Alpha-invariants and purely log terminal blow-ups

We prove that the sum of the $\alpha$-invariants of two different Koll\'ar components of a Kawamata log terminal singularity is less than $1$.

Let V be a normal irreducible projective variety of dimension n 1, and let ∆ V be an effective Q-divisor on V . Write where each ∆ i is a prime divisor, and each a i is a positive rational number. Suppose that the log pair (V, ∆ V ) has at most Kawamata log terminal singularities. Then, in particular, each a i is less than 1. Suppose also that the divisor −(K V + ∆ V ) is ample, so that (V, ∆ V ) is a log Fano variety. Finally, suppose that V is faithfully acted on by a finite group G such that the divisor ∆ V is G-invariant. Let α G (V, ∆ V ) be the real number defined by sup    λ ∈ Q the pair V, ∆ V + λD V has Kawamata log terminal singularities for every G-invariant and effective Q-divisor This number is known as the α-invariant of the log Fano variety (V, ∆ V ), or its global log canonical threshold (see [12,Definition 3.1]). If G is trivial, we put α(V, ∆ V ) = α G (V, ∆ V ).
Example 2. A finite group G acting faithfully on P 1 is one of the following finite groups: the alternating group A 5 , the symmetric group S 4 , the alternating group A 4 , the dihedral group D 2m of order 2m, or the cyclic group µ m of order m (including the case m = 1, that is, the trivial group). The number α G (P 1 ) 2 is equal to the length of the smallest G-orbit in P 1 , which gives We assume that all varieties are defined over the field C.
If both ∆ V = 0 and G is trivial, we put α(V ) = α G (V, ∆ V ). This is the most classical case. Namely, if V is a smooth Fano variety, then by [11,Theorem A.3] the number α(V ) coincides with the α-invariant of V defined by Tian in [45]. Its values were found or estimated in many cases. For example, in the toric case the explicit formula for α(V ) is given by Cheltsov and Shramov in [11,Lemma 5.1]. It gives α(P n ) = 1 n+1 , which can also be verified by an easy explicit computation.
The α-invariant plays an important role in Kähler geometry. If V is a smooth Fano variety, then V admits a G-invariant Kähler-Einstein metric provided that This was proved by Tian in [45]. In [19], this result was improved by Fujita. He proved that V admits a Kähler-Einstein metric if it is smooth and α(V ) In particular, all smooth hypersurfaces in P d of degree d are Kähler-Einstein, because their α-invariants are at least d−1 d by [1,8]. The K-stability of the log Fano variety (V, ∆ V ) crucially depends on α(V, then the log Fano variety (V, ∆ V ) is K-unstable by [22,Theorem 3.5] and [21,Lemma 5.5]. This bound is sharp, since P n is K-semistable and α(P n ) = 1 The α-invariant also plays an important role in birational geometry. It was first observed by Park in [35], where he proved a theorem that evolved into the following: Theorem 3 ([4, Theorem 1.5]). Let X be a variety with at most terminal Q-factorial singularities. Suppose that there is a proper morphism φ : X → Z such that Z is a smooth curve, and −K X is φ-ample. Let P be a point in Z, and let E X be the scheme fiber of φ over P . Suppose that E X is irreducible, reduced, normal, and has at most Kawamata log terminal singularities, so that E X is a Fano variety by the adjunction formula. Suppose also that there is a commutative diagram such that Y is a variety with at most terminal Q-factorial singularities, ψ is a proper morphism, the divisor −K Y is ψ-ample, and ρ is a birational map that induces an isomorphism where E Y is the scheme fiber of ψ over P . Suppose, in addition, that E Y is irreducible. Then ρ is an isomorphism provided that α(E X ) 1. Moreover, if E Y is reduced, normal and has at most Kawamata log terminal singularities, then ρ is an isomorphism provided that α(E X ) + α(E Y ) > 1.
Theorem 3 gives a necessary condition in terms of α-invariants for the existence of a nonbiregular fiberwise birational transformation of a Mori fibre space over a curve. It follows from [29, Theorem 1.1] that this condition is not a sufficient condition. Nevertheless, the bound is sharp (one can find many examples in [35,36]).
