aa r X i v : . [ m a t h . AG ] J a n UNIFORM K-STABILITY AND PLT BLOWUPS OFLOG FANO PAIRS
KENTO FUJITA
Abstract.
We show relationships between uniform K-stabilityand plt blowups of log Fano pairs. We see that it is enough toevaluate certain invariants defined by volume functions for all pltblowups in order to test uniform K-stability of log Fano pairs.We also discuss the uniform K-stability of two log Fano pairs un-der crepant finite covers. Moreover, we give another proof of K-semistability of the projective plane.
Contents
1. Introduction 12. Uniform K-stability 53. Plt blowups 94. Applications 14References 171.
Introduction
In this paper, we work over an arbitrary algebraically closed field k of characteristic zero. Let ( X, ∆) be a log Fano pair , that is, X is a normal projective variety over k and ∆ is an effective Q -divisorsuch that ( X, ∆) is a klt pair and − ( K X + ∆) is an ample Q -Cartier Q -divisor. (For the minimal model program, we refer the readers to[KM98] and [BCHM10].) We are interested in the problem whether( X, ∆) is uniformly K-stable (resp., K-semistable ) or not (see [Tia97,Don02, Sz´e06, Sz´e15, Der15, BHJ15, BBJ15, Fjt16b] and referencestherein). In [Li16, Theorem 3.7] and [Fjt16b, Theorem 6.5], we haveseen that the uniform K-stability (and the K-semistability) of ( X, ∆) isequivalent to measure the positivity of certain invariants associated to Date : January 3, 2017.2010
Mathematics Subject Classification.
Primary 14J45; Secondary 14E30.
Key words and phrases.
Fano varieties, K-stability, Minimal model program. divisorial valuations on X . Before recalling those results, we preparesome definitions. Definition 1.1.
Let ( V, Γ) be a log pair , that is, V is a normal varietyand Γ is an effective R -divisor on V such that K V + Γ is R -Cartier. Let F be a prime divisor over V , that is, there exists a projective birationalmorphism σ : W → V with W normal such that F is a prime divisor on W .(1) The log discrepancy A ( F ) := A ( V, Γ) ( F ) of ( V, Γ) along F is de-fined to be A ( F ) := 1 + ord F ( K W − σ ∗ ( K V + Γ)). We remarkthat the value A ( F ) does not depend on the choice of σ . More-over, let c V ( F ) ⊂ V be the center of F on V , that is, the imageof F on V .(2) ([Ish04]) The divisor F is said to be primitive over V if thereexists a projective birational morphism τ : T → V with T nor-mal such that F is a prime divisor on T and − F on T is a τ -ample Q -Cartier divisor. We call the morphism τ the associatedprime blowup . (We do not assume that F is exceptional over V . We remark that the associated prime blowup is uniquelydetermined from F [Ish04, Proposition 2.4].) Moreover, let Γ T be the R -divisor on T defined by K T + Γ T + (1 − A ( F )) F = τ ∗ ( K V + Γ) . We often denote the associated prime blowup by τ : ( T, Γ T + F ) → ( V, Γ) . (3) ([Sho96, 3.1], [Pro00, Definition 2.1]) Assume that F is a prim-itive prime divisor over V and τ : ( T, Γ T + F ) → ( V, Γ) is theassociated prime blowup. The divisor F is said to be plt-type (resp., lc-type ) over ( V, Γ) if the pair ( T, Γ T + F ) is plt (resp.,lc). Under the situation, we call the associated morphism τ the associated plt blowup (resp., the associated lc blowup ). Definition 1.2 ([Fjt16b, Definition 6.1]) . Let ( X, ∆) be a log Fanopair of dimension n , L := − ( K X + ∆), and let F be a prime divisorover X .(1) For arbitrary k ∈ Z ≥ with kL Cartier and x ∈ R ≥ , let H ( X, kL − xF ) be the sub k -vector space of H ( X, kL ) de-fined by the set of global sections vanishing at the generic pointof F at least x times. -STABILITY AND PLT BLOWUPS 3 (2) The divisor F is said to be dreamy over ( X, ∆) if the graded k -algebra M k,i ∈ Z ≥ H ( X, krL − iF )is finitely generated over k for some (hence, for an arbitrary) r ∈ Z > such that rL is Cartier.(3) For an arbitrary x ∈ R ≥ , setvol X ( L − xF ) := lim sup k →∞ kL : Cartier dim k H ( X, kL − kxF ) k n /n ! . By [Laz04a, Laz04b], the function vol X ( L − xF ) is continuousand non-increasing on x ∈ [0 , ∞ ). Moreover, the limsup isactually a limit.(4) We define the pseudo-effective threshold τ ( F ) of L along F by τ ( F ) := τ ( X, ∆) ( F ) := sup { x ∈ R > | vol X ( L − xF ) > } . Note that τ ( F ) ∈ R > .(5) We set β ( F ) := β ( X, ∆) ( F ) := A ( F ) · ( L · n ) − Z ∞ vol X ( L − xF ) dx. Moreover, we set ˆ β ( F ) := β ( F ) A ( F ) · ( L · n ) . (6) We set j ( F ) := j ( X, ∆) ( F ) := Z τ ( F )0 (( L · n ) − vol X ( L − xF )) dx. Obviously, we have the inequality j ( F ) > Remark 1.3. (1) ([Fjt16c, Lemma 3.1 (2)]) If F is a dreamy primedivisor over ( X, ∆), then F is primitive over X .(2) ([Xu14]) If F is a plt-type prime divisor over a klt pair ( V, Γ)and c V ( F ) = { o } , then the divisor F is said to be a Koll´arcomponent of the singularity o ∈ ( V, Γ).The following is the valuative criterion for uniform K-stability (andK-semistability) of log Fano pairs introduced in [Li16] and [Fjt16b].
