BOUNDEDNESS OF Q-FANO VARIETIES WITH DEGREESAND ALPHA-INVARIANTS BOUNDED FROM BELOW
CHEN JIANG
Abstract. We show that Q-Fano varieties of fixed dimension with
anti-canonical degrees and alpha-invariants bounded from below forma bounded family. As a corollary, K-semistable Q-Fano varieties of
fixed dimension with anti-canonical degrees bounded from below form abounded family.

1. Introduction
Throughout the article, we work over an algebraically closed field of characteristic zero. A Q-Fano variety is defined to be a normal projective variety
X with at most klt singularities such that the anti-canonical divisor −KX
is an ample Q-Cartier divisor.
When the base field is the complex number field, an interesting problem for Q-Fano varieties is the existence of Kähler–Einstein metrics whichis related to K-(semi)stability of Q-Fano varieties. It has been known
that a Fano manifold X (i.e., a smooth Q-Fano variety over C) admitsKähler–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]. K-stability is stronger than K-polystability, andK-polystability is stronger than K-semistability. Hence K-semistable QFano varieties are interesting for both differential geometers and algebraic
geometers.
It also turned out that Kähler–Einstein metrics and K-stability play crucial roles for construction of nice moduli spaces of certain Q-Fano varieties.
For example, compact moduli spaces of smoothable Kähler–Einstein Q-Fano
varieties have been constructed (see [OSS16] for dimension two case and[LWX14, SSY16, Oda15] for higher dimensional case). In order to consider
the moduli space of certain (singular) Q-Fano varieties, the first step is toshow the boundedness property, which is the motivation of this paper. We
show the boundedness of K-semistable Q-Fano varieties of fixed dimension
with anti-canonical degrees bounded from below, which gives an affirmative answer to a question asked by Yuchen Liu during the AIM workshop“Stability and moduli spaces” in January 2017.
Date: December 19, 2018.
2010 Mathematics Subject Classification. 14J45, 14C20.
Key words and phrases. Fano varieties, boundedness, K-stability, degrees, alphainvariants.
The author was supported by JSPS KAKENHI Grant Number JP16K17558 and World
Premier International Research Center Initiative (WPI), MEXT, Japan.
12Chen Jiang

Theorem 1.1. Fix a positive integer d and a real number δ > 0. Thenthe set of d-dimensional K-semistable Q-Fano varieties X with (−KX )d > δforms a bounded family.
Note that the assumption that (−KX )d is bounded from below is necessary, by Example 1.4(2) later.
As mentioned before, one might have further applications of Theorem 1.1
such as constructing moduli spaces of d-dimensional K-semistable Q-Fano
varieties with bounded anti-canonical degrees. An interesting corollary ofTheorem 1.1 is the discreteness of the anti-canonical degrees of K-semistable
Q-Fano varieties.
Corollary 1.2. Fix a positive integer d. Then the set of (−KX )d for ddimensional K-semistable Q-Fano varieties X is finite away from 0.
Here a set P of positive real numbers is finite away from 0 if for any
δ > 0, P ∩ (δ, ∞) is a finite set. We remark that Corollary 1.2 might berelated to the conjectural discreteness of minimal normalized volumes of klt
singularities, cf. [LX17, Question 4.3].
The idea of proof of Theorem 1.1 comes from birational geometry. According to Minimal Model Program, Q-Fano varieties form a fundamental
class in birational geometry, and the boundedness property for Q-Fano varieties are also interesting from the point view of birational geometry. For
example, Kollár, Miyaoka, and Mori [KMM92] proved that smooth Fano varieties form a bounded family. The most celebrated progress recently is theproof of Borisov–Alexeev–Borisov Conjecture due to Birkar [Bir16a, Bir16b],
which says that given a positive integer d and a real number  > 0, the setof -lc Q-Fano varieties of dimension d forms a bounded family.
In this paper, inspired by Birkar’s work, in order to show Theorem 1.1,
we show the following theorem.
Theorem 1.3. Fix a positive integer d and a real number δ > 0. Then theset of d-dimensional Q-Fano varieties X with (−KX )d > δ and α(X) > δ
forms a bounded family.
Here α(X) is the alpha-invariant of X defined by Tian [Tia87] (see also
[Dem08]) in order to investigate the existence of Kähler–Einstein metrics on
Fano manifolds. Recall that Fujita and Odaka [FO16, Theorem 3.5] provedthat the alpha-invariant of a K-semistable Q-Fano variety of dimension d isalways not less than 1/(d + 1), so Theorem 1.3 implies Theorem 1.1 naturally. The advantage to consider Theorem 1.3 is that we can then apply
methods from birational geometry, instead of dealing with K-semistable QFano varieties.
The point of Theorem 1.3 is that we replace the -lc condition in Borisov–Alexeev–Borisov Conjecture by the condition on lower bound of anti-canonicaldegrees and alpha-invariants, which are global invariants.
We remark that if one take δ = 1, then Theorem 1.3 is a consequenceof [Bir16a, Theorem 1.3], which says that the set of exceptional Q-Fano
varieties (i.e., Q-Fano varieties X with α(X) > 1) of fixed dimension forms
a bounded family. Note that in this case we even do not need to assume(−KX )d is bounded from below. But in general we need to assume both(−KX )d and α(X) are bounded from below, by the following examples.

