i ( W, O W ( mpG − n E −· · ·− n r E r )) = 0 for all i, m ∈ N + and all ( n , · · · , n r ) ∈ N with P ri =1 n i ≤ M m (this is possible by Fujita vanishing); and(3) O Y ( mpG ) ⊗ I MmW is globally generated and H i ( Y, O Y ( mpG ) ⊗ I mMW ) = 0 for all i, m ∈ N + .Working inductively along the filtration (4.3), it is then not hard to see that O Y ( mpG ) ⊗I is globally generated and H i ( Y, O Y ( mpG ) ⊗ I ) = 0 for any i, m ∈ N + and any ideal sheaf I ⊆ O Y that appears in the sequence (4.3). In particular, O Y ( mpG ) ⊗ a m is globallygenerated for all m ∈ N , which implies that pρ ∗ G − A X, ∆ ( F ) · F is nef. In other words, ε v ≥ p and we are done since p does not depend on the valuation v . (cid:3) Proof of Theorem 4.7.
The result now follows from a combination of Lemmas 4.10, 4.11and 4.12. (cid:3)
We now have all the ingredients to prove Theorem 4.2 and Theorem 4.1.
Proof of Theorem 4.2.
By Lemma 4.3 we already have (1) ⇒ (2), so it remains to prove(2) ⇒ (1). Let v be a valuation that satisfies (2), let Σ ⊆ QM(
Y, E ) ∩ DMR ( X, ∆ + Γ)be the smallest rational PL subspace containing v , and let U ⊆ Σ be a small openneighbourhood of v such that the closure of U is contained in the interior of the simplicialcone in QM( Y, E ) that contains v . By Theorem 4.4, it is enough to show that the set { ( X w , ∆ w ) | w ∈ U ( Q ) } is bounded. By Theorem 2.12, this is true if α ( X w , ∆ w ) ≥ α for some constant α > w ∈ U ( Q ), which then follows fromTheorem 4.7. (cid:3) Proof of Theorem 4.1.
This follows immediately from Theorem 4.2 and Corollary 3.4. (cid:3)
INITE GENERATION FOR VALUATIONS AND K-STABILITY 25 Applications
In this section we present some applications of the finite generation results from theprevious section. As we mentioned, combining with earlier works, Theorem 1.1 solves anumber of major questions on the study of K-stability of Fano varieties.
Theorem 5.1 (Optimal Destabilization Conjecture) . Let ( X, ∆) be a log Fano pair ofdimension n such that δ ( X, ∆) < n +1 n , then δ ( X, ∆) ∈ Q and there exists a divisorialvaluation E over X such that δ ( X, ∆) = A X, ∆ ( E ) S X, ∆ ( E ) .In particular, if δ ( X, ∆) ≤ , then there exists a non-trivial special test configuration ( X , ∆ X ) with a central fiber ( X , ∆ ) such that δ ( X, ∆) = δ ( X , ∆ ) , and δ ( X , ∆ ) iscomputed by the G m -action induced by the test configuration structure.Proof. Let v be a valuation on X that computes δ ( X, ∆). By Lemma 3.1, there existssome complement Γ of ( X, ∆) such that v ∈ DMR ( X, ∆ + Γ). Let Σ ⊆ DMR ( X, ∆ + Γ)be the smallest rational PL subspace containing v . By Theorem 4.4, the S -invariantfunction w S X, ∆ ( w ) on Σ is linear in a neighbourhood of v . As v computes δ ( X, ∆),we have A X, ∆ ( v ) = δ ( X, ∆) S X, ∆ ( v ) . Since the log discrepancy function w A X, ∆ ( w ) is linear in a neighbourhood of v ∈ Σand by the definition of stability thresholds we have A X, ∆ ( w ) ≥ δ ( X, ∆) S X, ∆ ( w )for all w ∈ Σ, we see that A X, ∆ ( w ) = δ ( X, ∆) S X, ∆ ( w )in a neighbourhood U ⊆ Σ of v . In