Theorem 1.4.
Let ( X, ∆) be a log Fano pair. (1) ([Li16, Theorem 3.7] , [Fjt16b, Theorem 6.6]) The following areequivalent:
KENTO FUJITA (i) ( X, ∆) is K-semistable ( see for example [Fjt16b, Definition6.4]) . (ii) For any prime divisor F over X , the inequality β ( F ) ≥ holds. (iii) For any dreamy prime divisor F over ( X, ∆) , the inequality β ( F ) ≥ holds. (2) ([Fjt16b, Theorem 6.6]) The following are equivalent: (i) ( X, ∆) is uniformly K-stable ( see for example [Fjt16b, Def-inition 6.4]) . (ii) There exists δ ∈ (0 , such that for any prime divisor F over X , the inequality β ( F ) ≥ δ · j ( F ) holds. (iii) There exists δ ∈ (0 , such that for any dreamy primedivisor F over ( X, ∆) , the inequality β ( F ) ≥ δ · j ( F ) holds. The purpose of this paper is to simplify the above theorem. More pre-cisely, we see relationships between the above criterion and plt blowupsof log Fano pairs. The following is the main theorem in this paper.
Theorem 1.5 (Main Theorem) . Let ( X, ∆) be a log Fano pair. (1) The following are equivalent: (i) ( X, ∆) is K-semistable. (ii) For any plt-type prime divisor F over ( X, ∆) , the inequality ˆ β ( F ) ≥ holds. (2) The following are equivalent: (i) ( X, ∆) is uniformly K-stable. (ii) There exists ε ∈ (0 , such that for any prime divisor F over X , the inequality ˆ β ( F ) ≥ ε holds. (iii) There exists ε ∈ (0 , such that for any dreamy primedivisor F over ( X, ∆) , the inequality ˆ β ( F ) ≥ ε holds. (iv) There exists ε ∈ (0 , such that for any plt-type primedivisor F over ( X, ∆) , the inequality ˆ β ( F ) ≥ ε holds. Remark 1.6. (1) In Theorem 1.4 (2), we need to evaluate j ( F ) inorder to test uniform K-stability. It seems relatively difficult toevaluate j ( F ) than to evaluate ˆ β ( F ) since the value τ ( F ) is noteasy to treat. It is one of the remarkable point that we do notneed to evaluate j ( F ) in Theorem 1.5 (2).(2) Theorem 1.5 claims that we can check uniform K-stability andK-semistability by evaluating ˆ β ( F ) for plt-type prime divisors F over ( X, ∆). The theory of plt blowups is important for thetheory of minimal model program and singularity theory (see[Pro00, Pro01]). It is interesting that such theories will relate -STABILITY AND PLT BLOWUPS 5 K-stability via Theorem 1.5. Moreover, Theorem 1.5 seems torelate with [LX16, Conjecture 6.5].As an easy consequence of Theorem 1.5, we get the following result.The proof is given in Section 4.1. We remark that Dervan also treatedsimilar problem. See [Der16].
Corollary 1.7 (see also Example 4.2) . Let ( X, ∆) and ( X ′ , ∆ ′ ) be logFano pairs. Assume that there exists a finite and surjective morphism φ : X ′ → X such that φ ∗ ( K X + ∆) = K X ′ + ∆ ′ . If ( X ′ , ∆ ′ ) is uniformlyK-stable ( resp., K-semistable ) , then so is ( X, ∆) . We can also show as an application of Theorem 1.5 that the pro-jective plane is K-semistable. The result is well-known (see [Don02]).Moreover, the result has been already proved purely algebraically (see[Kem78, RT07] and [Li16, Blu16, PW16]). However, it is worth writingthe proof since our proof is purely birational geometric. The proof isgiven in Section 4.2.
Corollary 1.8 (see also [Kem78, Don02, Li16, Blu16, PW16]) . Theprojective plane P is ( that is, the pair ( P , is ) K-semistable.
This paper is organized as follows. In Section 2, we see the equiv-alence between the conditions in Theorem 1.5 (2i), Theorem 1.5 (2ii),and Theorem 1.5 (2iii). For the proof, we use the log-convexity ofvolume functions and restricted volume functions. In Section 3, wesee how to replace a primitive divisor by a plt-type prime divisor withsmaller ˆ β -invariant. For the proof, we use techniques of minimal modelprogram. Theorem 1.5 follows from those observations. In Section 4,we prove Corollaries 1.7 and 1.8. Acknowledgments.
The author thank Doctor Atsushi Ito and Pro-fessor Shunsuke Takagi for discussions during the author enjoyed thesummer school named “Algebraic Geometry Summer School 2016” inTambara Institute of Mathematical Sciences. This work was supportedby JSPS KAKENHI Grant Number JP16H06885.2.
Uniform K-stability
In this section, we simplify the conditions in Theorem 1.4 (2). In thissection, we always assume that ( X, ∆) is a log Fano pair of dimension n , L := − ( K X + ∆), and F is a prime divisor over X .The proof of the following proposition is essentially same as theproofs of [FO16, Theorem 4.2] and [Fjt16c, Proposition 3.2]. KENTO FUJITA
Proposition 2.1.
We have the inequality nn + 1 τ ( F ) ≥ L · n ) Z ∞ vol X ( L − xF ) dx. Proof.