Boundedness of Q-Fano varieties with degrees and α-invariants bounded from below

3

Example 1.4. Fix a positive integer d.
(1) Consider the weighted projective space Xn = P(1d , n) which is a QFano variety of dimension d with (−KXn )d = (n + d)d /n > 1, but it
is clear that {Xn } does not form a bounded family.
(2) Consider Y8n+4 ⊂ P(2, 2n + 1, 2n + 1, 4n + 1), a general weighted
hypersurface of degree 8n + 4, which is a Q-Fano variety of dimension 2 with α(Y8n+4 ) = 1 (see [CPS10, Corollary 1.12] or [JK01]),
but it is clear that {Y8n+4 } does not form a bounded family. For
more interesting examples of Q-Fano varieties with α ≥ 1, we refer
to [CPS10, CS13] in dimension 2 and [CS11, CS14] in higher dimensions. Note that all examples with α ≥ 1 are K-semistable (in fact,
K-stable) by [OS12, Theorem 1.4] (or [Tia87]).
By [Bir16a, Proposition 7.13] or [Bir16b, Theorem 2.15], Theorem 1.3 isa consequence of the following theorem.
Theorem 1.5. Fix a positive integer d and a real number δ > 0. Then
there exists a positive integer m depending only on d and δ such that if X
is a d-dimensional Q-Fano variety with (−KX )d > δ and α(X) > δ, then| − mKX | defines a birational map.
To show Theorem 1.5, our main idea is to establish an inequality expressedin terms of the volume of −KX |G on a covering family of subvarieties G of
X and (−KX )d , α(X), see Lemma 3.1.
As a variation of Theorem 1.3, we can also show the following theorem.
Theorem 1.6. Fix a positive integer d and a real number θ > 0. Then theset of d-dimensional Q-Fano varieties X with α(X)d · (−KX )d > θ forms abounded family.
Logically, Theorem 1.3 is implied by Theorem 1.6. But we will show
Theorem 1.3 first in order to make the explanation more clear.
Remark 1.7. Note that the invariant α(X)d · (−KX )d appears naturally inbirational geometry, see for example [Kol97, Theorem 6.7.1]. It is not clear0
whether we can replace α(X)d · (−KX )d in Theorem 1.6 by α(X)d · (−KX )dfor some positive real number d0 < d. At least d0 ≤ d − 1 is not sufficient
to conclude the boundedness. For example, in Example 1.4(1), (−KXn )d =(n + d)d /n and α(Xn ) = 1/(n + d) (for computation of alpha-invariants oftoric varieties, see [Amb16, 6.3]), hence α(Xn )d−1 · (−KXn )d > 1.
Remark 1.8. We remark that the proof of both Theorems 1.3 and 1.6 works
under weaker assumption that X is a weak Q-Fano variety (i.e., X has at
most klt singularities and −KX is nef and big), see also Remark 2.5. But it
is not clear yet whether the log Fano pair versions hold or not.
Acknowledgment. The topic of this paper was brought to the author by
Yuchen Liu. The author would like to thank Yuchen Liu for inspiration andfruitful discussions. The author is grateful to Jingjun Han, Yusuke Nakamura, and Taro Sano for discussions and to Kento Fujita, Yujiro Kawamata,Ivan Cheltsov, Wenhao Ou, and the referee(s) for valuable comments andsuggestions. Part of this paper was written during the author enjoyed theworkshops “NCTS Workshop on Singularities, Linear Systems, and Fano

