K-SEMISTABLE FANO MANIFOLDS WITH THE
SMALLEST ALPHA INVARIANTCHEN JIANG

Abstract. In this short note, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singularcases are also investigated.

1. introduction
Throughout the article, we work over the complex number field C. A QFano variety is a normal projective variety X with log terminal singularities
such that the anti-canonical divisor −KX is an ample Q-Cartier divisor.
It has been known that a Fano manifold X (i.e., a smooth Q-Fano variety)admits Kähler–Einstein metrics if and only if X is K-polystable by the works[DT92, Tia97, Don02, Don05, CT08, Sto09, Mab08, Mab09, Ber16] and[CDS15a, CDS15b, CDS15c, Tia15].
On the other hand, the existence of Kähler–Einstein metrics and Kstability are related to the alpha invariants α(X) of X defined by Tian
[Tia87] (see also [TY87, Zel98, Lu00, Dem08]). Tian [Tia87] proved thatfor a Fano manifold X, if α(X) > dim X/(dim X + 1), then X admitsKähler–Einstein metrics. Odaka and Sano [OS12, Theorem 1.4] (see also its
generalizations [Der16, BHJ15, FO16, Fuj16c]) proved a variant of Tian’s
theorem: if a Q-Fano variety X satisfies that α(X) > dim X/(dim X + 1)
(resp. ≥ dim X/(dim X + 1)), then X is K-stable (resp. K-semistable). We
are interested in the relation of alpha invariants and K-semistability.
Recall that Fujita and Odaka proved that there exists a lower bound ofalpha invariants for K-semistable Q-Fano varieties.
Theorem 1.1 ([FO16, Theorem 3.5]). Let X be a K-semistable Q-Fano
variety of dimension n.
1.
Then α(X) ≥ n+1
It is natural and interesting to ask when the equality holds. For example,
1
it is well-known that Pn is K-semistable with α(Pn ) = n+1. The main
theorem of this paper is the following.
Theorem 1.2. Let X be a K-semistable Fano manifold of dimension n.
1
Then α(X) = n+1
if and only if X ∼= Pn .
This is an application of Birkar’s answer to Tian’s question [Bir16, Theorem 1.5], and Fujita–Li’s criterion for K-semistability [Li15, Fuj16b].
Date: January 31, 2017.
The author was supported by JSPS KAKENHI Grant Number JP16K17558 and World
Premier International Research Center Initiative (WPI), MEXT, Japan.
1

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CHEN JIANG

It is natural to ask whether the same statement holds true for K-semistable
Q-Fano varieties instead of manifolds. However, this is on longer true evenin dimension 2. We are grateful to Kento Fujita for kindly providing thefollowing example:
Example 1.3. Consider the cubic surface X = (x30 = x1 x2 x3 ) ⊂ 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 Kähler–Einstein metric (cf. [DT92]), hence is K-semistable. On
the other hand, α(X) = 31 (cf. [PW10]).
In fact, by the classification of possible du Val singularities of K-semistablelog del Pezzo surfaces (cf. [Liu16, Corollary 6]) and explicit computation ofalpha 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) = 31 if and only if X ∼= P2 or X ⊂ P3 is a
cubic surface with at least 2 singularities of type A2 .
Moreover, by classification of Q-Fano 3-fold with Q-factorial terminal singularities 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 (−KX ) ≥ 22. Thenα(X) = 14 if and only if X ∼= P3 .
Finally, we propose the following much stronger conjecture. For someevidence in dimension 3, we refer to [CS08] and [Fuj16a].
Conjecture 1.6. Let X be a K-semistable Fano manifold.
Then α(X) < n1 if and only if X ∼= Pn .
Acknowledgments. The author would like to thank Professors Kento Fujita and Yoshinori Gongyo for effective discussions. The main part of thispaper was written during the author enjoyed the workshop “Higher Dimensional 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 thepair (X, αD) is log canonical for any effective Q-divisor D with D ∼Q −KX .
In other words,
α(X) := inf{lct(X; D)|0 ≤ D ∼Q −KX }.
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

3Theorem 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
−KX and lct(X; D) = α(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 andany prime divisor F on Y , that is, F is a prime divisor over X.
(1) Define the log discrepancy of F as A(F ) := multF (KY − σ ∗ KX ) + 1;
(2) Define volX (−KX − xF ) := volY (−σ ∗ KX − xF );(3) Define
Z ∞nvolX (−KX − xF ) dx.
β(F ) := A(F ) · (−KX ) −0

Note that the definitions do not depend on the choice of birational modelY.
Instead of recalling the original definition, we use the following criterion
to define K-semistability.
Definition-Proposition 2.4 ([Fuj16b, Corollary 1.5], [Li15, Theorem 3.7]).
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.
1
Assume that α(X) = n+1, then there exists a prime divisor E on X suchthat −KX ∼Q (n + 1)E and (X, E) is plt.
Proof. Let X be a K-semistable Q-Fano variety of dimension n with α(X) =1n+1 . By Theorem 2.2, there is a divisor D ∼Q −KX such that lct(X; D) =1
1n+1 . Take F to be a non-klt place of (X, n+1 D), then there is a resolution
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
multF KY − σ ∗ KX +
D
= −1,
n+1
which means thatµ.
n+1
By Definition-Proposition 2.4, β(F ) ≥ 0, which means that
A(F ) =1A(F )(−KX )n =(−KX )nn+1µZ
1 ∞volX (−KX − xF ) dx
≥
µ 0
Z ∞
=
volX (−KX − xµF ) dx
0

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CHEN JIANG

Z ∞volX (−KX − xD) dx

≥
0

Z 1
=

(1 − x)n (−KX )n dx

0=1(−KX )n .
n+1

The second equality holds since σ ∗ D ≥ µF . Hence all inequalities become
equalities. In particular,
volX (−KX − xµF ) = volX (−KX − xD) = (1 − x)n (−KX )nfor almost all x. By differentiability of volume functions ([BFJ09, CorollaryC]),µ · volY |F (−σ ∗ KX )1 d
volY (−σ ∗ KX − xµF )n dx x=01 d= −(1 − x)n (−KX )nn dx x=0= (−KX )n .

