LLOG K-STABILITY OF GIT-STABLE DIVISORS ON FANO MANIFOLDS
CHUYU ZHOU
Abstract.
For a given K-polystable Fano manifold X and a natural number l , we showthat there exists a rational number 0 < c < X , suchthat D ∈ | − lK X | is GIT-(semi/poly)stable under the action of Aut( X ) if and only if thepair ( X, (cid:15)l D ) is K-(semi/poly)stable for some rational 0 < (cid:15) < c . Contents
1. Introduction 1Acknowledgement 22. Preliminaries 22.1. K-stability of Q -Fano varieties 22.2. Valuative criterion 32.3. CM-line bundle 33. Q -Fano degenerations of a Q -Fano variety 54. Proof of the main result 6References 81. Introduction
In the past few years, people have made tremendous progress on the construction ofK-moduli space of Fano varieties, e.g [Jia20, BLX19, Xu20, BX19, ABHLX20, XZ20] etc.,and only properness is missed in this picture. However, with the help of analytic tools,e.g [CDS15a, CDS15b, CDS15c, Tia15, TW19] ect., people can construct proper K-modulifor smoothable Fano varieties with K¨ahler-Einstein metrics, e.g [LWX19, ADL19]. Withproperness in hand, moduli continuity method is now widely used in the literature to con-struct explicit K-moduli spaces of some special kinds of Fano varieties, such as K-modulifor cubic 3-folds ([LX19]) and cubic 4-folds ([Liu20]). It’s well known that GIT-stabilityand K-stability, although not the same, have close relationship via CM-line bundle, i.e. thegeneralized Futaki invariant of an one parameter subgroup can be identified with the cor-responding GIT-weight of CM-line bundle on the base (up to a positive multiple), thus inmany cases one can construct a morphism from a K-moduli space to a GIT-moduli space.The moduli continuity method aims to establish an isomorphism between these two spaces,which becomes a powerful way to confirm K-stability of explicit Fano varieties.In order to state the main theorem, we first fix some notation. Let X be a K-polystableFano manifold and l a sufficiently divisible natural number. As Aut( X ) is reductive (e.g[ABHLX20]), it’s natural to determine GIT-stability of elements in | − lK X | under theaction of Aut( X ). For a rational number 0 < (cid:15) <
1, we denote M KX,l,(cid:15) the Artin stack which a r X i v : . [ m a t h . AG ] F e b CHUYU ZHOU parametrizes all K-semistable pairs of the form ( Y, (cid:15)l ˜ D ), where ( Y, ˜ D ) is the degenerationof ( X, | − lK X | ) (see Section 4). We also denote M GITX,l := [( | − lK X | ) ss / Aut( X )]. It’srecognized that the GIT-stability of D ∈ | − lK X | is related to the K-stability of the logFano pair ( X, (cid:15)l D ) for small (cid:15) . For example, in the case X = P n , the work [ADL19, Theorem1.4] establishes such a correspondence between GIT-stability and K-stability. In this paper,we apply moduli continuity method to prove the following theorem, which is conjectured in[GMGS, Conjecture 1.3]. Theorem 1.1.
Let X be a K-polystable Fano manifold and l a positive integer. Assume D ∈ | − lK X | , there exists a rational number < c < depending only on the dimension of X such that the following two statements are equivalent.(1) D is GIT-(semi/poly)stable under the action of Aut( X ) ,(2) the pair ( X, (cid:15)l D ) is K-(semi/poly)stable for any rational ≤ (cid:15) < c .In general, there is a morphism φ (cid:15) : M KX,l,(cid:15) → M
GITX,l which admits an isomorphic descent φ (cid:48) (cid:15) : M KX,l,(cid:15) → M GITX,l for any rational < (cid:15) < c . We note here that the above theorem applies to any positive integer number l , not onlyfor sufficiently divisible ones. The original reason comes from the fact that GIT-stabilityof D ∈ | − lK X | doesn’t change up to a positive multiple of Aut( X )-linearisation. If l isnot big enough, then we can instead consider kD ∈ | − klK X | for some multiple k , andthe GIT-stability of D ∈ | − lK X | coincides with that of kD ∈ | − klK X | . The key for theproof of the main result is to confirm that, the K-semistable degeneration of ( X, (cid:15)l D ) forsmall (cid:15) and D ∈ | − lK X | preserves the ambient space. This is reduced to the fact that, if aK-polystable Q -Fano variety admits a K-semistable degeneration, then the degeneration isstill K-polystable (see Theorem 3.5). Here we say Y is a Q -Fano degeneration of X if thereis a Q -Gorenstein flat family X → C over a smooth pointed curve 0 ∈ C such that − K X /C is a relatively ample Q -Cartier divisor, X t ∼ = X for t (cid:54) = 0, and X ∼ = Y is a Q -Fano variety.In fact, if the degeneration is obtained by a test configuration, then the fact is well knownby [LWX18, Section 3] (see also Definition 2.2). Acknowledgement.
