Higher Codimensional Alpha Invariants and Characterization of Projective Spaces
aa r X i v : . [ m a t h . AG ] J a n HIGHER CODIMENSIONAL ALPHA INVARIANTS ANDCHARACTERIZATION OF PROJECTIVE SPACES
ZIWEN ZHU
Abstract.
We generalize the definition of alpha invariant to arbitrary codimension. Wealso give a lower bound of these alpha invariants for K-semistable Q -Fano varieties andshow that we can characterize projective spaces among all K-semistable Fano manifoldsin terms of higher codimensional alpha invariants. Our results demonstrate the rela-tion between alpha invariants of any codimension and volumes of Fano manifolds in thecharacterization of projective spaces. Introduction
We work over the complex number field C . A variety X is called Q -Fano if X is anormal projective variety with klt singularities such that the anti-canonical divisor − K X is an ample Q -divisor. A Fano manifold is a smooth Q -Fano variety. we will always use n to denote the dimension of the Q -Fano variety X .It is well-known that a Fano manifold admits a K¨ahler-Einstein metric if and only ifit is K-polystable due to [CDS15a, CDS15b, CDS15c, Tia15]. More generally, we wouldlike to study K-semistable Q -Fano varieties. Recent work of Kento Fujita, Yuji Odaka andChen Jiang shows that among K-semistable Fano manifolds, the projective space P n canbe characterized by either of the following two properties:(1) [Fuj18] ( − K X ) n ≥ ( n + 1) n ;(2) [FO16, Jia17] α ( X ) ≤ n + 1 .Here ( − K X ) n is the volume of X , and α ( X ) is the alpha invariant of X .The purpose of this paper is to show that the above two characterizations of projectivespaces are special cases of a more general one where cycles of intermediate codimensionsare considered.We first generalize the definition of alpha invariant: Definition 1.1.
Let X be a Q -Fano variety of dimension n . For 1 ≤ k ≤ n , the completeintersection ci( L , . . . , L k ) in X cut out by effective Cartier divisors L , . . . , L k is defined tobe the scheme-theoretic intersection of L , . . . , L k with the expected codimension k . Thenwe define the alpha invariant of codimension k for X to be α ( k ) ( X ) := inf r (cid:26) lct (cid:18) X, r Z (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Z = ci( L , . . . , L k ) , L , . . . , L k ∈ | − rK X | (cid:27) . Remark . When k = 1, the generalized alpha invariant α (1) ( X ) is just the usual alphainvariant α ( X ). We will use α (1) ( X ) to denote the usual alpha invariant for the rest partof the paper. Research partially supported by NSF Grant DMS-1402907.
Tian proved in [Tia87] that a Fano manifold X of dimension n admits a K¨ahler-Einsteinmetric if α (1) ( X ) > n/ ( n + 1). Fujita improved the theorem in [Fuj17] by showing that aFano manifold X of dimension n is K-stable if α (1) ( X ) ≥ n/ ( n + 1). A recent related resultby Stibitz and Zhuang in [SZ18] shows that a birationally superrigid (or more generally logmaximal singularity free) Fano variety X is K-stable (resp. K-semistable) if α (1) ( X ) > / α (1) ( X ) ≥ / α (2) . (See Theorem 3.7 inSection 3).For K-semistable Q -Fano varieties, we can give a lower bound of higher codimensionalalpha invariants as our first main result: Theorem A.
Let X be a K-semistable Q -Fano variety of dimension n . Then α ( k ) ( X ) ≥ kn + 1 . (1.1)Note that when k = 1, the inequality (1.1) is proved by Fujita and Odaka in [FO16].It is well-known that P n is K-semistable. Therefore by considering the log canonicalthresholds of linear subspaces of P n , together with Theorem A, we know that the equalityholds in (1.1) when X ∼ = P n . Then we have our second main result about characterizationof projective spaces: Theorem B.