Take any log resolution σ : Y → X of ( X, ∆) such that F ⊂ Y and there exists a σ -ample Q -divisor A Y on Y with γ := − ord F A Y > − A Y effective. Then, for any 0 < ε ≪ σ ∗ L + ( ε/γ ) A Y is ample.Hence B + ( σ ∗ L + ( ε/γ ) A Y ) = ∅ , where B + is the augmented base locus(see [ELMNP09]). Note that B + ( σ ∗ L − εF ) ⊂ B + ( σ ∗ L + ( ε/γ ) A Y ) ∪ Supp( − ( ε/γ ) A Y − εF ) . This implies that F B + ( σ ∗ L − εF ). Thus, by [ELMNP09, TheoremA] (and by [BFJ09, Theorem A and Corollary C]), the restricted volumevol Y | F ( σ ∗ L − xF ) on x ∈ [0 , τ ( F )) satisfies the log-concavity (in thesense of [ELMNP09, Theorem A]). In particular, for an arbitrary x ∈ (0 , τ ( F )), we have ( vol Y | F ( σ ∗ L − xF ) ≥ ( x/x ) n − · vol Y | F ( σ ∗ − x F ) if x ∈ [0 , x ] , vol Y | F ( σ ∗ L − xF ) ≤ ( x/x ) n − · vol Y | F ( σ ∗ − x F ) if x ∈ [ x , τ ( F )) . On the other hand, by [LM09, Corollary 4.27], for an arbitrary x ∈ [0 , τ ( F )], we have the equalityvol Y ( σ ∗ L − xF ) = n Z τ ( F ) x vol Y | F ( σ ∗ L − yF ) dy. Let us set b := R τ ( F )0 y · vol Y | F ( σ ∗ L − yF ) dy R τ ( F )0 vol Y | F ( σ ∗ L − yF ) dy . Obviously, b ∈ (0 , τ ( F )) holds. Moreover, we get0 = Z τ ( F ) − b − b y · vol Y | F ( σ ∗ L − ( y + b ) F ) dy ≤ Z τ ( F ) − b − b y · (cid:18) y + bb (cid:19) n − · vol Y | F ( σ ∗ L − bF ) dy = vol Y | F ( σ ∗ L − bF ) · τ ( F ) n n · b n − (cid:18) nn + 1 τ ( F ) − b (cid:19) . -STABILITY AND PLT BLOWUPS 7 Thus b ≤ ( n/ ( n + 1)) τ ( F ). On the other hand, we have b = n R τ ( F )0 R τ ( F ) x vol Y | F ( σ ∗ L − yF ) dydx ( L · n )= R τ ( F )0 vol Y ( σ ∗ L − xF ) dx ( L · n ) . Thus we have proved Proposition 2.1. (cid:3)
The following lemma is nothing but a logarithmic version of [FO16,Lemma 2.2]. We give a proof just for the readers’ convenience.
Lemma 2.2 ([FO16, Lemma 2.2]) . We have the inequality τ ( F ) n + 1 ≤ L · n ) Z ∞ vol X ( L − xF ) dx. Proof.
By [LM09, Corollary 4.12], we havevol X ( L − xF ) ≥ (cid:18) − xτ ( F ) (cid:19) n · ( L · n ) . Lemma 2.2 follows immediately from the above. (cid:3)
Now we are ready to prove the following theorem.
Theorem 2.3.
Let ( X, ∆) be a log Fano pair. Then the following areequivalent: (i) ( X, ∆) is uniformly K-stable. (ii) There exists ε ∈ (0 , such that for any prime divisor F over X , the inequality ˆ β ( F ) ≥ ε holds. (iii) There exists ε ∈ (0 , such that for any dreamy prime divisor F over ( X, ∆) , the inequality ˆ β ( F ) ≥ ε holds.Proof. Let F be an arbitrary prime divisor over X . Firstly, we observethat the condition β ( F ) ≥ δ · j ( F ) for some δ ∈ (0 ,
1) is equivalent tothe condition(1) (1 + δ ′ ) A ( F ) − δ ′ τ ( F ) ≥ L · n ) Z ∞ vol X ( L − xF ) dx, where δ ′ := δ/ (1 − δ ) ∈ (0 , ∞ ).We also observe that the condition ˆ β ( F ) ≥ ε for some ε ∈ (0 ,
1) isequivalent to the condition(2) A ( F ) ≥ ε ′ ( L · n ) Z ∞ vol X ( L − xF ) dx, where ε ′ := ε/ (1 − ε ) ∈ (0 , ∞ ). KENTO FUJITA
Claim 2.4 (see [FO16, Theorem 2.3]) . If the inequality (2) holds forsome ε ′ ∈ (0 , ∞ ) , then the inequality (1) holds for δ ′ = ε ′ / ( n + 1) .Proof of Claim 2.4. By Lemma 2.2, the inequality (2) implies that A ( F ) ≥ L · n ) Z ∞ vol X ( σ ∗ L − xF ) dx + ε ′ n + 1 τ ( F ) . Thus the inequality (1) holds for δ ′ = ε ′ / ( n + 1) since A ( F ) > (cid:3) Claim 2.5.
If the inequality (1) holds for some δ ′ ∈ (0 , ∞ ) , then theinequality (2) holds for ε ′ = min ( δ ′ − θθ − δ ′ − θθ , n + 1 ) ∈ (0 , , where θ := max (cid:26) n n + 1 , δ ′ δ ′ + 1 (cid:27) ∈ (0 , . Proof of Claim 2.5.
