K-semistable Fano manifolds with the smallest alpha invariant

TL;DR

This paper proves that among K-semistable Fano manifolds, those with the minimal alpha invariant are necessarily projective spaces, and also explores related singular cases.

Contribution

It establishes a characterization of K-semistable Fano manifolds with minimal alpha invariant as projective spaces, including singular cases.

Findings

K-semistable Fano manifolds with smallest alpha invariant are projective spaces

Singular cases of such manifolds are also characterized

Provides a classification result in Fano geometry

Abstract

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

Equations

$$\alpha (X):=\inf \{\text{lct}(X;D)|0\le D{\sim }_{\mathbb{Q}}-K_X\}.$$

$$\beta (F):=A(F)\cdot (-K_X{)}^n-{\int }_0^{\infty }{\text{vol}}_X(-K_X-xF)\text{d}x.$$

$${\text{\rm mult}}_{F}\bigg{(}K_{Y}-\sigma^{*}\bigg{(}K_{X}+\frac{1}{n+1}D\bigg{)}\bigg{)}=-1,$$

$$A(F)=\frac{\mu }{n+1}.$$

$$\frac{1}{n+1}(-K_X{)}^n=$$

$$\ge$$

$$=$$

$$\ge$$

$$=$$

$${\text{vol}}_X(-K_X-x\mu F)={\text{vol}}_X(-K_X-xD)=(1-x{)}^n(-K_X{)}^n$$

$$\mu \cdot {\text{vol}}_{Y|F}(-{\sigma }^{\ast }{K}_X)$$

$$=$$

$${\text{vol}}_{Y|F}(-{\sigma }^{\ast }{K}_X)=(-{\sigma }^{\ast }{K}_X{)}^{n-1}\cdot F=(-K_X{)}^{n-1}\cdot {\sigma }_{\ast }F.$$

$$(-K_X{)}^{n-1}(D-\mu {\sigma }_{\ast }F)=(-K_X{)}^n-\mu \cdot {\text{vol}}_{Y|F}(-{\sigma }^{\ast }{K}_X)=0.$$

$$(-K_X{)}^n=(n+1{)}^n(E{)}^n\ge (n+1{)}^n.$$

$$(-K_X{)}^n=\frac{(n+1{)}^n}{n^{n-1}}(-(K_X+E){)}^{n-1}\cdot E=\frac{(n+1{)}^n}{n^{n-1}}(-K_E{)}^{n-1}=(n+1{)}^n.$$

$$(-K_X{)}^3\le \frac{(r+2)(4+4r{)}^3}{(3r{)}^3}.$$

$$K_Y={\pi }^{\ast }{K}_X+\sum _i{a}_i{E}_i$$

$${\pi }^{-1}{\mathfrak{m}}_p\cdot {\mathcal{O}}_Y={\mathcal{O}}_Y(-\sum _i{b}_i{E}_i).$$

$$\text{lct}(X;{\mathfrak{m}}_p)\le \frac{1+a_{i_0}}{b_{i_0}}\le 1+\frac{1}{r}.$$

$$(-K_{X})^{3}\leq\bigg{(}1+\frac{1}{3}\bigg{)}^{3}{\text{\rm lct}}(X;{\mathfrak{m}}_{p})^{3}{\text{\rm mult}}_{p}X\leq\frac{(r+2)(4+4r)^{3}}{(3r)^{3}}.$$

$$\text{qW}(X)$$

$$\text{q}\mathbb{Q}(X)$$

$$\text{qW}(X),\text{q}\mathbb{Q}(X)\in \{1,\dots ,11,13,17,19\}.$$

$$\begin{array}[]{ccc}g(X)&\mathbf{B}&A^{3}\\ \hline\cr 21&\{3\}&2/3\\ 20&\{5,7\}&22/35\end{array}$$

Full text

The author was supported by JSPS KAKENHI Grant Number JP16K17558 and World Premier International Research Center Initiative (WPI), MEXT, Japan.

1. introduction

Throughout the article, we work over the complex number field ${\mathbb{C}}$. A $\mathbb{Q}$-Fano variety is a normal projective variety $X$ with log terminal singularities such that the anti-canonical divisor $-K_{X}$ is an ample Q-Cartier divisor. It has been known that a Fano manifold $X$ (i.e., a smooth $\mathbb{Q}$-Fano variety) admits Kähler–Einstein metrics if and only if $X$ is K-polystable by the works [DT92, Tia97, Don02, Don05, CT08, Sto09, Mab08, Mab09, Ber16] and [CDS15a, CDS15b, CDS15c, Tia15].

