Delta invariants of projective bundles and projective cones of Fano type

Abstract

In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.

1 Introduction

1.1 Motivation

Given an arbitrary Fano manifold V, it is often the case that V does not admit any Kähler–Einstein (KE) metric. But still, V always admits twisted KE or conical KE metrics. To study these metrics and their degenerations, some analytic and algebraic thresholds play important roles. For instance, the greatest Ricci lower bound $\beta (V)$ of Tian [43] measures how far V is away from a KE manifold. As shown in [2, 14], $\beta (V)$ is equal to the algebraic $\delta$-invariant introduced in [6, 18], which serves as the right threshold for V to be Ding-stable (cf. [2, 5, 9]).

So an important problem in algebraic geometry is to compute the $\delta$-invariants of Fano varieties. Although some efforts to tackle this problem have been made in low dimensions (see e.g. [14, 16, 37]), overall this is still a highly nontrivial problem. However as we shall see in this paper, when the variety enjoys certain symmetry, the difficulty could be substantially reduced.

More precisely, we will investigate projective bundles and projective cones of Fano type. These objects enjoy a natural $\mathbb {C}^*$-action in the fiber direction. On the analytic side, this torus action allows us to carry out the momentum construction due to Calabi, using which we can control the greatest Ricci lower bound from below. On the algebraic side, by using this torus action we can easily bound delta invariant from above. Surprisingly, the bounds coming from these two directions coincide and hence give us the precise formula for the $\delta$-invariant. This generalizes the example considered in [40, Section 3.1]. More generally we will also consider the (singular) log Fano setting and derive formulas for the corresponding $\delta$-invariants. In the log case, one can still carry out the Calabi trick as in [29]. But we will take a purely algebraic approach, again making use of the $\mathbb {C}^*$-action. Indeed, by [6, 21], to compute $\delta$, it is enough to consider $\mathbb {C}^*$-invariant divisorial valuations, which can greatly simplify the computation.

1.2 Main results

We work over complex number $\mathbb {C}$. We will first deal with the smooth projective bundles of Fano type, and then consider the more general log Fano case.

Let V be an n-dimensional Fano manifold with Fano index $I(V)\ge 2$. So we can find an ample line bundle L such that

$$\begin{aligned} L=-\frac{1}{r} K_V\text { for some rational number }r>1\text {.} \end{aligned}$$

(1)

We put

$$\begin{aligned} {\tilde{Y}}:=\mathbb {P}_V(L^{-1}\oplus {\mathcal {O}}_V)\xrightarrow {p}V, \end{aligned}$$

which is the $\mathbb {P}^1$-bundle over V with respect to $L^{-1}\oplus \mathcal {O}_V$. Let $V_0$ denote the zero section and $V_\infty$ the infinity section of ${\tilde{Y}}$. Then

$$\begin{aligned} -K_{{\tilde{Y}}}=p^*(-K_V)+V_0+V_\infty \sim _\mathbb {Q}(r+1)V_\infty -(r-1)V_0 \end{aligned}$$

is ample and hence ${\tilde{Y}}$ is an $(n+1)$-dimensional Fano manifold. We put

$$\begin{aligned} \beta _0:=\bigg (\frac{n+1}{n+2}\cdot \frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}-(r-1)\bigg )^{-1}. \end{aligned}$$

(2)

Using binomial formula, one can easily verify the following elementary fact:

$$\begin{aligned} \beta _0\in (1/2,1). \end{aligned}$$

(3)

Our first main result is stated as follows.

Theorem 1.1

One has

$$\begin{aligned} \delta ({\tilde{Y}})=\min \bigg \{\frac{\delta (V)r\beta _0}{1+\beta _0(r-1)},\beta _0\bigg \}. \end{aligned}$$

(4)

In particular, ${\tilde{Y}}$ cannot admit KE metrics. But as we shall see in Section A, ${\tilde{Y}}$ does admit a family of twisted conical KE metrics. When $\delta (V)\ge 1/r+\beta _0(1-1/r)$ (this holds for example when V is K-semistable), we deduce that

$$\begin{aligned} \delta (V)=\beta _0. \end{aligned}$$

(5)

As we will show, in this case $V_0$ computes $\delta ({\tilde{Y}})$. This generalizes the example ${\tilde{Y}}=Bl_1\mathbb {P}^2$ treated in [40]. Indeed, when ${\tilde{Y}}=Bl_1\mathbb {P}^2$, one has $V=\mathbb {P}^1$, $n=1$ and $r=2$, so that $\delta ({\tilde{Y}})=\beta _0=6/7$, which agrees with the result obtained in [27, 40]. In the case of $\delta (V)\le 1/r+\beta _0(1-1/r)$, Theorem 1.1 gives

$$\begin{aligned} \delta ({\tilde{Y}})=\frac{\delta (V)r\beta _0}{1+\beta _0(r-1)}. \end{aligned}$$

(6)

In this case, there always exists a prime divisor F over V computing $\delta (V)$ (see [9, Theorem 6.7]). This divisor naturally induces a divisor ${\overline{F}}$ over ${\tilde{Y}}$, and we will show that $\delta ({\tilde{Y}})$ is computed by ${\overline{F}}$.

Remark 1.2

In [44], Zhuang computed the $\delta$-invariants of product spaces. In particular, let ${\tilde{Y}}=V\times \mathbb {P}^1$ be the trivial $\mathbb {P}^1$-bundle over V, then

$$\begin{aligned} \delta ({\tilde{Y}})=\min \{\delta (V),1\}. \end{aligned}$$

So to some extent, Theorem 1.1 generalizes this product formula.