Example 4. Let S be a P 1 -bundle over a curve. Then we have an elementary transformation to another P 1 -bundle over the same curve. Note that the α(P 1 ) = 1 2 by Example 2. Example 5 ([18, Example 5.8]). Let S be a smooth cubic surface in P 3 with an Eckardt point O. Denote by L 1 , L 2 , L 3 the lines in S passing through O. Put X = S × A 1 , and let φ be the natural projection X → A 1 . Let us identify S with a fiber of φ. Then there is a commutative diagram . Let X and Y be subvarieties in A 1 ×P 3 given by eqautions x 3 + y 2 z + z 2 w + t 12 w 3 = 0 and x 3 + y 2 z + z 2 w + w 3 = 0, respectively, where t is a coordinate on A 1 , and (x : y : z : w) are homogeneous coordinates on P 3 . Then the projections φ : X → A 1 and ψ : Y → A 1 are fibrations into cubic surfaces, and the map (t, x, y, z, w) → (t, t 2 x, t 3 y, z, t 6 w) gives a non-biregular birational fiberwise map ρ : X Y . The fiber of φ over the point t = 0 is a cubic surface that has one Du Val singular point of type E 6 , so that its αinvariant is 1 6 (see [4,Theorem 1.4]), and the fiber of ψ over the point t = 0 is a smooth cubic surface with an Eckardt point, so that its α-invariant is 2 3 (see [2,Theorem 1.7]). The α-invariant also plays an important role in singularity theory. Let U ∋ P be a germ of a Kawamata log terminal singularity. Then it follows from [47, Lemma 1] (cf. [39,Proposition 2.12]) that there is a birational morphism φ : X → U such that its exceptional locus consists of a single prime divisor E X such that φ(E X ) = P , the log pair (X, E X ) has purely log terminal singularities, and the divisor −(K X + E X ) is φ-ample. Then for some positive rational number δ X . Recall from [39, Definition 2.1] that the birational morphism φ : X → U is a purely log terminal blow-up of the singularity U ∋ P .
By [26,Theorem 7.5], the divisor E X is a normal variety that has rational singularities. Moreover, it can be naturally equipped with a structure of a log Fano variety. Let R 1 , . . . , R s be all the irreducible components of the locus Sing(X) of codimension 2 that are contained in E X . Put Furthermore, the singularities of the log pair (E X , Diff E X (0)) are Kawamata log terminal by Adjunction ( see [44, 3.2] or [27, 17.6]). This means that (E X , Diff E X (0)) is a log Fano variety with Kawamata log terminal singularities, because −E X is φ-ample.
Let us show how to compute α(E X , Diff E X (0)) in three simple cases.
. Let U ∋ P be a germ of a Du Val singularity, and f : W → U be the minimal resolution of this singularity. Then the exceptional curves of f are smooth rational curves whose self-intersections are −2, and their dual graph is of type A m , D m , E 6 , E 7 , or E 8 . Let E W be one of the exceptional curves that is chosen as follows. If U ∋ P is not a singularity of type A m , let E W be the only f -exceptional curve that intersects three other f -exceptional curves, i.e., E W is the "fork" of the dual graph. If U ∋ P is a singularity of type A m , choose E W to be the k-th curve in the dual graph.
In this case, we may assume that k m+1 2 . In all the cases, there exists a commutative diagram where h is the contraction of all f -exceptional curves except E W , and g is the contraction of the proper transform of E W on the surface Y . Denote the g-exceptional curve by E Y . Then Y has at most Du Val singularities of type A, the curve E Y is smooth, and it contains all the singular points of the surface Y , if any. One can check that the log pair (Y, E Y ) has purely log terminal singularities (see [28,Theorem 4.15(3)]). Also, the divisor −(K Y +E Y ) is g-ample. Thus, the curve E Y is a Kollár component of the singularity U ∋ P . Moreover, where P i , Q j , and R ℓ are singular points of Y that lie on E Y . The singular point P i (resp. Q j and R ℓ ) is a Du Val singular point of type A i (resp. A j and A ℓ ). Since E Y ∼ = P 1 , it follows from Example 1 that Let Q be a point on one of the f -exceptional curves. We consider two cases, one is the case where the point Q belongs to one of the two exceptional curves that correspond to "tails" of the dual graph but it is not contained in any other exceptional curve, the other is the case where Q is the intersection point of the k-th and (k + 1)-th f -exceptional curves, 1 k m 2 . Let ξ : W → W be the blow up at Q, and ζ be the contraction of the proper transforms of all the f -exceptional curves. Thus, there exists a commutative diagram x q q q q q q q q q q q q q U Denote the g-exceptional curve by E Y . It is a smooth rational curve. The dual graphs of the exceptional curves of the minimal resolution of singularities ζ : W → Y are chains such that the self-intersection numbers of the exceptional curves are −3, −2, . . . , −2, and the proper transform of E Y intersects only the "tail" components of these chains. In the former case, Y has a unique singular point O, which is a quotient of C 2 by the cyclic group µ 2m+1 . In the latter case, it contains two singular points P 1 and P 2 , which are quotients of C 2 by the cyclic groups µ 2k+1 and µ 2(m−k)+1 , respectively.