particular, any divisorial valuation w ∈ U ( Q ) alsocomputes δ ( X, ∆). Since w is a divisorial lc place of a complement, it induces a weaklyspecial test configuration of ( X, ∆) by [BLX19, Theorem A.2]. By [Li17, Proof of Theorem3.7] or [Fuj19b, Theorem 5.2] we know that β X, ∆ ( w ) = A X, ∆ ( w ) − S X, ∆ ( w ) is rational.Since A X, ∆ ( w ) is clearly rational, we see that δ ( X, ∆) is also rational.For the last part, it follows from [BLZ19, Theorem 1.1] as the conjectural assumptionthere is verified by the first part. (cid:3) Theorem 5.2 (Yau-Tian-Donaldson conjecture) . A log Fano pair ( X, ∆) is uniformlyK-stable if and only if it is K-stable; and it is reduced uniformly K-stable if and onlyif it K-polystable. In particular, ( X, ∆) admits a weak KE metric if and only if it isK-polystable.Proof. Suppose first that δ ( X, ∆) ≤
1. Then by Theorem 5.1, the stability threshold iscomputed by some divisor E over X . By [BX19, Theorem 4.1], this implies that ( X, ∆)is not K-stable. In other words, if ( X, ∆) is K-stable, then δ ( X, ∆) >
1, i.e. ( X, ∆) isuniformly K-stable.Suppose next that ( X, ∆) is K-polystable. Let T ⊆ Aut( X, ∆) be a maximal torus.We show that δ T ( X, ∆) >
1. Suppose not, then by [XZ20, Appendix A], we know that δ T ( X, ∆) = 1 and δ ( X, ∆) is computed by some T -invariant quasi-monomial valuation v that is not of the form wt ξ for any ξ ∈ Hom( G m , T ) ⊗ Z R . Moreover, v is an lc place of a complement. Let m ∈ N be sufficiently divisible and consider the T -invariant linearsystem M := { s ∈ H ( − m ( K X + ∆)) | v ( s ) ≥ m · A X, ∆ ( v ) } . Let D ∈ |M| be a general member and let D = m D . Then ( X, ∆ + m M ) has the sameset of lc places as ( X, ∆ + D ) and thus by construction v ∈ DMR ( X, ∆ + D ). Since T is a connected algebraic group, every lc place of the T -invariant pair ( X, ∆ + m M ) isautomatically T -invariant. In particular, DMR ( X, ∆ + D ) consists only of T -invariantvaluations.By the same argument as in the proof of Theorem 5.1, we see that δ ( X, ∆) is alsocomputed by some divisorial valuations w ∈ DMR ( X, ∆ + D ) that are sufficiently closeto v (in particular, w = wt ξ as well). Since w is T -invariant, by [BX19, Theorem 4.1], w induces a T -equivariant special test configuration ( X , D ) of ( X, ∆) with Fut( X , D ) = 0.Since T ⊆ Aut( X, ∆) is a maximal torus and w = wt ξ for any ξ ∈ Hom( G m , T ) ⊗ Z R ,we deduce that ( X , D ) is not a product test configuration. But this contradicts the K-polystability assumption of ( X, ∆). Therefore, we must have δ T ( X, ∆) > X, ∆) isreduced uniformly K-stable.The existence of KE metric now follows from this equivalence and [Li19, Theorem 1.2](see also [BBJ18, LTW19]). (cid:3) Theorem 5.3 (K-moduli conjecture) . The K-moduli space M Kps n,V,C is proper, and the CMline bundle on M Kps n,V,C is ample.Proof.
The properness follows from Theorem 5.1 and [BHLLX20, Corollary 1.4]. Theampleness of the CM line bundle follows from Theorem 5.2 and [XZ20, Theorem 1.1]. (cid:3)
For the proof of the next theorem, we need a simple lemma.
Lemma 5.4.
