K-STABILITY OF FANO VARIETIES: AN ALGEBRO-GEOMETRIC APPROACHmath.mit.edu/~cyxu/Kstability.pdf ·...

K-STABILITY OF FANO VARIETIES: AN ALGEBRO-GEOMETRICAPPROACH

CHENYANG XU

Abstract. We give an survey of the recent progress on the study of K-stability ofFano varieties by algebro-geometric approach.

Contents

1. Introduction 1

Part 1. What is K-stability? 42. Definition of K-stability by degeneration and minimal model program 43. Two invariant: β and δ 94. Local stability of klt singularities 16

Part 2. K-stable Fano varieties 245. K-Moduli of Fano varieties 246. Explicit examples 34References 39

This is a note which is still under construction!

1. Introduction

Throughout, we work over an algebraically closed field k of characteristic 0.The concept of K-stability is one of the more precious gifts differential geometers

brought to algebraic geometers. It was first introduced in [Tia97] as a criterion tocharacterise the existence of Kahler-Einstein metrics on Fano manifolds, which is acentral topic in complex geometry. It was defined by looking at an analytic invariant,namely the generalized Futaki invariant, of all possible normal C∗-degenerations ofa Fano manifold X. Later, in [Don01], the notion of K-stability was extended togeneral polarized manifolds (X,L), and defined in completely algebraic terms, whichthen allow arbitrary degenerations.

Date: October 7, 2019.CX is partially supported by a Chern Professorship of the MSRI (NSF No. DMS-1440140) and by

the National Science Fund for Distinguished Young Scholars (NSFC 11425101) ‘Algebraic Geometry’.1

2 CHENYANG XU

In this survey, we will discuss the recent progress on the algebraic study of K-stability of Fano varieties, using the techniques developed in higher dimensional ge-ometry, especially the ideas centered around the minimal model program (MMP).While there is a long history for complex geometers to study Kahler-Einstein metricson Fano varieties, algebraic geometers only started to look at the K-stability questionrelatively recently. One possible reason is that only until the necessary ingredient fromthe minimal model program was settled (see [BCHM10]), such kind of study couldrise above the horizon. Nevertheless, there has been spectacular progress from thealgebro-geometric side in the last a few years, which we aim to survey in this note.

1.1. A short history on K-stability in algebraic geometry. Unlike many otherstability notions, when K-stability was first defined in [Tia97] and later formulatedusing purely algebraic geometric terms in [Don01], it was not immediately clear toalgebraic geometers what this really means and whether it is going to be useful inalgebraic geometry. Comparing it to the geometric invariant theory (GIT) (see [RT07])makes algebraic geometers be more familiar with the concept. However, an apparentlysimilar nature to the asymptotical GIT stability also makes one daunted, since thelatter is notoriously known to be hard to check. Then people found examples of K-stable but not asymptotically GIT varieties, which made the picture even less clear.

However, it is remarkable that in [Oda13a], concepts and technicals from the min-imal model program were first noticed to be closely related to K-stability question.Then after [LX14], it became clearer that to study K-stability of Fano varieties, theMMP would play a prominent role. In fact, as a refinement of Tian’s original per-spective, in [LX14] it was realised that in the study of K-stability of Fano varieties,we should focus on Kawamata log terminal (klt) Fano varieties, since this class ofvarieties is equipped with the necessary compactness. Klt Fano varieties is a themethat has been investigated in higher dimensional geometry for decades, however suchcompactness was not noticed before by birational geometers. It was then foreseeablethat an intensive interplay between two originally disconnected subjects was not farfrom the future.

Nevertheless, it was only until a series of works (see e.g. [BHJ17, Fuj19b, Fuj18,Li18, Li17, BJ17] etc.) which replaced the original definition of K-stability using oneparameter subgroup degenerations by a new one using valuations, a circle of powerfulMMP techniques can be fully embedded in the study of K-stability of Fano varieties.Especially, the introductions of β-invariants in [Fuj18] and normalised volumes in[Li18] were two remarkable conceptual progress.

Since then, there has been a huge progress in the study of algebraic K-stability,including the construction of K-moduli, the local stability problem and the verificationof K-stability for explicit Fano varieties etc..

Remark 1.1. There is a huge body of complex geometry study on this topic. Wedeliberately avoid any detailed discussion on them, except occasionally referring as abackground. For readers who are interested, one could look at [Sze18] for a recentsurvey.

K-STABILITY OF FANO VARIETIES: AN ALGEBRO-GEOMETRIC APPROACH 3

While we try to explain various aspects of the recent progress on the algebro-geometric theory of K-stability of Fano varieties, the choice of the materials is ofcourse based on the author’s knowledge and taste.

1.2. Organisation of the paper. The paper are divided into two parts.In Part 1, we will discuss algebraic geometers’ gradually evolved understanding

of K-stability. As we mentioned, although in [Don01], the formulation was alreadyalgebraic, the more recent equivalent characterisation using valuations turns out tofit much better into higher dimensional geometry. Therefore, we focus on explainingthis new characterisation of K-stability. In Section 2, we will first briefly review thedefinition of K-stability given in [Tia97,Don01], then we will discuss the main resulton special degenerations in [LX14]. In Section 3 and Section 4, we will prove thevaluation criterions of K-(semi)stability, established by [Fuj19b, Li17] and others. InSection 3, we will first introduce Fujita’s β-invariant. It is then easy to deduce fromthe special degeneration theorem discussed in Section 2 that the positivity (resp.non-negativity) of β implies K-stability (resp. K-semistability). We will also discusstwo equivalent definitions of δ(X) for a Fano variety X from [FO18, BJ17], whichis an invariant precisely telling whether a given Fano is K-(semi)stable. To finishthe converse direction that K-stability (resp. K-semistability) implies β > 0 (resp.β ≥ 0), we follow Chi Li’s approach in [Li17] with some improvements from [LX16],which uses the concept introduced by him, called the normalized volume (see [Li18]).What is behind is a local stability theory on klt singularities, which we will brieflydiscuss in Section 4.

In Part 2, we will focus on two central questions of K-stability of Fano varieties,and explain how to use the new interpretation of K-stability discussed in Part 1 toattack them. The first one is the K-moduli Conjecture on the existence of a projectivespace which parametrises all K-polystable Fano varieties with the fixed numericalinvariants. This will be discussed in Section 5. The fact that a large class of Fanovarieties can be parametrised by a projective moduli space is probably an astonishingfact to birational geometers. In Section 6, we will survey some new classes of naturalexamples of K-stable Fano varieties. We verify the K-stability either by studyingthe singularities of the Q-linear system | −KX |Q or by establishing explicit K-modulispaces. Both of these two methods have older roots in works like [Tia87,Tia90,MM93]etc.. Nevertheless, to obtain these new examples, the recent remarkable progress ofour understanding on K-stability is indispensable.

1.3. Conventions. We will follow the terminology in [KM98,Kol13]. A variety X isQ-Fano if it is projective, has klt singularities, and −KX is ample. A pair (X,∆) islog Fano if X is projective, (X,∆) is klt, and −KX −∆ is ample.

Acknowledgement: We are grateful to for many helpful conversations with HaroldBlum, Chi Li, Yuchen Liu, Xiaowei Wang, Ziquan Zhuang. The paper was writtenwhile the author enjoyed the hospitality of the MSRI, which is gratefully acknowl-edged.

4 CHENYANG XU

Part 1. What is K-stability?

Unlike smooth projective varieties which are canonically polarized or Calabi-Yautype, Fano manifolds do not necessarily have a Kahler-Einstein (KE) metric. It hadbeen speculated for a long time that the existence of KE metric on a Fano manifoldshould be equivalent to certain algebraic stability. After searching for a few decades,the concept of K-stability was eventually formulated in [Tia97] and made in algebraicterms [Don01]. Since then the Yau-Tian-Donaldson Conjecture, which predicts theexistence of a KE metric on a Fano manifold X is equivalent to X being K-polystable,prevailed in complex geometry. Eventually, it was solved in [CDS15, Tia15] (see also[???BBJ18]), though the corresponding version for singular Fano varieties is still open,and has attracted lots of recent interests.

In this part, we will discuss the purely algebraic theory of K-stability of Fanovarieties. We started with the original definitions introduced in [Tia97, Don01], byconsidering all C∗-degenerations of a Fano variety X. Then we will explain that,one can apply the minimal model program to put a strong restriction on the alloweddegenerations, namely we only need to consider Q-Fano degenerations. With all thesepreparations, we will introduce a new (but equivalent) definition of K-stability ofFano varieties using valuations over X. This is the central topic in Part 1. Finally,we will briefly discuss a corresponding local notion of K-stability, which yields deepunexpected properties of Kawamata log terminal (klt) singularities.

2. Definition of K-stability by degeneration and minimal modelprogram

2.1. One parameter group degeneration. In this section, we will introduce theoriginal definition of K-stability in [Tia97, Don01]. It was related to the geometricinvariant theory (GIT) stability, or more precisely the asymptotical version. SeeRemark 2.4. In GIT theory, by Hilbert-Mumford criterion, we know to test GITstability, it suffices to computes the weight of the linearisation on all possible oneparameter subgroup degenerations.

Here we first consider an abstract one parameter subgroup degeneration, which iscalled a test configuration.

Definition 2.1. Let X be an n-dimensional normal Q-Gorenstein variety such that−KX is ample. Assume that −rKX is Cartier for some fixed r ∈ N. A test configu-ration of (X,−rKX) consists of

· a variety X tc with a Gm-action,· a Gm-equivariant ample line bundle Ltc → X tc,· a flat Gm-equivariant map π : (X tc,Ltc) → A1, where Gm acts on A1 by

multiplication in the standard way (t, a)→ ta,

such that over A1 \ {0}, there is an isomorphism

φ : (X tc,Ltc)×A1 (A1 \ {0})→ (X,−rKX)× (A1 \ {0}).


For any Q-test configuration, we can define the generalised Futaki invariant. Firstby Riemann-Roch theorem, for sufficiently divisible k ∈ N, we have

dk = dimH0(X,OX(−krKX)) = a0kn + a1k

n−1 +O(kn−2)

for some rational numbers a0 and a1. Let (X tc0 ,Ltc

0 ) be the restriction of (X tc,Ltc)over {0}. Since Gm acts on (X tc

0 ,Ltc⊗k0 ), Gm also acts on H0(X tc0 , kLtc

0 ). We denotethe total weight of this action by wk. By the equivariant Riemann-Roch Theorem,

wk = b0kn+1 + b1k

n +O(kn−1).

So we can expandwkkdk

= F0 + F1k−1 +O(k−2).

Definition 2.2. Under the above notion, the generalised Futaki invariant of the testconfiguration (X tc,Ltc) is defined to be

Fut(X tc,Ltc) = −F1 =a1b0 − a0b1

a20

(1)

We easily see for any a ∈ N, Fut(X tc,Ltc⊗a) = Fut(X tc,Ltc), therefore when Ltc

is only a Q-line bundle, we can still define Fut(X tc,Ltc) := Fut(X tc,Ltc⊗a) for somesufficiently divisible a.

Remark 2.3. Test configurations were first introduced in [Tia97], where the specialfiber was required to be normal, and there the generalised Futaki invariant was definedin analytic terms.

Later in [Don01], any degeneration was allowed (indeed instead of a Fano variety,[Don01] considered test configurations of any given polarised variety), and the gener-alised Futaki invariant was defined in algebraic terms as above. Therefore, in someliterature, the generalised Futaki invariant is also called a Donaldson-Futaki invariant.Since in our current note, we will mostly restrict ourselves to a even smaller class (seeDefinition 2.11) than in Tian’s setting, to avoid any confusion, we will only use theterminology in [Tia97].

In different literatures, the definition of the generalised Futaki invariant may differby a (positive) constant. As we will see, in Definition 2.6 of K-stability, it is only thesign of the generalised Futaki invariant that matters.

Remark 2.4. The most important feature for testing K-stability is that we have tolook at all r. In other words, we need to consider higher and higher re-embeddingsgiven by | − rKX | and all their one parameter group degenerations. This is similarto the asymptotical Chow stability. See [RT07, Section 3] for a detailed discussion,especially for the implications among different notions of stability.