I would like to thank Chen Jiang, Yuchen Liu and Ziquan Zhuang forhelpful discussions and beneficial comments.2.
Preliminaries
In this section, we provide necessary preliminaries. We work over complex number field C . We say ( X, ∆) is a log pair if X is a projective normal variety and ∆ is an effective Q -divisor on X such that K X + ∆ is Q -Cartier. We say a log pair ( X, ∆) is log Fano ifit admits klt singularities and − K X − ∆ is ample. If ∆ = 0, we just say a log Fano pair( X, ∆) is a Q -Fano variety. For the concepts of singularities in birational geometry such asklt singularities, we refer to [KM98, Kol13].2.1. K-stability of Q -Fano varieties. Let ( X, ∆) be a log Fano pair of dimension n , wedenote L := − K X − ∆, which is an ample Q -line bundle. Definition 2.1.
We say a triple π : ( X , ∆ tc ; L ) → A is a test configuration of ( X, ∆; L ) ifthe following conditions are satisfied: OG K-STABILITY OF GIT-STABLE DIVISORS ON FANO MANIFOLDS 3 (1) π is a flat projective morphism from a normal variety X and ∆ tc ⊂ X is a Q -divisorflat over A ,(2) L is a relatively ample Q -line bundle on X with a C ∗ -action induced by the naturalmultiplication on A ,(3) ( X ∗ , ∆ ∗ tc ; L ∗ ) is C ∗ -equivariantly isomorphic to ( X × C ∗ , ∆ × C ∗ ; L × C ∗ ), where X ∗ := X \ X .The compactification of the test configuration is denoted by ( ¯ X , ¯∆ tc ; ¯ L ) → P , which isobtained by gluing ( X , ∆ tc ) and ( X × ( P \ , ∆ × ( P \ X × C ∗ , ∆ × C ∗ ). Definition 2.2.
Let ( X , ∆ tc ; L ) → A be a test configuration of ( X, ∆; L ), then the gener-alized Futaki invariant of this test configuration is defined as follows:Fut( X , ∆ tc ; L ) := n ¯ L n +1 ( n + 1)( − K X − ∆) n + ¯ L n ( K ¯ X / P + ¯∆ tc )( − K X − ∆) n . We say ( X, ∆) is K-semistable if Fut( X , ∆ tc ; L ) ≥ X, ∆) is K-polystable if it’s K-semistable and for any test configuration of ( X, ∆; L ) whosecentral fiber is K-semistable, we have an isomorphism between ( X, ∆) and the central fiber.2.2. Valuative criterion.
Let ( X, ∆) be a log Fano pair of dimension n . We say E isa prime divisor over X if there is a proper birational morphism from a normal variety φ : Y → X such that E is a prime divisor on Y . We define A X, ∆ ( E ) := ord E ( K Y − φ ∗ ( K X + ∆)) + 1 ,S X, ∆ ( E ) := 1( − K X − ∆) n (cid:90) ∞ vol( − φ ∗ ( K X + ∆) − tE )dt . Definition 2.3.
The beta-invariant of a prime divisor E over a log Fano pair ( X, ∆) isdefined as follows: β X, ∆ ( E ) := A X, ∆ ( E ) − S X, ∆ ( E ) . The delta invariant of a log Fano pair ( X, ∆) is defined as follows: δ ( X, ∆) := inf E A X, ∆ ( E ) S X, ∆ ( E ) , where E runs through all prime divisors over X .We have the following well-known theorem due to [Li17, Fuj19, FO18, BJ20]. Theorem 2.4.
Let ( X, ∆) be a log Fano pair of dimension n , then(1) ( X, ∆) is K-semistable if and only if β X, ∆ ( E ) ≥ for any prime divisor E over X .(2) ( X, ∆) is K-semistable if and only if δ ( X, ∆) ≥ . CM-line bundle.