By [28,Theorem 4.15(3)] the log pair (Y, E Y ) has purely log terminal singularities. Also, the divisor −(K Y + E Y ) is g-ample. Thus, the curve E Y is a Kollár component of the singularity U ∋ P . Moreover, in the former case, 2k 2k+1 P 1 + 2(m−k) 2(m−k)+1 P 2 in the latter case. Therefore, 2 in the former case, 2k+1 2m+2 1 2 in the latter case. In particular, in the latter case we see that α(E Y , Diff E Y (0)) = 1 2 if and only if m is even, and Q is the "central point" of the configuration of the f -exceptional curves.
It is easy to see from [28,Theorem 4.15] that if U ∋ P is a Du Val singularity of type D or E, and the exceptional curve E W in Example 8 is not the one corresponding to the "fork" of the dual graph, then the curve E Y is not a Kollár component (see [39,Example 4.7]). We will see later that in these cases the singularity U ∋ P has a unique Kollár component, which is described in Example 8. This is not true in general, i.e., a Kollár component of a singularity U ∋ P may not be unique, as one can see from Examples 8 and 9. Nevertheless, Li and Xu established in [31, Theorem B] the following: The K-semistable Kollár components of two-dimensional Du Val singularities are described in our Examples 8 and 9. They are precisely the Kollár components whose αinvariants are at least 1 2 (cf. [32,Example 4.7]). Note that Du Val singularities are two-dimensional rational quasi-homogeneous isolated hypersurface singularities. The K-semistable Kollár components of many threedimensional rational quasi-homogeneous isolated hypersurface singularities have been described in [9,15]. Similarly, the K-semistable Kollár components of many four-dimensional rational quasi-homogeneous isolated hypersurface singularities have been described in [23].
The purpose of this paper is to prove the following analogue of Theorem 3.
Theorem 11. Let U ∋ P be a germ of a Kawamata log terminal singularity. Suppose that there is a commutative diagram then ρ is an isomorphism.
Before proving this result, let us consider its applications. Suppose that By Theorem 11, this inequality implies that the α-invariant of another Kollár component of the singularity U ∋ P , if any, must be less than 1 dim(U ) , so that it should be K-unstable. Of course, this also follows from Theorem 10, because the inequality (12) implies that the log Fano variety (E X , Diff E X (0)) is K-semistable.
Theorem 11 also implies  [39,Definition 4.1] that the singularity U ∋ P is said to be weakly exceptional if it has a unique purely log terminal blow-up. This is equivalent to the condition that there is a Kollár component E X of U ∋ P such that α(E X , Diff E X (0)) 1 (see [39,Theorem 4.3], [30,Theorem 2.1], and [12]). It follows from Example 8 that Du Val singularities of types D and E are weakly exceptional. On the other hand, Du Val singularities of type A are not weakly exceptional, since each of them admits several Kollár components (see Examples 8 and 9), and thus has several purely log terminal blow ups.