Let X be a K-semistable Fano manifold of dimension n . Consider thefollowing three statements: (1) X ∼ = P n ; (2) α ( k ) ( X ) = kn + 1 , and it is realized by some complete intersection Z ; (3) ( − K X ) k is rationally equivalent to lZ ′ for some rational number l ≥ ( n + 1) k and Z ′ an integral ( n − k ) -cycle.We have (1) ⇒ (2) ⇒ (3) . Moreover, if we assume that k divides n , then (3) ⇒ (1) andtherefore all three statements are equivalent.Remark . Note that in the second statement of Theorem B, we say α ( k ) ( X ) is realizedby a complete intersection Z if Z is the complete intersection of L . . . , L k ∈ | − rK X | ,such that lct( X, r Z ) = α ( k ) ( X ). In this case, the infimum in the definition of α ( k ) ( X ) isin fact a minimum. When α (1) ( X ) ≤
1, Birkar shows in [Bir16] that α (1) ( X ) is realized bysome D ∈ | − K X | Q . In general for higher codimensional alpha invariants, it is not clearwhether they can always be realized by some complete intersection.When k = 1, Theorem B reduces to the main result in [Jia17] that characterizes projec-tive spaces among all K-semistable Fano manifolds in terms of the usual alpha invariant.In [Jia17], Jiang proves that (2) implies (3) first. (3) ⇒ (1) follows from the main resultof [KO73]. Also note that in this case, due to Birkar’s result, we do not need to assume α (1) ( X ) is realized in the second statement of Theorem B.When k = n , using the inequality in [dFEM04], a quick proof of Theorem B is given inSection 3 (See Corollary 3.3). In this case, it it also not necessary to assume α ( n ) ( X ) isrealized. Note that (3) ⇒ (1) when k = n is implied by the following theorem: IGHER CODIMENSIONAL ALPHA INVARIANTS 3
Theorem 1.2 ([Fuj18]) . Let X be a K-semistable Q -Fano variety of dimension n . Thenwe have ( − K X ) n ≤ ( n + 1) n . Moreover if X is smooth and ( − K X ) n = ( n + 1) n , then weknow that X ∼ = P n . In fact we use Theorem 1.2 to prove the last part of Theorem B for any k that divides n . We would also like to comment that in both [Liu16] and [LZ17], Liu and Zhuang proveda stronger version of Theorem 1.2 without assuming smoothness. Acknowledgements.
The author would like to thank his advisor Tommaso de Fernex forguiding his research and providing insightful thoughts throughout the project. He wouldalso like to thank Harold Blum, Yuchen Liu, Ziquan Zhuang and Ivan Cheltsov for effectivediscussions. 2.
Preliminaries
Cycles, rational equivalence and numerical equivalence.
Let X be a schemeand Z a k -dimensional subscheme of X . Let Z , . . . , Z t be the irreducible components of Z .Then following the notation of [Ful12], the k-cycle of a subscheme Z is [ Z ] = P ti =1 a i [ Z i ],where a i = l ( O Z,Z i ) is the length of O Z,Z i as an O Z,Z i -module.For any two k -cycles Z and W on a proper scheme, we write Z ∼ rat W if Z is rationallyequivalent to W ; and write Z ≡ num W if Z is numerically equivalent to W . We refer to[Ful12] for definitions.In particular, we know that rational equivalence is the same as linear equivalence fordivisors, and we write D ∼ lin D ′ if D is linear equivalent to D ′ . Also on any smooth properscheme X of dimension n , two k -cycles Z and W are numerically equivalent if and only ifwe have the equality of intersection numbers Z · T = W · T for any ( n − k )-cycle T .2.2. Multiplicity of ideals.
Let X be a scheme of dimension n , and Z ⊂ X a closedsubscheme corresponding to the ideal sheaf I Z ⊂ O X . Let V be an irreducible componentof Z with codimension k . Then the multiplicity of X along Z at the generic point of V isdefined as e ( I Z · O X,V ) := lim t →∞ l ( O X,V /I tZ · O X,V ) t k /k ! . Suppose in addition Z is irreducible. Let σ : Y → X be a proper birational morphism suchthat σ − I Z · O Y = O Y ( − E ). Then we have the following equality of ( n − k )-cycles: e ( I Z · O X,V )[ V ] = ( − k − σ ∗ ( E k ) . (2.1)Refer to [Ful12] for more details about multiplicity.For the multiplicity of an ideal corresponding to a complete intersection, we have thefollowing property (See Example 4.3.5 of [Ful12] for a proof): Proposition 2.1.
Let ( R, m ) be a Noetherian local ring of dimension k and a = ( x , . . . , x k ) an m -primary ideal. If R is Cohen-Macaulay, then e ( a ) = l ( R/ a ) . Singularities and log canonical threshold.