On the other hand, the existence of Kähler–Einstein metrics and K-stability are related to the alpha invariants $\alpha(X)$ of $X$ defined by Tian [Tia87] (see also [TY87, Zel98, Lu00, Dem08]). Tian [Tia87] proved that for a Fano manifold $X$, if $\alpha(X)>\dim X/(\dim X+1)$, then $X$ admits Kähler–Einstein metrics. Odaka and Sano [OS12, Theorem 1.4] (see also its generalizations [Der16, BHJ15, FO16, Fuj16c]) proved a variant of Tian’s theorem: if a $\mathbb{Q}$-Fano variety $X$ satisfies that $\alpha(X)>\dim X/(\dim X+1)$ (resp. $\geq\dim X/(\dim X+1)$), then $X$ is K-stable (resp. K-semistable). We are interested in the relation of alpha invariants and K-semistability.

Recall that Fujita and Odaka proved that there exists a lower bound of alpha invariants for K-semistable $\mathbb{Q}$-Fano varieties.

Theorem 1.1** ([FO16, Theorem 3.5]).**

Let $X$ be a K-semistable $\mathbb{Q}$-Fano variety of dimension $n$.

Then $\alpha(X)\geq\frac{1}{n+1}$.

It is natural and interesting to ask when the equality holds. For example, it is well-known that $\mathbb{P}^{n}$ is K-semistable with $\alpha(\mathbb{P}^{n})=\frac{1}{n+1}$. The main theorem of this paper is the following.

Theorem 1.2**.**

Let $X$ be a K-semistable Fano manifold of dimension $n$.

Then $\alpha(X)=\frac{1}{n+1}$ if and only if $X\cong\mathbb{P}^{n}$.

This is an application of Birkar’s answer to Tian’s question [Bir16, Theorem 1.5], and Fujita–Li’s criterion for K-semistability [Li15, Fuj16b].

It is natural to ask whether the same statement holds true for K-semistable $\mathbb{Q}$-Fano varieties instead of manifolds. However, this is on longer true even in dimension $2$. We are grateful to Kento Fujita for kindly providing the following example:

Example 1.3**.**

Consider the cubic surface $X=(x_{0}^{3}=x_{1}x_{2}x_{3})\subset\mathbb{P}^{3}$, which is a toric log del Pezzo surface (i.e, a $\mathbb{Q}$-Fano variety of dimension $2$) with $3$ du Val singularities of type $A_{2}$. On one hand, it is well-known that $X$ admits a Kähler–Einstein metric (cf. [DT92]), hence is K-semistable. On the other hand, $\alpha(X)=\frac{1}{3}$ (cf. [PW10]).

In fact, by the classification of possible du Val singularities of K-semistable log del Pezzo surfaces (cf. [Liu16, Corollary 6]) and explicit computation of alpha invariants (cf. [Par03, PW10, CK14]), we have the following theorem.

Theorem 1.4**.**

Let $X$ be a K-semistable log del Pezzo surface with at worst du Val singularities. Then $\alpha(X)=\frac{1}{3}$ if and only if $X\cong\mathbb{P}^{2}$ or $X\subset\mathbb{P}^{3}$ is a cubic surface with at least 2 singularities of type $A_{2}$.

Moreover, by classification of $\mathbb{Q}$-Fano 3-fold with $\mathbb{Q}$-factorial terminal singularities and $\rho(X)=1$ with large Fano index due to Prokhorov [Pro10, Pro13], we prove the following:

Theorem 1.5**.**

Let $X$ be a K-semistable $\mathbb{Q}$-Fano 3-fold with $\mathbb{Q}$-factorial terminal singularities and $\rho(X)=1$. Assume that $h^{0}(-K_{X})\geq 22$. Then $\alpha(X)=\frac{1}{4}$ if and only if $X\cong\mathbb{P}^{3}$.

Finally, we propose the following much stronger conjecture. For some evidence in dimension $3$, we refer to [CS08] and [Fuj16a].

Conjecture 1.6**.**

Let $X$ be a K-semistable Fano manifold.

Then $\alpha(X)<\frac{1}{n}$ if and only if $X\cong\mathbb{P}^{n}$.

Acknowledgments. The author would like to thank Professors Kento Fujita and Yoshinori Gongyo for effective discussions. The main part of this paper was written during the author enjoyed the workshop “Higher Dimensional Algebraic Geometry, Holomorphic Dynamics and Their Interactions” at Institute for Mathematical Sciences, National University of Singapore. The author is grateful for the hospitality and support of IMS.

2. Preliminaries

We adopt the standard notation and definitions in [KM98] and will freely use them.