Let ( X, ∆) be a log Fano pair and let D ∼ Q − ( K X + ∆) be an effective Q -divisor such that ( X, ∆ + D ) is klt. Assume that ( X, ∆ + tD ) is K-semistable for some t ∈ (0 , . Then ( X, ∆ + sD ) is uniformly K-stable for all s ∈ ( t, .Proof. This simple interpolation result is well known. We include it here for the sake ofcompleteness.By assumption, for any valuation v on X with A X, ∆ ( v ) < ∞ we have A X, ∆+ tD ( v ) = A X, ∆ ( v ) − t · v ( D ) > (1 − t ) v ( D )and A X, ∆+ tD ( v ) ≥ S X, ∆+ tD ( v ) = (1 − t ) S X, ∆ ( v ). Thus for any s ∈ ( t, A X, ∆+ sD ( v ) = A X, ∆+ tD ( v ) − ( s − t ) v ( D ) > − s − t A X, ∆+ tD ( v ) ≥ (1 − s ) S X, ∆ ( v ) = S X, ∆+ sD ( v )for any valuation v on X . Hence ( X, ∆ + sD ) is uniformly K-stable. (cid:3) Theorem 5.5.
Let ( X, ∆) be a log Fano pair such that δ := δ ( X, ∆) < . Then ( X, ∆ +(1 − δ ) D ) is K-semistable for any sufficiently divisible integer m ∈ N and any general D ∈ m | − m ( K X + ∆) | . In particular, ( X, ∆ + (1 − δ ′ ) D ) is uniformly K-stable for any ≤ δ ′ < δ . INITE GENERATION FOR VALUATIONS AND K-STABILITY 27
Proof.
By [Zhu20b, Theorem 1.5], there exists a closed subvariety W of X such that W is contained in C X ( v ) for any valuation v computing δ ( X, ∆). Thus we know thatif m ∈ N is sufficiently large and divisible and D m ∈ m | − m ( K X + ∆) | is general,then W is not contained in the support of D m , which implies that v ( D m ) = 0 for anyvaluation v that computes δ ( X, ∆). We claim that for any sufficiently divisible m , thepair ( X, ∆ + (1 − δ ) D m ) is K-semistable, which will finish the proof by Lemma 5.4.To see this, we first note that ( X, ∆ + mD m ) is lc by Bertini’s theorem. Thus A X, ∆ ( v ) ≥ A X, ∆+(1 − δ ) D m ( v ) = A X, ∆ ( v ) − (1 − δ ) v ( D m ) ≥ A X, ∆ ( v ) − − δm A X, ∆ ( v )for any valuation v on X . Since S X, ∆+(1 − δ ) D m ( v ) = δ · S X, ∆ ( v ), this implies that1 − − δm = (cid:18) − − δm (cid:19) δ − · δ ( X, ∆) ≤ δ ( X, ∆ + (1 − δ ) D m ) ≤ δ − · δ ( X, ∆) = 1 . By Theorem 5.1, there is a special degeneration,( X, ∆ + (1 − δ ) D m ) ( X m , ∆ m + (1 − δ ) G m )induced by some special divisorial valuation v m , such that(5.1) δ ( X m , ∆ m + (1 − δ ) G m ) = δ ( X, ∆ + (1 − δ ) D m ) ≥ − − δm . This implies A X m , ∆ m ( v ) ≥ A X m , ∆ m +(1 − δ ) G m ≥ (cid:18) − − δm (cid:19) S X m , ∆ m +(1 − δ ) G m ( v )= (cid:18) − − δm (cid:19) δ · S X m , ∆ m ( v )for all valuation v over X m and hence(5.2) δ ( X m , ∆ m ) ≥ (cid:18) − − δm (cid:19) δ is bounded from below by some constants that only depend on δ . By Theorem 2.12, wesee that ( X m , ∆ m ) belongs to a bounded family.Since the δ -invariant function (more precisely min { δ ( X, ∆) , } ) is constructible in abounded family [BLX19, Theorem 1.1], we have δ ( X m , ∆ m ) ≥ δ when m is sufficientlylarge by (5.2). By [BLZ19, Theorem 1.1] again, this implies that v m computes δ ( X, ∆)and hence by our choice of D m we get v m ( D m ) = 0. But then as v m also computes δ ( X, ∆ + (1 − δ ) D m ), we get δ ( X, ∆ + (1 − δ ) D m ) = A ∆+(1 − δ ) D m ( v m ) S X, ∆+(1 − δ ) D m ( v m ) = A X, ∆ ( v m ) δ · S X, ∆ ( v m ) = 1 , i.e. ( X, ∆ + (1 − δ ) D m ) is K-semistable. (cid:3) Examples