= −Here volY |F is the restricted volume, we refer to [ELMNP09] for definition and properties. Since volY |F (−σ ∗ KX ) > 0, F 6⊆ B+ (−σ ∗ KX ) by
[ELMNP09, Theorem C]. Hence by [ELMNP09, Corollary 2.17],
volY |F (−σ ∗ KX ) = (−σ ∗ KX )n−1 · F = (−KX )n−1 · σ∗ F.
In other words, we have(−KX )n−1 (D − µσ∗ F ) = (−KX )n − µ · volY |F (−σ ∗ KX ) = 0.
This implies that D = µσ∗ F since D ≥ µσ∗ F and −KX is ample. In
particular, F is not σ-exceptional and σ∗ F is a prime divisor on X. Denote
11
E := σ∗ F . Moreover, since F is a non-klt place of (X, n+1
D), multE n+1
D=1, that is, µ = n + 1. In particular, −KX ∼Q D = (n + 1)E. Finally, thisargument shows that F is the only non-klt place of (X, E), which meansthat (X, E) is plt.
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) If X is factorial, then E is a Cartier divisor. In particular, (E)n ≥1. Hence this is a special case of (2).
(2) If (E)n ≥ 1, then(−KX )n = (n + 1)n (E)n ≥ (n + 1)n .
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,(KX + E)|E = KE , and(−KX )n =(n + 1)n(n + 1)n
n−1
(−(K
+
E))
·
E
=
(−KE )n−1 = (n + 1)n .
X
nn−1
nn−1

Again by [Liu16, Theorem 1.1] or [LZ16, Theorem 9], X ' Pn .



Proof of Theorem 1.2. It follows directly from Proposition 3.1 and Corollary
3.2(1) (or [KO73]).

4. Singular surfaces
Recall the following theorem on classification of possible du Val singularities 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 , orA3 ;(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 (−KX )2 ≥ 5, then X is smooth.
We remark that in [Liu16, Corollary 6], log del Pezzo surfaces are assumed to be admitting Kähler–Einstein metrics, but the proof works wellfor K-semistable log del Pezzo surfaces. The only part that the existence ofKähler–Einstein metrics is needed is to exclude the case that (−KX )2 = 1
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, Theorem 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) = 31 if and only if
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 )0 }, 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 withat worst du Val singularities and α(X) = 31 . If X is smooth, then X ' P2
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]).
6

CHEN JIANG

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 singularity with local index r. Then(−KX )3 ≤(r + 2)(4 + 4r)3.
(3r)3

Proof. Denote by mp the maximal ideal at p. We may take a log resolutionof (X, mp ), namely π : Y → X such that π is an isomorphism away fromp and π −1 mp · OY is an invertible ideal sheaf on Y . Let Ei be exceptional
divisors of π. We define the numbers ai and bi by
X
KY = π ∗ KX +
ai Ei
i

and
π −1 mp · OY = OY (−X

bi Ei ).

ii
It is clear that lct(X; mp ) = mini 1+abi . Since π is an isomorphism away fromp, we have bi ≥ 1 for any i. Since X is terminal at p, by [Kaw93], there
exists an index i0 such that ai0 = 1r . Hencelct(X; mp ) ≤1 + ai0
1
≤1+ .
bi0
r

On the other hand, by [Kak00] (see also [TW04, Proposition 3.10]),multp X ≤ r + 2. Hence by [Liu16, Theorem 16],

1 3
(r + 2)(4 + 4r)3
3(−KX ) ≤ 1 +lct(X; mp )3 multp X ≤
.
3
(3r)3

Now let X be a K-semistable Q-Fano 3-fold with Q-factorial terminal
singularities and ρ(X) = 1 with α(X) = 41 . By Proposition 3.1, there exists
a prime divisor E on X such that −KX ∼Q 4E.
Recall that we may define ([Pro10])qW(X) := max{q | −KX ∼ qA, A is a Weil divisor},qQ(X) := max{q | −KX ∼Q qA, A is a Weil divisor}.
It is known by [Suz04 , Pro10] thatqW(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 (−KX ) ≥ 22. Define the genus g(X) := h0 (−KX ) −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). But in either case,K-SEMISTABLE FANO MANIFOLDS WITH THE SMALLEST ALPHA INVARIANT

7−KX ∼ 8A where A is an effective divisor, which implies that α(X) ≤ 81
since (X, A) is not klt, a contradiction.
Now assume that qQ(X) = 4, by [Pro13, Lemma 8.3], Cl(X) is torsionfree and qW(X) = qQ(X), hence there is a Weil divisor A such that −KX ∼4A. If g(X) ≥ 22, then by [Pro13, Theorem 1.2(vi)], X ' P3 or X4 ⊂P(1, 1, 1, 2, 3). The latter is absurd, since it has a singularity of index 3, and(−KX4 )3 = 128/3, which contradicts to Theorem 5.1. If 20 ≤ g(X) ≤ 21,then we have the following possibilities due to computer computation (see[GRD], or [BS07, Pro10, Pro13]):
g(X)
B
A3
21
{3}
2/3
20
{5, 7} 22/35
Here B is the set local indices of singular points. It is easy to see that bothcases contradict to Theorem 5.1.
In summary, Theorem 1.5 is proved.
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Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 2778583, Japan.
E-mail address: chen.jiang@ipmu.jp