Let X be a normal variety such that K X is Q -Cartier and Z ⊂ X a closed subscheme. Denote by I Z the ideal sheaf defining Z . We say E is a divisor over X if E is a prime divisor on some normal variety Y with aproper birational morphism σ : Y → X , and σ ( E ) is called the center of E on X . We saythat E is exceptional if the center of E on X has dimension smaller than E . Use A X ( E )to denote the log discrepancy of E . For a >
0, we say that the pair (
X, aZ ) has klt (orlog canonical) singularities if A X ( E ) − a ord E ( I Z ) > ≥
0) for any prime divisor E ZIWEN ZHU over X . Let H be a linear system on X and Z be the base scheme Bs H . Singularities ofthe pair ( X, a H ) are defined to be the same as the singularities of the pair ( X, aZ ). Inthe definition of klt and log canonical singularities, it is enough to examine finitely manydivisors on a fixed log resolution. We refer to [KM08] and [Laz04] for more details.Let X have klt singularities and assume Z is non-empty. For any nonnegative number a , the log canonical threshold of the formal power I aZ is defined to be the log canonicalthreshold of the pair ( X, aZ ), which is a nonnegative number computed bylct( I aZ ) := lct( X, aZ ) = inf E A X ( E ) a ord E ( I Z ) , (2.2)where the infimum runs through all prime divisors E over X . Note that immediately fromthe definition, the log canonical threshold of ( X, aZ ) can also be viewed as the largestnumber t >
X, taZ ) is log canonical. Then by fixing a log resolution of(
X, Z ), we see that the infimum in (2.2) is in fact a minimum running through finitelymany divisors. In particular, log canonical thresholds of pairs are rational numbers. Alsofrom the definition, for a >
0, we have lct(
X, aZ ) := a lct( X, Z ).2.4.
K-stability.
The β -invariant of a divisor over a Q -Fano variety is defined in thefollowing way: Definition 2.2 ([Fuj16]) . Let X be a Q -Fano variety of dimension n , and F a prime divisorover X . Suppose σ : Y → X is a projective birational morphism such that F is a divisoron Y . Then we define the β -invariant of F to be β ( F ) := A X ( F ) vol X ( − K X ) − Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx. Instead of recalling the original definition of K-stability via normal test configurationsand Donaldson-Futaki invariants, we use a valuative criterion of K-semistability in termsof β -invariants developed by Fujita and Li. Theorem 2.3 ([Fuj16, Li17]) . Let X be a Q -Fano variety. Then X is K-semistable if andonly if β ( F ) ≥ for any divisor F over X . We can also define β -invariants for proper closed subschemes of X in a similar way. Definition 2.4 ([Fuj18]) . Let X be a Q -Fano variety of dimension n and Z ⊂ X asubscheme of X defined by the ideal sheaf I Z ⊂ O X . Take a projective birational morphism σ : Y → X that factors through the blow-up of X along Z , and write σ − I Z ·O Y = O Y ( − F ).Then we define the β -invariant of Z to be β ( Z ) := lct( X, Z ) vol X ( − K X ) − Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx. Note that β ( F ) and β ( Z ) do not depend on the choice of the birational morphism σ .An immediate consequence of Theorem 2.3 is the following result. Theorem 2.5 ([Fuj18]) . Let X be a Q -Fano variety. If X is K-semistable, then β ( Z ) ≥ for any proper closed subscheme Z ( X . IGHER CODIMENSIONAL ALPHA INVARIANTS 5
Characterizations of Projective Spaces.
We recall some results about character-ization of projective spaces among Fano manifolds. They are all in some sense related tothe divisibility of the canonical divisor.
Theorem 2.6 ([KO73]) . Let X be a smooth Fano manifold of dimension n and H anample divisor on X . If − K X ∼ lin lH with l ≥ n + 1 , then X ≃ P n , l = n + 1 , and O X ( H ) = O P n (1) .Remark . Note that in the theorem it is enough to assume − K X ≡ num lH . Indeed, if K X + lH is numerically trivial, then we know that χ ( K X + lH ) = χ ( O X ) = 1. By Kodairavanishing, we have h i ( X, K X + lH ) = 0 for i >
0. Therefore h ( X, K X + lH ) = 1, whichimplies that K X + lH ∼ lin n , we can characterize P n by the volume vol X ( − K X ) = ( − K X ) n instead, which is exactly the second part ofTheorem 1.2. Note that the condition ( − K X ) n = ( n + 1) n in Theorem 1.2 can be viewedas the divisibility of the 0-cycle ( − K X ) n by ( n + 1) n , comparing to the divisibility of thedivisor − K X by n + 1 in Theorem 2.6.By considering the divisibility of cycles with intermediate codimension, we have thefollowing immediate corollary of Theorem 1.2: Corollary 2.7.
Let X be a K-semistable Fano manifold of dimension n . Suppose k divides n . If ( − K X ) k ≡ num lZ for some rational number l ≥ ( n + 1) k and Z an integral ( n − k ) -cycle, then X ≃ P n .Proof. because k divides n , we can write( − K X ) n = (cid:16) ( − K X ) k ) (cid:17) nk = ( lZ ) nk ≥ l nk ≥ ( n + 1) n . Since X is K-semistable, we know from Theorem 1.2 that ( − K X ) n = ( n + 1) n and X ≃ P n . (cid:3) For any dimension n , we at least know that k divides n when k = 1 or k = n . These arethe two cases discussed in Theorem 2.6 and Theorem 1.2 respectively. Along with somemild assumptions if necessary, we expect that the divisibility condition of the cycle ( − K X ) k described in Corollary 2.7 can characterize projective spaces among all K-semistable Fanomanifolds. More precisely, we would like to ask the following question: Question 2.8.