We firstly assume that case A ( F ) < θ · τ ( F ). Thenthe inequality (1) implies that1( L · n ) Z ∞ vol X ( L − xF ) dx < (cid:18) − δ ′ − θθ (cid:19) A ( F ) . Note that δ ′ − θθ ∈ (0 , / ⊂ (0 , . Thus the inequality (2) holds for such ε ′ .We secondly consider the remaining case A ( F ) ≥ θ · τ ( F ). In thiscase, by Proposition 2.1, we have1( L · n ) Z ∞ vol X ( L − xF ) dx ≤ nn + 1 τ ( F ) ≤ nn + 1 1 θ A ( F ) . Note that n + 1 n θ − ≥ n + 1 . Thus the inequality (2) holds for such ε ′ . (cid:3) Theorem 2.3 immediately follows from Theorem 1.4 (2), Claims 2.4and 2.5. (cid:3)
Remark 2.6.
Let ( X, ∆) be a log Fano pair and F be a prime divisorover X . If τ ( F ) ≤ A ( F ), then ˆ β ( F ) ≥ / ( n + 1) by Proposition 2.1.Thus it is enough to consider prime divisors F over X with τ ( F ) >A ( F ) in order to check the conditions in Theorem 1.5. -STABILITY AND PLT BLOWUPS 9 Plt blowups
The following theorem is inspired by [Xu14, Lemma 1].
Theorem 3.1.
Let ( X, ∆) be a quasi-projective klt pair with ∆ effective R -divisor. Let F be a primitive prime divisor over X and σ : ( Y, ∆ Y + F ) → ( X, ∆) be the associated prime blowup. Assume that F is notplt-type ( resp., not lc-type ) over ( X, ∆) . Then there exists a plt-typeprime divisor G over ( X, ∆) such that the inequality A ( Y, ∆ Y + F ) ( G ) ≤ resp., A ( Y, ∆ Y + F ) ( G ) < holds.Proof. Let π : V → Y be a log resolution of ( Y, ∆ Y + F ) and let E , . . . , E k be the set of π -exceptional divisors on V . We set F V := π − ∗ F and ∆ V := π − ∗ ∆ Y . We may assume that there exists a ( σ ◦ π )-ample Q -divisor A V = − k X i =1 h i E i − h F F V on V with h , . . . , h k ∈ Q > and ( h F ∈ Q > if F is exceptional over X,h F = 0 otherwise . Take a sufficiently ample Cartier divisor L on X such that σ ∗ L − F isample. Let L Y be a general effective Q -divisor with small coefficientssuch that L Y ∼ Q σ ∗ L − F . Set L V := π − ∗ L Y . Then L V = π ∗ L Y and the pair ( V, ∆ V + F V + P ki =1 E i + L V ) is log smooth by generality.Moreover, the Q -divisor σ ∗ ( L Y + F ) is Q -Cartier with σ ∗ ( L Y + F ) ∼ Q L and σ ∗ σ ∗ ( L Y + F ) = L Y + F .Let us set π ∗ F =: F V + k X i =1 c i E i ( c i ∈ Q ≥ ) ,a F := A ( X, ∆) ( F ) ∈ R > ,a i := A ( X, ∆) ( E i ) ∈ R > ,b i := A ( Y, ∆ Y + F ) ( E i ) ∈ R . Of course, we have the inequality h i > h F c i (from the negativitylemma) and the equality a i = b i + a F c i for any 1 ≤ i ≤ k . By assump-tion, the inequality b i ≤ b i <
0) holds for some 1 ≤ i ≤ k . Inparticular, the inequality c i > ≤ i ≤ k . By changing E , . . . , E k and by perturbing the coefficients of A V if necessary, we canassume that the following conditions are satisfied: • There exists 1 ≤ l ≤ k such that c i > ≤ i ≤ l . • b /c = min ≤ i ≤ l { b i /c i } . • The inequality h j /c j < h /c holds for any 2 ≤ j ≤ l with b /c = b j /c j .Take a rational number 0 < ε ≪ t := ( a − εh ) /c ∈ R > .Since ε is very small, we get the following properties: • a − ( tc + εh ) = 0, • a i − ( tc i + εh i ) > ≤ i ≤ k , • − a F + t ∈ (1 − a F ,
1) and 1 − a F + t + εh F <
1, and • π ∗ ( L Y + F ) + εA V is ample on V .Take a general effective R -divisor L ′ with small coefficients such that L ′ ∼ R π ∗ ( L Y + F ) + εA V . Moreover, we set b F := ( F is exceptional over X, − a F + t otherwise . Then K V + ∆ V + b F F V + tL V + k X i =1 E i + L ′ is dlt. (We remark that 1 − a F + t ≥ t > F is a divisor on X . We also remark that the coefficients of tL V can be less than 1 bythe definition of L V .) Moreover, we have K V + ∆ V + b F F V + tL V + k X i =1 E i + L ′ ∼ R ( σ ◦ π ) ∗ ( K X + ∆) + tπ ∗ ( L Y + F ) + ( a F + b F − F V − tπ ∗ F + k X i =1 a i E i + π ∗ ( L Y + F ) + εA V ∼ R ,X ( b F − (1 − a F + t ) − εh F ) F V + k X i =2 ( a i − ( tc i + εh i )) E i . The right-hand side is effective and its support is equal to the union of( σ ◦ π )-exceptional prime divisors other than E . Furthermore, K V + ∆ V + b F F V + tL V + k X i =1 E i + L ′ ∼ R ,X K V + ∆ V + b F F V + tL V + k X i =1 E i + (1 − δ ) L ′ + δεA V -STABILITY AND PLT BLOWUPS 11 is klt for 0 < δ ≪