The smoothness assumption of ${\tilde{Y}}$ in Theorem 1.1 is only used for a simpler exposition of our argument in Section A. As we now show, one can consider the following more general singular setting.

Let V be a normal projective variety of dimension n and L an ample line bundle on V. We define the affine cone over V associated to L to be

$$\begin{aligned}X:= {\mathrm{Spec}}\oplus _{k\in \mathbb {N}}H^0(V,kL) ,\end{aligned}$$

and the projective cone over X associated to L to be

$$\begin{aligned}Y:={\mathrm{Proj}}\oplus _{k,j\in \mathbb {N}} H^0(V,kL)s^j.\end{aligned}$$

It is clear that both X and Y are normal varieties of dimension $n+1$ (to see X is a normal variety, we first focus on an integrally closed sub-ring $\oplus _{k\in \mathbb {N}}H^0(V,kmL)$ where m is sufficiently divisible such that mL is very ample, then $\oplus _{k\in \mathbb {N}}H^0(V,kL)$ is integral over $\oplus _{k\in \mathbb {N}}H^0(V,kmL)$), and Y is obtained by adding an infinite divisor $V_\infty$ to X, where $V_\infty$ is the infinite divisor on Y defined by $s=0$.

We provide another viewpoint of X and Y. Let $p:{{\tilde{Y}}}:=\mathbb {P}_V(L^{-1}\bigoplus \mathcal {O}_V)\rightarrow V$ be the projective $\mathbb {P}^1$-bundle over V associated to $L^{-1}\bigoplus \mathcal {O}_V$. Let $V_\infty$ and $V_0$ be the infinite and zero sections of p respectively. Then we know there is a contraction $\phi :{\tilde{Y}}\rightarrow Y$ which only contracts divisor $V_0$, and ${\tilde{Y}}{\setminus } V_0\cong Y{\setminus } o$, where o is the cone vertex of X.

If the base V is a $\mathbb {Q}$-Fano variety and $L\sim _\mathbb {Q}-\frac{1}{r}K_V$ is an ample line bundle on V for some positive rational number r, then it is not hard to see that ${\tilde{Y}}$ (for $r>1$) and Y (for $r>0$) are $\mathbb {Q}$-Fano varieties. Moreover, both $({\tilde{Y}}, aV_0+bV_\infty )$ and $(Y,cV_\infty )$ are log Fano varieties for any rational $0\le a<1,0\le b<1,0\le c<1$ if $r>1$, and $1-r<a<1, 0\le b<1, 0\le c<1$ if $0<r\le 1$. The main results of this paper is about the computation of delta invariants of these log Fano varieties.

Theorem 1.3

Let V be a $\mathbb {Q}$-Fano variety of dimension n and $L\sim _\mathbb {Q}-\frac{1}{r}K_V$ an ample line bundle on V for some positive rational number r. Let a, b be rational numbers such that $0\le a<1,0\le b<1$ if $r>1$, and $1-r<a<1, 0\le b<1$ if $0<r\le 1$. Then we have the following formula computing delta invariants,

$$\begin{aligned}\delta ({\tilde{Y}},aV_0+bV_\infty )= \min \bigg \{\frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}, \frac{1-a}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A}, \frac{1-b}{B-\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\bigg \},\end{aligned}$$

where $A=r-(1-a)$ and $B=r+(1-b)$. In particular, when V is a Fano manifold and $a=b=0$ (in this case, $r>1$ automatically), we have

$$\begin{aligned} \delta ({\tilde{Y}})= \min \bigg \{\frac{r\delta (V)}{\frac{n+1}{n+2}\frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}}, \frac{1}{\frac{n+1}{n+2}\frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}-(r-1)}\bigg \}. \end{aligned}$$

Note that when V is a Fano manifold, we naturally have $r\le n+1$, e.g [26, Chapter 5], thus one can easily have the comparison

$$\begin{aligned}\frac{1}{\frac{n+1}{n+2}\frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}-(r-1)}\le \frac{1}{r+1-\frac{n+1}{n+2}\frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}}.\end{aligned}$$

This is precisely what we get in Theorem 1.1. If we write $\beta _{a,b}=\frac{1-a}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A}$ and $\beta _{a,b}'=\frac{1-b}{B-\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}$, then the above formula can be simplified as

$$\begin{aligned}&\delta ({\tilde{Y}},aV_0+bV_\infty )= \min \bigg \{\frac{r\delta (V)}{1-a+A\beta _{a,b}}\beta _{a,b}, \beta _{a,b},\frac{(1-b)\beta _{a,b}}{(B-A)\beta _{a,b}-(1-a)}\bigg \}, \\&\delta ({\tilde{Y}})=\min \bigg \{\frac{r\delta (V)\beta _{0,0}}{1+(r-1)\beta _{0,0}}, \beta _{0,0}, \frac{\beta _{0,0}}{2\beta _{0,0}-1}\bigg \}.\end{aligned}$$

We just note here that $\beta _{a,b}$ is the delta invariant computed by the divisor $V_0$, while $\beta _{a,b}'$ is computed by the divisor $V_\infty$, and $\frac{r\delta (V)}{1-a+A\beta _{a,b}}\beta _{a,b}$ is computed by some divisor arising from V.

The following formula tells us the delta invariant of the projective cone.

Theorem 1.4

Let V be a $\mathbb {Q}$-Fano variety of dimension n and $L\sim _\mathbb {Q}-\frac{1}{r}K_V$ an ample line bundle on V for some positive rational number r, then we have the following formula computing delta invariants

$$\begin{aligned}\delta (Y,cV_\infty )= \min \bigg \{\frac{(n+2)r}{(n+1)(r+1-c)}\delta (V), \frac{(n+2)r}{(n+1)(r+1-c)}, \frac{(n+2)(1-c)}{r+1-c}\bigg \},\end{aligned}$$

where $0\le c<1$ is a rational number.