It is still natural to ask which lc places of a given Q -complement induce finitely gen-erated associated graded rings. Unlike the divisorial case, where the finite generation isguaranteed essentially by [BCHM10], in the higher rank case, the associated graded ringcould generally be non-finitely generated. In fact, it was first discovered in [AZ20, Theo-rem 4.16] that on every smooth cubic surface there exist lc places of complements whoseassociated graded rings are not finitely generated.In this section, we give a complete picture of the finite generation problem in an explicitexample, that is, lc places of ( P , C ) where C is an irreducible nodal cubic curve. Asone will see, even in this simple set-up, the locus of finitely generated lc places is fairlycomplicated, and there are infinitely many special degenerations of P . In addition, bothnon-finitely generated and finitely generated non-divisorial rational rank two lc placesappear in the same simplex (given by a dlt modification of ( P , C )). This also illustratesthe importance of considering special complements in previous discussions. It will be aninteresting question to understand better how to locate the valuations on a dual complexwith a finitely generated associated graded ring.We fix the following notation. Let o be the node of C . Choose an analytic coordinate( z, w ) of P at o such that the analytic local equation of C is given by ( zw = 0). For t ∈ R > , let v t be the monomial valuation of weights (1 , t ) in the coordinate ( z, w ). Since C is normal crossing, we know that any lc place of ( P , C ) is a multiple of v t or ord C . Let R := R ( P , O P (3)) be the anti-canonical ring of P . We also denote X := P . Theorem 6.1.
With the above notation, the associated graded ring gr v t R is finitely gen-erated if and only if t ∈ Q > ∪ ( − √ , √ ) . Moreover, Proj gr v t R is a Q -Fano varietyif and only if t ∈ ( − √ , √ ) . For a detailed description of these special degenerationsof P , see Remark 6.6. The proof of Theorem 6.1 is divided into several parts. We first recall a useful lemma.
Lemma 6.2 ([Fuj19, Claim 4.3]) . Suppose a, b are coprime positive integers. Let µ : e X → X = P be an ( a, b ) -weighted blow up at a smooth point with exceptional divisor E . Let ǫ X ( E ) := max { λ ∈ R ≥ | µ ∗ ( − K X ) − λE is nef } . Then we have (6.1) T X ( E ) · ǫ X ( E ) = 9 ab and S X ( E ) = T X ( E ) + ǫ X ( E )3 . Proposition 6.3. If t ≥ √ and t Q , then gr v t R is not finitely generated.Proof. By Theorem 4.4, it suffices to show that the function t S X ( v t ) is not linear in anysub-interval of [ √ , + ∞ ). We first compute the S -invariant for v t when t = ba > √ where a, b are coprime positive integers. Let µ : e X → X = P be the ( a, b )-weighted blowup at o in the analytic coordinates ( z, w ). Let E be the µ -exceptional divisor. Then easycomputation shows that e C := µ − ∗ C ∼ π ∗ ( − K X ) − ( a + b ) E and ( e C ) = 9 − ( a + b ) ab < T X ( E ) = ord E ( C ) = a + b which implies S X ( E ) = ( a + b ) +9 ab a + b ) by (6.1). Since INITE GENERATION FOR VALUATIONS AND K-STABILITY 29 v t = a ord E , for any rational t > √ we have(6.2) S X ( v t ) = 1 a S X ( E ) = t + 11 t + 13( t + 1) . Since the S -invariant is continuous in the dual complex [BLX19, Proposition 2.4], (6.2)holds for any t ∈ [ √ , + ∞ ). Thus t S X ( v t ) is not linear in any sub-interval. (cid:3) Next we turn to proving finite generation of gr v t R for t ∈ [1 , √ ). The idea is tofind a sequence of increasing rational numbers 1 = t < t < · · · < t n < · · · withlim n →∞ t n = √ such that the function t S X ( v t ) is linear in each interval [ t n , t n +1 ].Let ( d n ) n ≥ be the following sequence of integers, where d = 1, d = 1, d = 2, and d n +1 = 3 d n − d n − . The sequence of ( d n ) n ≥ goes as 1 , , , , , , , · · · . It is easy tosee that ( d n ) satisfies the following properties. • Each d n is not divisible by 3; • d n = F n − where ( F n ) is the Fibonacci sequence; • (1 , d n , d n +1 ) is a Markov triple, i.e. 1 + d n + d n +1 = 3 d n d n +1 ; • d n + 1 = d n − d n +1 .Let t n := d n +1 d n − for n ≥ t := 1. Then it is easy to see that ( t n ) is a strictlyincreasing sequence whose limit is √ . Proposition 6.4.