1. Thus, by [BCHM10, Corollary 1.4.2], we can runand terminate a ( K V + ∆ V + b F F V + tL V + P ki =1 E i + L ′ )-MMP withscaling L ′ over X . Let V ψ / / ❴❴❴❴❴❴❴ σ ◦ π ❅❅❅❅❅❅❅❅ W φ ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ X be the output of this MMP. The MMP does not contract E . Let G W ⊂ W be the image of E . Moreover, by the negativity lemma, any( σ ◦ π )-exceptional prime divisor other than E is contracted by ψ . Inparticular, we get ψ ∗ ( K V + ∆ V + b F F V + tL V + k X i =1 E i + L ′ ) ∼ R ,X . Furthermore, by the definition of MMP with scaling, the R -divisor ψ ∗ ( K V + ∆ V + b F F V + tL V + k X i =1 E i + (1 + λ ) L ′ ) ∼ R ,X λψ ∗ L ′ ∼ R ,X − λεh G W is nef over X for any 0 < λ ≪
1. Moreover, by the base point freetheorem, the above R -divisor admits the ample model over X . Let W µ / / φ ❇❇❇❇❇❇❇❇ Z τ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ X be the model and we set G := µ ∗ G W . Since − G is τ -ample, the mor-phism µ is a small morphism. We remark that ψ ∗ ( K V + ∆ V + b F F V + tL V + k X i =1 E i + L ′ )= K W + φ − ∗ ∆ + tψ ∗ ( F V + L V ) + G W + ψ ∗ L ′ is dlt, R -linearly equivalent to zero over X , and G W is the unique primedivisor whose coefficient is equal to one. Thus this is plt and K Z + τ − ∗ ∆ + t ( µ ◦ ψ ) ∗ ( F V + L V ) + G + ( µ ◦ ψ ) ∗ L ′ is also plt. Note that ( µ ◦ ψ ) ∗ L ′ is effective R -Cartier. Moreover, since τ ∗ ( µ ◦ ψ ) ∗ ( F V + L V ) is R -Cartier and τ ∗ τ ∗ ( µ ◦ ψ ) ∗ ( F V + L V ) − ( µ ◦ ψ ) ∗ ( F V + L V ) is equal to some multiple of G , the R -divisor ( µ ◦ ψ ) ∗ ( F V + L V )is also effective R -Cartier. This implies that the pair ( Z, τ − ∗ ∆ + G ) is also plt. By construction, − G is τ -ample, G is exceptional over X (since E is exceptional over Y ), and A ( Y, ∆ Y + F ) ( G ) = b ≤ < (cid:3) Corollary 3.2.
Let ( X, ∆) be a log Fano pair and F be a primitiveprime divisor over X with the associated prime blowup σ : ( Y, ∆ Y + F ) → ( X, ∆) . If F is not plt-type over ( X, ∆) , then there exists aplt-type prime divisor G over ( X, ∆) with A ( Y, ∆ Y + F ) ( G ) ≤ such thatthe inequality ˆ β ( F ) > ˆ β ( G ) holds.Proof. We set n := dim X and L := − ( K X +∆). By Theorem 3.1, thereexists a plt-type prime divisor G over ( X, ∆) with A ( Y, ∆ Y + F ) ( G ) ≤ τ : ( Z, ∆ Z + G ) → ( X, ∆) be the associated plt blowup. Let V π ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ρ ❅❅❅❅❅❅❅❅ Y σ ❅❅❅❅❅❅❅❅ Z τ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ X be a common log resolution of ( Y, ∆ Y + F ) and ( Z, ∆ Z + G ). Since( X, ∆) is klt and A ( Y, ∆ Y + F ) ( G ) ≤
0, we have c Y ( G ) ⊂ F . Set f :=ord π − ∗ F ( ρ ∗ G ) and g := ord ρ − ∗ G ( π ∗ F ). Since c Y ( G ) ⊂ F , we have theinequality g >
0. Moreover, we have the following equalities: A ( X, ∆) ( F ) = A ( Z, ∆ Z + G ) ( F ) + f · A ( X, ∆) ( G ) ,A ( X, ∆) ( G ) = A ( Y, ∆ Y + F ) ( G ) + g · A ( X, ∆) ( F ) . Claim 3.3. (1)
For any x ∈ R ≥ , we have the inequality vol X ( L − xF ) ≤ vol X ( L − gxG ) . (2) If A ( Y, ∆ Y + F ) ( G ) = 0 , then, for any < x ≪ , we have theinequality vol X ( L − xF ) < vol X ( L − gxG ) .Proof of Claim 3.3. The assertion (1) is trivial since we know that H ( X, kL − jF ) ⊂ H ( X, kL − gjG )for any sufficiently divisible k , j ∈ Z > . We see the assertion (2). Weassume that A ( Y, ∆ Y + F ) ( G ) = 0. Then we have A ( X, ∆) ( G ) = gA ( X, ∆) ( F )= gA ( Z, ∆ Z + G ) ( F ) + f gA ( X, ∆) ( G ) > f gA ( X, ∆) ( G ) . This implies the inequality 1 > f g . -STABILITY AND PLT BLOWUPS 13 Fix any 0 < x ≪ σ ∗ L − xF and τ ∗ L − gxG areample. Note that vol X ( L − xF ) = (( σ ∗ L − xF ) · n ) and vol X ( L − gxG ) =(( τ ∗ L − gxG ) · n ). Since − x ( π ∗ F − gρ ∗ G ) = π ∗ ( σ ∗ L − xF ) − ρ ∗ ( τ ∗ L − gxG )is ρ -nef and ρ ∗ ( π ∗ F − gρ ∗ G ) = (1 − f g ) ρ ∗ π − ∗ F ≥ , we have π ∗ F ≥ gρ ∗ G by the negativity lemma. Thus, for any 0 ≤ i ≤ n −
