K-SEMISTABLE FANO MANIFOLDS WITH THE SMALLEST ALPHA INVARIANT CHEN JIANG 7 1 0 Abstract. In this short note, we show that K-semistable Fano mani- 2 folds with the smallest alpha invariant are projective spaces. Singular cases are also investigated. n a J 1. introduction 1 3 Throughout the article, we work over the complex number field C. A Q- Fano variety is a normal projective variety X with log terminal singularities ] G such that the anti-canonical divisor −KX is an ample Q-Cartier divisor. A It has been known that a Fano manifold X (i.e., a smooth Q-Fano variety) admits Ka¨hler–Einstein metrics if andonly if X is K-polystable bythe works . h [DT92, Tia97, Don02, Don05, CT08, Sto09, Mab08, Mab09, Ber16] and t a [CDS15a, CDS15b, CDS15c, Tia15]. m On the other hand, the existence of Ka¨hler–Einstein metrics and K- [ stability are related to the alpha invariants α(X) of X defined by Tian 1 [Tia87] (see also [TY87, Zel98, Lu00, Dem08]). Tian [Tia87] proved that v for a Fano manifold X, if α(X) > dimX/(dimX + 1), then X admits 5 Ka¨hler–Einstein metrics. Odaka and Sano [OS12, Theorem 1.4] (see also its 8 0 generalizations [Der16, BHJ15, FO16, Fuj16c]) proved a variant of Tian’s 9 theorem: if a Q-Fano variety X satisfies that α(X) > dimX/(dimX +1) 0 (resp. ≥ dimX/(dimX +1)), then X is K-stable (resp. K-semistable). We . 1 are interested in the relation of alpha invariants and K-semistability. 0 Recall that Fujita and Odaka proved that there exists a lower bound of 7 1 alpha invariants for K-semistable Q-Fano varieties. : v Theorem 1.1 ([FO16, Theorem 3.5]). Let X be a K-semistable Q-Fano i X variety of dimension n. r Then α(X) ≥ n+11. a It is natural and interesting to ask when the equality holds. For example, it is well-known that Pn is K-semistable with α(Pn) = 1 . The main n+1 theorem of this paper is the following. Theorem 1.2. Let X be a K-semistable Fano manifold of dimension n. Then α(X) = 1 if and only if X ∼= Pn. n+1 This is an application of Birkar’s answer to Tian’s question [Bir16, The- orem 1.5], and Fujita–Li’s criterion for K-semistability [Li15, Fuj16b]. Date: February 1, 2017. TheauthorwassupportedbyJSPSKAKENHIGrantNumberJP16K17558andWorld Premier International Research Center Initiative (WPI),MEXT, Japan. 1 2 CHENJIANG ItisnaturaltoaskwhetherthesamestatementholdstrueforK-semistable Q-Fano varieties instead of manifolds. However, this is on longer true even in dimension 2. We are grateful to Kento Fujita for kindly providing the following example: Example 1.3. Consider the cubic surface X = (x30 = x1x2x3) ⊂ P3, which is a toric log del Pezzo surface (i.e, a Q-Fano variety of dimension 2) with 3 du Val singularities of type A2. On one hand, it is well-known that X admits a Ka¨hler–Einstein metric (cf. [DT92]), hence is K-semistable. On 1 the other hand, α(X) = (cf. [PW10]). 