We have S X ( v t ) = d n +1 d n + d n d n +1 t for t ∈ [ t n , t n +1 ] and n ≥ . In order to prove Proposition 6.4, in the following lemma we find a sequence of verysingular plane curves D n of degree d n such that they compute the T -invariant for v t n .Note that D is precisely the singular plane quintic with an A -singularity (see e.g.[ADL19, Section 7.1]). Lemma 6.5.
For each n > , there exists an integral plane curve D n such that deg( D n ) = d n and the Newton polygon of the defining function of D n in ( z, w ) at o is the line segmentjoining ( d n +1 , and (0 , d n − ) .Proof. Let µ n : e X n → X be the ( d n − , d n +1 )-weighted blow-up in ( z, w ) at o . Let E n bethe µ n -exceptional divisor, so ord E n = d n − v t n . It is clear that h ( P , O ( d n )) = d n +3 d n +22 .Using Pick’s theorem, it is easy to compute that colength( a d n − d n +1 (ord E n )) = d n +3 d n .Thus we have h ( P , O ( d n )) > colength( a d n − d n +1 (ord E n )), which implies the existence ofa plane curve D n of degree d n with ord E n ( D n ) ≥ d n − d n +1 .Next, we show that the curve D n is integral. Assume to the contrary, then there existsan integral plane curve D of degree d < d n , such that ord En ( D ) d ≥ ord En ( D n ) d n ≥ d n − d n +1 d n . Infact, we always have ord En ( D ) d > d n − d n +1 d n since d n − d n +1 d n = d n + d n and d < d n . Clearly D = C since ord En ( C )3 = d n − + d n +1 = d n < d n − d n +1 d n . Computing local intersection numbers,yields ( D · C ) o ≥ ord E n ( D ) (cid:16) d n − + d n +1 (cid:17) > d ( d n − + d n +1 ) d n = 3 d . On the other hand,Bezout’s theorem implies ( D · C ) o ≤ ( D · C ) = 3 d , a contradiction. Thus D n is integral.Finally, we show the Newton polygon statement. Suppose the Newton polygon of D n passes through ( p,
0) and (0 , q ). Then by computing local intersection numbers, we know that 3 d n = ( D n · C ) ≥ ( D n · C ) o = p + q . On the other hand, ord E n ( D n ) ≥ d n − d n +1 implies that p ≥ d n +1 and q ≥ d n − . Hence we must have p = d n +1 and q = d n − . (cid:3) Proof of Proposition 6.4.