Nevertheless, the directions which were not addressed in [RT07] were more subtle.While it is proved in [Don02] that any polarized manifold (X,L) with a cscK metricand finite automorphism group is asymptotically Chow stable, however, this is knownto not hold for either X has an infinite automorphism group (see [OSY12]) or X issingular (see [Oda12a]).

6 CHENYANG XU

Remark 2.5. The notions of K-stability can be defined for a polarized log pair(X,D;L) (See [Don12, (30)]). In fact, though for the purpose of making the ex-position simpler, we will only discuss Q-Fano varieties, all the K-stability results wediscussed in this survey can be generalised from a Q-Fano variety X to a log Fanopair (X,D), and for most of the time the generalisation is merely a book-keeping.

Definition 2.6. Let X be an n-dimensional normal Q-Gorenstein variety such that−KX is ample, then

(1) X is K-semistable if for any test configuration (X tc,Ltc) of (X,−KX), we haveFut(X tc,Ltc) ≥ 0.

(2) X is K-stable (resp. K-polystable) if for any test configuration (X tc,Ltc) of(X,−KX), we have Fut(X tc,Ltc) ≥ 0, and the equality holds only if (X tc,Ltc)is trivial (resp. only if X tc and X ×A1 are isomorphic) outside a codimension2 locus on X tc.

Example 2.7. Consider a smooth Fano variety X, such that there is an effectivetorus action T (∼= Gr

m) on X. Then for any (integral) coweight Gm → T , we can definea test configuration X tc ∼= X × A1 and Ltc ∼= −KX×A1 with the Gm-action given by

t · (x, a)→ (t(x), t · a).

This kind of test configuration is called a product test configuration. Since if we reversethe action of Gm, the total weight will change the sign, we conclude that if X is K-semistable, then Fut(X tc,Ltc) = 0 for all product test configurations. This conditionwas first introduced in [Fut83].

There is an intersection formula description of the generalised Futaki invariants (see[Wan12,Oda13b]) for any given test configuration (X tc,Ltc).

Lemma 2.8 ([Wan12, Oda13b]). If glue (X tc,Ltc) with (X × (P1 \ {0}), p∗1(−rKX))over A1 \ {0} by φ to get a proper family (X tc, Ltc) over P1, we have the followingequality:

Fut(X tc,Ltc) =1

2(n+ 1)(−KX)n

(n(

1

rLtc)n+1 + (n+ 1)KX tc/P1 · (

1

rLtc)n

)(2)

Proof. See e.g. [LX14, Page 224-225]. �

2.2. MMP on family of Fano varieties. In this section, we will start to uncoverthe connection between K-stability and MMP.

The following theorem proved in [Oda13a] was probably the first intersection ofK-stability theory and the minimal model program in algebraic geometry.

Theorem 2.9 ([Oda13a]). Let X be an n-dimensional normal Q-Gorenstein varietysuch that −KX is ample. If X is K-semistable, then X has (at worst) klt singularities.

The proof of the above theorem is a combination of (2) and a MMP constructioncalled the lc modification whose existence relies on the relative minimal model program([OX12]).


It was probably not a big surprise that the notion of K-stability should have somerestriction on the singularities, however, it was really a remarkable observation thatthe right category of singularities should be the one from the minimal model pro-gram theory. From now on, we will only consider the K-stability problem for Q-Fanovarieties.

Remark 2.10. It is natural to ask whether we can restrict ourselves to an even smallercategory of singularities than klt singularities. The answer is likely to be negative (seeSection 4). However, the global invariant of the volume (−KX)n for the K-semistableFano variety will post more restrictive conditions on the possible local singularities(see Theorem 5.5).

Now we introduce a smaller class of test configurations called special test configu-ration, which will play a crucial role in our study.

Definition 2.11 (Special test configurations). A test configuration (X tc,Ltc) of (X,−KX)is called a special test configuration if Ltc ∼Q −rKX tc and the special fiber X0 is aQ-Fano variety. By inversion of adjunction, this is equivalent to saying X tc is Q-Gorenstein and −KX tc is ample and (X tc, X0) is plt.

The next theorem shows the difference in the definition of K-stability for Fanovarieties in [Tia97] and in [Don01] (see Remark 2.3) does not really play any role.

Theorem 2.12 ([LX14]). Let (X tc,Ltc) → A1 be a test configuration (X,−rKX),then there exists a special test configuration (X st,Lst) → A1 which is birational to(X tc,Ltc)×A1,z→zd A1 over A1, such that Fut(X st,Lst) ≤ d · Fut(X tc,Ltc).

Moreover, the equality holds if and only if the birational map

(X st,Lst) 99K (X tc,Ltc)×A1,z→zd A1

is an isomorphism outside codimension 2.

Sketch of the proof. Started from any test configuration X , we will use the the minimalmodel program to modify X such that at the end we obtain a special test configu-ration X s, and during the process the generalized Futaki invariants decrease. Themodification consists of a few steps.Step 0: If we replace X by the normalisation n : X n → X ×z→zd A1, then we have

d · Fut(X ,L) ≥ Fut(X n,Ln := n∗L)

with the equality holds if and only if X n → X ×z→zd A1 is an isomorphism outsidecodimension at least 2. This can be seen by directly applying the intersection formula(2).

Key idea 2.13. After Step 0, we can always assume the special fiber is reduced.The following steps will all involve minimal model program constructions. The mainobservation is the following calculation: denote by L the polarization on X , such thatL|Xt ∼ −rKXt for t 6= 0, so we can write L+ rKXt ∼ E which is a divisor supported

8 CHENYANG XU

over 0. Then let t > 0 such that Lt = L + tE is still ample, then applying (2), wehave

d

dtFut(X ,Lt) =

n

2(−KX)n(1

rLt)n−1 · (1

rE)2 ≤ 0 . (3)

Step 1: From X n, we can construct the log canonical modification of f lc : X lc →(X , Xn

0 ) (see [Kol13, Theorem 1.32]), where Xn0 is the special fiber. By a suitable

base change, we can assume that the special fiber X lc0 of X lc is also reduced. Let F

be the reduced exceptional divisor. Then by the definition of the lc modification,

KX lc + f lc∗ (Xn

0 ) + F = KX lc +X lc0 ∼ KX lc

is ample over X n. So E = 1rf lc∗Ln + KX lc is ample over Xn and Llc

t := 1rf lc∗Ln + tE

is ample for some t� 0. Therefore, (3) implies Fut(X lc,Llct ) ≤ Fut(X n,Ln), and the

equality holds if and only if (X n, Xn0 ) is log canonical.

Step 2: Replacing Llc by its power, we can assume that H := Llc−KX lc is ample. Thenwe run KX lc-MMP with the scaling of H, which is automatically Gm-equivariant ineach step. Thus we get a sequence of numbers t0 = 1 > t1 ≥ t2 ≥ ... ≥ tm−1 > tm = 1

r,

with a sequence of models

X lc = Y0 99K Y1 99K · · · 99K Ym−1

such that if we let Hi be the pushforward of H on Yi, KYi + sHi is nef for anys ∈ [ti−1, ti]. Moreover, we have KYm−1 + tmHm−1 ∼Q 0. Thus

KYm−1 + tm−1Hm−1 ∼Q (tm−1 − tm)Hm−1

is big and nef. Let Xan be the ample model of Hm−1 and Lan the ample Q-divisorinduced by Hm−1. If we use (2) to define the generalized Futaki invariant when L isa big and nef line Q-bundle, then (3) implies

Fut(X lc,Llc) = Fut(Y0, KY0 +H)

≥ Fut(Y0, KY0 + t1H0)

= Fut(Y1, KY1 + t1H1)

≥ · · ·= Fut(Ym−1, KY1 + tm−1Hm−1)

= Fut(X an,Lan),

and the equality holds if and only if X lc = X an. We note that since −KX an is propor-tional to Lan, we indeed have

Fut(X an,Lan) =−1

2(n+ 1)(−KX)n(KX an/P1

)n+1. (4)

Step 3: In the last step, by a tie-breaking argument, we can show after a possible basechange, by running a suitable minimal model program, we can construct a model such


that (X s, Xs0) is plt and the discrepancy of Xs

0 with respect to (X an, Xan0 ) is −1. By

an intersection number calculation, we have

−1

2(n+ 1)(−KX)n(KX an/P1

)n+1 ≥ −1

2(n+ 1)(−KX)n(KX s/P1

)n+1

and the equality holds if and only if X an = X s. �

2.14. However, even simple Fano varieties could have many special degenerations.An easy example is in Example 2.15. One could think of the stack X of all klt Fanovarieties with fixed numerical invariants similar to the stack Shf of all coherent sheaveswith the fixed Hilbert polynomial.

Example 2.15. The family (x2 + y2 + z2 + tw2 = 0) ⊂ P2 × A1, gives a specialdegeneration of P1 × P1 to the cone over a conic.

We indeed have a characterization on which Q-Fano varieties only have trivial spe-cialization.

Example 2.16. A Q-Fano does not have any nontrivial special test configuration ifand only if X is weakly exceptional, that is

α(X) := inf{lct(X,D)| D ∼Q −KX} ≥ 1.

However, it is known for a Q-Fano variety X

α(X) >dim(X)

dim(X) + 1

is sufficient to imply that X is K-stable.So it is clear that Theorem 2.12 alone is not strong enough to verify the K-stability

of a general Fano variety.

3. Two invariant: β and δ

In this section, we will discuss the characterization of K-stability using valuations.The viewpoint of using valuations to reinterpret a one parameter group degenerationwas probably first introduced in [BHJ17]. Then in the remarkable work [Fuj18], theinvariant β(D), defined over all divisorial valuations D, was introduced. This invariantand its variants will be the main topic for our discussions. Soon in [Fuj19b,Li17], theequivalence of the definition of K-stability using valuations with the original one wasaddressed, and some further improvements were contained in [BJ17,BX19].

3.1. K-stability with divisorial valuations. Let X be a normal variety such thatKX is Q-Cartier, we define the log discrepancy for any divisor E over X

AX(E) = a(E,X) + 1

where a(E,X) is the discrepancy (see [KM98, Definition 2.25]). So X being klt isequivalent to saying that AX is positive for any E.

10 CHENYANG XU

3.1.1. β-invariant. Let X be an n-dimensional Q-Fano variety and E a divisor overX. The β-invariant βX(E) is first defined in [Fuj16,Fuj18] as follows,

βX(E) = (−KX)nAX(E)−∫ ∞

0

vol(µ∗(−KX)− tE) dt.

where E arises a prime divisor on a proper normal model µ : Y → X. In the rest ofthis survey, β-invariant and its variant will play a prominent role. Its importance canbe seen from the following theorem.

Theorem 3.1 (The valuation criterion for K-(semi)stability, [Fuj19b, Li17]). A Q-Fano variety X is

(1) K-semistable if and only if βX(E) ≥ 0 for all divisors E over X;(2) ([BX19]) K-stable if and only if βX(E) > 0 for all divisors E over X.

The rest of the section will be devoted to prove one direction of Theorem 3.1. Theproof of another direction will be completed in Section 4, after we put it in a broadercontext (see Theorem 4.5).

Consider a special test configuration X , and denote by its special fiber X0. Thenthe restriction of ordX0 on K(X ) = K(X × A1) to K(X) yields a valuation v. It iseasy to see when X is a trivial test configuration, then v is trivial, we also have

Lemma 3.2 ([BHJ17, Lemma 4.1]). If X is not a trivial test configuration, then v isa divisorial valuation, i.e. v = c · ordE for some c ∈ Q>0 and E over X.

Proof. Since tr.deg(K(X )/K(X)) = 1, by Abhyankar’s inequality, we know that

tr.deg(K(v)) + rankQ(v)

≥ tr.deg(K(ordX0)) + rankQ(ordX0)− 1

= dim(X).

This implies v is an Abhyankar valuation, whose value group is nontrivial and con-tained in Z, which implies the assertion. �

Lemma 3.3. For a nontrivial special test configuration X of a Q-Fano variety X,denote by v the valuation defined as above. Then we have

2(−KX)n · Fut(X ) = βX(v) := c · βX(ordE).