4

Chen Jiang

Varieties” at National Center for Theoretical Sciences and “BICMR-Tokyo
Algebraic Geometry Workshop” at Beijing International Center for Mathematical Research. The author is grateful for the hospitality and support ofthese institutes.
2. Preliminaries
2.1. Notation and conventions. We adopt the standard notation anddefinitions in [KMM87] and [KM98], and will freely use them.
A pair (X, B) consists of a normal projective variety X and an effective
R-divisor B on X such that KX + B is R-Cartier.
Let f : Y → X be a log resolution of the pair (X, B), write
X
KY = f ∗ (KX + B) +ai Fi ,
where {Fi } are distinct prime divisors. Take a real number  ≥ 0. The pair
(X, B) is called(a) kawamata log terminal (klt, for short) if ai > −1 for all i;(b) log canonical (lc, for short) if ai ≥ −1 for all i;(c) -log canonical (-lc, for short) if ai ≥ −1 +  for all i.
Usually we write X instead of (X, 0) in the case B = 0.
Fi is called a non-klt place of (X, B) if ai ≤ −1. A subvariety V ⊂ X iscalled a non-klt center of (X, B) if it is the image of a non-klt place.
Let (X, B) be an lc pair and D ≥ 0 be a R-Cartier R-divisor. The log
canonical threshold of D with respect to (X, B) is defined bylct(X, B; D) = sup{t ≥ 0 | (X, B + tD) is lc}.
If X is a Q-Fano variety, the alpha-invariant of X is defined byα(X) = inf{lct(X; D) | D ∼Q −KX , D ≥ 0}.
A collection of varieties {Xt }t∈T is said to be bounded if there exists
h : X → S a projective morphism between schemes of finite type such thateach Xt is isomorphic to Xs for some s ∈ S.
2.2. Volumes. Let X be a d-dimensional normal projective variety and D
be a Cartier divisor on X. The volume of D is the real number
h0 (X, OX (mD)).
volX (D) = lim sup
md /d!
m→∞
Note that the limsup is actually a limit. Moreover by the homogenous property and continuity of volumes, we can extend the definition to R-Cartier
R-divisors. Note that if D is a nef R-Cartier R-divisor, then volX (D) = Dd .
For more background on volumes, see [Laz04, 2.2.C, 11.4.A].
2.3. Potentially birational divisors. Let X be a normal projective variety and D be a big Q-Cartier Q-divisor on X. We say that D is potentiallybirational (see [HMX14, Definition 3.5.3]) if for any two general points x
and y of X, possibly switching x and y, we can find an effective Q-divisor∆ ∼Q (1−)D for some 0 <  < 1 such that (X, ∆) is not klt at y but (X, ∆)is lc at x and {x} is a non-klt center. Note that if D is potentially birational,
then |KX + dDe| defines a birational map ([HMX13, Lemma 2.3.4]).