Definition 2.1**.**

Let $X$ be a $\mathbb{Q}$-Fano variety. The alpha invariant $\alpha(X)$ of $X$ is defined by the supremum of positive rational numbers $\alpha$ such that the pair $(X,\alpha D)$ is log canonical for any effective $\mathbb{Q}$-divisor $D$ with $D\sim_{\mathbb{Q}}-K_{X}$. In other words,

[TABLE]

Tian [Tia90] asked whether whether the infimum is a minimum, which is answered by Birkar affirmatively.

Theorem 2.2** ([Bir16, Theorem 1.5]).**

Let $X$ be a $\mathbb{Q}$-Fano variety. Assume that $\alpha(X)\leq 1$. Then there exists an effective $\mathbb{Q}$-divisor $D$ such that $D\sim_{\mathbb{Q}}-K_{X}$ and ${\text{\rm lct}}(X;D)=\alpha(X)$.

Definition 2.3** ([Fuj16b]).**

Let $X$ be a $\mathbb{Q}$-Fano variety of dimension $n$. Take any projective birational morphism $\sigma:Y\to X$ with $Y$ normal and any prime divisor $F$ on Y , that is, $F$ is a prime divisor over $X$.

1. (1)

Define the log discrepancy of $F$ as $A(F):={\text{\rm mult}}_{F}(K_{Y}-\sigma^{*}K_{X})+1$; 2. (2)

Define ${\text{vol}}_{X}(-K_{X}-xF):={\text{vol}}_{Y}(-\sigma^{*}K_{X}-xF)$; 3. (3)

Define

[TABLE]

Note that the definitions do not depend on the choice of birational model $Y$.

Instead of recalling the original definition, we use the following criterion to define K-semistability.

Definition-Proposition 2.4** ([Fuj16b, Corollary 1.5], [Li15, Theorem 3.7]).**

Let $X$ be a $\mathbb{Q}$-Fano variety. $X$ is K-semistable if $\beta(F)\geq 0$ for any divisor $F$ over $X$.

Note that K-semistability is known to be equivalent to Ding-semistability by [BBJ15].

3. Proof of main theorem

Proposition 3.1**.**

Let $X$ be a K-semistable $\mathbb{Q}$-Fano variety of dimension $n$. Assume that $\alpha(X)=\frac{1}{n+1}$, then there exists a prime divisor $E$ on $X$ such that $-K_{X}\sim_{\mathbb{Q}}(n+1)E$ and $(X,E)$ is plt.

Proof.

Let $X$ be a K-semistable $\mathbb{Q}$-Fano variety of dimension $n$ with $\alpha(X)=\frac{1}{n+1}$. By Theorem 2.2, there is a divisor $D\sim_{\mathbb{Q}}-K_{X}$ such that ${\text{\rm lct}}(X;D)=\frac{1}{n+1}$. Take $F$ to be a non-klt place of $(X,\frac{1}{n+1}D)$, then there is a resolution $\sigma:Y\to X$ such that $F$ is a divisor on $Y$.

Denote $\mu$ to be the multiplicity of $F$ in $\sigma^{*}D$. Note that $\mu>0$ since $X$ is klt. By assumption,

[TABLE]

which means that

[TABLE]

By Definition-Proposition 2.4, $\beta(F)\geq 0$, which means that

[TABLE]

The second equality holds since $\sigma^{*}D\geq\mu F$. Hence all inequalities become equalities. In particular,

[TABLE]

for almost all $x$. By differentiability of volume functions ([BFJ09, Corollary C]),

[TABLE]

Here ${\text{vol}}_{Y|F}$ is the restricted volume, we refer to [ELMNP09] for definition and properties. Since ${\text{vol}}_{Y|F}(-\sigma^{*}K_{X})>0,$ $F\not\subseteq\mathbf{B}_{+}(-\sigma^{*}K_{X})$ by [ELMNP09, Theorem C]. Hence by [ELMNP09, Corollary 2.17],

[TABLE]

In other words, we have

[TABLE]

This implies that $D=\mu\sigma_{*}F$ since $D\geq\mu\sigma_{*}F$ and $-K_{X}$ is ample. In particular, $F$ is not $\sigma$-exceptional and $\sigma_{*}F$ is a prime divisor on $X$. Denote $E:=\sigma_{*}F$. Moreover, since $F$ is a non-klt place of $(X,\frac{1}{n+1}D)$, ${\text{\rm mult}}_{E}\frac{1}{n+1}D=1$, that is, $\mu=n+1$. In particular, $-K_{X}\sim_{\mathbb{Q}}D=(n+1)E$. Finally, this argument shows that $F$ is the only non-klt place of $(X,E)$, which means that $(X,E)$ is plt. ∎

Corollary 3.2**.**

Let $(X,E)$ as in Proposition 3.1. Then $X\simeq\mathbb{P}^{n}$ if one of the following condition holds:

1. (1)

$X$* is factorial;* 2. (2)

$(E)^{n}\geq 1$; 3. (3)

$E$* is Cartier in codimension two and $E\simeq\mathbb{P}^{n-1}$.*

Proof.