Let X be a K-semistable Fano manifold of dimension n . If ( − K X ) k ≡ num lZ for some rational number l ≥ ( n + 1) k and Z an integral ( n − k ) -cycle, then is X isomorphic to the projective space P n ? Corollary 2.7 answers Question 2.8 when k divides n . If the answer to Question 2.8 is yesin general, then the three statements in Theorem B will be equivalent for all codimension k . 3. Higher Codimensional Alpha Invariants
In this section, we will discuss some examples and properties of higher codimensionalalpha invariants. We first recall the definition of higher codimensional alpha invariants.
ZIWEN ZHU
Definition 3.1.
Let X be a Q -Fano variety of dimension n . For 1 ≤ k ≤ n , denoteby ci( L , . . . , L k ) the complete intersection in X cut out by effective Cartier divisors L , . . . , L k . Then we define the alpha invariant of codimension k for X to be α ( k ) ( X ) := inf r (cid:26) lct (cid:18) X, r Z (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Z = ci( L , . . . , L k ) , L , . . . , L k ∈ | − rK X | (cid:27) . Remark . Note that it follows immediately from the definition that α ( k ) ( X ) can also becomputed in terms of linear systems: α ( k ) ( X ) := inf r (cid:26) lct (cid:18) X, r H (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) H ⊂ | − rK X | , codim Bs H ≥ k (cid:27) . A first basic property of alpha invariants is that all α ( k ) ( X )’s form an increasing se-quence in terms of the codimension k : α (1) ( X ) ≤ α (2) ( X ) ≤ · · · ≤ α ( n ) ( X ). This followsimmediately from the definition of alpha invariant and log canonical threshold.The following proposition gives a lower bound for the top codimensional alpha invariant α ( n ) ( X ). Proposition 3.2 (Proposition 2.4, [Zhu19]) . Let X be a Fano manifold of dimension n .Then α ( n ) ( X ) ≥ n n √ ( − K X ) n . Combining Proposition 3.2 and Theorem 1.2, we immediately have the following resultwhich is a special case of Theorem B when k = n . Corollary 3.3.
Let X be a K-semistable Fano manifold of dimension n . If α ( n ) ( X ) ≤ nn +1 ,then X ∼ = P n . Example 3.4.
Let X be a del Pezzo surface of degree 1, then α (2) ( X ) = 2. Indeed, byProposition 3.2, we have α (2) ( X ) ≥
2. By computing the log canonical threshold of ageneral pencil inside | − K X | , we see that α (2) ( X ) = 2. Example 3.5.
Let X be a Fano manifold with Fano index l . Assume that − K X ∼ lin lD such that the linear system | D | is base point free. Then we know that α ( k ) ( X ) ≤ k/l .Indeed we can take k sufficiently general smooth elements L , . . . , L k ∈ | D | such that Z is the transversal intersection of L , . . . , L k . Then lct( X, Z ) = k , and therefore α ( k ) ( X ) ≤ lct( X, lZ ) = k/l . This gives an upper bound of higher codimensional alpha invariantsfor smooth Fano hypersurfaces in the projective spaces. In particular, for the projectivespace P n , we know that α ( k ) ( P n ) ≤ k/ ( n + 1). On the other hand, because P n is K-semistable, we know from Theorem A that α ( k ) ( P n ) ≥ k/ ( n + 1). Consequently we have α ( k ) ( P n ) = k/ ( n + 1), and it is realized by any linear subspaces of codimension k . Example 3.6.
Cheltsov computed in [Che08] that α (1) ( P × P ) = 1 /
2. However, even insimple examples, it seems hard to compute α ( k ) when k ≥
2. For P × P , we only knowthat 2 / < α (2) ( P × P ) ≤ /
4. Indeed, we first notice that − K P × P is of type (2 , L and L be two lines of type (1 ,
0) and type (0 ,
1) respectively that are symmetric withrespect to each other, and ∆ be the diagonal in P × P . Then L + L and ∆ are bothlinearly equivalent to − K P × P . Let Z be the complete intersection of L + L and ∆, andwe have lct( X, Z ) = 3 /
2. Therefore α (2) ( P × P ) ≤ lct( X, Z ) = 3 /
4. We also know that P × P is K-semistable (for example refer to [PW17]). Then by Theorem A and TheoremB, we have α (2) ( P × P ) > / IGHER CODIMENSIONAL ALPHA INVARIANTS 7
Recent results of [SZ18, Zhu18] can be reinterpreted from the point of view of highercodimensional alpha invariants in the following way.