Boundedness of Q-Fano varieties with degrees and α-invariants bounded from below

5

2.4. Non-klt centers. We recall the following proposition in [Bir16a] which
is proved by standard techniques for constructing families of non-klt centers,
see e.g. [Kol97, HMX14, Bir16a].
Proposition 2.1 (cf. [Bir16a, 2.31(2), page 22]). Let X be a normal projective variety of dimension d and D, A be two ample Q-divisors. Assume
Dd > (2d)d .
Then there is a bounded family of subvarieties of X such that for two
general points x, y in X, there is a member G of the family and an effective
Q-divisor ∆ ∼Q D + (d − 1)A such that
• (X, ∆) is lc near x with a unique non-klt place whose center contains
x, that center is G,• (X, ∆) is not klt at y, and• either dim G = 0 or (A|G )dim G ≤ dd .
2.5. Birkar’s results. We recall several theorems from [Bir16a]. The following theorem provides a criterion of boundedness of certain Q-Fano varieties, which is one of the key ingredients of [Bir16a, Bir16b].
Theorem 2.2 ([Bir16a, Proposition 7.13]). Let d, m, v be positive integers
and tl be a sequence of positive real numbers. Let P be the set of projective
varieties X such that
• X is a Q-Fano variety of dimension d,• KX has an m-complement,• | − mKX | defines a birational map,
• (−KX )d ≤ v, and• for any l ∈ N and any L ∈ | − lKX |, the pair (X, tl L) is klt.
Then P is a bounded family.
Here KX has an m-complement means that there exists an effective divisor
1M ∼ −mKX , such that (X, m
M ) is lc. For definition of complements ingeneral setting, we refer to [Bir16a]. The boundedness of complements isproved by Birkar as the following theorem.
Theorem 2.3 ([Bir16a, Theorem 1.1]). Let d be a positive integer. Then
there exists a positive integer n depending only on d such that if X is a
Q-Fano variety of dimension d, then KX has an n-complement.
Recall that a Q-Fano variety X is exceptional if (X, D) is klt for any
effective Q-divisor D ∼Q −KX . This is equivalent to say that α(X) > 1,
because, if α(X) > 1 then clearly X is exceptional by the definition (notethat we only need this direction of the implication in this paper); on theother hand, if X is exceptional, then it is easy to see that α(X) ≥ 1, but
one can use [Bir16b, Theorem 1.5] to exclude the case α(X) = 1.
Birkar proved the boundedness of exceptional Q-Fano varieties.
Theorem 2.4 ([Bir16a, Theorem 1.3]). Let d be a positive integer. Then theset of exceptional Q-Fano varieties of dimension d forms a bounded family.
Remark 2.5. All theorems in this subsection hold for weak Q-Fano varieties.

6

Chen Jiang

3. Proof of the theorems
The idea of proof of Theorem 1.5 is to construct isolated non-klt centers
by −KX , that is, for a general point x ∈ X, we need to construct an effective
Q-divisor ∆ ∼Q −mKX where m is fixed, so that (X, ∆) has an isolated
non-klt center at x, see e.g. [AS95, HMX13, Bir16a]. From the lower boundof (−KX )d , it is easy to construct some non-klt center G containing x. In
order to cut down the dimension of G, we need to bound the volume of−KX |G . The main point of this paper is to show that the volume of −KX |G
is bounded from below by an expression in terms of (−KX )d and α(X), as
the following lemma.
Lemma 3.1. Fix two positive integers d > k. Let X be a Q-Fano variety
of dimension d. Assume there is a contraction f : Y → T of projective
varieties with a surjective morphism φ : Y → X. Assume that a generalfiber F of f is of dimension k and is mapped birationally onto its image G
in X, φ is smooth over φ(ηF ), and φ(ηF ) is a smooth point of X. Then(−KX )k · G ≥α(X)d−k(−KX )d .

