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ON BERMAN-GIBBS STABILITY AND K-STABILITY OF Q-FANO VARIETIES 5 1 KENTO FUJITA 0 2 n a J Abstract. The notion of Berman-Gibbs stability was originally introduced by Robert Berman for Q-Fano varieties X. We show 1 that the pair (X,−KX) is K-stable (resp. K-semistable) provided ] that X is Berman-Gibbs stable (resp. semistable). G A . Contents h t a 1. Introduction 1 m 2. Preliminaries 3 [ 2.1. K-stability 3 1 2.2. Multiplier ideal sheaves 5 v 3. The projective line case 6 8 4 4. Key propositions 7 2 5. Proof of Theorem 1.4 11 0 References 13 0 . 1 0 5 1 1. Introduction : v One of the most important problem for the study of Q-Fano varieties i X X (i.e., projective log-terminal varieties with −K ample Q-Cartier) X r a is to determine whether the pairs (X,−KX) are K-stable or not (for the notion of K-stability, see Section 2.1). Recently, Robert Berman introduced a new stability of X, which he calls Gibbs stability, and its variants. The main purpose of this paper is to show that, slightly modifying the definition (we rename it as Berman-Gibbs stability), it impliestheK-stabilityinDonaldson’s[Don02]andTian’s[Tia97]sense. In particular, by [CDS12a, CDS12b, CDS13, Tia12], it implies the ex- istence of Ka¨hler-Einstein metric if X is smooth and the base field is Date: January 5, 2015. 2010 Mathematics Subject Classification. Primary 14L24; Secondary 14J17. Key words and phrases. Fano varieties, K-stability, multiplier ideal sheaves, Kähler-Einsteinmetrics. 1 2 KENTO FUJITA the complex number field. We remark that Robert Berman showed in [Ber13, Theorem 7.3] that strongly Gibbs stable Fano manifolds defined over the complex number field admit Ka¨hler-Einstein metrics, where the notion of strong Gibbs stability is stronger than the notion of Berman-Gibbs stability. Now we define the notion of Berman-Gibbs stability. (We remark that the notion of Berman-Gibbs stability is slightly weaker than the notion of uniform Gibbs stability. For detail, see [Ber13, Section 7].) Definition 1.1. Let X be a projective variety and L be a globally generated Cartier divisor on X. Set N := h0(X,O (L)) and φ := X φ : X → PN−1, where φ is a morphism defined by the complete |L| |L| linear system |L|. Consider the morphism Φ: XN → (PN−1)N defined by the copies of φ, that is, Φ(x ,...,x ) := (φ(x ),...,φ(x )) for 1 N 1 N x ,...,x ∈ X. Let Det ⊂ (PN−1)N be the divisor defined by the 1 N N equation det(x ) = 0, where ij 1≤i,j≤N (x : ··· : x ;······ ;x : ··· : x ) 11 1N N1 NN are the multi-homogeneous coordinates of (PN−1)N. We set the divisor D ⊂ XN defined by D := Φ∗Det . X,L X,L N Remark1.2. ThedivisorD ⊂ XN isdefineduniquelybyX andthe X,L linearequivalenceclassofL. Inparticular, thedefinitionisindependent of the choice of the basis of H0(X,O (L)). X Definition 1.3 ([Ber13, (7.2)]). Let X be a Q-Fano variety. For