Theorem 3.7.
Let X be a Q -factorial Q -Fano variety of Picard number 1 and dimension n . If α (2) ( X ) > n − n + 1 and α (1) ( X ) ≥ α (2) ( X )( n + 1) α (2) ( X ) − n + 1 resp. α (1) ( X ) > α (2) ( X )( n + 1) α (2) ( X ) − n + 1 ! , then X is K-semistable ( resp. K-stable ) .Remark . The alpha invariant of codimension 2 is related to the notion of log maximalsingularity. Recall that a Q -factorial Q -Fano variety X of Picard number 1 has a logmaximal singularity if there is a movable linear system H on X such that H ≡ num − rK X and ( X, r H ) is not log canonical. X is called log maximal singularity free if X does nothave a log maximal singularity. Note that if a Q -Cartier divisor L ≡ num − K X , then L ∼ Q − K X by similar arguments in the smooth case as in Remark 2.1. Therefore we seethat the linear system r H we consider in the pair ( X, r H ) is Q -linear equivalent to − K X .Then X is log maximal singularity free if and only if α (2) ( X ) ≥ α (2) ( X ) ≥
1, we have α (2) ( X )( n + 1) α (2) ( X ) − n + 1 ≤ . Therefore Theorem 3.7 reduces to the following theorem by Stibitz and Zhuang:
Theorem 3.8 ([SZ18]) . Let X be a Q -factorial Q -Fano variety of Picard number 1. If X is log maximal singularity free and α (1) ( X ) ≥ / resp. > / , then X is K-semistable ( resp. K-stable ) . Theorem 3.7 is an immediate consequence of the following theorem of Zhuang, whichprovides a more precise result compared to Theorem 3.8.
Theorem 3.9 ([Zhu18]) . Let X be a Q -factorial Q -Fano variety of Picard number 1 anddimension n . If for every effective divisor D ∼ Q − K X and every movable linear system M ∼ Q − K X , we have that the pair ( X, n +1 D + n − n +1 M ) is log canonical ( resp. klt ) , then X is K-semistable ( resp. K-stable ) . Indeed, pick any effective divisor D ∼ Q − K X and movable linear system M ∼ Q − K X .Then we know that the pairs ( X, α (1) ( X ) D ) and ( X, α (2) ( X ) M ) are log canonical. We canwrite n +1 D + n − n +1 M as a linear combination of α (1) ( X ) D and α (2) ( X ) M as follows:1 n + 1 D + n − n + 1 M = 1( n + 1) α (1) ( X ) α (1) ( X ) D + n − n + 1) α (2) ( X ) α (2) ( X ) M Note that the conditions α (2) ( X ) > n − n + 1and α (1) ( X ) ≥ α (2) ( X )( n + 1) α (2) ( X ) − n + 1 ZIWEN ZHU are equivalent to 1( n + 1) α (1) ( X ) + n − n + 1) α (2) ( X ) ≤ . (3.1)Then the pair ( X, n +1 D + n − n +1 M ) is also log canonical and by Theorem 3.9, X is K-semistable . Note that when α (1) ( X ) > α (2) ( X )( n + 1) α (2) ( X ) − n + 1we will have strict inequality in (3.1) instead. Then ( X, n +1 D + n − n +1 M ) is klt and X isK-stable. Remark . The above theorems are of course related to the well-known result of Tian in[Tia87], which was later improved by Fujita in [Fuj17] stating that a Fano manifold X ofdimension n is K-stable if α (1) ( X ) ≥ n/ ( n + 1). Theorem 3.7 does not require smoothness,and the lower bound of α (1) ( X ) is smaller at the cost of an additional assumption on α (2) ( X ) and the Picard number. Remark . In [Tia91, Mac14], they give a definition of higher alpha invariants. Althoughin general these higher alpha invariants are different from α ( k ) ( X )’s defined in this paper,they come up with similar results as Theorem 3.7.4. Proof of Theorem A
In this section, we prove Theorem A which gives a lower bound of higher codimensionalalpha invariants for K-semistable Q -Fano varieties. We first state 2 lemmas that will beused in later computation. Lemma 4.1.