d
d−k
(d
−
k)
k

Proof. Taking normalizations and resolutions, we may assume Y and T aresmooth. We may pick a general fiber F over t ∈ T such that f is smoothover t. Cutting by general smooth hyperplane sections of T containing t, we
may assume φ is generically finite, here note that all the assumptions arepreserved according to [Bir16a, Lemma 2.28]. In particular, dim Y = d.
Fix any rational number l such thats d (−KX )k · G
l > d−k,k(−KX )dtake an integer m such that ml is an integer and −lmKX is Cartier.
hmi
Note that there is a natural injection OX /IG → φ∗ (OY /IFm ) by comparing the order of local regular functions since φ is étale over the generichmipoint of G. Here IF denotes the ideal sheaf of F and IG is the ideal of
regular functions vanishing along a general point of G to order at least m.
By projection formula, this implies thathmi

h0 (X, OX (−lmKX ) ⊗ OX /IG )≤ h0 (X, OX (−lmKX ) ⊗ φ∗ (OY /IFm ))= h0 (Y, φ∗ OX (−lmKX ) ⊗ OY /IFm ).
On the other hand, since F is a general fiber of f and Y is smooth, the⊕(d−k)
conormal sheaf of F is trivial, that is, IF /IF2 ' OF
, also we have⊕(i+d−k−2
d−k−1 )

IFi−1 /IFi ' S i−1 (IF /IF2 ) ' OF
for i ≥ 1 (see [Har77, II. Theorem 8.24]). Hence

h0 (Y, φ∗ OX (−lmKX ) ⊗ OY /IFm )
≤

m
X
i=1

h0 (Y, φ∗ OX (−lmKX ) ⊗ IFi−1 /IFi )Boundedness of Q-Fano varieties with degrees and α-invariants bounded from below

7


m 
X
i+d−k−2 0=h (F, φ∗ OX (−lmKX )|F )
d−k−1
i=1


m+d−k−1 0=
h (F, φ∗ OX (−lmKX )|F ).
d−kHere for the last equality, we use the formula that for positive integers a and
b,
 

b 
X
i+a−1
a+b
=
.
a
a+1
i=1

This can be shown by induction on b.
By the exact sequencehmi

0 → OX (−lmKX )⊗IG

hmi

→ OX (−lmKX ) → OX (−lmKX )⊗OX /IG→ 0,

we havehmi

h0 (X, OX (−lmKX ) ⊗ IG )hmi

≥ h0 (X, OX (−lmKX )) − h0 (X, OX (−lmKX ) ⊗ OX /IG )m+d−k−1 0≥ h0 (X, OX (−lmKX )) −h (F, φ∗ OX (−lmKX )|F ).
d−kNote that by definition of volumes,
d! 0
h (X, OX (−lmKX )) = volX (OX (−lKX )) = (−lKX )d ,lim
m→∞ mdand
d! m + d − k − 1 0
h (F, φ∗ OX (−lmKX )|F )limm→∞ mdd−k
d! (m + d − k − 1) · · · (m + 1)m 0= lim·
h (F, φ∗ OX (−lmKX )|F )m→∞ md(d − k)!
d!
= lim
h0 (F, φ∗ OX (−lmKX )|F )km→∞ m (d − k)!
 d=volF (φ∗ OX (−lKX )|F )
k
 
d
=
(−φ∗ (lKX )|F )k
k
 
d
=
(−φ∗ (lKX ))k · F
k
 
d
=
(−lKX )k · G.
kHere for the last step we use the fact that F → G is birational (cf. [KM98,
Proposition 1.35(6)]). Note that by choice of l,
 
 
k
d
d
d
d
k d (−KX ) · G
d(−lKX ) = l (−KX ) > l(−KX ) =(−lKX )k · G.
dkk(−KX )Hencehmi

h0 (X, OX (−lmKX ) ⊗ IG ) > 0

8Chen Jiang

for m sufficiently large. This implies that there exists an effective Q-divisor
D ∼Q −lKX such that multG D ≥ 1. In particular, (X, (d − k)D) is not klt
along G as G interests the smooth locus of X (cf. [KM98, Lemma 2.29]).
Hence


1
(d − k)l ≥ lct X; D ≥ α(X).
lSince we can take l to be an arbitrary rational number such thats d (−KX )k · G,
l > d−kk(−KX )dit follows thats d (−KX )k · G(d − k) d−k≥ α(X),k(−KX )dthat is,(−KX )k · G ≥α(X)d−k(−KX )d .
dd−kk (d − k)

The proof is completed.