Remark 14. Du Val singularities are special examples of two-dimensional quotient singularities. Note that quotient singularities are always Kawamata log terminal. For each of them, it is easy to describe one Kollár component. Let G be a finite subgroup in GL n+1 (C). Suppose that U ∋ P is a quotient singularity C n+1 / G. By the Chevalley-Shephard-Todd theorem, we may assume that the group G does not contain any quasi-reflections (cf. [13,Remark 1.16]). Let η : C n+1 → U be the quotient map. Then there is a commutative where π is the blow up at the origin, the morphism ω is the quotient map that is induced by the action of G lifted to the variety W , and ψ is a birational morphism. Denote by E the exceptional divisor of π, and denote by E Y the exceptional divisor of ψ. Then E ∼ = P n , and E Y is naturally isomorphic to the quotient P n /G, where G is the image of the group G in PGL n+1 (C). Moreover, the log pair (Y, E Y ) has purely log terminal singularities, and the divisor −(  [12, Proof of Theorem 3.16]). Thus, if α G (P n ) 1, then this Kollár component is unique by Corollary 13. One can find many subgroups G ⊂ PGL n+1 (C) with α G (P n ) 1 in [33,12,13,41,14,42,16]. Note also that one always has α G (P n ) 1184036 by [46].
In the remaining part of the paper, we prove Theorem 11. Let us use its assumptions and notations. We have to show that ρ is an isomorphism. Suppose that this is not the case. Let us seek for a contradiction.
We may assume that U is affine. There exists a commutative diagram such that W is a smooth variety, and f and g are birational morphisms. Denote by E W X and E W Y the proper transforms of the divisors E X and E Y on the variety W , respectively. Then E W X is g-exceptional, and E W Y is f -exceptional. We may assume that E W X , E W Y and the remaining exceptional divisors of f and g form a divisor with simple normal crossings. Observe then ρ is small, which is impossible, because −E X is φ-ample, and −E Y is ψ-ample (see [17,Proposition 2.7]). Let F 1 , . . . , F m be the prime divisors on W that are contracted by both f and g. Then for some rational numbers a, a 1 , . . . , a m . Since the log pair (X, E X ) has purely log terminal singularities, all numbers a, a 1 , . . . , a m are strictly less than 1. Also, we have Then M X does not have any base points. Denote its proper transforms on Y and W by M Y X and M W X , respectively. Then for some rational numbers α, α 1 , . . . , α m . Similarly, let M Y be the base point free linear system |−nE Y |. Denote by M X Y and M W Y its proper transforms on X and W , respectively. Then there is a positive rational number for some rational numbers β, β 1 , . . . , β m .
Lemma 15. One has α > 1 and β > 1. In particular, the singularities of the log pairs are not log canonical. Proof. It is enough to show that α > 1. We have This gives where t X = 1 + α−1 δ X . On the other hand, from (16) we obtain

Now we let
so that −B is f -nef. Then B is effective if and only if f * (B) = (1 − α)E X is effective by Negativity Lemma (see [28,Lemma 3.39]). Since a < 1, the divisor B is not effective, which implies that α > 1.
As in the proof of Lemma 15, put Now take any positive rational numbers λ and µ such that λ + µ 1. One has are also log canonical. On the other hand, we have for some rational numbers c, c 1 , . . . , c m that do not exceed 1. Similarly, we have where d, d 1 , . . . , d m are rational numbers that do not exceed 1. Denote by D W the boundary Moreover, the log pair (W, D W ) is log canonical, since W is smooth, the linear systems M W Y and M W X are free from base points, and the divisors E W X , E W Y , F 1 , . . . , F m form a simple normal crossing divisor. Since K X + E X + λǫ Y M X Y + µǫ X M X is φ-ample, it follows from [28,Corollary 3.53] that the log pair (X, E X + λǫ Y M X Y + µǫ X M X ) is the canonical model of the log pair (W, D W ). Similarly, the log pair (Y, E Y + λǫ Y M Y + µǫ X M Y X ) is also the canonical model of the log pair (W, D W ), because K Y + E Y + λǫ Y M Y + µǫ X M Y X is ψ-ample. Since the canonical model is unique by [28,Theorem 3.52], we see that ρ is an isomorphism. Since ρ is not an isomorphism by assumption, we obtain a contradiction. This completes the proof of the lemma.
Let λ = α(E X , Diff E X (0)) and µ = α(E Y , Diff E Y (0)). We may assume that the log pair (X, E X + λǫ Y M X Y ) is not log canonical. Then (E X , Diff E X (0) + λǫ Y M X Y | E X ) is not log canonical by Inversion of adjunction, see [27, 17.6]. On the other hand, we have This is impossible by the definition of the α-invariant α(E X , Diff E X (0)).