Let X be a normal projective variety of dimension n with klt singularities,and L an ample divisor on X . Let Z be a complete intersection of X cut out by k elements L , . . . , L k in the linear system | L | with ideal sheaf I Z ⊂ O X . Let σ : Y → X be any properbirational morphism that factors through the blow-up of X along Z , and write σ − I Z ·O Y = O Y ( − F ) . Let H be the sub linear system of | L | with base ideal I Z , and σ ∗ H = | M | + F bethe decomposition into the moving part and the fixed part of the linear system σ ∗ H . Thenwe have σ ∗ ( M i − · F ) = (cid:26) [ Z ] , i = k ;0 , i = k, and in particular, σ ∗ L n − i · ( M i − · F ) = (cid:26) deg L ( Z ) = L n , i = k ;0 , i = k. Proof.
First note that the linear system | M | corresponds to the global sections of O Y ( σ ∗ L − F ) and is base point free. Write deg L ( Z ) asdeg L ( Z ) = L n = n X i =1 σ ∗ L n − i · ( M i − F ) + M n . For any i , we know that σ ∗ L n − i · ( M i − F ) ≥ M n ≥
0. Therefore, we only need toshow [ Z ] = σ ∗ ( M k − F ).We write [ Z ] = P a i [ Z i ] with a i = l ( O Z,Z i ) and Z i ’s to be irreducible componentsof Z . Localizing at one Z i and let U = Spec O X,Z i . Then we know that σ ∗ L is linear IGHER CODIMENSIONAL ALPHA INVARIANTS 9 equivalent to zero over U . Then over U , we have that M ∼ lin − F , and σ ∗ ( M k − F ) =( − k − σ ∗ ( F k ) = e ( I Z · O X,Z i )[ Z i ] by formula (2.1). Note that X is klt. Then in particular X is Cohen-Macaulay. Therefore we can apply Proposition 2.1 to the complete intersection Z so that we know e ( I Z · O X,Z i ) can be computed by the length of O X,Z i / ( I Z · O X,Z i ) = O Z,Z i . Therefore we have e ( I Z · O X,Z i ) = a i . Now by localizing at all Z i ’s, we see that σ ∗ ( M k − F ) = P a i [ Z i ] = [ Z ]. (cid:3) Proof of Theorem A.
Let Z = ci( L , . . . , L k ) where L , . . . , L k ∈ | − rK X | . Assume σ : Y → X is a proper birational morphism that factors through the blow-up X along Z .Write σ − I Z · O Y = O Y ( − F ) for some Cartier divisor F on Y . Let ǫ ( Z, − K X ) be theSeshadri constant of Z with respect to − K X . By the construction of Z we know that I Z · O X ( − rK X ) is globally generated. Hence ǫ ( Z, − K X ) ≥ /r . In particular for x < /r ,we know that σ ∗ ( − K X ) − xF is the pullback of an ample line bundle on the blow-up of X along Z , and hence nef and big. Then for 0 < x < /r , we havevol Y ( σ ∗ ( − K X ) − xF ) = ( σ ∗ ( − K X ) − xF ) n . Applying Theorem 2.5 to Z , we have β ( X, Z ) := lct(
X, Z ) vol X ( − K X ) − Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx ≥ . Consequently, we know thatlct(
X, Z ) ≥ X ( − K X ) Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx ≥ X ( − K X ) Z r vol Y ( σ ∗ ( − K X ) − xF ) dx = 1 r ( − rK X ) n Z ( σ ∗ ( − rK X ) − xF ) n dx. (4.1)Integrating by parts k times, we get that Z n − k X i =0 (cid:18) n − − ik − (cid:19) (1 − x ) n − k − i x k ! dx = 1 − kn + 1 . Therefore, using formula (4.2) in Lemma 4.2 below, we have Z ( σ ∗ ( − rK X ) − xF ) n dx = ( − rK X ) n · kn + 1 . Then by (4.1) we know that lct( X, r Z ) = r lct( X, Z ) ≥ kn + 1 for any choice of Z , whichimplies that α ( k ) ( X ) ≥ kn + 1 . (cid:3) We close this section with the following lemma which we have used in the proof above.The lemma also includes formula (4.3) which will be used in the next section for the proofof Theorem B.
Lemma 4.2.
Under the setting of Lemma 4.1, we have the following two formulas for thepolynomial ( σ ∗ L − xF ) n : ( σ ∗ L − xF ) n = L n − n − k X i =0 (cid:18) n − − ik − (cid:19) (1 − x ) n − k − i x k ! (4.2)( σ ∗ L − xF ) n = L n n X i = k ( − i + k − (cid:18) ni (cid:19) (cid:18) i − k − (cid:19) x i ! . (4.3) Proof.