k ∈ Z with −kK Cartier and globally generated, we set N := N := >0 X k h0(X,O (−kK )) and D := D ⊂ XN. Set X X k X,−kKX 1 γ(X) := liminf lct XN, D , k→∞ ∆X k k −kKX:Cartier(cid:18) (cid:18) (cid:19)(cid:19) where ∆ (≃ X) is the diagonal, that is, X ∆ := {(x,...,x) ∈ XN | x ∈ X} ⊂ XN, X andlct (XN,(1/k)D )is thelog-canonicalthreshold (see [Laz04, §9]) ∆X k of the pair (XN,(1/k)D ) around ∆ , that is, k X 1 c log-canonical lct XN, D := sup c ∈ Q XN, D : . ∆X k k >0 k k around ∆ X (cid:18) (cid:19) (cid:26) (cid:27) (cid:12)(cid:16) (cid:17) WesaythatX isBerman-Gibbsstable(cid:12)(resp.Berman-Gibbssemistable) (cid:12) if γ(X) > 1 (resp. γ(X) ≥ 1). WeshowinthispaperthatBerman-GibbsstabilityimpliesK-stability for any Q-Fano variety. More precisely, we show the following: BERMAN-GIBBS STABILITY 3 Theorem 1.4 (Main Theorem). Let X be a Q-Fano variety. If X is Berman-Gibbs stable (resp. Berman-Gibbs semistable), then the pair (X,−K ) is K-stable (resp. K-semistable). X Nowweexplainhowthisarticleisorganized. InSection2.1, werecall the notion and basic properties of K-stability. In Section 2.2, we recall the notion and basic properties of multiplier ideal sheaves, which is a powerful tool to determine how much the singularities of given divisors or given ideal sheaves are mild. In Section 3, we determine whether the projective line P1 is Berman-Gibbs stable or not. We will see that P1 is Berman-Gibbs semistable but is not Berman-Gibbs stable. In Section 4, we prove the key propositions in order to prove Theorem 1.4. We will prove in Proposition 4.2 that Berman-Gibbs stability of X implies that the singularity of a given certain ideal sheaf on X ×A1 is somewhat mild. The strategy of the proof of Proposition 4.2 is to see their multiplier ideal sheaves in detail. In Section 5, we prove Theorem 1.4. By combining the results in [OS12], Proposition 4.2, and by some numerical arguments, we can prove Theorem 1.4. Acknowledgments. The author would like to thank Professor Robert Berman and Doctor Yuji Odaka for his helpful comments. Especially, DoctorYujiOdakainformedhimtheinteresting paper[Ber13]andPro- fessor Robert Berman pointed out Remark 5.2. The author is partially supported by a JSPS Fellowship for Young Scientists. Throughout this paper, we work in the category of algebraic (sepa- rated and of finite type) scheme over a fixed algebraically closed field k of characteristic zero. A variety means a reduced and irreducible algebraic scheme. For the theory of minimal model program, we refer the readers to [KM98]; for the theory of multiplier ideal sheaves, we refer the readers to [Laz04]. For varieties X ,...,X , let p : X → 1 N j 1≤i≤N i X be the j-th projection morphism for any 1 ≤ j ≤ N. j Q 2. Preliminaries In this section, we correct some definitions. 