Proof. Consider a section s ∈ H0(−mKX) for m sufficiently divisible. Let Ds be the

closure of (s)× A1 on X × A1. Fix a common log resolution X of X and X × A1

X

X × A1 X .

ψ′ψ

φ′


Denote by X0 the special fiber of X . So

ψ∗(Ds) = Ds + (ordX0(s))X0 + E ∈ H0(mKX −m · a(X0, X × A1)X0) (5)

where Ds and X0 are the birational transforms of Ds and X0 on X and Supp(E) doesnot contain the birational transform of X × {0} and X0. We know

a(X0, X × A1) = A(X0, X × A1)− 1 = c · AX(E) and ordX0(s) = c · ordE(s),

so if we denote by D′s = ψ′∗(Ds), the right hand side of (5) becomes

D′s + (c · ordE(s))X0 ∈ H0(mKX −mc · AX(E)X0).

We choose a basis of s1, ..., sr be a basis H0(−mKX) which is compatible with thefiltration, that is, if we let ji = dimF i(H0(−mKX)), then {s1, ..., sji} form a basisof F i(H0(−mKX)). The above computation says the total weight wm of Gm onH0(−mKX ) is

wm =∑i

c · (ji−1 − ji)i− rmc,

and if we divide by mn+1

n!and let m→∞, we have

limm→∞

wmmn+1/n!

= c ·∫ ∞

0

vol(−KX − t · E)dt− cAX(E)(−KX)n = −cβX(E).

On the other hand, a simple Riemann-Roch calculation implies

limm→∞

wmmn+1/n!

=1

n+ 1(−KX/P1)n+1 = −2(−KX)n · Fut(X )

by (4). �

The above interpretation of the total weight using the filtration was first introducedin [BHJ17]. A direct consequence of Lemma 3.3 is one direction of Theorem 3.1.

Corollary 3.4. If βX(E) ≥ 0 (resp. βX(E) > 0) for any divisor E over X, then Xis K-semistable (resp. K-stable).

3.5. We still need to prove that the K-(semi)-stability implies the (semi)-positivity ofβ. Note that the main reason we can prove Corollary 3.4 is that instead of consideringall test configurations, which are not easy to connect to β-invariants, Theorem 2.12allows us to only look at special test configurations. The difficulty for the reversedirection of course lies on the fact that not every divisor E over X arises from aspecial test configuration. In [Fuj19c] and [Li17], using a limit process, it was shownthat β ≥ 0 is implied by Ding-semistability (introduced in [Ber16]). In the definition,Ding-semistability is stronger than K-semistability, however, they coincides on specialtest configurations, and it was shown in [BBJ18] and [Fuj19c] that a similar processas in Theorem 2.12 can be applied to Ding-semistability to prove that to test Ding-semistability we only need to test on special test configurations, therefore it is indeedthe same as K-semistability.

12 CHENYANG XU

In Theorem 4.5, by relating it with Chi Li’s definition of normalized volume on acone singularity, and applying the ‘regularization’ argument developed in [LX16], wewill show that to prove β(E) ≥ 0 for all divisors E, it suffices to look at those inducedby special test configurations. This allows us to skip discussing Ding-semistability fora general degeneration, and only stick to special degeneration where everything is thesame.

Example 3.6 (Boundedness of volume, [Fuj18]). The first remarkable applicationof β-invariant is given in [Fuj18], which says that the volume of an n-dimensionalK-semistable Q-Fano variety is at most (n+ 1)n.

Let X be a K-semistable Q-Fano variety. Pick up a smooth point x ∈ X, and blowup x, we get µ : Y → X with an exceptional divisor E. Since

0→ mkx ⊗O(−mKX)→ O(−mKX)→ O(−mKX)⊗ (OX/mk

x)→ 0,

we know that

h0(OY (µ∗(−mKX)− kE)) ≥ h0(O(−mKX))− h0(O(−mKX)⊗ (OX/mkx).

Thus we have

0 ≤ β(E) = n · (−KX)n −∫ ∞

0

vol(µ∗(−KX)− tE)dt

≤ n · (−KX)n −∫ ((−KX)n)

1n

0

((−KX)n − tn),

which immediately implies (−KX)n ≤ (n+ 1)n.If X is singular, and we choose x to be a singularity, then an even stronger restriction

on the volume in terms of the local volume of the singularity is obtained in [Liu18].See Theorem 5.5.

3.1.2. Stability threshold. It is also natural to consider a variant,

δX(E) =defn(−KX)n · AX(E)∫∞

0vol(−KX − tE)dt

.

It is easy to see δ(v) = δ(λ · v) for any λ > 0.

Definition 3.7 ([FO18,BJ17]). For a Q-Fano variety, we define the stability thresholdof (X,∆) to be

δ(X) := infE

δX(E),

where the infumum through all divisors E over X.

In fact, the δ-invariant δ(X) was first defined in [FO18] in the following way: LetX be a Q-Fano variety. Given a sufficiently divisible m ∈ N, we say D ∼Q −KX

is an m-basis type Q-divisor of −KX − ∆ if there exists a basis {s1, . . . , sNm} ofH0(X,OX(m(−KX)) such that

D =1

mNm

({s1 = 0}+ · · ·+ {sNm = 0}

).


We set

δm(X,∆) := min{lct(X;D) |D ∼Q −KX is m-basis type}.The original definition of δ(X) in [FO18] is lim sup

m→∞δm(X). Then it is shown in [BJ17,

Theorem C] that the limit exists and

limm→∞

δm(X) = infE

δX(E). (6)

Theorem 3.1(1) can be translated into the following theorem

Theorem 3.8 ([FO18, BJ17]). A Q-Fano variety X is K-semistable if and only ifδ(X) ≥ 1.

3.9 (Uniform K-stability). We call X is uniformly K-stable if δ(X) > 1. This conceptwas first introduced in [BHJ17,Der16b]. The equivalence between the original defini-tion and the current one follows from the work of [Fuj19b,BJ17]. Theorem 3.1(2) saysuniform K-stability implies K-stability. However, the converse is much subtler, as wedo not know the infimum is attained by a divisorial valuation. This is Conjecture 3.16in the case δ = 1.

A divisor Q-divisor D ∼Q −KX such that (X,D) is log canonical is called a comple-ment. This is a well studied topic in birational geometry. Combining with the descrip-tion using the basis type divisor (see (6)), Theorem 3.8 relates the K-semistability toa special type of complements.

Even more crucially, the importance of δ will only fully appear after we enlarge thedefinition to all valuations. See next section.

Remark 3.10 (Twisted Kahler-Einstein metric). It turns out that the invariant δ(X)also has an older origin from the differential geometry in terms of twisted Kahler-Einstein metrics: For a Fano manifold X, in [Tia92] and then in [Rub08, Rub09], aninvariant called the greatest Ricci lower bound of X was first defined and studied as

sup{t ∈ [0, 1] | a Kahler metric ω ∈ c1(X) such that Ric(ω) > tω}.Later this invariant was further studied in [Sze11, Li11, SW16] etc.. It is shown in

[BBJ18,CRZ18] that for a Fano manifold X, the greatest Ricci lower bound is equalto min{1, δ(X)}.

3.2. General valuations. In this section, we will first extend the above definitionsfrom divisors to all valuations over X. The key point is that this enlargement allowsus to study our minimising question in a space with certain compactness.

Let X be a variety. A valuation on X will mean a valuation v : K(X)× → R thatis trivial on the ground field and has center cX(v) on X. We denote by ValX the setof valuations on X, equipped with the weak topology.

To any valuation v ∈ ValX and t ∈ R, there is an associated valution ideal sheafat(v): For an affine open subset U ⊆ X, at(v)(U) = {f ∈ OX(U) | v(f) ≥ t} ifcX(v) ∈ U and at(v)(U) = OX(U) otherwise.

14 CHENYANG XU

Example 3.11 (Divisors over X). Let X be a variety and π : Y → X be a properbirational morphism, with Y normal. A prime divisor E ⊂ Y defines a valuationordE : K(X)× → Z given by order of vanishing at E. Note that cX(ordE) is thegeneric point of π(E) and, assuming X is normal, ap(v) = π∗OX(−pE).

Example 3.12 (Quasi-monomial valuations). Denote Z → X a log resolution withsimple normal crossing divisors E1,...., Er on Z. Denote by α = (α1,...., αr) ∈ Rr

≥0.Assume

⋂ri=1Ei 6= ∅; and there exists a component C ⊂ ∩Ei, such that around the

generic point η of C, Ei is given by the equation zi in OZ,η(C). We define a valuationvα to be

vα(f) = min{∑

αiβi| cβ(η) 6= 0},

and all such valuations are called quasi-monomial valuations. It is precisely the val-uations satisfying the equality case in Abhyankar’s inequality, therefore, it is anotherdescription of Abhyankar valuation.

We call the dimension of the Q-vector space spanned by {α1,...., αr} the rationalrank of vα, and one can show vα is a rescaling of a divisorial valuation if and only ifr = 1. For fixed C ⊂ (Z,E) as above, the valuations vα for all α gives a simplicialcone which is a natural subspace in ValX .

Example 3.13. Given a valuation v, and a simple normal crossing (but possiblynon-proper) model (Z,E =

∑Ei) over X such that the center of v on Z is non-

empty, we can define a valuation vα = ρ(Z,E)(v), where the corresponding componentαi defined to be v(zi). Started from a simple normal crossing model, by successivelyblowing up the center of v and (possibly shrinking), we get a sequence of modelsφi : Zi → Zi−1 where Z0 = X such that the center of v on Zi is not empty. DefineEi = φ−1

i∗ (Ei−1) + Ex(φi). Denote by vα,i = ρZi,Ei(v), then v = limi→∞(vα,i).

Definition 3.14 (Log discrepancy function on Val(X), see [JM12,BdFFU15]). WhenX is klt, the log discrepancy function AX can be extended to a function

AX : Val(X)→ (0,+∞]

in the following way: we have already defined AX(E) for a divisorial valuation. For aquasi-monomial valuation as in Example 3.12, we define AX(vα) =

∑i αiAX(E). And

for a general valuation v, we define AX(v) = supY,E AX(ρY,E(v)).

For any t ∈ R≥0, we can also define a volume function:

vol(−KX − tv) = limk→∞

dimH0(OX(−kKX)⊗ atk)

kn/n!,

where atk is the associated ideal sheaf of v.Then for any valuation v with AX,∆(v) < +∞ we can similarly define,

βX(v) := (−KX)n · AX(v)−∫ ∞

0

vol(−KX − tv)dt.


and

δX(v) =defn(−KX)n · AX(v)∫∞

0vol(−KX − tv)dt

.

The advantage of extending the definition to all valuations can be seen by the nexttheorem.

Theorem 3.15. We have the following two facts:

(1) [BJ17]For a Q-Fano variety X,

δ(X) := infv

δX(v),

where the infimum runs through all valuations v with AX(v) < +∞.(2) [BLX19] When δ(X) ≤ 1, then the infimum is attained by a quasi-monomial

valuation.

One crucial conjecture remaining is the following.

Conjecture 3.16. Let X be a Q-Fano variety. If δ(X) ≤ 1, then there is a specialtest configuration X with the induced divisorial valuation D (see Lemma 3.2) suchthat δ(X) = δX(D) = δ(X0), where X0 is the degeneration.

Remark 3.17 (Yau-Tian-Donaldson Conjecture). Consider the case δ(X) = 1. Inthis case, Conjecture 3.16 says that a Fano variety is K-stable if and only if it isuniformly K-stable. We can also formulate an equivariant version of the conjecturewhich predicts that a Fano variety K-polystable if and only if it is T -uniformly K-stable (see ?? for the definition). It has been proved recently that any T -uniformlyK-stable Fano variety has a Kahler-Einstein metric ([BBJ18,LTW19,Li19]). ThereforeConjecture 3.16 (resp. its equivariant version) would imply that any K-stable (resp.K-polystable) Q-Fano variety admits a Kahler-Einstein metric, hence provide a newproof of Yau-Tian-Donaldson Conjecture which works even for singular Fano varieties.