Let π : ( X , D ; L ) → T be a flat family of projective normal varietiesof dimension n over a normal base T , where D is an effective Q -divisor on X whose compo-nents are all flat over T , and L is a relative ample Q -line bundle. By the work of Mumford-Knudsen([KM76]) there exist Q -line bundles λ i , i = 0 , , ..., n + 1 and ˜ λ i , i = 0 , , ...n, on T such that we have the following expansions for all sufficiently large k ∈ N :det π ∗ ( L k ) = λ ( kn +1 ) n +1 ⊗ λ ( kn ) n ⊗ ... ⊗ λ ( k ) ⊗ λ , det π ∗ ( L| k D ) = ˜ λ ( kn ) n ⊗ ˜ λ ( kn − ) n − ⊗ ... ⊗ ˜ λ . CHUYU ZHOU
By Riemann-Roch formula, cf [CP18, Appendix], we have c ( π ∗ L k ) = π ∗ ( L n +1 )( n + 1)! k n +1 + π ∗ ( − K X /T L n )2 n ! k n + ...,c ( π ∗ L| k D ) = π ∗ ( L n D ) n ! k n + .... From above formulas, it’s not hard to see λ n +1 = π ∗ ( L n +1 ) , λ n = n π ∗ ( L n +1 ) + 12 π ∗ ( − K X /T L n ) and ˜ λ n = π ∗ ( L n D ) . By flatness of π and π D , we write h ( X t , k L t ) = a k n + a k n − + o ( k n − ) and h ( D t , k L t | D t ) = ˜ a k n − + o ( k n − ) , which don’t depend on the choice of t ∈ T . Then we have a = L nt n ! , a = − K X t L tn − n − and ˜ a = L n − t D t ( n − . Definition 2.5.
We define the CM-line bundles for the family π : ( X , D ; L ) → T as follows: λ CM, ( X , L ; π ) := λ a a + n ( n +1) n +1 ⊗ λ − n +1) n ,λ CM, ( X , D , L ; π ) := λ a − ˜ a a + n ( n +1) n +1 ⊗ λ − n +1) n ⊗ ˜ λ n +1 n . Now we assume π : ( X , D ; L ) → P to be a test configuration of a log pair ( X, D ; L ) where L is an ample Q -line bundle on X . As π ∗ L k is a C ∗ -equivariant vector bundle on P , wewrite the total weights to be: w (det( π ∗ L k )) = b k n +1 + b k n + o ( k n ) and w (det( π ∗ L| k D )) = ˜ b k n + o ( k n ) . It’s not hard to compute that(1) b = L n +1 ( n +1)! = w ( λ n +1 )( n +1)! ,(2) b = − K X / P L n n ! = w ( λ n ) n ! − n ( n +1)2( n +1)! w ( λ n +1 ),(3) ˜ b = L n D n ! = w (˜ λ n ) n ! . Definition 2.6.
Notation as above, we define(1) The generalized Futaki invariant of ( X , L ):Fut( X , L ) := 2( b a − b a ) a = 1( n + 1) L n w ( λ a a + n ( n +1) n +1 ⊗ λ − n +1) n ) , (2) The Chow weight of π : ( X , D ; L ) → T :Chow( X , D ; L ) := ˜ b a − ˜ a b a = 1( n + 1) L n w ( λ − ˜ a a n +1 ⊗ ˜ λ n +1 n ) , (3) The generalized Futaki invariant of ( X , D ; L ):Fut( X , D ; L ) :=Fut( X , L ) + Chow( X , D ; L )= 1( n + 1) L n w ( λ a − ˜ a a + n ( n +1) n +1 ⊗ λ − n +1) n ⊗ ˜ λ n +1 n ) . OG K-STABILITY OF GIT-STABLE DIVISORS ON FANO MANIFOLDS 5
Remark 2.7.
We have a few remarks for the above definition.(1) When π : ( X , D ; L ) → T is a test configuration of a log Fano pair, then the general-ized Futaki invariants here coincide with the definition in Section 2.1.(2) In the case of test configurations, we see that the generalized Futaki invariantscoincide with the GIT-weights of CM-line bundles up to a multiple. We also notehere that CM-line bundles are in fact Q -line bundles on the base.(3) Suppose π : ( X , D ; L ) → T is a flat family of log Fano varieties such that L ∼ Q − K X /T − D , then λ CM, ( X , D , L ; π ) = − π ∗ ( L n +1 ) ∼ Q − π ∗ ( − K X /T − D ) n +1 . Example 2.8.