Remark 1.5

In fact, [30, Remark 4.5] has essentially obtained above result for $0<r\le 1$.

Assume $c=0$. If $\delta (V)\ge 1$, we naturally have $r\le n+1$, see [19, 28]. If $\delta (V)<1$ then we have $r\delta (V)\le n+1$ by [6, Theorem D]. Hence the following result is deduced.

Corollary 1.6

Let V be a $\mathbb {Q}$-Fano variety of dimension n and $L\sim _\mathbb {Q}-\frac{1}{r}K_V$ an ample line bundle on V for some positive rational number r.

• (1) If V is K-semistable, i.e. $\delta (V)\ge 1$, then $\delta (Y)$ is achieved by $V_0$, that is, $\delta (Y)=\frac{A_Y(V_0)}{S_Y(V_0)}=\frac{(n+2)r}{(n+1)(r+1)}\le 1$,

• (2) If V is not K-semistable, i.e. $\delta (V)<1$, then $\delta (Y)=\frac{(n+2)r}{(n+1)(r+1)}\delta (V)<1$.

As we have mentioned in the beginning of this paper, there may not have any KE metric on an arbitrarily given Fano manifold, however, there may be conical KE metrics along some smooth divisors. Before we state next theorem, we first fix some notation. Let V be a projective Fano manifold of dimension n, and S is a smooth divisor on V such that $S\sim _\mathbb {Q}-\lambda K_V$ for some positive rational number $\lambda$. Write

$$\begin{aligned}E(V,S):=\{a\in [0,1)| (V,aS)\textit{ is K-semistable} \}.\end{aligned}$$

As an application of Theorem 1.4, we prove the following theorem on optimal angle of K-stability.

Theorem 1.7

Notation as above, suppose V and S are both K-semistable and $0<\lambda <1$, then $E(V,S)= [0,1-\frac{r}{n}]$, where $r=\frac{1}{\lambda }-1$. In particular, if V and S are both K-polystable, then (V, aS) is K-polystable for any $a\in [0, 1-\frac{r}{n})$.

This result has been essentially known to experts (cf. [29, 31, 35] etc.), however, according to the authors’ knowledge, it has not been explicitly written down. Since it can be derived by Theorem 1.4, we just put it here and provide a complete algebraic proof. As a corollary, we directly have following result, which provides an answer to the question posed in [17, Remark 3.6]

Corollary 1.8

For the pair $(\mathbb {P}^n, S_d)$ where $S_d$ is a smooth hypersurface of degree $1\le d\le n$. If $S_d$ is K-polystable (this is expected to be true), then we have $E(\mathbb {P}^n, S_d)=[0, 1-\frac{r}{n}]$, where $r=\frac{n+1-d}{d}$.

The paper is organized as follows. In Sect. 2, we give a brief introduction to delta invariant and greatest lower Ricci bound, and also include some results on projective bundles and cones of Fano type. In Sects. 3 and 4, we prove Theorem 1.3 and Theorem 1.4 respectively, by purely algebraic method. In Sect. 5, we give a complete proof of Theorem 1.7. In the last section, as an application of our main results, we give some examples on computing delta invariants of some special hypersurfaces. Finally in Section A, we use Calabi ansatz to prove a weaker version of Theorem 1.1.

2 Preliminaries

In this section, we will collect some fundamental results on projective bundles and projective cones, then we give a quick introduction of delta invariant which will play a central role in the subsequent contents, and finally we briefly recall the definition of the greatest Ricci lower bound. We say $(V,\Delta )$ is a log pair if V is a projective normal variety and $\Delta$ is an effective $\mathbb {Q}$-divisor on V such that $K_V+\Delta$ is $\mathbb {Q}$-Cartier. Suppose $f: W\rightarrow V$ is a proper birational mophism between normal varieties and E is a prime divisor on W, then we define

$$\begin{aligned}A_{V,\Delta }(E):={\mathrm{ord}}_E(K_W-f^*(K_V+\Delta ))\end{aligned}$$

to be the log discrepancy of the divisor E associated to the pair $(V,\Delta )$. The log pair $(V,\Delta )$ is called a log Fano variety if it admits klt singularities and $-(K_V+\Delta )$ is ample. If $\Delta =0$, we just say V is a $\mathbb {Q}$-Fano variety. For the concepts of klt singularities, please refer to [24, 25].

2.1 Projective bundles and projective cones

Throughout, (V, L) will be a polarized pair where V is a normal projective variety of dimension n and L an ample line bundle on V. Just as in the introduction, we fix some notation below,

$$\begin{aligned}&Y:={\mathrm{Proj}}\oplus _{k\in \mathbb {N}}\oplus _{l\in \mathbb {N}}H^0(V,kL)s^l, {\tilde{Y}}:=\mathbb {P}_V(L^{-1}\oplus \mathcal {O}_V), \\&X:={\mathrm{Spec}}\oplus _{k\in \mathbb {N}} H^0(V,kL)= Y{\setminus } V_\infty , {\tilde{X}}:={\tilde{Y}}{\setminus } V_\infty ,\end{aligned}$$

where $V_\infty$ is the infinite section of ${\tilde{Y}}$. There is a natural contraction $\phi : {\tilde{Y}}\rightarrow Y$ (resp. $\phi : {\tilde{X}}\rightarrow X$) which contracts $V_0$, where $V_0$ is the zero section of ${\tilde{Y}}$. We just list the properties of ${\tilde{Y}}$ and Y in the following lemma.