3 Infact,bytheclassificationofpossibleduValsingularitiesofK-semistable log del Pezzo surfaces (cf. [Liu16, Corollary 6]) and explicit computation of alpha invariants (cf. [Par03, PW10, CK14]), we have the following theorem. Theorem 1.4. Let X be a K-semistable log del Pezzo surface with at worst du Val singularities. Then α(X) = 1 if and only if X ∼= P2 or X ⊂ P3 is a 3 cubic surface with at least 2 singularities of type A2. Moreover, by classification of Q-Fano 3-fold with Q-factorial terminal sin- gularities and ρ(X) = 1 with large Fano index due to Prokhorov [Pro10, Pro13], we prove the following: Theorem 1.5. Let X be a K-semistable Q-Fano 3-fold with Q-factorial terminal singularities and ρ(X) = 1. Assume that h0(−K ) ≥ 22. Then X α(X) = 1 if and only if X ∼= P3. 4 Finally, we propose the following much stronger conjecture. For some evidence in dimension 3, we refer to [CS08] and [Fuj16a]. Conjecture 1.6. Let X be a K-semistable Fano manifold. Then α(X) < 1 if and only if X ∼= Pn. n Acknowledgments. The author would like to thank Professors Kento Fu- jita and Yoshinori Gongyo for effective discussions. The main part of this paper was written during the author enjoyed the workshop “Higher Dimen- sional Algebraic Geometry, Holomorphic Dynamics and Their Interactions” at Institute for Mathematical Sciences, National University of Singapore. The author is grateful for the hospitality and support of IMS. 2. Preliminaries We adopt the standard notation and definitions in [KM98] and will freely use them. Definition 2.1. Let X be a Q-Fano variety. The alpha invariant α(X) of X is defined by the supremum of positive rational numbers α such that the pair (X,αD) is log canonical forany effective Q-divisor D withD ∼ −K . Q X In other words, α(X) := inf{lct(X;D)|0 ≤ D ∼ −K }. Q X Tian [Tia90] asked whether whether the infimum is a minimum, which is answered by Birkar affirmatively. K-SEMISTABLE FANO MANIFOLDS WITH THE SMALLEST ALPHA INVARIANT 3 Theorem 2.2 ([Bir16, Theorem 1.5]). Let X be a Q-Fano variety. Assume that α(X) ≤ 1. Then there exists an effective Q-divisor D such that D ∼ Q −K and lct(X;D) = α(X). X Definition 2.3 ([Fuj16b]). Let X be a Q-Fano variety of dimension n. Take any projective birational morphism σ : Y → X with Y normal and any prime divisor F on Y , that is, F is a prime divisor over X. (1) Define the log discrepancy of F as A(F) := mult (K −σ∗K )+1; F Y X (2) Define vol (−K −xF):= vol (−σ∗K −xF); X X Y X (3) Define ∞ β(F) := A(F)·(−K )n − vol (−K −xF)dx. X X X Z0 Note that the definitions do not depend on the choice of birational model Y. Instead of recalling the original definition, we use the following criterion to define K-semistability. Definition-Proposition 2.4([Fuj16b,Corollary1.5],[Li15,Theorem3.7]). Let X be a Q-Fano variety. X is K-semistable if β(F) ≥ 0 for any divisor F over X. Note that K-semistability is known to be equivalent to Ding-semistability by [BBJ15]. 