3.18 (Filtration). A useful further generalisation is considering graded multiplicativedecreasing filtrations F t (t ∈ R) where R is the section ring

R = R(X) =⊕m∈N

Rm =⊕m∈N

H0(X,OX(−mKX))

satisfying FλRm = ∩λ′<λFλ′Rm (e.g. FλRm = F dλeRm) for all λ, F0Rm = Rm and

FλRm = 0 for λ � 0. All the filtrations we consider are linearly bounded, that is tosay there exists C > 0 so that FCmRm = 0 for all m ∈ N.

A valuation induces a filtration

F tvRm := {s ∈ H0(−mKX) | v(s) ≥ t}, ∀m ∈ Z≥0

and it is linearly bounded if AX(v) < +∞.We can define the base ideal sequence a•(F) for a given filtration as following: ap is

the base ideal of the linear system FpH0(−mKX) for sufficiently divisible m. Then

16 CHENYANG XU

we can define

βX(F) := lct(X; a•)(−KX)n −∫ ∞

0

vol(F t)dt.

In [BL18], it was shown that if X is K-semistable, then βX(F t) ≥ 0 for any linearbounded multiplicative decreasing filtrations F t (t ∈ R). Moreover, δ can be computedas

δ(X) = inflct(X; a•)(−KX)n∫∞

0vol(F t)dt

, (7)

where the infimum runs through all filtrations as above.

4. Local stability of klt singularities

In this section, we will discuss a (conjectural) local K-stability picture. This isestablished using Chi Li’s definition on normalized volume. For read who wants toknow more background on this, see the survey paper [LLX18]. In Secion 4.1, wegive the definition of normalised volumes as well as a very quick sketch of the StableDegeneration Conjecture 4.2. Then in Section 4.2, we will discussion its connectionto the global question, which is only a small (known) part of the Stable DegenerationConjecture. See Theorem 4.5, which implies Theorem 3.1.

4.1. Normalized volume. We discuss an invariant called the normalised volumewhich is similar to the δ-invariant, but defined in a local setting. It was first introducedin [Li18] and closely related to the K-semistability of log Fano pairs.

Let (Y,Γ) be an n-dimensional klt pair and x ∈ Y a closed point. The non-archimedean link of Y at x is defined as

ValY,x := {v ∈ ValX | cY (v) = {x} } ⊂ ValY .

Definition 4.1 ([Li18]). The normalized volume function

vol(Y,Γ),x : ValY,x → (0,+∞]

is defined by

vol(Y,Γ),x(v) =

{A(Y,Γ)(v)n · vol(v) if A(Y,Γ)(v) < +∞;

+∞ if A(Y,Γ)(v) = +∞.

Here, AY,Γ(v) is the log discrepancy of v and vol(v) is the volume of v. The volumeof the singularity (x ∈ (Y,Γ)) is defined as

vol(x, Y,Γ) := infv∈ValY,x

vol(Y,Γ),x(v).

The previous infimum is a minimum by the main result in [Blu18].

In the recent study of the normalized volume function, the guiding question is thefollowing (see [Li18, Conjecture 7.1] and [LX19a, Conjecture 1.2]).


Conjecture 4.2 (Stable Degeneration Conjecture). Given any arbitrary klt singular-ity x ∈ (Y = Spec(R),Γ). The minimizer v is unique up to a rescaling. Moreover,it is quasi-monomial with a finitely generated associated graded ring, and the induceddegeneration

(Y0 = Spec(R0),Γ0, ξv)

is K-semistable.

Here (Y0,Γ0, ξv) being K-semistable is in the sense of a Fano cone singularity. See[CS18] or [LLX18, Section 2.5] for the definition of a Fano cone singularity and itsK-semistability.

We call a valuation in Valx,X to be a K-semistable valuation if v is quasi-monomialwith a finitely generated associated graded ring, and the induced degeneration

(Y0 = Spec(R0),Γ0, ξv)

is K-semistable. It was proved in [Li17,LX16,LX19a] that any K-semistable valuationmust be a minimizer of the normalized volume function. So we can rephrase the mainpart of Conjecture 4.2 to be the following.

Conjecture 4.3. Any klt singularity x ∈ (Y,Γ) always has a unique K-semistablevaluation up to rescaling.

There has been a lot of progress on the stable degeneration conjecture (see [LLX18]).In particular, it is shown

Theorem 4.4. For any klt singularity x ∈ (Y,Γ),

(1) [Xu19] The infimum of the normalized function vol(Y,Γ),x(v) is attained by aquasi-monomial minimizer;

(2) [Li17,LX16] if the minimizer is divisorial, then it is a K-semistable valuation;and

(3) [LX16,LX19a] any K-semistable valuation, if exists, is unique (up to rescaling).

This answers the K-semistable part of [DS17, Conjecture 3.22].

4.2. Cone construction. For a Q-Fano variety X, we define the cone

Y = Spec⊕m∈Z≥0

Rm, where Rm = H0(−mrKX) for some r ∈ N.

In [Li17], by considering the normalized volume function of the vertex, this con-struction was related to the study of K-stability question. At first sight, using thenormalized volume function to study the K-stability of Q-Fano varieties may seemindirect. However, working on Y encodes all information of the anti-canonical ring,and we believe this is indeed a useful way to think about X. A number of new resultsare yielded through this approach, e.g. Theorem 4.5 and 4.12. We will now explainthe main ideas.

18 CHENYANG XU

Blowing up the vertex x of the cone Y , we get an exceptional divisor X∞ ∼= X.Since the degeneration induced by X∞ is just Y itself, one established part of the sta-ble degeneration conjecture implies the following statement, which contains Theorem3.1(1).

Theorem 4.5 ([Li17,LL19,LX16]). We have the following equivalence

(X is K-semistable)⇐⇒ (v0 := ordX∞ is a minimiser)⇐⇒ (βX(E) ≥ 0,∀E/X).

We already see (β(E) > 0,∀E) =⇒ (X is K-semistable) (see Corollary 3.4). Nowwe discuss the other implications Proposition 4.6 and Proposition 4.11.

4.2.1. v0 minimising implies β ≥ 0.

Proposition 4.6 ([Li17]). (v0 is a minimiser) =⇒ (βX(E) ≥ 0,∀E).

Proof. Let E be a divisor over X that arises on a proper normal model µ : Z → X.Following [Li17,LX16], E gives rise to a ray of valuations

{vt | t ∈ [0,∞) ⊂ ValY,x}. (8)

Since the blowup of Y at 0 is canonically isomorphic to the total space of the linebundle OX(−L), there is a proper birational map from ZL−1 → Y , where ZL−1 denotethe total space of µ∗OX(−L). Now,

v0 = ordX∞ and v∞ = ordE∞ ,

where E∞ denotes the pullback of E under the map ZL−1 → Y and X∞ denote thezero section of ZL−1 . Furthermore, vt is defined to be the quasimonomial valuationwith weights (1, t) along X∞ and E∞. Denote by a = AX(E).

We have

AY (vt) = AY (ordX∞) + tAY (ordE∞) = 1/r + at.

For t > 0,

ap(vt) = ⊕m≥0FE(p−m)/tRm ⊆ R and ap(v0) = ⊕

m≥pRm ⊆ R.

When k ∈ N, v 1k

= 1kordEk

, where Ek is a divisor over X.

Since v0 is a minimizer, d vol(vt)dt

∣∣t=0+

≥ 0, which implies β(E) ≥ 0 by Lemma4.7. �

Lemma 4.7 (C. Li’s derivative formula).

d

dtvol(vt)

∣∣∣∣t=0+

= (n+ 1)βX(E). (9)

Proof. Let at,p := ap(vt). So at,p contains ⊕m≥pRm. We claim the following hold:

mult(at,•) = rn(−KX)n − (n+ 1)∫∞

0vol(FER(x)) t dx

(1+tx)n+2 . (10)


This follows from the argument in [Li17, (18)-(25)]. For the reader’s convenience,we give a brief proof. For t ∈ R>0, we have

mult(at,•) = limp→∞

(n+ 1)!

pn+1dimk(R/at,p)

= limp→∞

(n+ 1)!

pn+1

∞∑m=0

dimk(Rm/F (p−m)/tE Rm)

= limp→∞

(n+ 1)!

pn+1

p∑m=0

(dimk Rm − dimk F (p−m)/t

E Rm

)= vol(L)− lim

p→∞

(n+ 1)!

pn+1

p∑m=0

dimk F (p−m)/tE Rm.

Then we can identify the limit of the summation with the integral in (10), where achange of the variable is needed (see [Li17, (25)], where one chooses c1 = 0, α = β = 1

t).

Computing the derivative, we have

d

dtvol(vt)

∣∣∣t=0+

= a(n+ 1)

(1

r

)nmult(a0,•) +

1

rn+1

d

dt(mult(at,•))

∣∣∣t=0+

.

From (10), we know mult(a0,•) = rn(−KX −∆)n and

d

dt(mult(at,•))

∣∣∣t=0+

= −(n+ 1)

∫ ∞0

(vol(FER(x))

(1− tx(n+ 1)

(1 + tx)n+3

))∣∣∣t=0+

dx.

Since the latter simplifies to −(n+ 1)∫∞

0vol(FER(x))dx, thus

d

dt(mult(at,•))

∣∣∣t=0+

= (n+ 1)βX(E).

�

4.2.2. K-semistable implies v0 minimising. For the last implication that v0 is a min-imizer if X is K-semistable, we will take the proof from [LX16]. There we try to‘regularize’ the minimizing valuation to conclude that we only need to compare thenormalised volume v0 := ordX∞ and those arisen from a special test configuration.

Recall that a prime divisor E over a klt point x ∈ Y is called a Kollar componentif there is a birational morphism µ : Y ′ → Y , such that E = Ex(µ), (Y ′, E) is plt and−KY ′ −E is ample over Y . Such a morphism µ is called a plt blow-up, and it can beconsidered as a birational version of special degenerations.

Proposition 4.8 ([LX16, Proposition 4.4]). We have vol(x, Y ) = infE volY,x(E) whereE run through all Gm-equivariant Kollar components.

Proof. By [Liu18, Theorem 7], we know that

vol(x, Y ) = infa

mult(a) · lct(Y, a)n, (11)

20 CHENYANG XU

where the right hand side runs through all mx-primary ideals. For any a, if we considerthe initial degeneration bk of ak, then b• forms an ideal sequence, and mult(a) =

mult(b•) = lim mult(bk)kn

, and lct(Y ; a) ≥ 1klct(Y ; bk). Therefore,

mult(a) · lct(Y, a) ≤ infk

mult(bk) · lctn(Y, bk).

Therefore, we can restrict the right hand side of (11) by only running through allGm-equivariant mx-primary ideals.

Started from any Gm-equivariant mx-primary ideal a, let c = lct(Y ; a). Let Y ′ → Ybe a dlt modification of (X, c · a) with reduced exceptional divisor Γ. We can mimicthe last step in the proof of Theorem 2.12, to show that there is a Gm-equivariantKollar component E (with the model YE → Y ), such that a(E,X, c · a) = −1, and

vol(E) = ((−KYE − E)|E)n−1 ≤ ((−KY ′ − Γ)|Γ)n−1 ≤ mult(a) · lct(Y, a)n.

(see [LX16, Section 3.1]). �

4.9. Next we will explain the relationship between Kollar components over x ∈ Y andspecial degenerations, following [Li17,LWX18b], we will show they are essentially thesame up to some normalisation.

The following general construction is from [BHJ17, §2]:Let v be a valuation onX which yields a filtration Fv onR(X) =

⊕m∈Z≥0

H0(−mKX).

Then we can construct the Rees algebra of F

Rees(F) :=⊕m∈N

⊕p∈Z

t−pFpRm ⊆ R[t, t−1].

as a k[t]-algebra.The associated graded ring of F is

grFR :=⊕m∈N

⊕p∈Z

grpFRm, where grpFRm =FpRm

Fp+1Rm

.