2.1. K-stability. We quickly recall the definition and basic properties of K-stability. For detail, for example, see [Odk13] and references therein. Definition 2.1 (see [Tia97, Don02, RT07, Odk13, LX14]). Let X be a Q-Fano variety of dimension n. (1) A flag ideal I is an ideal sheaf I ⊂ O of the form X×A1 t I = I +I t+···+I tM−1 +(tM) ⊂ O , M M−1 1 X×A1 t 4 KENTO FUJITA where O ⊃ I ⊃ ··· ⊃ I is a sequence of coherent ideal X 1 M sheaves. (2) Let I be a flag ideal and let s ∈ Q . A normal Q-semi test >0 configuration (B,L)/A1 of (X,−K ) obtained by I and s is X defined by the following datum: • Π: B → X ×A1 is the blowing up along I and let E be the exceptional divisor, that is, O (−E) := IO , B B • L := Π∗p∗(−K )−sE, 1 X and we require the following conditions: • B is normal and the morphism Π is not an isomorphism, • L is semiample over A1. (3) Let π: (B,L) → A1 be a normal Q-semi test configuration of (X,−K ) obtained by I and s. For a sufficiently divisible X positiveintegerk, themultiplicative groupG naturallyactson m (B,O (kL)) and the morphism π is G -equivariant, where the B m action G ×A1 → A1 is in a standard way (a,t) 7→ at. Let w(k) m be the total weight of the induced action on (π O (kL))| and ∗ B {0} set N := h0(X,O (−kK )). Then w(k)k′N −w(k′)kN is a k X X k′ k polynomial in variables k and k′ for k, k′ sufficiently divisible positive integers. Let DF(B,L) be its coefficient in kn+1k′n, and is called the Donaldson-Futaki invariant of (B,L)/A1. We set DF := 2((n+1)!)2DF(B,L)/((−K )·n) for simplicity. 0 X (4) The pair (X,−K ) is said to be K-stable (resp. K-semistable) X if DF(B,L) > 0 (resp. DF(B,L) ≥ 0) holds for any normal Q- semi test configuration (B,L)/A1 of (X,−K ) obtained by I X and s. The following is a fundamental result. Theorem 2.2 ([OS12, Odk13]). Let X be a Q-Fano variety of dimension n, (B,L)/A1 be a normal Q-semi test configuration of (X,−K ) X obtained by I and s, and (B¯,L¯)/P1 be its natural compactification to P1, that is, Π: B¯ → X × P1 be the blowing up along I and L¯ := Π∗p∗(−K )−sE on B¯. Then the following holds: 1 X (1) For a sufficiently divisible positive integer k, we have w(k) = χ(B¯,O (kL¯))−χ(B¯,Π∗p∗O (−kK ))+O(kn−1). B¯ 1 X X In particular, we have w(k) (L¯·n+1) lim = . k→∞ kNk (n+1)((−KX)·n) BERMAN-GIBBS STABILITY 5 (2) We have n DF = (L¯·n+1)+(L¯·n ·K ) 0 n+1 B¯/P1 1 = − (L¯·n+1)+(L¯·n ·K −sE). n+1 B¯/X×P1 (3) We have (L¯·n ·E) > 0. (4) If K −sE ≥ 0, then DF > 0. B¯/X×P1 0 Proof. (1) and (2) follow from [Odk13, Proof of Theorem 3.2], (3) follows from [OS12, Lemma 4.5], and (4) follows from [OS12, Proposition (cid:3) 4.4]. 