The statement is clear when t ∈ [1 ,
2] since v t is toric, so wemay assume n ≥
1. By Lemma 6.5, we know that ( e D n ) = ( D n ) + ( d n +1 d n − ) ( E n ) = − e D n := ( µ n ) − ∗ D n . Hence T X (ord E n ) = d n ord E n ( D n ) = d n − d n +1 d n . Thus Lemma 6.2implies that S X ( v t n ) = d n − S X ( E n ) = d n +1 d n + d n d n − . So t = t n satisfies the statement ofProposition 6.4.Let t ′ n := d n +1 d n for n ≥
1, then we have t n < t ′ n < t n +1 . Let µ ′ n : e X ′ n → X be the( d n , d n +1 )-weighted blow-up in ( z, w ) at o with exceptional divisor E ′ n . Then using Lemma6.5, similar computation shows ( e D ′ n ) = 0 where e D ′ n := ( µ ′ n ) − ∗ D n . Thus T X (ord E ′ n ) = d n ord E ′ n ( D n ) = 3 d n d n +1 , and Lemma 6.2 implies that t = t ′ n also satisfies the statementof Proposition 6.4. Since S -invariant is concave by Lemma 4.5, the proof is finished. (cid:3) Proof of Theorem 6.1.
Choose a suitable projective coordinates [ x , x , x ] of P such that C = ( x x = x + x x ). Then the automorphism σ of ( P , C ) given by σ ([ x , x , x ]) :=[ x , x , − x ] interchanges the two analytic branches of C at o . Thus σ ∗ v t = t · v t − whichimplies that v t − and v t have isomorphic associated graded rings after a grading shift.So we may assume t ∈ [1 , + ∞ ) from now on. The non-finite generation of gr v t R when t ∈ [ √ , + ∞ ) \ Q is proven in Proposition 6.3. For t ∈ ( √ , + ∞ ) ∩ Q , Theorems 2.24and 2.25 imply that gr v t R is finitely generated whose Proj is not klt as A X ( v t ) = T X ( v t ).The finite generation of gr v t R for t ∈ [1 , √ ) = ∪ n ≥ [ t n , t n +1 ] follows from Theorem 4.4and Proposition 6.4.Finally, we show that Proj gr v t R is a Q -Fano variety for t ∈ [ t n , t n +1 ]. Since v t is toricfor t ∈ [1 , n ≥
1. From computations in the proof of Proposition 6.4,we know that A X ( E n ) = ǫ X ( E n ) < T X ( E n ) and A X ( E ′ n ) < ǫ X ( E ′ n ) = T X ( E ′ n ). Sinceboth e X n and e X ′ n are of Fano type, we know that − K e X n − E n and − K e X ′ n − E ′ n are nefand hence semiample. Thus by Bertini’s theorem we can find Q -complements G n and G ′ n of ( e X n , E n ) and ( e X ′ n , E ′ n ) respectively, such that ( e X n , E n + G n ) and ( e X ′ n , E ′ n + G ′ n )are plt. Hence Theorem 2.25 implies that both E n and E ′ n are special divisors as theysatisfy A < T , and the desired Q -complements of plt type are given by ( µ n ) ∗ G n and( µ ′ n ) ∗ G ′ n . Since v t n is a rescaling of ord E n , it induces a special degeneration of P . By thelast paragraph of the proof of Lemma 4.6, we know that gr v t R ∼ = gr v t ′ n R ∼ = gr E ′ n R for any t ∈ ( t n , t n +1 ). Thus Proj gr v t R is a Q -Fano variety as E ′ n is special. (cid:3) Remark . Using similar arguments to [ADL19, Proof of Proposition 7.4], one can showthat Proj gr v tn R for n ≥ x x = x d n +1 + x d n − ) ⊂ P (1 , d n − , d n +1 , d n ) . Such a Manetti surface is a common partial smoothing of P (1 , d n − , d n ) and P (1 , d n , d n +1 ).Similarly, for any t ∈ ( t n , t n +1 ) one can show that Proj gr v t R ∼ = Proj gr v t ′ n R ∼ = P (1 , d n , d n +1 ).This provides infinitely many special degenerations of P which are unbounded. INITE GENERATION FOR VALUATIONS AND K-STABILITY 31
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Department of Mathematics, Princeton University, Princeton, NJ 08544, USA.
Email address : [email protected] Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
Email address : [email protected] Department of Mathematics, MIT, Cambridge, MA 02139, USA
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