Note that

Rees(F)⊗k[t] k[t, t−1] ' R[t, t−1] andRees(F)

tRees(F)' grFR. (12)

Hence, Rees(F) is said to give a degeneration of R to the associated graded ring of F .We say that an N-filtration F is finitely generated if Rees(F) is a finitely generatedk[t]-algebra. Equivalently, ⊕

(m,p)∈Z≥0×Z

FpRm

is finitely generated over k.If F is finitely generated, we set X := ProjA1 (Rees(F)). By (12),

XA1\{0} ' X × (A1 \ {0}) and X0 ' Proj(grFR).


If we start with a nontrivial special degeneration X of X, then we can take the in-duced valuation as in Lemma 3.2, the Rees algebra construction gives

⊕m≥0H

0(−mKX )as a k[t]-algebra. Since there are two gradings given by m and p, we indeed have aT = (Gm)2-actions on

⊕m≥0H

0(−mKX ). The relative cone structure correspondsto the action by the coweight (1, 0), and the Gm-action from the test configurationcorresponds to the coweight (0, 1).

Now the divisorial valuation Ek given by k · v 1k

(see the proof of Proposition 4.6) is

given by

Projk(grFvR) = Proj

⊕d∈N

(⊕

mk+p=d

grpFvRm),

and the different divisor ∆Ekis given by the orbifold divisor for the cone. Therefore,

(Ek,∆Ek) is log Fano for any k ≥ 0.

Conversely, if we start with a Gm-equivariant Kollar component E over x ∈ Y , wecan also form the the filtration by ordE, and hence obtain the Rees algebra Rees(FE),where the associated graded ring is

⊕m∈N

⊕p∈Z grpERm. Let a be the vanishing order

of X∞ along E, which is a rational number rd, since X∞ is only Q-Cartier. Thus E

corresponds to the coweight vector ( rd, 1). Taking a base change A1 → A1 by z → zd

simply corresponds to change (0, 1) to (0, d),

To summarise, the above discussion gives the following correspondence.

Proposition 4.10. There is a one to one correspondence between{Special degenerations of X

}/ ∼ ←→

{Gm-equivariant Kollarcomponents E 6= X∞

}/ ∼,

where two special degeneration Xi are equivalent if and only if after base changesz → zdi they become isomorphic, and two Gm-equivariant Kollar components areequivalent if they are on the same ray vt for some vt ∈ Valx,Y (t ∈ [0,+∞)).

Proposition 4.11. (X is K-semistable) =⇒ (v0 is a minimiser).

Proof. Assume X is K-semistable. To show v0 = ordX∞ is a minimizer of volY,x, byProposition 4.8, it suffices to show that for any Kollar component E over x ∈ Y , we

have vol(E) ≥ vol(v0). By Proposition 4.10, we know that E will induces a ray vtcontaining a · ordE for some a ∈ Q, which corresponds to a special test configurationX . We can rescale t such that vt is defined to be the quasimonomial valuation withweights (1, t) along X∞ and c · E∞, where c · E∞ is the pull back of the divisorialvaluation induced by X (see Lemma 3.2).

Since X is K-semistable, by Lemma 3.3 and 4.7, we know that

d

dtvol(vt)|t=0 = (n+ 1) · Fut(X ) ≥ 0

Since vol(vt) is a convex function on t by (10) (or see [LX19a, Section 3.2]),

vol(ordE) = vol(a · ordE) ≥ vol(v0).

22 CHENYANG XU

�

4.2.3. K-stability and β. To finish the proof of Theorem 3.1(2), it suffices to show ifX is K-stable, and E is a divisor such that βX(E) = 0, then E induces a special testconfiguration. This is a special case of Conjecture 3.16. By the discussion in 4.9, acrucial thing is to prove that the double graded ring⊕

m,n∈Z≥0

H0(−mµ∗KX − nE)

is finitely generated. Such a divisor is called dreamy in [Fuj19c, Definition 1.2(2)].The following argument appearing in [BX19, Section 4] based on [LWX18b, Section3.1] uses a combination of the derivative formula and the minimal model program toshow that E is always dreamy, and thus establishes Theorem 3.1(2).

Theorem 4.12. Let X be a K-semistable log Fano pair and E a divisor over X. If

1 = δ(X) = δ(E),

then E is dreamy and induces a non-trivial special test configuration X such thatFut(X ) = 0. In particular, X is not K-stable.

Proof. The argument again uses the cone construction. Fix a positive integer r so thatL := −rKX is a Cartier divisor and set R = R(X,L). Consider the cone Y := C(X;L)with vertex x. As before, we have the ray of valuations

{vt | t ∈ [0,∞)} ⊂ ValY,x,

with v0 = ordX∞ the canonical valuation over the cone and E∞ the divisor over Ydefined via pulling back E.

Recall that, for k ∈ Z>0, v 1k

= 1kordEk

where Ek is a divisor over X. By (9), we

haved

dtvol(vt)

∣∣∣t=0+

= (n+ 1)βX(E) = 0.

Defining f(t) := vol(vt), a Taylor expansion gives

f(t) = f(0) +O(t2) for 0 ≤ t� 1.

For a fixed positive integer k, set

ak,• := a•(ordEk) and ck := lct(Y ; ak,•).

Note that ck ≤ AY (Ek). This implies

f(0) ≤ cnk ·mult(ak,•) ≤ f

(1

k

)= AY (Ek)

n ·mult(ak,•),

where the first inequality follows from [Liu18, Theorem 7] (see (11)) and the assump-tion that X is K-semistable. Therefore,(

f(0)

f(1/k)

)1/n

≤ ckAY (Ek)

≤ 1.


Since (1 +O(1/k2))1/n is also of the order 1 +O(1/k2), we see

1−O(

1

k2

)=

ckAY (Ek)

≤ 1.

We also know AY (Ek) = kAY (v1/k) = kAY (v0) + AY (E∞). Hence,

limk→∞

(AY (Ek)− ck) = limk→∞

(AY (Ek)

(1− ck

AY (Ek)

))= 0.

By the previous limit, we may apply [BCHM10, Corollary 1.4.3] to extract Ek fork � 0. Specifically, there exists a proper birational morphism µk : Yk → Y such thatEk ⊂ Y and −Ek is ample over Y . Hence,

⊕p∈N µk∗(OYk(−pEk)) is finitely generated.

Since µk∗(OYk(−pEk)) = ap(kv1/k), we see⊕p∈N

µk∗(OYk(−pEk)) =⊕p∈N

(⊕m∈N

Fp−mkE Rm

).

Therefore, Rees(FE) is finitely generated as well and E is dreamy.Since Rees(FE) is finitely generated, FE induces a test configuration X of X. Fur-

thermore Fut(X ) = 0, since it is a multiple of βX(E) [Fuj19b, 6.12]. As X is K-semistable and X is a normal test configuration with Fut(X ) = 0, [LX14, 1] impliesX is special. �

Remark 4.13. The previous theorem can be proved without the above cone con-struction argument. Specifically, one can use the characterization δ(X,∆) in termsof m-basis type divisors to show that there exists an extraction µ : Y → X of Eso that Y is log Fano type. Since the Cox ring of a log Fano type variety is finitelygenerated [BCHM10], this implies E is dreamy. However, we chose to write down thecone construction argument, since it better illuminates Step 3 in Theorem 5.12.

24 CHENYANG XU

Part 2. K-stable Fano varieties

In this part, we will discuss some questions about K-(semi,polystable)stable Fanovarieties. In the author’s opinion, currently there are four central questions in alge-braic K-stability theory. We have already mentioned Conjecture 3.16, which wouldfinish the Yau-Tian-Donaldson Conjecture in the singular case, as well as the StableDegeneration Conjecture 4.2.

Another two questions are using K-stability to construct a project moduli space,called K-moduli; and verifying a given example of Q-Fano variety is K-stable or not.This will be respectively discussed in Section 5 and Section 6. Naturally, the methodwe discussed in Part 1 will significantly advance our understanding of these two ques-tions.

5. K-Moduli of Fano varieties

To give a general framework for intrinsically constructing moduli spaces of Fanovarieties is a challenging question in algebraic geometry, especially if one wants tofind a compactification. So when the definition of K-stability from complex geome-try (see [Tia97]) and its algebraic formulation (see [Don02]), first appeared in frontof algebraic geometers, though the connection with the existence of Kahler-Einsteinmetric provides a philosophic justification, technically it seemed bold to expect sucha notion would be a key ingredient in constructing moduli spaces of Fano varieties, asit is remote from any known approaches of constructing moduli.

There are two successful moduli constructions that one can consult. The first one isthe moduli space which parametrizes Kollar-Shepherd-Barron (KSB) stable varieties,that is projective varieties X with semi-log-canonical singularities (slc) and ample ωX(see [?, Definition 23]). The main tools involved in the construction is the MinimalModel Program. However, there are two main differences between moduli of Fanovarieties and moduli of KSB-stable varieties: first, Minimal Model Program oftenprovides more than one limit for a family of Fano varieties over a punctured curve,thus it is unclear how to find a MMP theory that picks the right limit; second, if we aimto find a compact moduli space of Fano varieties, we often have to add Fano varietieswith infinite automorphism group, which is a phenomena that does not happen in theKSB moduli case.

The second moduli space is the one parametrizing (Gieseker) semistable sheaves ona polarized projective scheme (X,OX(1)) with fixed Hilbert polynomial (see [HL10,Section 4]). This moduli space is given by the Geometric Invariant Theory which, aswe have noted, is not clear how to apply to our case. Nevertheless, there have beena lot of recent works (see e.g. [Alp13], [AFS17], [AHLH18]) to give a stack Y a goodmoduli space π : Y → Y , such that the morphism π has the properties shared by themorphism from the stack of GIT-semistable locus [Xss/G] to its GIT-quotient X//G.We will use this framework to construct the moduli of K-(semi,projective)stable Fanovarieties.


History: Using Donaldson’s theorem that all Fano manifold with KE metric and finiteautomorphism are asymptotically GIT stable, it was noted GIT implies such Fanomanifolds can be parametrized by a quasi-projective variety (see [Oda12b, Don15]).Then the K-moduli conjecture for all smoothable K-(semi,poly)stable Fano varietieswas systematically studied in [LWX19], and it was confirmed except the projectivity(also see [SSY16, Oda15]). All these results heavily depend on the analytic toolsdeveloped in the solution of Yau-Tian-Donaldson Conjecture.

Using purely algebraic geometry to prove more results are only achieved after thevaluation criterion was established (see Part 1). Since then there has been a dramaticprogress (see [Jia17, BL18, BX19, ABHLX19] etc.). The algebro-geometric method,which we will focus in the below, has the advantage that it often treats Fano manifoldsas well as Q-Fano varieties without any difference.

5.1. Main Conjecture. The main theorem is the following.

Theorem 5.1 (K-moduli). The moduli functor XKssn,V of n-dimensional K-semistable

Q-Fano varieties of volume V , which sends S ∈ Schk to

XKssn,V (S) =

Flat proper morphisms X → S, whose fibers aren-dimensional K-semistable klt Fano varieties with

volume V , satisfying Kollar’s condition

is represented by an Artin stack XKss

n,V of finite type and admits a good moduli space

φ : XKssn,V → XKps

n,V (in the sense of [Alp13]), whose closed points are in bijection withn-dimensional K-polystable Q-Fano varieties of volume V .

We call such moduli spaces to be K-moduli of K-polystable Q-Fano varieties. Themain remain part is the following.

Conjecture 5.2. The moduli good space XKpsn,V is proper and projective.

5.2. Artin stack X kssn,V .

5.2.1. Kollar condition. The Kollar condition is a condition on families. When B isnormal, a flat family satisfying the Kollar’s condition is just the same as asking thetotal family to be Q-Gorenstein.

Definition 5.3. A Q-Gorenstein family of log Fano pairs π : X → B over a normalbase B is composed of a flat proper morphism π : X → B satisfying:

(1) π has normal, connected fibers (hence, X is normal as well)(2) −KX is a π-ample Q-Cartier divisor, and(3) Xt is klt for all t ∈ C.