2.2. Multiplier ideal sheaves. We recall the definition and basic properties of multiplier ideal sheaves. Definition 2.3. Let Y be a normal Q-Gorenstein variety, a ,...,a ⊂ 1 l O be coherent ideal sheaves and c ,...,c ∈ Q . The multiplier ideal Y 1 l ≥0 sheaf I(Y,ac1···acl) ⊂ O of the pair (Y,ac1···acl) is defined by the 1 l Y 1 l following. Take a common log resolution µ: Yˆ → Y of a ,...a , i.e., 1 l Yˆ is smooth, a O = O (−F ) and Exc(µ), Exc(µ) + F are i Yˆ Yˆ i 1≤i≤l i divisors with simple normal crossing supports. Then we set P I(Y,ac1···acl) := µ O (⌈K − c F ⌉), 1 l ∗ Yˆ Yˆ/Y i i 1≤i≤l X where ⌈K − c F ⌉ is the smallest Z-divisor which contains Yˆ/Y 1≤i≤l i i K − c F . Yˆ/Y 1≤i≤l Pi i The following proposition can be proved essentially same as the P proofs in [Laz04, §9]. We omit the proof. Proposition 2.4 (see [Laz04, §9]). We have the following: (1) I(Y,ac1···acl) does not depend on the choice of µ. 1 l (2) For an effective Cartier divisor D on Y, we have I(Y,O (−D)1ac1···acl) = I(Y,ac1···acl)⊗O (−D). Y 1 l 1 l Y (3) If coherent ideal sheaves b ,...,b ⊂ O satisfy that a ⊂ b for 1 l Y i i all 1 ≤ i ≤ l, then I(Y,ac1···acl) ⊂ I(Y,bc1···bcl). 1 l 1 l (4) Let Y′ be another normal Q-Gorenstein variety, b ,...,b ⊂ 1 l′ O be coherent ideal sheaves and c′,...,c′ ∈ Q . Then we Y′ 1 l′ ≥0 have I(Y ×Y′,p−1ac1···p−1acl ·p−1bc′1···p−1bc′l′) 1 1 1 l 2 1 2 l′ = p−1I(Y,ac1···acl)·p−1I(Y′,bc′1···bc′l′). 1 1 l 2 1 l′ 6 KENTO FUJITA The following theorem is a singular version of Musta¸ta˘’s summation formula [Mus02, Corollary 1.4] due to Shunsuke Takagi. Theorem2.5 ([Tak06,Theorem3.2]). LetY be anormalQ-Gorenstein variety, let a ,a ,...,a ⊂ O be coherent ideal sheaves and let c , 0 1 l Y 0 c ∈ Q . Then we have ≥0 l l c I Y,ac0· a = I Y,ac0 · aci . 0 i 0 i ! ! (cid:16)Xi=1 (cid:17) cc11,+...X·,·c·+l∈cQl=≥c0 Yi=1 3. The projective line case In this section, we see whether the projective line P1 is Berman- Gibbs stable or not. For any k ∈ Z , we have N = 2k + 1 and >0 k the morphism associated to the complete linear system | − kK | is P1 the (2k)-th Veronese embedding P1 → P2k. If the multi-homogeneous coordinates of (P1)2k+1 are denoted by (t : t ;··· ;t : t ), 1,0 1,1 2k+1,0 2k+1,1 then the divisor D ⊂ (P1)2k+1 corresponds to the following section: k t2k t2k−1t1 ··· t1 t2k−1 t2k 1,0 1,0 1,1 1,0 1,1 1,1 . . . . . . . . . . ··· . . det . . . . . . . . . . . ··· . .   t2k t2k−1 t1 ··· t1 t2k−1 t2k   2k+1,0 2k+1,0 2k+1,1 2k+1,0 2k+1,1 2k+1,1   The above matrix is so-called the Vandermonde matrix. Thus, around 0 ∈ A2k+1 ⊂ (P1)2k+1, the divisor D ⊂ A2k+1 is defined by u1,...,u2k+1 k u1,...,u2k+1 the polynomial f ∈ k[u ,...,u ], where k 1 2k+1 f := (u −u ). k i j 1≤i<j≤2k+1 Y By Lemma 3.1, lct (A2k+1,(f = 0)) = 2/(2k +1). Thus 0 k lct ((P1)N,(1/k)D ) = 2k/(2k+1). ∆P1 k Hence γ(P1) = 1. As a consequence, the projective line P1 is Berman- Gibbs semistable but is not Berman-Gibbs stable. Lemma 3.1 ([Mus06]). For g ≥ 2, we have lct Ag , (u −u ) = 0 = 2/g. 0 u1,...,ug i j !! 