Lemma 2.1

Notation as above, let $p: {\tilde{Y}}:=\mathbb {P}_V(L^{-1}\oplus \mathcal {O}_V)\rightarrow V$ be the natural projection, then we have

• (1) $V_\infty -V_0\sim p^*L, V_\infty |_{V_\infty }\sim L, V_0|_{V_0}\sim L^{-1}.$

• (2) $\mathcal {O}_{{\tilde{Y}}}(1)\sim V_\infty ,$

• (3) $\phi :{\tilde{Y}}\rightarrow Y$ is the blowup of vertex point $o\in X\subset Y$, and ${\tilde{Y}}{\setminus } V_0\cong Y{\setminus } o$, where o is the cone vertex of X.

• (4) If V is $\mathbb {Q}$-Gorenstein, then $K_{{\tilde{Y}}}=p^*K_V\otimes \mathcal {O}_{{\tilde{Y}}}(-2)\otimes p^*L\sim p^*K_V-V_\infty -V_0.$

• (5) If V is a $\mathbb {Q}$-Fano variety and $L\sim _\mathbb {Q}-\frac{1}{r}K_V$ is an ample line bundle on V for some positive rational number r, then both $({\tilde{Y}},aV_0+bV_\infty )$ and $(Y,cV_\infty )$ are log Fano pairs, where $0\le a<1,0\le b<1,0\le c<1$ if $r>1$, and $1-r<a<1, 0\le b<1,0\le c<1$ if $0<r\le 1$.

Proof

By the construction we see that

$$\begin{aligned}{\tilde{Y}}\cong {\mathrm{Proj}}_V S(L\oplus \mathcal {O}_V)={\mathrm{Proj}}_V \oplus _{i,j\in \mathbb {N}} L^is^j \rightarrow {\mathrm{Proj}}\oplus _{i,j\in \mathbb {N}} H^0(V, iL)s^j:=Y\end{aligned}$$

is the natural blowup of the vertex $o\in Y$, thus $\mathcal {O}_{{\tilde{Y}}}(1)$ is defined by $s=0$, which is the infinite section $V_\infty$, and $V_0|_{V_0}\sim L^{-1}$. It’s clear that $V_\infty -V_0$ is a relative trivial line bundle for the projection map $p: {\tilde{Y}}\rightarrow V$, thus there is a line bundle M on V such that $p^*M\sim V_\infty -V_0$. Restrict it to $V_0$ we see $M\sim L$, and restrict it to $V_\infty$ we the see that $V_\infty |_{V_\infty }\sim L$. By now we complete the proof of the first three statements. The fourth one follows directly from (1) and (2). The only issue is to make sure $(Y,cV_\infty )$ are indeed log Fano varieties, specially with klt singularities. This follows from the following Theorem 2.3. $\square$

Let $\Delta$ be an effective $\mathbb {Q}$-divisor on V such that $K_V+\Delta$ is $\mathbb {Q}$-Cartier, and $\Delta _X$ and $\Delta _{{\tilde{X}}}$ be the corresponding extending divisors on X and ${\tilde{X}}$ respectively. We have following results [25].

Lemma 2.2

Notation as above, we have

• (1) ${\mathrm{Cl}}({\tilde{X}})\cong {\mathrm{Cl}}(V).$

• (2) ${\mathrm{Pic}}(X)=0.$

• (3) ${\mathrm{Cl}}(X)\cong {\mathrm{Cl}}(V)/\mathbb {Z}[L].$

• (4) $K_{{\tilde{X}}}+\Delta _{{\tilde{X}}}+V_0\sim p^*(K_V+\Delta ),$ where we also use p to denote the projection ${\tilde{X}}\rightarrow V$.

• (5) We assume more that $K_X+\Delta _X$ is $\mathbb {Q}$-Cartier, then there is some rational number r such that $L\sim _\mathbb {Q}-\frac{1}{r}(K_V+\Delta )$.

• (6) Conversely, if $K_V+\Delta$ is proportional to L, then $K_X+\Delta _X$ is $\mathbb {Q}$-Cartier.

• (7) We still assume $K_X+\Delta _X$ is $\mathbb {Q}$-Cartier, then $K_{{\tilde{X}}}+\Delta _{{\tilde{X}}}+V_0=\phi ^*(K_X+\Delta _X)+rV_0,$ where r is as above and $r=A_{X,\Delta _X}(V_0).$

Proof

For (1), one only needs to note that ${\tilde{X}}$ is an $\mathbb {A}^1$-bundle over V. For (2), we consider $V\cong V_0\hookrightarrow {\tilde{X}}\rightarrow X$, where the last arrow denoted by $\phi$ means the blowup of the cone vertex. For any line bundle M on X, it is pulled back to ${\tilde{X}}$ to be a trivial line bundle, thus M is also a trivial line bundle. For (3), consider the exact sequence

$$\begin{aligned}\mathbb {Z}[V_0]\rightarrow {\mathrm{Cl}}({\tilde{X}})\rightarrow {\mathrm{Cl}}(X{\setminus } o)\rightarrow 0,\end{aligned}$$

then the facts ${\mathrm{Cl}}(X{\setminus } o)\cong {\mathrm{Cl}}(X)$ and $V_0|_{V_0}\sim L^{-1}$ conclude the result. For (4), it is directly implied if we write down the differential form. For (5), as $m(K_X+\Delta _X)$ is a trivial line bundle for a divisible m due to (2), $m(K_{X{\setminus } o}+\Delta _{X{\setminus } o})$ is also trivial, then by the exact sequence above, one sees there is a rational number r such that $p^*(K_V+\Delta )\sim _\mathbb {Q}rV_0$. Hence (5) is finished by restriction. For (6), one only needs to note that $m(K_X+\Delta _X)$ is Cartier if and only if $m(K_X+\Delta _X)$ is a trivial line bundle if and only if $m(K_{X{\setminus } o}+\Delta _{X{\setminus } o})$ is a trivial line bundle, then apply the above exact sequence. For (7), we apply (5) to get a rational number r such that $L\sim _\mathbb {Q}-\frac{1}{r}(K_V+\Delta )$, so $K_{{\tilde{X}}}+\Delta _{{\tilde{X}}}+V_0\sim _{\mathbb {Q},\phi } rV_0$ by (4). $\square$

The above lemma directly implies the following result on cone singularities, also see [25].