Let X be a smooth Fano manifold of dimension n and l a sufficiently divisiblenatural number. We denote P N := | − lK X | and D ⊂ X × P N the universal divisor corre-sponding to the linear system | − lK X | . For a rational number 0 < (cid:15) <
1, we compute theCM-line bundle for the family π : ( X × P N , (cid:15)l D ; L ) → P N , where L ∼ Q − K X × P N / P N − (cid:15)l D .As the base is of Picard number 1, it suffices to consider a base change for a general rationalcurve P (cid:44) → P N , thus we get a family ( X × P , (cid:15)l D P ; L P ) → P , still denoted by π forconvenience. By Remark 2.7 we have λ CM, ( X × P , (cid:15)l D P , L P ; π ) ∼ Q − π ∗ ( − K X × P / P − (cid:15)l D P ) n +1 ∼ Q (1 − (cid:15) ) n (cid:15) ( − K X ) n O P (1) . Therefore λ CM, ( X × P N , (cid:15)l D , L ; π ) is an ample Q -line bundle on P N . Then the GIT-stability of D ∈ | − lK X | under the action of Aut( X ) with respect to λ CM, ( X × P N , (cid:15)l D , L ; π ) is the same asthat with respect to O P N (1).3. Q -Fano degenerations of a Q -Fano variety In this section, we fix X to be a Q -Fano variety of dimension n . Definition 3.1.
We say a variety Y is a Q -Fano degeneration of X if there is a Q -Gorensteinflat family X → C over a smooth pointed curve 0 ∈ C such that(1) − K X /C is a relative ample Q -line bundle,(2) for t (cid:54) = 0, X t ∼ = X ,(3) X ∼ = Y is a Q -Fano variety.Fix a rational number 0 < (cid:15) <
1, we consider the set F of Q -Fano varieties such that Y ∈ F if and only if(1) Y is a Q -Fano degeneration of X ,(2) the pair ( Y, cD ) is K-semistable for some rational 0 < c < − (cid:15) and D ∼ Q − K Y .We have the following lemma. Lemma 3.2.
Notation as above, F is contained in a bounded family, and there is a rationalnumber < η < depending only X such that Y is K-semistable once δ ( Y ) ≥ η .Proof. For Y ∈ F , by valuative criterion (see Section 2.2), we have A Y,cD ( E ) S Y,cD ( E ) ≥ E over Y . Note that S Y,cD ( E ) = (1 − c ) S Y ( E ), thus one sees A Y ( E ) S Y ( E ) ≥ − c > (cid:15) , whichimplies that δ ( Y ) ≥ (cid:15) . As vol( − K Y ) = vol( − K X ), combining the work [Jia20], we knowthat F is contained in a bounded family. By [BLX19, Xu20], the set { min { δ ( Y ) , }| Y ∈ F } CHUYU ZHOU is finite, then there exists a rational number 0 < η < Y is K-semistable if δ ( Y ) ≥ η for any Y ∈ F . The proof is finished. (cid:3) We are ready to prove the following theorem.
Theorem 3.3.
Notation as above, assume Y ∈ F and D ∼ Q − K Y , there exists a ratio-nal number < c < depending only on X such that the following two statements areequivalent:(1) the pair ( Y, c D ) is K-semistable,(2) the pair ( Y, (cid:15)D ) is K-semistable for any rational ≤ (cid:15) ≤ c .Moreover, if X is K-polystable, then the following two statements are equivalent:(1) the pair ( Y, c D ) is K-polystable,(2) the pair ( Y, (cid:15)D ) is K-polystable for any rational ≤ (cid:15) ≤ c .Proof. We choose c < min { − (cid:15) , − η } . We first look at the K-semistable part. Thedirection (2) ⇒ (1) is obvious. For the converse direction, it suffices to show that δ ( Y ) ≥ η , which is directly concluded by valuative criterion. For the K-polystable part, assume( Y, c D ) is K-polystable, then Y is K-semistable. It remains to show that Y is K-polystable,which follows from next result. (cid:3) Remark 3.4.
If we assume X is smoothable, then vol( − K X ) automatically admits a positivelower bound which doesn’t depend on X , thus c only depends on the dimension of X . Theorem 3.5.