1≤i<j≤g Y BERMAN-GIBBS STABILITY 7 Proof. Set D := ( (u − u ) = 0) ⊂ Ag. Let τ: V → Ag 1≤i<j≤g i j be the blowing up along the line (u = ··· = u ) and let F be its 1 g Q exceptional divisor. For c ∈ Q , the discrepancy a(F,Ag,cD) is equal >0 to g −2−cg(g −1)/2. Thus lct (Ag,D) ≤ 2/g. Hence it is enough to 0 show that lct(Ag,D) ≥ 2/g. Let H ⊂ Ag be the hyperplane defined by u − u = 0 and set ij i j A := {H } . We set ij 1≤i,j≤g,i6=j L(A) := W ⊂ Ag ∃A′ ⊂ A;W = H . n (cid:12) H\∈A′ o (cid:12) For W ∈ L(A), set s(W) := (cid:12)#{H ∈ A|W ⊂ H} and r(W) := codim W. By [Mus06, Corollary 0.3], Ag r(W) lct(Ag,D) = min . W∈L(A)\{Ag} s(W) (cid:26) (cid:27) Pick any W ∈ L(A) \ {Ag} and set r := r(W). It is enough to show that s(W) ≤ r(r + 1)/2. If r = 1, then s(W) = 1. Thus we can assume that r ≥ 2. There exist distinct H ,...,H ∈ A such that i1j1 irjr W = H ∩···∩H . i1j1 irjr Assume that i ,j 6∈ {i ,j ,...,i ,j }. For any H ∈ L(A), if W ⊂ 1 1 2 2 r r ij H then H = H or H ∩ ··· ∩ H ⊂ H . Thus s(W) = ij i1j1 ij i2j2 irjr ij 1+s(H ∩···∩H ) ≤ 1+r(r−1)/2 < r(r+1)/2 by induction on i2j2 irjr r. Hence we can assume that (i :=)i = i . 0 1 2 Assume that i ,j ,j 6∈ {i ,j ,...,i ,j }. For any H ∈ L(A), if 0 1 2 3 3 r r ij W ⊂ H then H ∩H ⊂ H or H ∩···∩H ⊂ H . Thus ij i0j1 i0j2 ij i3j3 irjr ij s(W) = s(H ∩H )+s(H ∩···∩H ) ≤ 2·3/2+(r−1)(r−2)/2 < i0j1 i0j2 i3j3 irjr r(r+1)/2by induction onr. Hence we canassume thati ∈ {i ,j ,j }. 3 0 1 2 If i = j , then H ∩H = H ∩H . By replacing H to H , 3 1 i0j1 j1j3 i0j1 i0j3 j1j3 i0j3 we can assume that (i =)i = i = i . 0 1 2 3 We repeat this process. (We note that, for any 1 ≤ j ≤ r − 1, j(j +1)/2+(r −j)(r −j +1)/2 < r(r +1)/2.) We can assume that (i =)i = ··· = i . For any H ∈ L(A), the condition W ⊂ H 0 1 r ij ij is equivalent to the condition {i,j} ⊂ {i ,j ,...,j }. Thus s(W) = 0 1 r r(r+1)/2. Therefore we have proved that s(W) ≤ r(r+1)/2. (cid:3) 4. Key propositions Inthissection, weseethekey propositionsinordertoproveTheorem 1.4. Throughout the section, let X be a Q-Fano variety of dimension n and let (B,L)/A1, I, s, and so on are as in Section 2.1. Lemma 4.1. Let k be a sufficiently divisible positive integer. 8 KENTO FUJITA (1) (cf. [RT07, §3–4]) Set I := O . We also set 0 X I˜ := I ···I j j1 jks j1+·X··+jks=j 0≤j1,...,jks≤M for all 0 ≤ j ≤ Mks. Then Iks = I˜ + I˜ t + ··· + Mks Mks−1 I˜tMks−1+(tMks). Consider the filtration 1 H0(X,O (−kK )) = F ⊃ F ⊃ ··· ⊃ F ⊃ 0 X X 0 1 Mks definedbyF := H0(X,O (−kK )·I˜). Setm := MksdimF . j X X j j=1 j Then m = NMks+w holds, where w = w(k) and N = N are k P as in Definition 2.1 (3). (2) Let I˜ ⊂ O be the copies of I˜ ⊂ O (X := X) for all i,j Xi j X i 1 ≤ i ≤ N and set J := p−1I˜ ···p−1I˜ ⊂ O j 1 1,j1 N N,jN XN j1+·X··+jN=j 0≤j1,...,jN≤Mks for all 0 ≤ j ≤ NMks. Then O (−D ) ⊂ J holds. XN k m Proof. (1) By [RT07, §3–4], (π O (kL))| is equal to ∗ B {0} H0(X ×A1,O(−kK )·Iks)/t·H0(X ×A1,O(−kK )·Iks) t X×A1 t X×A1 and is also equal to Mks F ⊕ tj · F /F . Mks Mks−j Mks−j+1 Mj=1 (cid:16) (cid:17) Thus w = Mks(−j)(dimF −dimF ) = −MksdimF + j=1 Mks−j Mks−j+1 0 MksdimF . This implies that m = NMks+w. j=1 Pj (2) Choose a basis s ,...,s ∈ H0(X,O (−kK )) along the filtra- 1 N X X P tion {F } . For 1 ≤ j ≤ N, set j 0≤j≤Mks f(j) := max{0 ≤ i ≤ Mks|s ∈ F }. j i Let s ,...,s ∈ H0(X ,O (−kK )) be the i-th copies of s ,...,s i1 iN i Xi Xi 1 N for all 1 ≤ i ≤ N. Then the divisor D ⊂ XN corresponds to the k section sgnσ ·s ···s ∈ H0(XN,O (−kK )), 1σ(1) Nσ(N) XN XN σX∈SN where S is the N-th symmetric group. Take any σ ∈ S . Since N N s ∈ p−1I˜ , we have i,j i i,f(j) s ···s ∈ p−1I˜ ···p−1I˜ . 1σ(1) Nσ(N) 1 1,f(σ(1)) N N,f(σ(N)) BERMAN-GIBBS STABILITY 9 Note that N f(σ(i)) = N f(i) = Mksj(dimF −dimF ) = i=1 i=1 j=0 j j+1 m, where F := 0. Thus O (−D ) ⊂ J . (cid:3) Mks+1 XN k m P P P Proposition 4.2. Assume that a positive rational number γ ∈ Q >0 satisfies that, for a sufficiently divisible positive integer k, the pair (XN,(γ/k)D ) is log-canonical around ∆ . Then for any ε ∈ (0,1)∩Q k X and any sufficiently big positive integer P, the structure sheaf O X×A1 is contained in the sheaf I X ×A1,(t)(1−ε)(1+γw/(kN))+P ·I(1−ε)γs ⊗O (P ·(t = 0)) X×A1 (that(cid:0)is, the pair (X ×A1,(t)(1+γw/(kN)) ·Iγ(cid:1)s) is “sub-log-canonical”), where w = w(k) and N = N are as in Definition 2.1 (3). k Proof. We set Θ := ~j = (j ,...,j ) j1+···+jN=m, , 1 N 0≤j1,...,jN≤Mks A := nα~ = (α ) P~j(cid:12)(cid:12)∈Θα~j=(1−ε)γ/k, o, ~j ~j∈Θ (cid:12) ∀α~j∈Q≥0 B := nβ~ = (β ,...,(cid:12)(cid:12)β ) β0,...,βMkso∈Q≥0, , 0 (cid:12)Mks PMj=k0sβj=(1−ε)γ/k n (cid:12)(cid:12) ξ0,...,ξMks∈Q≥0o, Ξ := ξ~= (ξ0,...,ξMks) (cid:12) PMj=k0sξj=(1−ε)γ/k, ( (cid:12) PMj=k0sjξj≥(1−ε)γm/(kN)) (cid:12) for simplicity. (cid:12) (cid:12) Claim 4.3. We have the equality Mks O = I X, I˜ξi . X i ! Xξ~∈Ξ Yi=0 Proof of Claim 4.3. By Proposition 2.4, Theorem 2.5 and Lemma 4.1, around ∆ , we have X O = I(XN,O (−D )(1−ε)γ/k) XN XN k ⊂ I(XN,J(1−ε)γ/k) m (1−ε)γ/k = I XN, p−1I˜ ···p−1I˜ 1 1,j1 N N,jN (cid:18) (cid:16)~Xj∈Θ (cid:17) (cid:19) = I XN, (p−11I˜1,j1···p−N1IÑ,jN)α~j αX~∈A (cid:16) ~jY∈Θ (cid:17) = p−1I X , I˜α~j ···p−1I X , I˜α~j . 1 1 1,j1 N N N,jN αX~∈A (cid:0) ~jY∈Θ (cid:1) (cid:0) ~jY∈Θ (cid:1) 10 KENTO FUJITA Restricts to ∆ , we have X O = I X, I˜α~j ···I X, I˜α~j . X j1 jN Xα~∈A (cid:0) ~jY∈Θ (cid:1) (cid:0) ~jY∈Θ (cid:1) Fix an arbitrary α~ ∈ A. Since α j +···+ α j = (1−ε)γm/k, ~j 1 ~j N ~Xj∈Θ ~Xj∈Θ we have α j ≥ (1−ε)γm/(kN) for some 1 ≤ q ≤ N. We set ~j∈Θ ~j q P ξ := α i ~j ~j∈ΘX;jq=i ~ for 0 ≤ i ≤ Mks. Then ξ := (ξ ,...,ξ ) ∈ Ξ and 0 Mks Mks I X, I˜α~j = I X, I˜ξi . jq i (cid:16) ~jY∈Θ (cid:17) (cid:16) Yi=0 (cid:17) (cid:3) Therefore we have proved Claim 4.3. By Proposition 2.4 (4) and Claim 4.3, we have Mks O (−P ·(t = 0)) = I X ×A1,(t)1−ε+P · I˜ξi . X×A1 i Xξ~∈Ξ (cid:16) Yi=0 (cid:17) For any ξ~∈ Ξ, since (1−ε)(1+γm/(kN))+P − Mksiξ ≤ 1−ε+P, i=0 i we have P Mks I X ×A1,(t)1−ε+P · I˜ξi i (cid:16) Yi=0 (cid:17) Mks ⊂ I X ×A1,(t)(1−ε)(1+γm/(kN))+P−PMi=k0siξi · I˜ξi . i (cid:16) Yi=0 (cid:17)