3. Proof of main theorem Proposition 3.1. Let X be a K-semistable Q-Fano variety of dimension n. Assume that α(X) = 1 , then there exists a prime divisor E on X such n+1 that −K ∼ (n+1)E and (X,E) is plt. X Q Proof. LetX beaK-semistableQ-Fanovariety of dimensionnwithα(X) = 1 . By Theorem 2.2, there is a divisor D ∼ −K such that lct(X;D) = n+1 Q X 1 1 . Take F to be a non-klt place of (X, D), then there is a resolution n+1 n+1 σ : Y → X such that F is a divisor on Y. Denote µ to be the multiplicity of F in σ∗D. Note that µ >0 since X is klt. By assumption, 1 mult K −σ∗ K + D = −1, F Y X (cid:18) (cid:18) n+1 (cid:19)(cid:19) which means that µ A(F) = . n+1 By Definition-Proposition 2.4, β(F) ≥ 0, which means that 1 A(F) (−K )n = (−K )n X X n+1 µ 1 ∞ ≥ vol (−K −xF)dx X X µ Z0 ∞ = vol (−K −xµF)dx X X Z0 4 CHENJIANG ∞ ≥ vol (−K −xD)dx X X Z0 1 = (1−x)n(−K )ndx X Z0 1 = (−K )n. X n+1 The second equality holds since σ∗D ≥ µF. Hence all inequalities become equalities. In particular, vol (−K −xµF) = vol (−K −xD)= (1−x)n(−K )n X X X X X for almost all x. By differentiability of volume functions ([BFJ09, Corollary C]), µ·vol (−σ∗K ) Y|F X 1 d = − vol (−σ∗K −xµF) Y X ndx(cid:12) (cid:12)x=0 (cid:12) 1 d = − (cid:12) (1−x)n(−K )n X ndx(cid:12) (cid:12)x=0 = (−K )n(cid:12). X (cid:12) Here vol is the restricted volume, we refer to [ELMNP09] for defini- Y|F tion and properties. Since volY|F(−σ∗KX) > 0, F 6⊆ B+(−σ∗KX) by [ELMNP09, Theorem C]. Hence by [ELMNP09, Corollary 2.17], vol (−σ∗K ) = (−σ∗K )n−1·F = (−K )n−1·σ F. Y|F X X X ∗ In other words, we have (−K )n−1(D−µσ F) = (−K )n−µ·vol (−σ∗K )= 0. X ∗ X Y|F X This implies that D = µσ F since D ≥ µσ F and −K is ample. In ∗ ∗ X particular, F is not σ-exceptional and σ F is a prime divisor on X. Denote ∗ 1 1 E := σ F. Moreover, sinceF isanon-kltplaceof(X, D),mult D = ∗ n+1 En+1 1, that is, µ = n+1. In particular, −K ∼ D = (n+1)E. Finally, this X Q argument shows that F is the only non-klt place of (X,E), which means that (X,E) is plt. (cid:3) Corollary 3.2. Let (X,E) as in Proposition 3.1. Then X ≃ Pn if one of the following condition holds: (1) X is factorial; (2) (E)n ≥ 1; (3) E is Cartier in codimension two and E ≃ Pn−1. Proof. (1)IfX isfactorial, thenE isaCartierdivisor. Inparticular, (E)n ≥ 1. Hence this is a special case of (2). (2) If (E)n ≥ 1, then (−K )n = (n+1)n(E)n ≥ (n+1)n. X By [Liu16, Theorem 1.1] or [LZ16, Theorem 9], X ≃ Pn. K-SEMISTABLE FANO MANIFOLDS WITH THE SMALLEST ALPHA INVARIANT 5 (3) If E is Cartier in codimension two and E ≃ Pn−1, then by adjunction, (K +E)| = K , and X E E (n+1)n (n+1)n (−K )n = (−(K +E))n−1·E = (−K )n−1 = (n+1)n. X nn−1 X nn−1 E Again by [Liu16, Theorem 1.1] or [LZ16, Theorem 9], X ≃ Pn. (cid:3) Proof of Theorem 1.2. ItfollowsdirectlyfromProposition3.1andCorollary 3.2(1) (or [KO73]). (cid:3) 4. Singular surfaces Recall the following theorem on classification of possible du Val singular- ities of a K-semistable log del Pezzo surface. Theorem 4.1 ([Liu16, Theorem 23, Proof of Corollary 6]). Let X be a K-semistable log del Pezzo surface with at worst du Val singularities. (1) If (−KX)2 = 1, then X has at worst singularities of type A1, A2, A3, A4, A5, A6, A7, A8, or D4; (2) If (−KX)2 = 2, then X has at worst singularities of type A1, A2, or A3; (3) If (−KX)2 = 3, then X has at worst singularities of type A1 or A2; (4) If (−KX)2 = 4, then X has at worst singularities of type A1; (5) If (−K )2 ≥ 5, then X is smooth. X We remark that in [Liu16, Corollary 6], log del Pezzo surfaces are as- sumed to be admitting Ka¨hler–Einstein