In (see [Kol16, Theorem 11.6]), it is shown that the above condition on −KX beingQ-Gorenstein is equivalent to the volume (−KXt)

n being a local constant on t.For a general base B, Kollar’s condition requires that for any m ∈ Z the reflexive

power ω[m]X/S commutes with arbitrary base change (see [Kol09, 24]). This condition

26 CHENYANG XU

first appeared in the study of families of KSB stable varieties, and now is well acceptedas a right local condition for a family of varieties with dimension at least two overa general (possibly non-reduced) base. So in this survey, when we say a family ofQ-Fano varieties, we always assume Kollar’s condition holds for this family.

In general, it is a quite subtle locally property. However, since Q-Fano varieties onlyhave klt singularities, which is a smaller class of singularities than the singularitiesKSB varieties allow to have, on our luck all the subtleties brought up by Kollar’scondition we have to face in the construction of MKss

n,V have already been addressed(see [Kol09,Kol19]).

5.2.2. Boundedness. In moduli problems, boundedness is often a deep property toestablish. Fortunately, since the BAB conjecture has been recently settled in [Bir16],we can apply it to obtain the boundedness for K-semistable Q-Fano varieties withvolume bounded from below. The implication was first settled in [Jia17].

Theorem 5.4 (Boundedness, [Jia17]). Fix n ∈ N and V > 0. All n-dimensionalK-semistable Q-Fano varieties with volume at least V , are contained in a boundedfamily.

Here we will discuss the argument in [LLX18, Section 6.2.2], which used the normal-ized volume (see Section 4). In the course of generalising Example 3.6, in [Liu18, The-orem 1], Yuchen Liu found that there is an inequality connect the local volume andglobal volume for a K-semistable Fano variety. Such an inequality was latter gener-alised to an arbitrary Q-Fano variety X in [BJ17].

Theorem 5.5 ([Fuj18,Liu18,BJ17]). Let X be a Q-Fano variety, then for any x ∈ X,we have

vol(x,X) ·(n+ 1

n

)n ≥ (−KX)n · δ(X)n

Proof. See [BJ17, Theorem D]. �

So if X is K-semistable and (−KX)n ≥ C, we know that for any x ∈ X, vol(x,X) ≥C ·(

nn+1

)n. Then an easy argument in [LLX18] shows that we can bound the minimal

log discrepancy mld(x,X) by the volume vol(x,X) mld(x,X)n ≥ vol(x,X). Thisimplies all x ∈ X on a K-semistable Q-Fano variety with (−KX)n ≥ C, we have

mld(x,X) ≥ C1n ·(

nn+1

). Therefore, we can apply the BAB conjecture (see [Bir16]),

which says that all n-dimensional Fano varieties with a positive lower bound of thelog discrepancy form a bounded family.

Remark 5.6. Presumably, the lower bound of the volume vol(x,X) should post muchstronger restrictions on the singularity than only on the log discrepancy. A specialimplication the stable degeneration conjecture indeed implies that the Cartier index

of x ∈ X is bounded from above by nn/vol(x,X).An even stronger conjecture predicts that x ∈ X with volume bounded from below,

always specialize to a bounded family of singularities.


5.2.3. Openness. To know that X kssn,V is an Artin stack of finite type, we need to show

the K-semistability is an open condition in a family. Very recently, in [Xu19] and[BLX19], two different proofs using the normalized volume and δ-invariants respec-tively are given. Both proofs use Birkar’s theorem on the existence of bounded com-plements [Bir19].

Theorem 5.7 (Openness, [Xu19, BLX19]). If X → B is a Q-gorenstein family ofQ-Fano varieties, then the locus where the fiber is K-semistable is an open set.

We have indeed shown two stronger statements

Theorem 5.8. For a point s ∈ B, denote by s is the geometric point over s ∈ B. Wehave the following:

(1) [BLX19] If X → B is a Q-gorenstein family of Q-Fano varieties, then thefunction

(s ∈ B)→ min{δ(Xs), 1}is a constructible, lower-semicontinous function, and

(2) [Xu19] for a family of klt singularities π : (B ⊂ X)→ B, the function

(s ∈ B)→ vol(s, Xs)

is a constructible, lower-semicontinous function.

Remark 5.9. In [BL18], it is showed that δ(Xt) is a lower semi-continuous functionin Zariski topology on t ∈ B for a Q-gorenstein family of Q-Fano varieties X → B.

Sketch of the idea. To be added. �

By combining Theorem 5.4 and 5.7 we know that the open locus of uniformly K-stable Q-Fano varieties admits a good moduli space.

Theorem 5.10 (Moduli of uniformly K-stable Q-Fano varieties). The functor X kssn,V

is represented by an Artin stack of finite type.

5.3. Separated quotient. As we have already seen in Example 2.15, a family ofQ-Fano varieties X◦ over a punctured curve C◦ = C \ {0} could have many differentfillings to be Q-Fano varieties X over C. Moreover, since a K-polystable Q-Fano couldhave an infinite automorphism group, i.e., in general we can not expect the extensionfamily is unique. What one should expect from separatedness of MKps

n,V is that any twoK-semistable fillings are S-equivalent.

Definition 5.11. Two K-semistable Q-Fano varieties X and X ′ are S -equivalent ifthey degenerate to a common K-semistable log Fano pair via special test configura-tions.

This has been quite challenging for a while. In [LWX18b], the case when X →C arises from a test configuration was solved, i.e., it was proved that any two K-semistable degenerations X and X ′ of a same K-semistable Q-Fano variety, are S-equivalent. Therefore, the orbit inclusion relation for an S-equivalence class of a K-semistable Q-Fano varieties behaves exactly the same as the GIT situation (see [???]).

28 CHENYANG XU

n particular, this gives the following description of K-polystable Q-Fano variety as theminimal element in the S-equivalence class: A Q-Fano variety Y is K-polystable if it isK-semistable and any special test configuration Y of Y with Y0 K-semistable satisfiesX ' X0.

Then the argument there was improved in [BX19] which shows that any two K-semistable fillings of a Q-Gorenstein families of Q-Fano varieties over a smooth curveare S-equivalent. We will discuss this in more details below.

In [ABHLX19], we put the argument in [BX19] in the context of [AHLH18], then theresults are enhanced, so that one can show any K-polystable Q-Fano has a reductiveautomorphism group, and moreover, any S-closed finite substack ofMKss

n,V has a goodmoduli space.

5.3.1. S-equivalence. The following statement is the main result in [BX19]. When Xand X ′ are given by two special test configuration of the same K-semistabe Q-Fanovariety, with Fut(X) = Fut(X ′) = 0, this is proved in [LWX18b]. For the more generalsituation, we have to elaborate the argument in [LWX18b] into a relative setting.

Theorem 5.12 ([BX19]). Let π : (X,∆)→ C and π′ : (X ′,∆′)→ C be Q-Gorensteinfamilies of log Fano pairs over a smooth pointed curve 0 ∈ C. Assume there exists anisomorphism

φ : (X,∆)×C C◦ → (X ′,∆′)×C C◦

over C◦ : = C \ {0}. If (X0,∆0) and (X ′0,∆′0) are K-semistable, then they are S-

equivalent.

Proof. We separate the proof into a few steps.

Step 1: Defining the filtrations.Let π : X → C and π′ : X ′ → C be Q-Gorenstein families of Q-Fano varieties over

a smooth pointed curve 0 ∈ C. Assume there exists an isomorphism

φ : X ×C C◦ → X ′ ×C C◦

over C◦ : C \ {0} that does not extend to an isomorphism X ' X ′ over C. Aftershrinking, we may assume C is affine and there exists a local uniformizer t.

From this setup, we will construct filtrations on the section rings of the specialfibers. Set

L := −rKX and L′ := −rKX′ ,

where r is a positive integer so that L and L′ are Cartier. For each non-negativeinteger m, set

Rm := H0(X,OX(mL)) R′m := H0(X ′,OX(mL′))

Rm := H0(X0,OX(mL0)) R′m := H0(X ′0,OX(mL0)).

Additionally, set

R := ⊕mRm, R := ⊕mRm, R′ := ⊕mR′m, and R′ := ⊕mR′m.

Fix a common log resolution X of X and X ′


X

X X ′

ψ′ψ

φ′

and write X0 and X ′0 for the birational transforms of X0 and X ′0 on X. Set

a := AX,X0(X′0) and a′ := AX′,X′0(X0). (13)

Observe that X0 6= X ′0, since otherwise φ would extend to an isomorphism over C as−KX and −KX′ are ample. Moreover, a, a′ > 0 since X0 and X ′0 are klt, and we canapply inversion of adjunction.

For each p ∈ Z and m ∈ N, set

FpRm := {s ∈ Rm | ordX′0

(s) ≥ p}, and F ′pR′m := {s ∈ Rm | ordX0(s) ≥ p}.

We define filtrations of R and R′ by setting

FpRm := im(FpRm → Rm) and F ′pR′m := im(F ′pR′m → R′m),

where the previous maps are given by restriction of sections. It is straightforward tocheck that F and F ′ are filtrations of R and R′.

Note that a section s ∈ Rm lies in FpRm if and only if there exists an extensions ∈ Rm of s such that s ∈ FpRm. The analogous statement holds for F ′.Step 2: Relating the filtrations.

Since p∗(X0) = q∗(X ′0) have multiplicity one along X0 and X ′0, we may write

KX = ψ∗(KX) + aX ′0 + F and KX = ψ′∗(KX′) + a′X0 + F ′,

where the components of Supp(F ) ∪ Supp(F ′) are both ψ- and ψ′-exceptional. Now,

FpRm ' H0(X,OX

(mψ∗L− pX ′0

))= H0

(X,OX

(mψ′∗L′ + (mra− p)X ′0 −mra′X0 +mr(F − F ′)

)).

Hence, for s ∈ FpRm, multiplying ψ∗s by tmra−p gives an element of

H0(X,OX

(mψ′∗L′ − (mr(a+ a′)− p)X0

)),

which can be identified with Fmr(a+a′)−pR′m.As described above, for each p ∈ Z and m ∈ N, there is a map

ϕp,m : FpRm −→ Fmr(a+a′)−pR′m,which, when Rm and R′m are viewed as submodules of K(X) = K(X ′), sends s ∈FpRm to tmra−p(φ−1)∗(s). Similarly, there is a map

ϕ′p,m : FpR′m −→ Fmr(a+a′)−pRm,

which sends s′ ∈ FpR′m to tmra′−pφ∗(s′).

30 CHENYANG XU

Lemma 5.13. The map ϕp,m is an isomorphism. Furthermore , given s ∈ FpRm

(1) s vanishes on X0 if and only if ϕp,m(s) ∈ F ′mr(a+a′)−p+1R′m, and(2) ϕp,m(s) vanishes on X ′0 if and only if s ∈ Fp+1Rm.

Proof. The map ϕ′mr(a+a′)−p,m is the inverse to ϕp,m, since ϕ′mr(a+a′)−p,m ◦ ϕp,m is mul-

tiplication by tmra′−(mr(a+a′)−p)tmra−p = 1. Hence, ϕp,m is an isomorphism.

For (1), fix s ∈ FpRm and note that s vanishes on X0 if and only if

ψ∗s ∈ H0(X,OX

(mψ∗L− pX ′0 − X0

)).

The latter holds precisely when

tmra−bψ∗s ∈ H0(X,OX

(mψ′∗L′ − (mr(a+ a′)− p+ 1)X0

)),

which is identified with F ′mr(a+a′)−p+1R;m. Statement (2) follows from a similar ar-gument. �

Proposition 5.14. The maps (ϕp,m) induce an isomorphism of graded rings⊕m∈N

⊕p∈Z

grpFRmϕ−→⊕m∈N

⊕p∈Z

grpF ′R′m, (14)

that sends the degree (m, p)-summand on the left to the degree (m,mr(a + a′) − p)-summand on the right. Hence, grpFRm and grpF ′R

′m vanish for p > mr(a+ a′).

Proof. Consider the map

FpRm −→ grmr(a+a′)−pF ′ R′m. (15)

defined as follows. Given an element of s ∈ FpRm, choose s ∈ FpRm such that s isan extension of s. Now, send s to the image of ϕp,m(s) under the composition of maps

F ′mr(a+a′)−pR′m → F ′mr(a+a′)−pR′m → grmr(a+a′)−pF ′ R′m.