Suppose X is a K-polystable Q -Fano variety and Y is a Q -Fano degenerationof X . If Y is K-semistable, then Y ∼ = X .Proof. Let f : X → C be a Q -Gorenstein flat family over a smooth pointed curve 0 ∈ C which produces the degeneration from X to Y . By [BX19, Theorem 1.1], we know thereis a degeneration from Y to X via a special test configuration, denoted by g : Y → A .We denote the origin 0 ∈ A , then Y ∼ = X . Choose a sufficiently large natural number r such that − rK X and − rK Y are both very ample, and X (resp. Y ) can be embedded into C × P N (resp. A × P N ). Suppose p ( m ) = χ ( − mK X ), we denote Hilb the Hilbert schemewhose points parametrize closed sub-varieties in P N with Hilbert polynomial p ( m ). Thenthe morphism f (resp. g ) induces a morphism C → Hilb (resp. A → Hilb), with C \ and 0 (resp. A \ and 0 ) being sent to [ X ] ∈ Hilb (resp. [ Y ] ∈ Hilb). Thus we seePGL( N +1)[ X ] ⊂ PGL( N + 1)[ Y ] and PGL( N +1)[ Y ] ⊂ PGL( N + 1)[ X ]. Suppose Y is notisomorphic to X , then the containments are strict, and we directly have dim PGL( n +1)[ Y ] < dim PGL( n + 1)[ X ] < dim PGL( n + 1)[ Y ], contradiction. (cid:3) Proof of the main result
In this section, we fix X to be a K-polystable Fano manifold of dimension n . Let l be asufficiently divisible natural number and 0 < (cid:15) < Definition 4.1.
We say the log Fano pair ( Y, (cid:15)l ˜ D ) is a degeneration of ( X, (cid:15)l | − lK X | ) ifthere is a Q -Gorenstein flat family ( X , (cid:15)l D ) → C over a smooth pointed curve 0 ∈ C suchthat(1) − K X − (cid:15)l D is a relative ample Q -line bundle,(2) for each t (cid:54) = 0, X t ∼ = X and D t ∈ | − lK X t | , OG K-STABILITY OF GIT-STABLE DIVISORS ON FANO MANIFOLDS 7 (3) ( X , D ) ∼ = ( Y, ˜ D ).We denote M KX,l,(cid:15) the Artin stack parametrizing all K-semistable log Fano pairs of theform ( Y, (cid:15)l ˜ D ), which is the degeneration of ( X, (cid:15)l |− lK X | ), and M GITX,l := [( |− lK X | ) ss / Aut( X )].By taking good moduli spaces, we denote M KX,l,(cid:15) (resp. M GITX,l ) the descents of M KX,l,(cid:15) (resp. M GITX,l ) whose closed points parametrize K-polystable (resp. GIT-polystable) elements. Wefirst have the following lemma.
Lemma 4.2.
Notation as above, for general D ∈ | − lK X | , the pair ( X, (cid:15)l D ) is K-polystablefor any rational number < (cid:15) < .Proof. As l is sufficiently divisible, we may assume − lK X is very ample. By Bertini typetheorem, for general D ∈ | − lK X | , the pair ( X, l D ) is a klt Calabi-Yau pair, thus being K-semistable by [BHJ17, Corollary 9.4] or [Oda13, Theorem 1.5]. By the interpolation propertyof K-stability we at once see that ( X, (cid:15)l D ) is K-polystable for any general D ∈ | − lK X | andany rational 0 < (cid:15) <
1. In particular, the stack M KX,l,(cid:15) is not empty. (cid:3)
We are ready to prove the main result.