Proof of Theorem 1.5. Take a d-dimensional Q-Fano variety X with (−KX )d >
δ and α(X) > δ. Take2d,q0 = √
d
δ
p0 =

max


s
 k d(d − k)d−k dd 1≤k≤d−1 kδ d−k+1.

Take roundups q = dq0 e and p = dp0 e. By definition, (−qKX )d > (2d)d .
Applying Proposition 2.1 for D = −qKX and A = −pKX , then there
is a bounded family of subvarieties of X such that for two general points
x, y in X, there is a member G of the family and an effective Q-divisor∆ ∼Q D + (d − 1)A such that (X, ∆) is lc near x with a unique non-klt place
whose center contains x, that center is G, and (X, ∆) is not klt at y, andeither dim G = 0 or (A|G )dim G ≤ dd . We will show that the latter case will
never happen, that is, dim G = 0 always holds for general x, y.
Note that since x, y are general, we can assume G is a general memberof the family. Recall from [Bir16a, 2.27] that this means the family is given
by finitely many morphisms Vj → Tj of projective varieties with surjective
morphisms Vj → X such that each G is a general fiber of one of thesemorphisms. If G is a general fiber of some Vj → Tj of dimension k > 0, as
Vj → Tj is constructed from the Hilbert scheme of subvarieties (cf. [Bir16a,
2.27, 2.31]), it satisfies the assumptions of Lemma 3.1, and then by applying
Lemma 3.1 to Vj → Tj and G, we get(−KX )k · G ≥α(X)d−k(−KX )d >

d
d−k
(d
−
k)
k

δ d−k+1dk (d − k)In particular, by the definition of p,(A|G )dim G = (−pKX )k · G > dd .

d−k.

Boundedness of Q-Fano varieties with degrees and α-invariants bounded from below

9This is a contradiction.
Hence dim G = 0 for general x, y, that is, G = {x}. Recall our construction, this means that for any two general points x, y ∈ X we can choose aneffective Q-divisor ∆ ∼Q D + (d − 1)A = −(q + p(d − 1))KX so that (X, ∆) islc near x with a unique non-klt place whose center is {x}, and (X, ∆) is not
klt at y. Hence −(q + p(d − 1) + 1)KX is potentially birational and hence|KX − (q + p(d − 1) + 1)KX | defines a birational map by [HMX13, Lemma
2.3.4]. We may take m = q + p(d − 1).
Proof of Theorem 1.3. By Theorem 2.2, it suffices to show that there exist
positive integers m, v and a sequence of positive real numbers tl depending
only on d and δ such that if X is a d-dimensional Q-Fano variety with(−KX )d > δ and α(X) > δ, then the conditions in Theorem 2.2 are satisfied,that is,(1) KX has an m-complement,(2) | − mKX | defines a birational map,(3) (−KX )d ≤ v, and(4) for any l ∈ N and any L ∈ | − lKX |, the pair (X, tl L) is klt.
Firstly, by Theorems 2.3 and 1.5, there exists a positive integer m depending only on d and δ such that KX has an m-complement and | − mKX |defines a birational map.
Secondly, it is well-known (cf. [Kol97, Theorem 6.7.1]) thatd.
α(X) ≤ pd(−KX )d
In fact, for any rational number s such that s > √dd, we have(−KX )d(−sKX )d > dd .
By [Kol97, Theorem 6.7.1], there exists an effective Q-divisor B ∼Q −sKX
such that (X, B) is not lc. Hence α(X) < s. By the arbitrarity of s, weconclude thatdα(X) ≤ p.
d(−KX )dHence
dd
dd(−KX )d ≤
<
α(X)d
δd
and we may take v = dd /δ d .
Finally, for any l ∈ N and any L ∈ | − lKX |, the pair (X, δl L) is klt sinceα(X) > δ. We may take tl = δ/l.
In summary, by Theorem 2.2, the set of d-dimensional Q-Fano varieties
X with (−KX )d > δ and α(X) > δ forms a bounded family.
Proof of Theorem 1.6. Take a d-dimensional Q-Fano variety X with α(X)d ·(−KX )d > θ. We want to apply Theorem 1.3 in this situation, that is, itsuffices to show that there exists a real number δ > 0 depending only on dand θ such that (−KX )d > δ and α(X) > δ.
Firstly, note that if α(X) > 1, X is an exceptional Q-Fano variety andhence belongs to a bounded family by Theorem 2.4.