This map can be easily seen well defined.Using Lemma 5.13, we see that (15) is surjective and has its kernel equal to Fp+1Rm.

Indeed, the surjectivity follows from the fact that ϕp,m is an isomorphism. The de-scription of the kernel is a consequence of Lemma 5.13.2. Therefore,

grpFRm → grmr(a+a′)−pF ′ R′m

is an isomorphism. The previous isomorphism induces an isomorphism of graded rings,since ϕp1,m1(s1)ϕp2,m2(s2) = ϕp1+p2,m1+m2(s1s2) for s1 ∈ Fp1Rm1 and s2 ∈ Fp2Rm2 .

To see the vanishing statement, observe that grpFRm and gr′pFR′m vanish for p < 0.

Hence, the isomorphism of graded rings yields the vanishing for p > mr(a+ a′). �

Remark 5.15. The above filtration defined in [BX19] was trying to extend the fil-tration defined in [BHJ17] (see the proof of Lemma 3.3) for test configurations into amore general relative setting. It is quite interesting to see that this indeed coincideswith the canonical filtration in [AHLH18], and therefore Proposition 5.14 holds in amore general context.


The connection between the above filtration with K-stability can be seen by thefollowing statement.

Proposition 5.16. Let

β = arn(−KX0)n −

∫ ∞0

vol(F tR)dt and β′ = arn(−KX′0)n −

∫ ∞0

vol(F ′tR′)dt

Then β + β′ = 0 .

Proof. Applying Proposition 5.14, we see

dimFpRm =

mr(a+a′)∑j=p

grjFRm =

mr(a+a′)−p∑j=0

grjF ′R′m

= dimRm − dimFmr(a+a′)−p+1,

for p ∈ {0, . . . ,mr(a+ a′) + 1}. Therefore,

mr(a+a′)∑p=0

dimFpRm +

mr(a+a′)∑p=0

dimF ′pR′m = mr(a+ a′) dimRm.

Then we conclude by dividing both sides by 1n!mn+1 and let m→∞.

Let b•(F) and a•(F) be the base ideal sequences for F on X and X0 respectively.then inversion of adjunction implies that lct(X,X0; b•(F)) = lct(X0; a•(F)). Since wehave a ≥ lct(X,X0; b•(F)), we have

a ≥ lct(X0; a•(F)), and similarly a′ ≥ lct(X0; a•(F ′)).Now since X0 and X ′0 are K-semistable, we have β, β ≥ 0 (see (7)). We can thenconclude that β = β′ = 0. �

Step 3: Finite generations.The remaining part of the proof is to show that the graded ring in (14) is finitely

generated, since then by [LX14], this will imply that the Proj of the graded ring yieldsa K-semistable Q-Fano variety. Not surprisingly, this is the most involving part ofthe proof. For instance, we do not know how to show that on a K-semistable Q-Fanovariety, the graded ring induced by a valuation v on R :=

⊕m∈Z≥0

H0(−mKX) with

β(v) = 0 is finitely generated.Here to proceed, we rely on the fact that our filtrations are induced by divisors

over our families. More precisely, we first use that β = 0 to show that there exists an

extraction Y → X of X ′0 and the fiber Y0 = V ∪W , where V and W are the birationaltransforms of X0 and X ′0. Now, we set E = W |V and observe that E induces afiltration FE on the section ring of X0. Furthermore, we show that F := Supp(E)is a prime divisor and βX0(F ) = 0. By Theorem 4.12, this yields that FE is finitelygenerated and the corresponding degeneration of X0 is a special test configurationwith generalized Futaki invariant zero.

32 CHENYANG XU

We still need to show that the filtrations F and FE induce the same degenerationof X0, i.e., F and FE are equal or at least equal on a Veronese subring. This state-ment is equivalent to the surjectivity of certain restriction maps. For this we take arelative cone C(X/C,−rKX/C) of X over C and run an analysis similar to the proofof Theorem 4.12. �

Remark 5.17. To general the above argument to the log Fano case, unlike mostof other results appearing in this survey, requires extra non-trivial arguments. Thisprobably reflects the subtly on defining a right functor for log pairs.

5.18 (S-completeness). In [AHLH18], the authors formulate two elegant criterions tocheck whether a stack of finite type admits a good moduli space. The first one iscalled S-completeness (see [AHLH18, Definition 3.35]), which we will explain that, asobserved in [ABHLX19], it essentially follows from Theorem 5.12.

Let R be a DVR with η = Spec(K) the generic point. Consider two families X andX ′ of K-semistable Q-Gorenstein Fano varieties over Spec(R) such that there is anisomorphism

φ : X ×Spec(R) Spec(K) ∼= X ′ ×Spec(R) Spec(K).

Fix a uniformizer π of R. Following [AHLH18, (3.6)], denote by

ST(R) := [Spec(R[s, t]/(st− π))/Gm],

where the action is (s, t) → (µ · s, µ−1 · t). Let 0 = [(0, 0)/Gm], then ST(R) \ 0 isisomorphic to the double points curve Spec(R) ∪Spec(K) Spec(R). Thus φ yields a

family π◦ : X◦ → ST(R) \ 0. We want to show this can be indeed extended to a familyπ : X → ST(R), which is precisely the claim of S-completeness for the functor of K-semistable Fano varieties with fixed numerical invariants (see [AHLH18, Definition3.35]).

Denote by i : ST(R) \ 0 ⊂ ST(R) the open inclusion, then since π◦∗(−rmKX◦) is avector bundle on ST(R)\0 and 0 in ST(R) is of codimension 2, then i∗(π

◦∗(−rmKX◦))

is a vector bundle over ST(R). Moreover, a calculation shows that for any m we have

i∗(π◦∗(−rmKX◦))|0 ∼=

⊕p∈Z

grpFRm,

(see Proposition 5.14). Thus we can define

X := Proj⊕m

(i∗(π

◦∗(−rmKX◦))

),

and the necessary finite generation results for such a definition to make sense is pre-cisely the one obtained in Step 3 of the proof of Theorem 5.12.

An important consequence of S-completeness is the following theorem.

Theorem 5.19 ([ABHLX19]). For any K-polystable Fano variety, Aut(X) is reduc-tive.


Proof. This follows from the S-completeness. In fact, if we apply the above discussionto Aut(X)(K), which can be used to glue two trivial families X × Spec(R), trackingthe a basis of sections which are defined over ST(R), what we get is exactly the Iwahoridecomposition

Aut(X)(K) = Aut(X)(R) · Hom(Gm,Aut(X)) · Aut(X)(R),

but this implies that Aut(X) is reductive. �

When X is smooth with a KE metric, the above theorem was proved by Matsushima[Mat57]. When X is a Q-Fano variety with a weak KE metric, this is an important stepin the proof of Yau-Tian-Donaldson Conjecture in [CDS15,Tia15] (also see [BBE+11]).Theorem 5.19 gives a completely algebraic treatment.

5.20 (Θ-reductivity). The second criterion in [AHLH18] is called Θ-reductivity. Thisasks that for any Q-Gorenstein family of K-semistable Fano varieties X/C over asmooth curve C, any K-semistable degeneration of the generic fiber Xη can be ex-tended to a family of K-semistable degenerations. This is proved in [ABHLX19] byusing the arguments developed in [LWX18b].

To summarize, applying the main theorem of [AHLH18], we conclude that

Theorem 5.21 ([ABHLX19]). The finite type artin stack X kssn,V admits a separated

good moduli space Xkpsn,V .

5.4. Open problems. In this section, we discuss unknown parts of Conjecture 5.2and some recent progress.

5.4.1. Properness. The following statement is equivalent to the properness of the goodquotient moduli space.

Conjecture 5.22 (Properness). Any family of K-semistable Fano varieties over apunctured curve C◦ = C \ {0}, after a possible finite base change, can be filled in over0 to a family of K-semistable Fano varieties over C.

This is a probably the most difficult step in the entire construction. When theK-polystable Fano varieties Xt are all smooth for t ∈ C◦, it is shown that the limitX0 for t→ 0 is the Chow limit of [Xt] ∈ Chow(PN) induced by Tian’s embedding, i.e.,embeddings [Xt] → PN given by | − mKXt | for m � 0 with the orthonormal basesunder the Kahler-Einstein metrics on Xt (see [CDS15,Tia15,LWX18b]).

5.4.2. Projectivity. OnXKpsn,V there is a natural Q line bundle, called the Chow-Mumford

line bundle or the CM-line bundle (see e.g. [PT09,LWX18b]).

Conjecture 5.23 (Projectivity). A sufficiently divisible multiple of the CM-line bun-

dle yields an ample line bundle on XKpsn,V .

34 CHENYANG XU

People are expecting this conjecture to be true, because when the family parametrisessmooth fibers, the curvature of the Weil-Petersson metric on the CM line bundle ispositive.

Significant progress on this problem was recently made in [CP18]. They show thatthe CM line bundle is nef; and it is big and nef, if the general fiber is uniformly K-stableand the family is of maximal variation. The proof essentially uses the characterisationof δ-invariants as the limit of the log canonical thresholds of basis type divisors (see(6)). In fact, for any family of uniformly K-stable Fano varieties X → B, in [CP18], byvarious clever reductions, they turn the question into positivity questions for bundleson the base of a family of KSB pairs, which has been addressed in [Fuj18,KP17,PX17].Here the family of KSB pairs is obtained by adding a boundary (1− ε)δ ·∆ where ∆is a family of basis type divisors. Then (Xt, (1− ε)δ ·∆t) is klt, and KXt + (1− ε)δ ·∆t

is ample for ε� 1 since δ > 1.

Remark 5.24 (Recent progress).

6. Explicit examples

Telling whether an explicit given Q-Fano variety is K-semistable is a quite chal-lenging question. The case of smooth surfaces was solved by Tian in [Tia90]. Butin higher dimension, the knowledge is much more limited. For instance, combing[Fuj19a, LX19b], it is only recently proved that all smooth hypersurfaces in P4 areK-stable (see Corollary 6.4 and Theorem 6.8). When dimension is larger than three,besides the simple case of degree 1 and 2, only degree n+ 1 smooth hypersurfaces inPn+1 are known to be all K-stable (see [Fuj19a,SZ19] or Corollary 6.4).

Since the Fermat hypersurfaces are known to have KE metrics by [Tia87]. We knowa general smooth hypersurface is K-stable. The following is a folklore conjecture.

Question 6.1. Are all smooth degree d hypersurfaces with 3 ≤ d ≤ n in Pn+1 K-stable?

In the below, we will discuss two ways of showing Fano varieties are K-(semi,poly)stable.The first one is by studying the singularity in | −KX |Q, and the second approach isconstructing explicit K-moduli spaces.

6.1. Singularities of |−KX |Q. For a long time, there has been a very limited numberof methods to prove that a Fano given manifold admits a KE metric. Among them,probably the most well known one is Tian’s α-invariant criterion which says if α(X) >nn+1

, then X is K-polystable. See [Tia87] (and also [OS12] for an algebraic treatment).It established the philosophy that if members in | −KX |Q is not ‘too singular’, thenX should be close to be K-stable.

By now, we have the new invariant δ(X), which is precisely computed by the infi-mum of log canonical thresholds of basis type divisor in | −KX |Q (see (6)). Althoughdirectly computing δ for general X is a quite hard problem, by using properties of thevolume function, one can indeed verify the K-stability in some special cases. Theseworks confirm K-stability of examples which people have speculated for a long time.


6.1.1. Fano manifolds with α = nn+1

. The first new class of examples provided bythe new approach are Fano manifolds with α(X) = n

n+1. It has been known for long

time that any Q-Fano variety X with α(X) > nn+1

is K-stable and α(X) ≥ nn+1

isK-semistable. In [Fuj19a], Fujita used the valuation criterion to study the equalitycase, and realised the following somewhat surprising fact.

Theorem 6.2 ([Fuj19a]). If X is an n-dimensional smooth Fano manifold, α(X) =nn+1

and β(E) = 0 for some divisor E over X, then X ∼= P1.