Theorem 2.3

Let $(V,\Delta )$ be a log pair of dimension n, then

• (1) If $(V,\Delta )$ is a log Fano pair and $L=-\frac{1}{r}(K_V+\Delta )$ is an ample line bundle on V for some rational $r>0$, then $(X,\Delta _X)$ admits klt singularities.

• (2) If $(V,\Delta )$ is a general type pair and $L=-\frac{1}{r}(K_V+\Delta )$ is an ample line bundle on V for some rational $r<0$, then $(X,\Delta _X)$ is not log canonical.

• (3) If $(V,\Delta )$ is a log canonical Calabi-Yau pair and L is an ample line bundle on V, then $(X,\Delta _X)$ is log canonical.

Corollary 2.4

Let V be a projective normal variety and L an ample line bundle on V. If $X:={\mathrm{Spec}}\oplus _{k\in bN} H^0(V,kL)$ admits klt singularities, specially, $K_X$ is $\mathbb {Q}$-Cartier, then there is a positive rational number $r>0$ such that $L\sim _\mathbb {Q}-\frac{1}{r}K_V$. In particular, V is a $\mathbb {Q}$-Fano variety.

2.2 Delta invariant

In this section we assume $(V,\Delta )$ is a log Fano variety of dimension n. Delta invariant is introduced in [18] to measure the singularities of the anti-canonical divisor of a $\mathbb {Q}$-Fano variety, which is proved to be a powerful K-stability threshold and has led to many progresses in the field of K-stability of Fano varieties. For various concepts of K-stability please refer to [6, 8, 10, 20] etc.. We first define m-basis type divisors of $(V,\Delta )$.

Definition 2.5

For a sufficiently divisible fixed natural number m such that $-m(K_V+\Delta )$ is an ample line bundle, one chooses any complete basis of $H^0(V,-m(K_V+\Delta ))$, say $\{s_1,s_2, \dots ,s_{r_m} \}$, where $r_m=\dim H^0(V,-m(K_V+\Delta ))$, define $D_m:=\frac{\sum _{i=1}^{r_m}\mathrm{div}(s_i=0)}{mr_m}$. Divisors of this form are called m-basis type divisors. It is clear that $D_m\sim _\mathbb {Q}-(K_V+\Delta )$. For a prime divisor E over V (i.e. there exists a proper birational morphism from a normal variety $W\rightarrow V$ such that E is a prime divisor on W), we define the following $S_m$-invariant and m-th delta invariant of $(V,\Delta )$,

$$\begin{aligned}&S_{m,(V,\Delta )}(E):=\sup _{D_m}{\mathrm{ord}}_E(D_m), \\&\delta _{m,(V,\Delta )}({\mathrm{ord}}_E):=\frac{A_{(V,\Delta )}(E)}{S_{m,(V,\Delta )}(E)}.\end{aligned}$$

The following lemma is due to [6, 18].

Lemma 2.6

Notation as in above definition,

• (1) For each sufficiently divisible m and a prime divisor E over V, there is an m-basis type divisor $D_m$ such that $S_{m,(V,\Delta )}(E)={\mathrm{ord}}_E(D_m)$.

• (2) Let m tend to infinity, then the limits in above definition indeed exist, denoted by $S_{(V,\Delta )}(E)$ and $\delta _{(V,\Delta )}({\mathrm{ord}}_E)$.

• (3) S-invariant can be computed by the following integration,

  $$\begin{aligned}&S_{(V,\Delta )}(E)=\frac{1}{{\mathrm{vol}}(-(K_V+\Delta ))}\int _0^{\infty }{\mathrm{vol}}(-(K_V+\Delta )-tE){\mathrm{dt}}, \\&\delta _{(V,\Delta )}({\mathrm{ord}}_E)=\frac{A_{(V,\Delta )}(E)}{S_{(V,\Delta )}(E)}.\end{aligned}$$

Definition 2.7

Delta invariant of the log Fano pair $(V,\Delta )$ is defined to be

$$\begin{aligned}\delta (V,\Delta ):=\inf _E\delta _{(V,\Delta )}({\mathrm{ord}}_E)=\inf _E \frac{A_{(V,\Delta )}(E)}{S_{(V,\Delta )}(E)},\end{aligned}$$

where E runs through all prime divisors over V.

We sometimes leave out $(V,\Delta )$ in the subscript if there is no confusion. It is clear that the above definition applies to any $\mathbb {Q}$-ample line bundle L on V, and we respectively get m-basis type divisors associated to L just by replacing $-(K_V+\Delta )$ by L. In this case, we use $S_{m,L}(E)$ and $S_L(E)$ (resp. $\delta _{m,L}({\mathrm{ord}}_E)$ and $\delta _L({\mathrm{ord}}_E))$ to denote the $S_m$-invariant and S-invariant (resp. m-th delta invariant and delta invariant), and $\delta (L):=\inf _E \frac{A_{(V,\Delta )}(E)}{S_L(E)}$.

The following result is well known by works [6, 18], we just state it here.

Theorem 2.8

The log Fano pair $(V,\Delta )$ is K-semistable if and only if $\delta (V,\Delta )\ge 1$.