10

Chen Jiang

Hence from now on, we may assume that α(X) ≤ 1. In particular,(−KX )d >

θ
≥ θ.
α(X)d

Take
2d
,
q0 = √
d
θ
p0 =

max

s 

 k d (d − k)d−k dd k1≤k≤d−1 θ.

Take roundups q = dq0 e and p = dp0 e. By definition,(−qα(X)KX )d ≥(2d)d· α(X)d · (−KX )d > (2d)d .
θApplying Proposition 2.1 for D = −qα(X)KX and A = −pα(X)KX ,then there is a bounded family of subvarieties of X such that for two generalpoints x, y in X, there is a member G of the family and an effective Q-divisor∆ ∼Q D + (d − 1)A such that (X, ∆) is lc near x with a unique non-klt place
whose center contains x, that center is G, and (X, ∆) is not klt at y, andeither dim G = 0 or (A|G )dim G ≤ dd . We will show that the latter case will
never happen, that is, dim G = 0 always holds for general x, y.
Note that since x, y are general, we can assume G is a general memberof the family. Recall from [Bir16a, 2.27] that this means the family is given
by finitely many morphisms Vj → Tj of projective varieties with surjective
morphisms Vj → X such that each G is a general fiber of one of thesemorphisms. If G is a general fiber of some Vj → Tj of dimension k > 0, as
Vj → Tj is constructed from the Hilbert scheme of subvarieties (cf. [Bir16a,
2.27, 2.31]), it satisfies the assumptions of Lemma 3.1, and then by applying
Lemma 3.1 to Vj → Tj and G, we get(−KX )k · G ≥α(X)d−k(−KX )d >

d
d−k
(d
−
k)
k

θ
d
d−k α(X)kk (d − k).

In particular, by the definition of p,(A|G )dim G = (−pα(X)KX )k · G > dd .
This is a contradiction.
Hence dim G = 0 for general x, y, that is, G = {x}. Recall our construction, this means that for any two general points x, y ∈ X we can choose aneffective Q-divisor ∆ ∼Q D + (d − 1)A = −(q + p(d − 1))α(X)KX so that(X, ∆) is lc near x with a unique non-klt place whose center is {x}, and(X, ∆) is not klt at y. This means that the non-klt locus (i.e. union of all
non-klt centers) Nklt(X, ∆) contains y and x such that x is an isolated point.
By Shokurov–Kollár connectedness lemma (see Shokurov [Sho93, Sho94] andKollár [Kol92, Theorem 17.4]), −(KX + ∆) can not be ample. On the other
hand,−(KX + ∆) ∼Q −(1 − (q + p(d − 1))α(X))KX .

Boundedness of Q-Fano varieties with degrees and α-invariants bounded from below

11

As −KX is ample, this implies that 1 − (q + p(d − 1))α(X) ≤ 0, that is,α(X) ≥

1.
q + p(d − 1)

Hence we may take δ = min{θ, 1/2(q + p(d − 1))} and apply Theorem 1.3
to conclude Theorem 1.6.

Proof of Theorem 1.1. Without loss of generality, we may assume δ < 1/(d+1). For a K-semistable Q-Fano variety X of dimension d, by [FO16, Theorem
3.5], α(X) ≥ 1/(d + 1) > δ. Hence Theorem 1.1 follows immediately from
Theorem 1.3.
Proof of Corollary 1.2. By Theorem 1.1, the set of d-dimensional K-semistable
Q-Fano varieties X with (−KX )d > δ forms a bounded family, hence (−KX )dcan only take finitely many possible values for such Q-Fano varieties.

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Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 2778583, Japan.
Email address: chen.jiang@ipmu.jp