1, we have0 ≤ (cid:0) π ∗ ( σ ∗ L − xF ) · i · ρ ∗ ( τ ∗ L − gxG ) · n − − i · π ∗ ( xF ) − ρ ∗ ( gxG ) (cid:1) = (cid:0) π ∗ ( σ ∗ L − xF ) · i · ρ ∗ ( τ ∗ L − gxG ) · n − i (cid:1) − (cid:0) π ∗ ( σ ∗ L − xF ) · i +1 · ρ ∗ ( τ ∗ L − gxG ) · n − − i (cid:1) . Moreover, we have (cid:0) π ∗ ( σ ∗ L − xF ) · n − · ρ ∗ ( τ ∗ L − gxG ) (cid:1) − ( π ∗ ( σ ∗ L − xF ) · n )= x (1 − f g ) (cid:0) ( σ ∗ L − xF ) · n − · F (cid:1) > . Therefore, we have(( τ ∗ L − gxG ) · n ) − (( σ ∗ L − xF ) · n )= n − X i =0 (cid:18)(cid:0) π ∗ ( σ ∗ L − xF ) · i · ρ ∗ ( τ ∗ L − gxG ) · n − i (cid:1) − (cid:0) π ∗ ( σ ∗ L − xF ) · i +1 · ρ ∗ ( τ ∗ L − gxG ) · n − − i (cid:1)(cid:19) > . Thus we get Claim 3.3. (cid:3)
From Claim 3.3, we get the inequalitiesˆ β ( F ) = 1 − R ∞ vol X ( L − xF ) dxA ( X, ∆) ( F ) · ( L · n ) ≥ − R ∞ vol X ( L − gxG ) dxA ( X, ∆) ( F ) · ( L · n ) = 1 − R ∞ vol X ( L − xG ) dxgA ( X, ∆) ( F ) · ( L · n ) ≥ − R ∞ vol X ( L − xG ) dxA ( X, ∆) ( G ) · ( L · n ) = ˆ β ( G ) . Moreover, at least one of the inequalities is the strict inequality. (cid:3)
Proof of Theorem 1.5.
This follows immediately from Remark 1.3 (1),Theorems 1.4, 2.3 and Corollary 3.2. (cid:3) Applications
In this section, we give several applications of Theorem 1.5.4.1.
Finite covers.
In this section, we prove Corollary 1.7. To beginwith, we show the following easy lemma.
Lemma 4.1.
Let ψ : W → V be a generically finite and surjectivemorphism between normal projective varieties. For any Cartier divisor D on V , we have the following inequality: vol W ( ψ ∗ D ) ≥ (deg ψ ) · vol V ( D ) . Proof.
We may assume that D is big. By [Fuj94, Theorem], for any ε >
0, there exists a projective birational morphism σ : V ′ → V with V ′ normal, ample Q -divisor A , and an effective Q -divisor E such that σ ∗ D ∼ Q A + E and vol V ( D ) ≤ vol V ′ ( A ) + ε hold. Let W ′ ψ ′ −−−→ V ′ σ ′ y y σ W −−−→ ψ V be the normalization of the fiber product. Then we getvol W ( ψ ∗ D ) = vol W ′ ( ψ ′∗ σ ∗ D ) ≥ vol W ′ ( ψ ′∗ A )= (deg ψ ) · vol V ′ ( A ) ≥ (deg ψ ) · (vol V ( D ) − ε ) . The assertion immediately follows from the above inequalities. (cid:3)
Proof of Corollary 1.7.
We set n := dim X , L := − ( K X + ∆), L ′ := − ( K X ′ + ∆ ′ ) and d := deg φ . From Theorem 1.5, there exists ε > ≥
0) such that ˆ β ( F ′ ) ≥ ε holds for any prime divisor F ′ over X ′ . Take any plt blowup σ : ( Y, ∆ Y + F ) → ( X, ∆). From Theorem1.5, it is enough to show the inequality ˆ β ( F ) ≥ ε . Let Y ′ ψ −−−→ Y σ ′ y y σ X ′ −−−→ φ X be the normalization of the fiber product. (Note that the morphism ψ is a finite morphism.) Let ψ ∗ F = m X i =1 r i F ′ i -STABILITY AND PLT BLOWUPS 15 be the irreducible decomposition of the pullback of F (see [KM98,Proposition 5.20]), where r i ∈ Z > . By [KM98, Proposition 5.20], wehave the equality A ( X ′ , ∆ ′ ) ( F ′ i ) = r i A ( X, ∆) ( F )for any 1 ≤ i ≤ m . Moreover, for any x ∈ R ≥ , we have ψ ∗ ( σ ∗ L − xF ) = σ ′∗ L ′ − x m X i =1 r i F ′ i ≤ σ ′∗ L ′ − xr F ′ . Therefore, from Lemma 4.1, we have the following inequalities:1 − ε ≥ − ˆ β ( F ′ ) = R ∞ vol Y ′ ( σ ′∗ L ′ − yF ′ ) dyA ( X ′ , ∆ ′ ) ( F ′ ) · ( L ′· n )= R ∞ vol Y ′ ( σ ′∗ L ′ − xr F ′ ) dxA ( X, ∆) ( F ) · d ( L · n ) ≥ R ∞ vol Y ′ ( ψ ∗ ( σ ∗ L − xF )) dxA ( X, ∆) ( F ) · d ( L · n ) ≥ R ∞ vol Y ( σ ∗ L − xF ) dxA ( X, ∆) ( F ) · ( L · n ) = 1 − ˆ β ( F ) . As a consequence, we have proved Corollary 1.7. (cid:3)
We remark that the converse of Corollary 1.7 is not true in general.See the following example.
Example 4.2.