2.3 The greatest Ricci lower bound

Let V be a Fano manifold. Then the greatest Ricci lower bound $\beta (V)$ of V is defined to be

$$\begin{aligned} \beta (V):=\sup \{\beta \in \mathbb {R}\ |\ \exists \text { K}\ddot{\text {a}}\text {hler form }\omega \in 2\pi c_1(V)\ \text {s.t. }{{\text {Ric}}}(\omega )\ge \beta \omega \}. \end{aligned}$$

(7)

This invariant was first implicitly studied by Tian [43] and then explicitly introduced in [38, 39]. Recently it is shown independently by [14] and [2] that

$$\begin{aligned} \beta (X)=\min \{1,\delta (X)\}. \end{aligned}$$

(8)

Finally we remark that, suppose in addition that there is a semipositive (1, 1)-current $\theta$ on X, then there are analogous results for the $\theta$-twisted $\delta$- and $\beta$-invariants (see [2] for more information).

3 Delta invariants of projective bundles of Fano type

In this section, we will prove Theorem 1.3. We start by the following lemma computing $\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_0})$ and $\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_\infty })$.

Lemma 3.1

Notation as in Theorem 1.3, we have

$$\begin{aligned}\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_0})= \frac{1-a}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A}=:\beta _{a,b},\end{aligned}$$

and

$$\begin{aligned}\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_\infty })= \frac{1-b}{B-\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}=:\beta _{a,b}'.\end{aligned}$$

Proof

It is clear that $A_{({\tilde{Y}},aV_0+bV_\infty )}(V_0)=1-a, A_{({\tilde{Y}},aV_0+bV_\infty )}(V_\infty )=1-b$, and by Lemma 2.1,

$$\begin{aligned} -(K_{{\tilde{Y}}}+aV_0+bV_\infty )\sim _\mathbb {Q}(r+1-b)p^*L+(2-a-b)V_0. \end{aligned}$$

(9)

Denote ${\mathrm{vol}}(t):={\mathrm{vol}}((r+1-b)p^*L+(2-a-b-t)V_0)$, then we have

$$\begin{aligned}&S_{({\tilde{Y}},aV_0+bV_\infty )}(V_0)\\&\quad =\frac{1}{{\mathrm{vol}}(-(K_{{\tilde{Y}}}+aV_0+bV_\infty ))}\int _0^{\infty }{\mathrm{vol}}(-(K_{{\tilde{Y}}}+aV_0+bV_\infty )-tV_0){\mathrm{dt}}\\&\quad =\frac{1}{{\mathrm{vol}}(t=0)}\int _0^{2-a-b}{\mathrm{vol}}((r+1-b)p^*L+(2-a-b-t)V_0)\\&\quad =\frac{L^n}{{\mathrm{vol}}(t=0)}\int _0^{2-a-b}\sum _{j=1}^{n+1}\left( {\begin{array}{c}n+1\\ j\end{array}}\right) (r+1-b)^{n+1-j}(2-a-b-t)^j(-1)^{j+1}{\mathrm{dt}}\\&\quad =\frac{L^n}{{\mathrm{vol}}(t=0)}\int _0^{2-a-b}-(r+1-b-(2-a-b-t))^{n+1}+(r+1-b)^{n+1}{\mathrm{dt}}\\&\quad =\frac{1}{B^{n+1}-A^{n+1}}\int _0^{B-A}(B^{n+1}-(A+t)^{n+1}){\mathrm{dt}}\\&\quad =\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A. \end{aligned}$$

Similarly we have

$$\begin{aligned}S_{({\tilde{Y}},aV_0+bV_\infty )}(V_\infty ) =B-\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}.\end{aligned}$$

The proof is finished. $\square$

Note that the above lemma is essentially a Futaki invariant computation under a fancy $\delta$-invariant guise for the natural $\mathbb {C}^*$-action on ${\tilde{Y}}$ (see the proof of Lemma A.4), and it tells us that $\delta ({\tilde{Y}},aV_0+bV_\infty )\le \min \{\beta _{a,b},\beta _{a,b}'\}$, we next show that $\delta ({\tilde{Y}},aV_0+bV_\infty )\le \frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}$.

Lemma 3.2

Notation as in Theorem 1.3, we have

$$\begin{aligned}\delta ({\tilde{Y}},aV_0+bV_\infty )\le \frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}.\end{aligned}$$

Proof

We choose sufficiently divisible m such that ma, mb and mr are all integers, and by Lemma 2.1 we have

$$\begin{aligned} -m(K_{{\tilde{Y}}}+aV_0+bV_\infty )\sim (mr-m+ma+j)p^*L+(2m-ma-mb-j)V_\infty +jV_0. \end{aligned}$$

(10)

Let ${\tilde{R}}_m$ denote $H^0({\tilde{Y}},-m(K_{{\tilde{Y}}}+aV_0+bV_\infty ))$, we define

$$\begin{aligned}{\tilde{R}}_{m,j}:=\{s\in {\tilde{R}}_m|{{\mathrm{ord}}_{V_0}(s)=j}\}.\end{aligned}$$

By (9), one sees that

$$\begin{aligned}{\tilde{R}}_m=\bigoplus ^{m(2-a-b)}_{j=0} {\tilde{R}}_{m,j},\end{aligned}$$

and by (10)

$$\begin{aligned} {\tilde{R}}_{m,j}\cong H^0(V, (mr-m+ma+j)L) . \end{aligned}$$

(11)

Let E be a prime divisor over V and $E_{{\tilde{Y}}}$ the natural extended divisor over ${\tilde{Y}}$, then we want to explore the relationship between $\delta _L({\mathrm{ord}}_E)$ and $\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{E_{{\tilde{Y}}}})$. As it is clear that $A_V(E)=A_{({\tilde{Y}},aV_0+bV_\infty )}(E_{{\tilde{Y}}})$, it suffices to explore the relationship between $S_L(E)$ and $S_{({\tilde{Y}},aV_0+bV_\infty )}(E_{{\tilde{Y}}})$. We first construct a special m-basis type divisor of $-(K_{{\tilde{Y}}}+aV_0+bV_\infty )$.