Proof. Consider the restricted volume function

Q := − 1

n

d

dtvol(−µ∗KX − tE)dt for t ∈ [0, τ),

where τ := τ(F ) is the pseudo-effective threshold of E with respect to −KX . Themain property we need about Q is that it is a concave function which follows from theBrunn-Minkowski Theorem. Moreover, Q is smooth on [0, τ) and can be extended toa continuous function on [0, τ ]. We have∫ τ

0tQdt∫ τ

0Qdt

=1n

∫ τ0

vol(µ∗(−KX)− tF )dt1n(−KX)n

= AX(F ) ≥ n

n+ 1τ, (16)

as β(F ) = 0 and α(X) = nn+1

. We denote by A := AX(F ).

As Q(0) = 0, by the concavity, we know that Q(t) ≥ ( tA

)n−1Q(A) for x ∈ [0, A] andQ(t) ≤ ( t

A)n−1Q(A) for t ∈ [A, τ ]. So

0 =

∫ τ

0

(t− A)Q(t)dt

≤ Q(A)

∫ τ

0

(t− A)(t

A)n−1dt

=Q(A)τn

An−1(

τ

n+ 1− A

n)

≤ 0.

This implies that A = nn+1

τ and Q(t) = ( tA

)n−1Q(A) for t ∈ [0, τ ].To proceed, by considering the ample model of µ∗(−KX) − εE for ε sufficiently

small, we know we can find a model µ : Y → X only extracting E, so we know thatfor t� 1,

Q = − 1

n

d

dtvol(−µ∗KX − tE)|t=t0 = E · (−µ∗KX − tE)n−1.

Compared to Q(t) = ( tA

)n−1Q(A), we know Ei · (µ∗KX)n−i = 0 and all 0 ≤ i ≤ n− 1,which in particular implies E is mapped to a point. Moreover, Q = tn−1(−E|E)n−1,which implies for t ∈ [0, τ ],

vol(−KX − tE) = n

∫ τ

t

Qdu = (τn − tn)E(−E)n−1 = (−KX)n − tnE(−E)n−1.

36 CHENYANG XU

This implies τ = ε, which is the nef threshold of E with respect to −KX .Since E is dreamy by Theorem 4.12, we know µ∗(−KX)−τE is semi-ample but not

big. Since µ∗(−KX)− τE is ample on E, we know a sufficiently divisible multiple ofµ∗(−KX)− τE will give a fibration struction ρ : Y → Z, whose restrict on E is finite.Thus a general fiber of ρ is a curve l. Since Y is normal, l is in the smooth locus.Thus KY · l = −2 and 0 = (µ∗(−KX) − τE) · l = (1 + 1

n+1τ)E · l − 2, which implies

E · l = 1, τ = n + 1 and A = n. So Y → X is the blow up of the smooth point, andwe know the Seshadri constant at this point is n + 1, which implies that X ∼= Pn by[LZ18]. Then α(Pn) = 1

n+1. �

Remark 6.3. The smooth assumption in Theorem 6.2 is indeed necessary. In [LZ19],they found a class of singular Q-Fano varieties X with α(X) = n

n+1, but they are only

strictly K-semistable.

Corollary 6.4. For n ≥ 2, smooth degree n+1 hypersurfaces in Pn+1 are all K-stable.

Proof. For any such hypersurface X, we have α(X) ≥ nn+1

(see e.g. [Che01]), thus wecan apply Theorem 6.2. �

6.1.2. Birationally Superrigid Q-Fano varieties. A special class of Fano varieties, calledbirationally superrigid Fano variety has been studied for a long time, dated back toGino Fano.

Definition 6.5. A Fano variety X is said to be birationally superrigid if it has terminalQ-factorial singularities, Picard number one and every birational map f : X 99K Yfrom X to a Mori fiber space Y is an isomorphism.

The Noether-Fano method is a criterion to determine a Fano variety to be bira-tionally superrigid. We have the following equivalent characterization of birationalsuperrigidity.

Theorem 6.6 (Noether-Fano method, see [KSC04, Section 5]). Let X be a terminalFano variety, which is Q-factorial and Picard number. Then it is birationally super-rigid if and only if for every movable boundary M ∼Q −KX on X, the pair (X,M)has canonical singularities.

By the above characterisation, we know being birationally superrigid posts a verystrong restriction on the mobile linear subsystem of | −KX |Q. So it is natural to askwhether all of them are K-stable. After some early partial result in [OO13], in a recentwork [SZ19], the question is affirmatively answered under a mild assumption.

Theorem 6.7 ([SZ19]). Let X be a birationally superrigid. Assume α(X) > 12

(resp.

α(X) ≥ 12), then X is K-stable (resp. K-semistable).

Proof. The analysis is similar to the one in the proof of Theorem 6.2. Assume Fsuch that β(F ) < 0. Then by Theorem 6.6, we know there is only one Q-divisorD ∼Q −KX with irreducible support, such that ordFD > AX(F ) =: A. Then we havethe pseudoeffective threshold τ = ordF (A).


We define Q the restricted volume function as in Theorem 6.2, and by our assump-tion, we have

b :=

∫ τ0tQdt∫ τ

0Qdt

> A.

Thus we have 0 < A < b < τ . Using the concavity again (see Theorem 6.2), we knowfor t ∈ [0, A], Q(t) ≥ ( t

A)n−1Q(A); and for t ∈ [A, τ ], since

−µ∗(KX)− tF =τ − tτ − A

(−µ∗(KX)− A · F ) +t− Aτ − A

(−µ∗(KX)− A · F ),

we know Q(t) =(τ−tτ−A

)n−1Q(A). Thus

0 =

∫ τ

0

(t− b)Q(t)dt

≤∫ A

0

(t− b)( tA

)n−1Q(A)dt+

∫ τ

A

(t− b)( τ − tτ − A

)n−1Q(A)dt

= Q(A)τ(τ − 2A

n(n+ 1)+A− bn

) < 0.

since Aτ≥ α(X) ≥ 1

2. And the K-stable case is proved in the same way. �

Combining with [dF16], this gives a different proof on Corollary 6.4. So far allknown birationally superrigid Fano varieties satisfy α(X) > 1

2.

6.2. Explicit K-moduli spaces. We can also prove a Q-Fano variety is K-(semi,poly)-stable by showing that they appear on a K-moduli. Note that although we have notyet known the algebraic construction of the compact K-moduli space, we can usethe analytic depended construction [LWX19] for the closed component parametrisingsmoothable Q-Fano varieties. Here the compactness is crucial, since often we prove Xis K-(semi,poly)stable by showing it is a limit of a family of K-stable Fano manifolds,and all other possible limits are not K-(semi,poly)stable.

More precisely, if we start with a Fano manifold Xt∗ which is a fiber of a familyX→ B over an irreducible base, and assume we know that Xt∗ is K-semistable. Thenwe know that there is a Zariski open set B0 ⊂ B such that for any t ∈ B0, thefiber Xt is K-semistable. However, usually we do not a priori know how large B0 is.Nevertheless, let C → B be a mapped from a curve whose image contains t0. Then bythe compactness of the K-moduli space, we know that for any s ∈ C, we can extendthe family of X× (B0 ∩C)→ (B0 ∩C) over s such that the fiber Xs is K-semistable.

Then in some cases, the local restriction could be useful to determine what Xt is:By Theorem 5.5, we know that any point x ∈ Xs is a smoothable singularity with

vol(x,Xs) ≥ (−KXs)n ·( n

n+ 1

)n.

When (−KXs)n is large, this local condition could post strong restrictions for pos-

sible Xs. For instance, because the stable degeneration conjecture is known for any

38 CHENYANG XU

singularity x ∈ Xs appearing on the Gromov-Hausdorff limit of Kahler-Einstein Fanomanifolds, it is proved in [LX19a] that

πloc1 (x,Xs) ≤

nn

vol(x,Xs)≤ (n+ 1)n

(−KXs)n, (17)

and the first inequality holds if and only if x ∈ Xs is a quotient singularity.In [OSS16] this approach has been used to identify the singular surfaces parametrised

by the boundary of the moduli space of smooth del Pezzo surfaces of degree 1, 2 or 3.In [SS17], it was also used to find the singular Q-Fano varieties parametrised by theboundary of the moduli space of the intersection of two quadrics.

In [LX19b], the explicit K-moduli spaces are used to find out new smooth K-stableFano manifolds.

Theorem 6.8 ([LX19b]). The GIT moduli of cubic threefolds is the same as theK-moduli.

Sketch of the proof. By the discussion above, the volume of any x ∈ Xs is at least 818

.

So by Theorem 6.9, it is either 12(1, 1, 0) or the local fundamental group πloc

1 (x,Xs) istrivial. In each case, we can easily show that the degeneration of the hyperplane classO(1) on Xs is still a Cartier divisor. Then a classical result by Takao Fujita showsthat Xs is indeed also a cubic threefold.

Then we know it is GIT (semi,poly)stable if it is K-(semi,poly)stable. �

As we explain, the key local input is the following

Theorem 6.9 ([LX19b]). For any three dimension singularity x ∈ X, vol(x,X) ≤ 16,and the equality holds if and only if x ∈ X is the rational double point.

The proof of Theorem 6.9 relies heavily on the classification of three dimensionalcanonical and terminal singularities, which is classical in MMP.

Remark 6.10. From a computational viewpoint, an explicit description of the bound-ary point of the K-moduli should often be easier to find than the KSB moduli case,since the limit has to be klt, in particular normal.

Remark 6.11. While the approach in Section 6.1 is mostly for Fano varieties witha small volume, the approach in Section 6.2 is mainly for Fano varieties with a bigvolume. However, it seems there is still a gap between these two approaches, and wedo not know how to deal with general Fano varieties with intermediate volumes. See6.12.

6.12 (Fano threefolds with ρ(X) = 1). The Kahler-Einstein problem for smooth delPezzo surfaces has been completely answered by Tian [Tia90]. So it is natural to takea look at all smooth Fano threefolds which are classified by Iskovskih when ρ(X) = 1and Mori-Mukai when ρ(X) ≥ 2.

When ρ(X) = 1, there are nineteen families. Let

r(X) := {r| −KX ∼ −rH, for H generator of Pic(X)}


be the index. There are seven families with r(X) ≥ 2 that we completely know theK-(semi,poly)stability. They are

• P3, which is K-polystable;• the smooth quadric Q ⊂ P4, which is K-polystable;• V5 a codimension 3 linear section of Gr(2, 5) ⊂ P9, which is K-polystable by

[CS09];• V4 a smooth intersections of two quadrics in P5, which is K-stable by [AGP06];• V3 a smooth cubic in P4, which are all K-stable by [LX19b];• V2 a double cover of P3 with ramification degree 4, which is K-stable by

[Der16a, Theorem 1.1].• V1 a degree 6 hypersurface in P(1, 1, 1, 2, 3). See ???.

There are four families with r(X) = 1 that we completely know the K-(semi,poly)stability.They are

• X2 a double cover of P3 with ramification degree 6, which is K-stable by[Der16a, Theorem 1.1];• X4 a double cover of a quadric in P4 ramified in a quartic section, which is

K-stable by [Der16a, Theorem 1.1];• X4 a quartic in P4, which is K-stable by [Fuj19c];• X6 a smooth intersection of a quadric and a cubic in P5, which is K-stable by

[Zhu18].

For any of the rest eight families on the classification list, we can not completelycharacterise the K-(semi,poly)stability for all smooth members. Among them, theone with genus 12 (i.e. (−KX)3 = 22) which contains the famous Mukai-Umemuramanifold has attract lots of interests (see [CS18] for recent progress). It is temptingto believe

Conjecture 6.13. All smooth Fano threefolds with ρ(X) = 1 are K-semistable.

It was originally conjectured in [OO13, Conjecture 5.1] that all smooth Fano man-ifolds in any dimension with Picard number 1 are K-semistable. However, later thiswas disproved in [Fuj17], which shows that a del Pezzo manifold X of degree five withdimension four or five are K-unstable.

For ρ(X) ≥ 2, in [Fuj16, Section 10], β-invariants for divisors on X itself arecomputed, and many cases are shown to be K-unstable.

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MIT, Cambridge, MA 02139, USAE-mail address: [email protected]

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