First of all, by Lemma 4.1, we have L n − ( σ ∗ L − xF ) n = xF · n − X i =0 σ ∗ L i (cid:18) (1 − x ) σ ∗ L + xM (cid:19) n − − i ! = L n · n − k X i =0 (cid:18) n − − ik − (cid:19) (1 − x ) n − k − i x k ! . This gives us formula (4.2).On the other hand, expand the intersection number ( σ ∗ L − xF ) n , we have( σ ∗ L − xF ) n = L n + n X i =1 ( − i (cid:18) ni (cid:19) (cid:0) σ ∗ L n − i · F i (cid:1) x i . Note that by Lemma 4.1, for i ≥ k , σ ∗ L n − i · F i = F · σ ∗ L n − i ( σ ∗ L − M ) i − = ( − k − (cid:18) i − k − (cid:19) L n , and σ ∗ L n − i · F i = 0 if i < k . Therefore, we get formula (4.3). (cid:3) Proof of Theorem B
We first prove the following proposition, which gives the implication (2) ⇒ (3) in Theo-rem B. Proposition 5.1.
Let X be a smooth K-semistable Fano variety of dimension n . Let Z bea complete intersection of L , . . . , L k ∈ | − rK X | such that lct( X, r Z ) = k ( n +1) . Then Z isirreducible. Furthermore, we have ( − K X ) k ∼ rat lZ ′ for some rational number l ≥ ( n + 1) k ,where Z ′ is the integral ( n − k ) -cycle corresponding to the support of Z with the reducedscheme structure.Proof. Let σ : Y → X be a log resolution of ( X, Z ), with σ − I Z · O Y = O Y ( − F ) for someCartier divisor F on Y . Write F = P a i E i , where a i = ord E i ( Z ). Because X is smooth,we have lct( X, Z ) = min i A X ( E i ) a i . Let E be a divisor that computes lct( X, Z ) and a = ord E ( Z ). We want to show firstthat the center of E on X has the same dimension as Z . By Theorem 2.3, we know that β ( E ) ≥
0, so A X ( E ) a ≥ a ( − K X ) n Z ∞ vol Y ( σ ∗ ( − K X ) − xE ) dx. IGHER CODIMENSIONAL ALPHA INVARIANTS 11
Then because all the equality holds in (4.1), we have1( − K X ) n Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx = lct( X, Z ) = A X ( E ) a . Consequently,1( − K X ) n Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx ≥ a ( − K X ) n Z ∞ vol Y ( σ ∗ ( − K X ) − xE ) dx. By a change of variable for the integral on the RHS of the above inequality, we get1( − K X ) n Z ∞ vol Y ( σ ∗ ( − K X ) − xF ) dx ≥ − K X ) n Z ∞ vol Y ( σ ∗ ( − K X ) − xaE ) dx. Also, because F ≥ aE , for all x we havevol Y ( σ ∗ ( − K X ) − xF ) ≤ vol Y ( σ ∗ ( − K X ) − xaE ) . Then there is an equality of volumes:vol Y ( σ ∗ ( − K X ) − xF ) = vol Y ( σ ∗ ( − K X ) − xaE ) , (5.1)for all x .Now let a m = σ ∗ O Y ( − mE ) for any integer m . Then a m defines a subscheme of X thesupport of which is σ ( E ). Combining the conclusions from [Blu16, Theorem 1.3, Theorem1.4, Proposition 1.5], we see that the graded sequence of ideals a • is finitely generated. Inaddition, suppose a • is generated in degree up to m . Then we have that π : W := Bl a m X → X is the blow-up of X along a m , and π − a m · O W = O W ( − mE W ), where E W is the primeexceptional divisor on W that induces the same divisorial valuation as E on Y .By replacing Y with a common log resolution of Y and W = Bl a m X , we may assumethat σ : Y → X factors through the blow-up π : W → X , and σ − a am · O Y = O ( − mD )for some divisor D on Y . Note that D is the pullback of aE W from W to Y . Also because aE ≤ F , we have σ ∗ O Y ( − mF ) ⊂ σ ∗ O Y ( − maE ) = a am , Then consider the inverse image sheaves of corresponding ideals under the map σ , we havethat mD ≤ mF . Therefore the divisor D satisfies the relation aE ≤ D ≤ F. Now that we have (5.1), we get the following equality:vol Y ( σ ∗ ( − K X ) − xF ) = vol Y ( σ ∗ ( − K X ) − xD ) = vol Y ( σ ∗ ( − K X ) − xaE ) . Since σ ∗ ( − K X ) − xD is the pullback of π ∗ ( − K X ) − xaE W from W to Y , we have theequality of the volumes:vol Y ( σ ∗ ( − K X ) − xF ) = vol W ( π ∗ ( − K X ) − xaE W ) . (5.2)Assume σ ( E ) = π ( E W ) is of codimension s