Write ${\tilde{r}}_{m,j}=\dim {\tilde{R}}_{m,j}=\dim H^0(V,(mr-m+ma+j)L)$, we can choose an $mr-m+ma+j$-basis type divisor ${\tilde{D}}_{m,j}$ of L such that ${\tilde{D}}_{m,j}$ computes $S_{mr-m+ma+j,L}(E)$. In fact, ${\tilde{D}}_{m,j}$ is created by the filtration induced by ${\mathrm{ord}}_E$ on $H^0(V,(mr-m+ma+j)L)$. Then we lift ${\tilde{D}}_{m,j}$ to be a divisor ${\tilde{\mathcal {D}}}_{m,j}$ over ${\tilde{Y}}$. By (10), it is not hard to see that

$$\begin{aligned} {\tilde{\mathcal {D}}}_m:=\frac{\sum _{j=0}^{m(2-a-b)}{\tilde{r}}_{m,j}((mr-m+ma+j){\tilde{\mathcal {D}}}_{m,j}+jV_0+(2m-am-bm-j)V_\infty )}{m\sum _{j=0}^{m(2-a-b)} {\tilde{r}}_{m,j}} \end{aligned}$$

(12)

is an m-basis type divisor of $-(K_{{\tilde{Y}}}+aV_0+bV_\infty )$. Then we have

$$\begin{aligned} S_{m,({\tilde{Y}},aV_0+bV_\infty )}(E_{{\tilde{Y}}})\ge {\mathrm{ord}}_{E_{{\tilde{Y}}}}({\tilde{\mathcal {D}}}_m)=\frac{\sum _{j=0}^{m(2-a-b)}(mr-m+ma+j){\tilde{r}}_{m,j}S_{mr-m+ma+j,L}(E)}{m\sum _{j=0}^{m(2-a-b)}{\tilde{r}}_{m,j}}. \end{aligned}$$

(13)

As m tends to infinity, by the computation of $\sum _{j=0}^{m(2-a-b)}j{\tilde{r}}_{m,j}$ and $\sum _{j=0}^{m(2-a-b)}m{\tilde{r}}_{m,j}$ in the next lemma, one directly obtains that

$$\begin{aligned} S_{({\tilde{Y}}, aV_0+bV_\infty )}(E_{{\tilde{Y}}})\ge \frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}\cdot S_L(E), \end{aligned}$$

(14)

where $B=r+(1-b)$ and $A=r-(1-a)$, hence we have

$$\begin{aligned} \delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{E_{{\tilde{Y}}}}) \le \frac{1}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\delta _L({\mathrm{ord}}_E). \end{aligned}$$

(15)

As the prime divisor E is arbitrarily chosen, we have

$$\begin{aligned}\delta ({\tilde{Y}}, aV_0+bV_\infty )\le \frac{1}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\delta (L)= \frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}. \end{aligned}$$

$\square$

Lemma 3.3

Notation as in the proof of Lemma 3.2, we have

$$\begin{aligned}\lim _{m\rightarrow \infty }\frac{\sum _{j=0}^{m(2-a-b)}j{\tilde{r}}_{m,j}}{m\sum _{j=0}^{m(2-a-b)}{\tilde{r}}_{m,j}} =\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A.\end{aligned}$$

Proof

The lemma is concluded by the following two computations

$$\begin{aligned}&\lim _{m\rightarrow \infty }\frac{\sum _{j=0}^{m(B-A)} jh^0(V,(mA+j)L)}{m^{n+2}/n!} =\lim _{m\rightarrow \infty }\sum _j\frac{j}{m}\frac{h^0(V,(mA+j)L)}{m^n/n!}\frac{1}{m}\\&\quad =\int _0^{B-A}{\mathrm{vol}}((A+t)L)t{\mathrm{dt}}=L^n\cdot \bigg \{\frac{B^{n+2}-A^{n+2}}{n+2}-A\frac{B^{n+1}-A^{n+1}}{n+1} \bigg \}, \end{aligned}$$

and

$$\begin{aligned}&\lim _{m\rightarrow \infty }\frac{\sum _{j=0}^{m(B-A)} mh^0(V,(mA+j)L)}{m^{n+2}/n!} =\lim _{m\rightarrow \infty }\sum _j\frac{h^0(V,(mA+j)L)}{m^n/n!}\frac{1}{m}\\&\quad =\int _0^{B-A}{\mathrm{vol}}((A+t)L){\mathrm{dt}}=L^n\cdot \frac{B^{n+1}-A^{n+1}}{n+1}. \end{aligned}$$

$\square$

Combine Lemmas 3.1 and 3.2 , we have the following result on upper bound.