Let X := P , X ′ := P and let us consider the morphism φ : X ′ → X with t t . Take any d ∈ (0 , ∩ Q and set ( ∆ := [0] + [ ∞ ] + d [1] on X, ∆ ′ := d [1] + d [ −
1] on X ′ . Then we know that ( X, ∆) and ( X ′ , ∆ ′ ) are log Fano pairs and theequality φ ∗ ( K X + ∆) = K X ′ + ∆ ′ holds from the ramification for-mula. By [Fjt16b, Example 6.6], ( X, ∆) is uniformly K-stable. How-ever, again by [Fjt16b, Example 6.6], ( X ′ , ∆ ′ ) is not uniformly K-stable(but K-semistable).4.2. K-semistability of the projective plane.
In this section, weshow Corollary 1.8. Take any plt blowup σ : ( Y, F ) → ( P , β ( F ) ≥ F is a divisor on P . Set d := deg P F . Thenvol P ( − K P − xF ) = (3 − dx ) for x ∈ [0 , /d ] and A ( F ) = 1. Thus we get ˆ β ( F ) = ( d − /d ≥ F is an exceptional divisor over P .Set { p } := c P ( F ). Of course, the Picard rank of Y is equal to two.We may assume that the divisor − ( K Y + F ) is big by Remark 2.6.By [Pro01, Proposition 6.2.6 and Remark 6.2.7], the morphism σ is aweighted blowup with weights a , b for some local parameters s , t of O P ,p , where a , b ∈ Z > , a ≥ b and a , b are mutually prime. (Notethat the variety Y is not a toric variety in general.) We know that( F · ) Y = − / ( ab ) and A ( F ) = a + b .Let π : ˜ Y → Y be the minimal resolution of Y and let E ⊂ ˜ Y bethe strict transform of the exceptional divisor of the ordinary blowupof p ∈ P . Then we can check that ord E π ∗ F = 1 /a . Let ˆ l ⊂ ˜ Y be thestrict transform of a general line on P passing though p ∈ P . Since ˆ l is movable, we have0 ≤ (cid:16) π ∗ ( σ ∗ ( − K P − τ ( F ) F )) · ˆ l (cid:17) = 3 − τ ( F ) · a . This implies that τ ( F ) ≤ a . We can write K ˜ Y = π ∗ K Y − E for someeffective and π -exceptional Q -divisor E on ˜ Y (see [KM98, Corollary4.3]). Thus − K ˜ Y is big. This implies that ˜ Y and Y are Mori dreamspaces in the sense of [HK00] by [TVAV11, Theorem 1]. In particular, F is dreamy over ( P ,
0) (see [ELMNP06, Lemma 4.8]).Let us set ε ( F ) := max { ε ∈ R ≥ | σ ∗ ( − K P ) − εF nef } . Then ε ( F ) ∈ (0 , τ ( F )]. Claim 4.3. (1)
We have the equality ε ( F ) τ ( F ) = 9 ab . (2) We get vol P ( − K P − xF ) = (cid:16) − x ε ( F ) τ ( F ) (cid:17) if x ∈ [0 , ε ( F )] , ( τ ( F ) − x ) τ ( F )( τ ( F ) − ε ( F )) if x ∈ ( ε ( F ) , τ ( F )] . (3) We have the equality ˆ β ( F ) = 1 − ε ( F ) + τ ( F )3( a + b ) . Proof of Claim 4.3.
The assertion (3) follows from (1) and (2). Weprove (1) and (2).We know thatvol P ( − K P − xF ) = (cid:0) ( σ ∗ ( − K P ) − xF ) · (cid:1) = 9 − x ab -STABILITY AND PLT BLOWUPS 17 for x ∈ [0 , ε ( F )]. If ε ( F ) = τ ( F ), then vol P ( − K P − ε ( F ) F ) = 0.Thus ε ( F ) = 9 ab . Hence we can assume that ε ( F ) < τ ( F ). In thiscase, ε ( F ) ∈ Q and the divisor σ ∗ ( − K P ) − ε ( F ) F gives a nontrivialbirational contraction morphism µ : Y → Z since F is dreamy over( P ,
0) (see [Fjt16c, Lemma 3.1 (4)]). Moreover, since the Picard rankof Z is one, µ ∗ ( σ ∗ ( − K P )) and µ ∗ F are numerically proportional. Thus,for x ∈ [ ε ( F ) , τ ( F )], we can writevol P ( − K P − xF ) = (cid:0) µ ∗ ( σ ∗ ( − K P ) − xF ) · (cid:1) Z = c ( τ ( F ) − x ) for some c ∈ R > . Note thatvol P ( − K P − ε ( F ) F ) = 9 − ε ( F ) ab = c ( τ ( F ) − ε ( F )) and, by [BFJ09, Theorem A], ddx (cid:12)(cid:12)(cid:12)(cid:12) x = ε ( F ) vol P ( − K P − xF ) = − ε ( F ) ab = − c ( τ ( F ) − ε ( F )) . This implies that ab = ε ( F ) τ ( F ) / c = 9 / ( τ ( F )( τ ( F ) − ε ( F ))). (cid:3) Since ε ( F ) ≤ τ ( F ) ≤ a and ε ( F ) τ ( F ) = 9 ab , we have τ ( F ) ∈ [3 √ ab, a ]. Moreover, ε ( F ) + τ ( F ) = τ ( F ) + 9 abτ ( F )and the function x +9 ab/x is monotonically increasing on x ∈ [3 √ ab, a ].Thus ε ( F ) + τ ( F ) ≤ ab/ (3 a ) + 3 a = 3( a + b ). This implies thatˆ β ( F ) = 1 − ε ( F ) + τ ( F )3( a + b ) ≥ . As a consequence, we have completed the proof of Corollary 1.8.