(1) If $X$ is factorial, then $E$ is a Cartier divisor. In particular, $(E)^{n}\geq 1$. Hence this is a special case of (2).

(2) If $(E)^{n}\geq 1$, then

[TABLE]

By [Liu16, Theorem 1.1] or [LZ16, Theorem 9], $X\simeq\mathbb{P}^{n}$.

(3) If $E$ is Cartier in codimension two and $E\simeq\mathbb{P}^{n-1}$, then by adjunction, $(K_{X}+E)|_{E}=K_{E}$, and

[TABLE]

Again by [Liu16, Theorem 1.1] or [LZ16, Theorem 9], $X\simeq\mathbb{P}^{n}$. ∎

Proof of Theorem 1.2.

It follows directly from Proposition 3.1 and Corollary 3.2(1) (or [KO73]). ∎

4. Singular surfaces

Recall the following theorem on classification of possible du Val singularities of a K-semistable log del Pezzo surface.

Theorem 4.1** ([Liu16, Theorem 23, Proof of Corollary 6]).**

Let $X$ be a K-semistable log del Pezzo surface with at worst du Val singularities.

1. (1)

If $(-K_{X})^{2}=1$, then $X$ has at worst singularities of type $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$, $A_{6}$, $A_{7}$, $A_{8}$, or $D_{4}$; 2. (2)

If $(-K_{X})^{2}=2$, then $X$ has at worst singularities of type $A_{1}$, $A_{2}$, or $A_{3}$; 3. (3)

If $(-K_{X})^{2}=3$, then $X$ has at worst singularities of type $A_{1}$ or $A_{2}$; 4. (4)

If $(-K_{X})^{2}=4$, then $X$ has at worst singularities of type $A_{1}$; 5. (5)

If $(-K_{X})^{2}\geq 5$, then $X$ is smooth.

We remark that in [Liu16, Corollary 6], log del Pezzo surfaces are assumed to be admitting Kähler–Einstein metrics, but the proof works well for K-semistable log del Pezzo surfaces. The only part that the existence of Kähler–Einstein metrics is needed is to exclude the case that $(-K_{X})^{2}=1$ and $X$ has singularities of type $A_{8}$.

Recall the following theorem on explicit computation of alpha invariants.

Theorem 4.2** ([Par03], [PW10, Theorems 1.4, 1.5, and 1.6], [CK14, Theorem 1.26, Example 1.27]).**

Let $X$ be a log del Pezzo surface with at worst du Val singularities. Assume that $X$ is singular, then $\alpha(X)=\frac{1}{3}$ if and only if one of the following holds:

1. (1)

$(-K_{X})^{2}=6$* and ${\text{\rm Sing}}(X)=\{A_{1}\}$;* 2. (2)

$(-K_{X})^{2}=5$* and ${\text{\rm Sing}}(X)=\{A_{2}\}$ or $\{2A_{1}\}$;* 3. (3)

$(-K_{X})^{2}=4$* and ${\text{\rm Sing}}(X)=\{A_{3}\}$ or ${\text{\rm Sing}}(X)\supseteq\{A_{1}+A_{2}\}$;* 4. (4)

$(-K_{X})^{2}=3$* and ${\text{\rm Sing}}(X)\supseteq\{A_{4}\}$, $\{2A_{2}\}$, or ${\text{\rm Sing}}(X)=\{D_{4}\}$;* 5. (5)

$(-K_{X})^{2}=2$* and ${\text{\rm Sing}}(X)\supseteq\{D_{5}\},\{(A_{5})^{\prime}\}$, or $\{A_{7}\}$;* 6. (6)

$(-K_{X})^{2}=1$* and ${\text{\rm Sing}}(X)\supseteq\{D_{8}\}$ or $\{E_{6}\}$.*

Proof of Theorem 1.4.