Theorem 4.3. (= Theorem 1.1)
Let X be a K-polystable Fano manifold and l a positiveinteger. Assume D ∈ | − lK X | , there exists a rational number < c < depending only onthe dimension of X such that the following two statements are equivalent.(1) D is GIT-(semi/poly)stable under the action of Aut( X ) ,(2) the pair ( X, (cid:15)l D ) is K-(semi/poly)stable for any rational ≤ (cid:15) < c .In general, there is a morphism φ (cid:15) : M KX,l,(cid:15) → M
GITX,l which admits an isomorphic descent φ (cid:48) (cid:15) : M KX,l,(cid:15) → M GITX,l for any rational < (cid:15) < c .Proof. We first assume l is sufficiently divisible such that − lK X is very ample. Suppose( X, (cid:15)l D ) is K-semistable for D ∈ |− lK X | , then D is naturally GIT-semistable with respect toan ample CM-line bundle by Remark 2.7 and Example 2.8. Take c as in Theorem 3.3. Fromnow on we assume 0 < (cid:15) < c is rational. For any K-semistable element [( Y, (cid:15)l ˜ D )] ∈ M KX,l,(cid:15) ,by Theorem 3.3 we know Y is K-semistable, and by Theorem 3.5 we have Y ∼ = X . Thusthere is a morphism φ (cid:15) : M KX,l,(cid:15) → M
GITX,l . By Lemma 4.2, for general D ∈ | − lK X | ,it’s clear that ( X, (cid:15)l D ) is K-semistable, then by taking good moduli spaces, the descent φ (cid:48) (cid:15) : M KX,l,(cid:15) → M GITX,l is an injective open immersion.To show φ (cid:48) (cid:15) is an isomorphism, it suffices to show that M KX,l,(cid:15) is proper. To see this,let π o : ( Y o , − (cid:15)l ˜ D o ) → C o be a Q -Gorenstein flat family of log Fano pairs over a smoothpunctured curve C o , with each fiber being K-semistable and of the form ( Y, (cid:15)l ˜ D ), which is thedegeneration of ( X, (cid:15)l | − lK X | ). By the last paragraph we know Y ot ∼ = X for t ∈ C o , thus thepair ( Y ot , al D ot ) is K-polystable for each t ∈ C o and rational 0 ≤ a < (cid:15) . We choose infinitelymany closed points { t i } ∞ i =1 on C o such that t i tends to the punctured point, and increasingrational numbers { (cid:15) i } ∞ i =1 with (cid:15) i → (cid:15) , then we consider the set { ( Y ot i , (cid:15) i l ˜ D ot i ) } ∞ i =1 . By takinga subsequence there exists a Gromov-Hausdorff limit ( Y, (cid:15)l ˜ D ) which is a K-polystable kltlog Fano pair (see [TW19]). Then one can fill the family π o by a K-polystable log Fanopair ( Y, (cid:15)l ˜ D ) up to a base change (e.g. [ADL19, Section 3.4]). By Theorem 3.3, Y is K-semistable, and by Theorem 3.5 we see Y ∼ = X . Therefore, the stack M KX,l,(cid:15) is proper and φ (cid:48) (cid:15) is an isomorphism, the proof is finished for sufficiently divisible l . CHUYU ZHOU
For small positive integer l , we just consider kD ∈ | − klK X | instead of D ∈ | − lK X | for a suitable multiple k . As GIT-stability doesn’t change up to a positive multiple ofAut( X )-linearization, we see that the GIT-stability of D ∈ | − lK X | coincides with that of kD ∈ | − klK X | . Thus we get the following equivalence:(1) D ∈ | − lK X | is GIT-(semi/poly)stable under the action of Aut( X ),(2) kD ∈ | − klK X | is GIT-(semi/poly)stable under the action of Aut( X ),(3) the pair ( X, (cid:15)kl kD ) = ( X, (cid:15)l D ) is K-(semi/poly)stable for any rational 0 ≤ (cid:15) < c .The proof is finished. (cid:3) Remark 4.4.
In fact, one can even allow l to be rational. In this case, we just identifyGIT-stability of D ∼ Q − lK X with that of kD ∈ | − klK X | for a sufficiently divisible k ∈ N . Remark 4.5.
In the proof of above theorem, we assume X to be a K-polystable Fano man-ifold, but not a K-polystable Q -Fano variety, since we need some analytic results. However,the analytic tools we take from [TW19] are only used to confirm the properness of M KX,l,(cid:15) .Once the optimal destabilization conjecture ([BLZ19, Conjecture 1.2]) is confirmed, thenthe properness of K-moduli follows from [BHLLX20]. Thus Theorem 1.1 can be directlygeneralized to the case of a K-polystable Q -Fano variety. However, in this case, the number c we choose depends on X , see Remark 3.4. References [ABHLX20] Jarod Alper, Harold Blum, Daniel Halpern-Leistner, and Chenyang Xu,
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On positivity of the CM line bundle on K-moduli spaces , Ann.of Math. (2) (2020), no. 3, 1005–1068. MR4172625 ↑ ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), MA C3 615, Station 8, 1015 Lausanne,Switzerland Email address ::