References [BBJ15] R. Berman, S. Boucksom and M. Jonsson,
A variational approach to theYau-Tian-Donaldson conjecture , arXiv:1509.04561v1.[BCHM10] C. Birkar, P. Cascini, C. D. Hacon and J. M c Kernan,
Existence ofminimal models for varieties of log general type , J. Amer. Math. Soc. (2010),no. 2, 405–468.[BFJ09] S. Boucksom, C. Favre and M. Jonsson, Differentiability of volumes ofdivisors and a problem of Teissier , J. Algebraic Geom. (2009), no. 2, 279–308.[BHJ15] S. Boucksom, T. Hisamoto and M. Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs ,arXiv:1504.06568v3; to appear in Ann. Inst. Fourier.[Blu16] H. Blum,
Existence of Valuations with Smallest Normalized Volume ,arXiv:1606.08894v2. [Der15] R. Dervan,
Uniform stability of twisted constant scalar curvature K¨ahlermetrics , Int. Math. Res. Notices., DOI: 10.1093/imrn/rnv291.[Der16] R. Dervan,
On K-stability of finite covers , Bull. Lond. Math. Soc. (2016), no. 4, 717–728.[Don02] S. Donaldson, Scalar curvature and stability of toric varieties , J. Differen-tial Geom. (2002), no. 2, 289–349.[ELMNP06] L. Ein, R. Lazarsfeld, M. Mustat¸˘a, M. Nakamaye and M. Popa, As-ymptotic invariants of base loci , Ann. Inst. Fourier (Grenoble) (2006), no.6, 1701–1734.[ELMNP09] L. Ein, R. Lazarsfeld, M. Mustat¸˘a, M. Nakamaye and M. Popa, Re-stricted volumes and base loci of linear series , Amer. J. Math. (2009), no.3, 607–651.[Fjt16a] K. Fujita,
On K-stability and the volume functions of Q -Fano varieties ,Proc. Lond. Math. Soc. (2016), no. 5, 541–582.[Fjt16b] K. Fujita, A valuative criterion for uniform K-stability of Q -Fano vari-eties , J. Reine Angew. Math., DOI: 10.1515/crelle-2016-0055.[Fjt16c] K. Fujita, K-stability of Fano manifolds with not small alpha invariants ,arXiv:1606.08261v1.[FO16] K. Fujita and Y. Odaka,
On the K-stability of Fano varieties and anti-canonical divisors , arXiv:1602.01305v2; accepted by Tohoku Math. J.[Fuj94] T. Fujita,
Approximating Zariski decomposition of big line bundles , KodaiMath. J. (1994), no. 1, 1–3.[HK00] Y. Hu and S. Keel, Mori dream spaces and GIT , Michigan Math. J. (2000), 331–348.[Ish04] S. Ishii, Extremal functions and prime blow-ups , Comm. Algebra (2004),no. 3, 819–827.[Kem78] G. Kempf, Instability in invariant theory , Ann. of Math. (1978), no.2, 299–316.[KM98] J. Koll´ar and S. Mori,
Birational geometry of algebraic varieties , With thecollaboration of C. H. Clemens and A. Corti. Cambridge Tracts in Math., ,Cambridge University Press, Cambridge, 1998.[Laz04a] R. Lazarsfeld,
Positivity in algebraic geometry, I: Classical setting: linebundles and linear series , Ergebnisse der Mathematik und ihrer Grenzgebiete.(3) , Springer, Berlin, 2004.[Laz04b] R. Lazarsfeld, Positivity in algebraic geometry, II: Positivity for VectorBundles, and Multiplier Ideals , Ergebnisse der Mathematik und ihrer Gren-zgebiete. (3) , Springer, Berlin, 2004.[Li16] C. Li, K-semistability is equivariant volume minimization ,arXiv:1512.07205v4.[LM09] R. Lazarsfeld and M. Mustat¸˘a,
Convex bodies associated to linear series ,Ann. Sci. ´Ec. Norm. Sup´er. (2009), no. 5, 783–835.[LX16] C. Li and C. Xu, Stability of Valuations and Koll´ar Components ,arXiv:1604.05398v3.[Pro00] Y. Prokhorov,
Blow-ups of canonical singularities , Algebra (Moscow, 1998),301–317, de Gruyter, Berlin, 2000.[Pro01] Y. Prokhorov,
Lectures on complements on log surfaces , MSJ Memoirs, .Mathematical Society of Japan, Tokyo, 2001. -STABILITY AND PLT BLOWUPS 19 [PW16] J. Park and J. Won, K-stability of smooth del Pezzo surfaces ,arXiv:1608.06053v1.[RT07] J. Ross and R. Thomas,
A study of the Hilbert-Mumford criterion for thestability of projective varieties , J. Algebraic Geom. (2007), no. 2, 201–255.[Sho96] V. V. Shokurov, 3 -fold log models , Algebraic geometry, 4. J. Math. Sci. (1996), no. 3, 2667–2699.[Sz´e06] G. Sz´ekelyhidi, Extremal metrics and K-stability , Ph.D Thesis,arXiv:math/0611002.[Sz´e15] G. Sz´ekelyhidi,
Filtrations and test-configurations , with an appendix by S.Boucksom, Math. Ann. (2015), no. 1-2, 451–484.[Tia97] G. Tian,
K¨ahler-Einstein metrics with positive scalar curvature , Invent.Math. (1997), no. 1, 1–37.[TVAV11] D. Testa, A. V´arilly-Alvarado and M. Velasco,
Big rational surfaces ,Math. Ann. (2011), no. 1, 95–107.[Xu14] C. Xu,
Finiteness of algebraic fundamental groups , Compos. Math. (2014), no. 3, 409–414.
Research Institute for Mathematical Sciences, Kyoto University,Kyoto 606-8502, Japan
E-mail address ::