in X . Now on the left hand side of (5.2),for x < /r , by a change of variable, we know from formula (4.3) thatvol Y ( σ ∗ ( − K X ) − xF ) = ( − K X ) n · n X i = k ( − i + k − (cid:18) ni (cid:19) (cid:18) i − k − (cid:19) ( rx ) i ! . (5.3) On the right hand side of (5.2), for sufficiently small x , the divisor π ∗ ( − K X ) − xaE W isample on the blow-up W . Therefore we havevol W ( π ∗ ( − K X ) − xaE W ) = ( π ∗ ( − K X ) − xaE W ) n . Expanding the intersection number, we get a polynomial in terms of x . Note that thedimension of π ( E W ) is n − s , so we have ( π ∗ ( − K X )) i E n − iW = 0 when n − s < i < n .Therefore( π ∗ ( − K X ) − xaE W ) n = ( − K X ) n + ( − s (cid:18) ns (cid:19) ( π ∗ ( − K X )) n − s a s E sW x s + O ( x s +1 ) . (5.4)Compare the above two polynomials on the right hand side of (5.3) and (5.4). We see that k = s , so dim σ ( E ) = n − k = dim Z . Therefore we know that the center of E on X is anirreducible component of Z .Next, we want to show that Z is irreducible. Using formula (4.3) for vol( σ ∗ ( − rK X ) − xF ),we see that the coefficient of x k in vol( σ ∗ ( − rK X ) − xF ) is − ( − rK X ) n (cid:18) nk (cid:19) . If we compareit with the coefficient of x k in the expansion of the polynomial vol( π ∗ ( − rK X ) − xaE W ) =( π ∗ ( − rK X ) − xaE W ) n , we get that( − rK X ) n = a k ( − k − π ∗ ( − rK X ) n − k E kW . (5.5)Now suppose Z is not irreducible. Write [ Z ] = a ′ Z ′ + P i a i [ Z i ], where Z ′ is the ( n − k )-cycledefined by the reduced irreducible component corresponding to the center of E . Note that a am = a am ⊃ σ ∗ O Y ( − mF ) = I mZ , where I mZ is the integral closure of the ideal I mZ . Since I mZ has the same multiplicity as I mZ ,we know that a k e ( a m · O X,Z ′ ) ≤ e ( I mZ · O X,Z ′ ) = m k e ( I Z · O X,Z ′ ) , By the computation in the proof of Lemma 4.1, we know that e ( I Z · O X,Z ′ ) = l ( O Z,Z ′ ) = a ′ . Therefore, we know that a ′ ≥ a k m k e ( a m · O X,Z ′ ) . By (2.1), we have ( − k − π ∗ E kW = 1 m k e ( a m · O X,Z ′ ) Z ′ for sufficiently divisible m . Together with (5.5), we have( − rK X ) n − k · (cid:18) a ′ − a k m k e ( a m · O X,Z ′ ) (cid:19) Z ′ + X i a i [ Z i ] ! = 0 . Since the cycle we intersect with ( − rK X ) n − k in the above equation is effective, we knowthat Z is irreducible supporting on Z ′ and( − rK X ) n = deg ( − rK X ) Z ′ a k m k e ( a m · O X,Z ′ ) . Note that A X ( E ) a = lct( X, Z ) = k ( n +1) r , so we have a = A X ( E )( n +1) rk . Consequently,( − K X ) n = deg ( − K X ) Z ′ (cid:18) n + 1 k (cid:19) k A X ( E ) k e ( a m · O X,Z ′ ) m k . (5.6) IGHER CODIMENSIONAL ALPHA INVARIANTS 13
The log discrepancy of E doesn’t change after we localize at Z ′ because Z ′ is the center of E . Working on Spec O X,Z ′ , we next want to show that A X ( E ) k e ( a m · O X,Z ′ ) m k ≥ k k . Note that by the definition of log canonical threshold and a m = σ ∗ O Y ( − mE ), we havelct( a m · O X,Z ′ ) ≤ A X ( E )ord E ( a m · O X,Z ′ ) = A X ( E ) m . Since Spec O X,Z ′ is smooth, we know that A X ( E ) k e ( a m · O X,Z ′ ) m k ≥ lct( a m · O X,Z ′ ) k e ( a m · O X,Z ′ ) ≥ k k , where the last inequality follows from [dFEM04]. Therefore from (5.6) we have the followinginequality: ( − K X ) n ≥ ( n + 1) k ( − K X ) n − k Z ′ , or equivalently, ( − K X ) n − k [( − K X ) k − ( n + 1) k Z ′ ] ≥ . Since ( − K X ) k ∼ rat r k Z , we may assume ( − K X ) k ∼ rat lZ ′ for some rational number l .Then by the ampleness of − K X , we know that l ≥ ( n + 1) k . This finishes the proof ofProposition 5.1. (cid:3) Proof of Theorem B.
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Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
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