metrics, but the proof works well for K-semistable log del Pezzo surfaces. The only part that the existence of Ka¨hler–Einstein metrics is needed is to exclude the case that (−K )2 = 1 X and X has singularities of type A8. Recall the following theorem on explicit computation of alpha invariants. Theorem 4.2 ([Par03], [PW10, Theorems 1.4, 1.5, and 1.6], [CK14, Theo- rem 1.26, Example 1.27]). Let X be a log del Pezzo surface with at worst du Val singularities. Assume that X is singular, then α(X) = 1 if and only if 3 one of the following holds: (1) (−KX)2 = 6 and Sing(X) = {A1}; (2) (−KX)2 = 5 and Sing(X) = {A2} or {2A1}; (3) (−KX)2 = 4 and Sing(X) = {A3} or Sing(X) ⊇ {A1 +A2}; (4) (−KX)2 = 3 and Sing(X) ⊇ {A4}, {2A2}, or Sing(X) = {D4}; (5) (−KX)2 = 2 and Sing(X) ⊇ {D5},{(A5)′}, or {A7}; (6) (−KX)2 = 1 and Sing(X) ⊇ {D8} or {E6}. Proof of Theorem 1.4. Let X be a K-semistable log del Pezzo surface with at worst du Val singularities and α(X) = 1. If X is smooth, then X ≃ P2 3 by Theorem 1.2. If X is singular, then (−KX)2 = 3 and Sing(X) ⊇ {2A2} by Theorems 4.1 and 4.2. To see the “if” part, one just notice that any cubic surface with at worst singularities of type A1 or A2 is K-semistable (cf. [OSS16, Theorem 4.3]). (cid:3) 6 CHENJIANG 5. Singular threefolds In this section, we prove Theorem 1.5. Recall the following theorem on the upper bound of volumes. Theorem 5.1 (cf. [Liu16, Theorem 25]). Let X be a K-semistable Q-Fano 3-fold with at worst terminal singularities. Let p ∈ X be an isolated singu- larity with local index r. Then (r+2)(4+4r)3 3 (−K ) ≤ . X (3r)3 Proof. Denote by m the maximal ideal at p. We may take a log resolution p of (X,m ), namely π : Y → X such that π is an isomorphism away from p p and π−1m ·O is an invertible ideal sheaf on Y. Let E be exceptional p Y i divisors of π. We define the numbers a and b by i i K = π∗K + a E Y X i i Xi and π−1m ·O = O (− b E ). p Y Y i i Xi Itisclear thatlct(X;m ) = min 1+ai. Sinceπ is an isomorphismaway from p i bi p, we have b ≥ 1 for any i. Since X is terminal at p, by [Kaw93], there i 1 exists an index i0 such that ai0 = r. Hence 1+a 1 lct(X;m ) ≤ i0 ≤ 1+ . p b r i0 On the other hand, by [Kak00] (see also [TW04, Proposition 3.10]), mult X ≤ r+2. Hence by [Liu16, Theorem 16], p 1 3 (r+2)(4+4r)3 (−K )3 ≤ 1+ lct(X;m )3mult X ≤ . X (cid:18) 3(cid:19) p p (3r)3 (cid:3) Now let X be a K-semistable Q-Fano 3-fold with Q-factorial terminal 1 singularities and ρ(X) = 1 with α(X) = . By Proposition 3.1, there exists 4 a prime divisor E on X such that −K ∼ 4E. X Q Recall that we may define ([Pro10]) qW(X) := max{q | −K ∼ qA,A is a Weil divisor}, X qQ(X) := max{q | −K ∼ qA,A is a Weil divisor}. X Q It is known by [Suz04 , Pro10] that qW(X),qQ(X) ∈ {1,...,11,13,17,19}. Moreover, by [Pro10, Lemma 3.2], in our case, 4|qQ(X). Hence there are 2 cases: (i) qQ(X) = 8; (ii) qQ(X) = 4. Now assume that h0(−K ) ≥ 22. Define the genus g(X) := h0(−K )− X X 2 ≥ 20. If qQ(X) = 8, since g(X) > 10, then by [Pro13, Theorem 1.2(ii)], either X ≃ X6 ⊂ P(1,2,3,3,5) or X ≃ X10 ⊂ P(1,2,3,5,7). 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