Let $X$ be a K-semistable log del Pezzo surface with at worst du Val singularities and $\alpha(X)=\frac{1}{3}$. If $X$ is smooth, then $X\simeq\mathbb{P}^{2}$ by Theorem 1.2. If $X$ is singular, then $(-K_{X})^{2}=3$ and ${\text{\rm Sing}}(X)\supseteq\{2A_{2}\}$ by Theorems 4.1 and 4.2. To see the “if” part, one just notice that any cubic surface with at worst singularities of type $A_{1}$ or $A_{2}$ is K-semistable (cf. [OSS16, Theorem 4.3]). ∎

5. Singular threefolds

In this section, we prove Theorem 1.5. Recall the following theorem on the upper bound of volumes.

Theorem 5.1** (cf. [Liu16, Theorem 25]).**

Let $X$ be a K-semistable $\mathbb{Q}$-Fano $3$-fold with at worst terminal singularities. Let $p\in X$ be an isolated singularity with local index $r$. Then

[TABLE]

Proof.

Denote by $\mathfrak{m}_{p}$ the maximal ideal at $p$. We may take a log resolution of $(X,\mathfrak{m}_{p})$, namely $\pi:Y\to X$ such that $\pi$ is an isomorphism away from $p$ and $\pi^{-1}\mathfrak{m}_{p}\cdot{\mathcal{O}}_{Y}$ is an invertible ideal sheaf on $Y$. Let $E_{i}$ be exceptional divisors of $\pi$. We define the numbers $a_{i}$ and $b_{i}$ by

[TABLE]

and

[TABLE]

It is clear that ${\text{\rm lct}}(X;\mathfrak{m}_{p})=\min_{i}\frac{1+a_{i}}{b_{i}}$. Since $\pi$ is an isomorphism away from $p$, we have $b_{i}\geq 1$ for any $i$. Since $X$ is terminal at $p$, by [Kaw93], there exists an index $i_{0}$ such that $a_{i_{0}}=\frac{1}{r}.$ Hence

[TABLE]

On the other hand, by [Kak00] (see also [TW04, Proposition 3.10]), ${\text{\rm mult}}_{p}X\leq r+2$. Hence by [Liu16, Theorem 16],

[TABLE]

∎

Now let $X$ be a K-semistable $\mathbb{Q}$-Fano 3-fold with $\mathbb{Q}$-factorial terminal singularities and $\rho(X)=1$ with $\alpha(X)=\frac{1}{4}$. By Proposition 3.1, there exists a prime divisor $E$ on $X$ such that $-K_{X}\sim_{\mathbb{Q}}4E$.

Recall that we may define ([Pro10])

[TABLE]

It is known by [Suz04, Pro10] that

[TABLE]

Moreover, by [Pro10, Lemma 3.2], in our case, $4|{{\text{\rm q}}\mathbb{Q}}(X)$. Hence there are 2 cases: (i) ${{\text{\rm q}}\mathbb{Q}}(X)=8$; (ii) ${{\text{\rm q}}\mathbb{Q}}(X)=4$.

Now assume that $h^{0}(-K_{X})\geq 22$. Define the genus $g(X):=h^{0}(-K_{X})-2\geq 20$.

If ${{\text{\rm q}}\mathbb{Q}}(X)=8$, since $g(X)>10$, then by [Pro13, Theorem 1.2(ii)], either $X\simeq X_{6}\subset\mathbb{P}(1,2,3,3,5)$ or $X\simeq X_{10}\subset\mathbb{P}(1,2,3,5,7)$. But in either case, $-K_{X}\sim 8A$ where $A$ is an effective divisor, which implies that $\alpha(X)\leq\frac{1}{8}$ since $(X,A)$ is not klt, a contradiction.

Now assume that ${{\text{\rm q}}\mathbb{Q}}(X)=4$, by [Pro13, Lemma 8.3], $\text{Cl}(X)$ is torsion-free and ${{\text{\rm qW}}}(X)={{\text{\rm q}}\mathbb{Q}}(X)$, hence there is a Weil divisor $A$ such that $-K_{X}\sim 4A$. If $g(X)\geq 22$, then by [Pro13, Theorem 1.2(vi)], $X\simeq\mathbb{P}^{3}$ or $X_{4}\subset\mathbb{P}(1,1,1,2,3)$. The latter is absurd, since it has a singularity of index $3$, and $(-K_{X_{4}})^{3}=128/3$, which contradicts to Theorem 5.1. If $20\leq g(X)\leq 21$, then we have the following possibilities due to computer computation (see [GRD], or [BS07, Pro10, Pro13]):

[TABLE]

Here $\mathbf{B}$ is the set local indices of singular points. It is easy to see that both cases contradict to Theorem 5.1.

In summary, Theorem 1.5 is proved.