Theorem 3.4

Notation as in Theorem 1.3, we have

$$\begin{aligned} \delta ({\tilde{Y}},aV_0+bV_\infty )\le \min \bigg \{\frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}, \frac{1-a}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A}, \frac{1-b}{B-\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A}\bigg \}. \end{aligned}$$

(16)

In particular, when $a=b=0$ (in this case, $r>1$ automatically), we have

$$\begin{aligned} \delta ({\tilde{Y}})\le \min \bigg \{\frac{r\delta (V)}{\frac{n+1}{n+2}\frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}}, \frac{1}{\frac{n+1}{n+2}\frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}}-(r-1)}\bigg \}. \end{aligned}$$

(17)

Now we turn to the converse direction. First recall that in the proof of Lemma 3.2, we construct a special m-basis type divisor ${\tilde{\mathcal {D}}}_m$ for $-(K_{{\tilde{Y}}}+aV_0+bV_\infty )$, however, we don’t know whether it is compatible with $E_{{\tilde{Y}}}$, so we only have $S_{m,({\tilde{Y}},aV_0+bV_\infty )}(E_{{\tilde{Y}}})\ge {\mathrm{ord}}_{E_{{\tilde{Y}}}}({\tilde{\mathcal {D}}}_m)$ in (13). Once this is indeed an equality, so are (14) and (15), which will lead to the final proof of Theorem 1.3.

Lemma 3.5

Notation as in the proof of Lemma 3.2, the m-basis divisor ${\tilde{\mathcal {D}}}_m$ we construct is compatible with $E_{{\tilde{Y}}}$, that is

$$\begin{aligned}S_{m,({\tilde{Y}},aV_0+bV_\infty )}(E_{{\tilde{Y}}})= {\mathrm{ord}}_{E_{{\tilde{Y}}}}({\tilde{\mathcal {D}}}_m),\end{aligned}$$

and

$$\begin{aligned}\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{E_{{\tilde{Y}}}}) = \frac{1}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\delta _L({\mathrm{ord}}_E)=\frac{r}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\delta _V({\mathrm{ord}}_E).\end{aligned}$$

Proof

Recall in the proof of Lemma 3.2, we obtained

$$\begin{aligned}{\tilde{R}}_m=H^0({\tilde{Y}},-m(K_{{\tilde{Y}}}+aV_0+bV_\infty ))=\bigoplus _{j=0}^{m(2-a-b)}{\tilde{R}}_{m,j},\end{aligned}$$

and

$$\begin{aligned}{\tilde{R}}_{m,j}\cong H^0(V,(mr-m+ma+j)L).\end{aligned}$$

As $E_{{\tilde{Y}}}$ induces a $\mathbb {C}^*$-invariant divisorial valuation over ${\tilde{Y}}$, then the filtration induced by $E_{{\tilde{Y}}}$ on ${\tilde{R}}_{m}$ is compatible with the filtration induced by E on $H^0(V,(mr-m+ma+j)L)$, concluded. $\square$

Proof of Theorem 1.3

By Lemma 3.5, we let m tend to infinity, then both (14) and (15) are equalities for any prime divisor E over V, that is

$$\begin{aligned} \delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{E_{{\tilde{Y}}}}) = \frac{1}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\delta _L({\mathrm{ord}}_E) \ge \frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}. \end{aligned}$$

(18)

Let $T:=\mathbb {C}^*$. As $\delta ({\tilde{Y}},aV_0+bV_\infty )$ can be approximated by T-equivariant divisorial valuations over ${\tilde{Y}}$, see [21, Section 4] or [6, Section 7], we need to deal with the T-invariant valuations over ${\tilde{Y}}$ whose centers lie in $V_0$ or $V_\infty$. Assume ${\tilde{F}}\ne V_0$ is a T-invariant prime divisor over ${\tilde{Y}}$ such that $c_{{\tilde{Y}}}({\tilde{F}})\subset V_0$, which means the center of ${\mathrm{ord}}_{{\tilde{F}}}$ lies in $V_0$, then there is a positive rational number c and a prime divisor F over V such that $r({\mathrm{ord}}_{{\tilde{F}}})=c\cdot {\mathrm{ord}}_F$, where $r({\mathrm{ord}}_{{\tilde{F}}})$ is the restriction of ${\mathrm{ord}}_{{\tilde{F}}}$ to K(V). Let $F_{{\tilde{Y}}}$ be the induced prime divisor over ${\tilde{Y}}$ extended by F, then by [4] ${\mathrm{ord}}_{{\tilde{F}}}$ is a quasimonomial valuation along $V_0$ and $F_{{\tilde{Y}}}$ with weights $({\mathrm{ord}}_{{\tilde{F}}}(V_0),c)$, that is ${\mathrm{ord}}_{{\tilde{F}}}={\mathrm{ord}}_{{\tilde{F}}}(V_0)\cdot {\mathrm{ord}}_{V_0}+c\cdot {\mathrm{ord}}_{F_{{\tilde{Y}}}}$, so by (10) and [23] we have

$$\begin{aligned} A_{({\tilde{Y}},aV_0+bV_\infty )}({\tilde{F}})&={\mathrm{ord}}_{{\tilde{F}}}(V_0)\cdot A_{({\tilde{Y}},aV_0+bV_\infty )}(V_0)+c\cdot A_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}})\\&={\mathrm{ord}}_{{\tilde{F}}}(V_0)(1-a)+c\cdot A_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}}), \end{aligned}$$

and

$$\begin{aligned} S_{({\tilde{Y}},aV_0+bV_\infty )}({\tilde{F}})&={\mathrm{ord}}_{{\tilde{F}}}(V_0)\lim _{m\rightarrow \infty }\frac{\sum _{j=0}^{m(B-A)}j{\tilde{r}}_{m,j}}{\sum _{j=0}^{m(B-A)}m{\tilde{r}}_{m,j}}+c\cdot S_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}})\\&= {\mathrm{ord}}_{{\tilde{F}}}(V_0)\bigg \{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}-A \bigg \}+ c\cdot S_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}}), \end{aligned}$$

so by Lemma 3.1 the following holds,

$$\begin{aligned} \delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{{\tilde{F}}})\ge \min \{\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_0}),\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{F_{{\tilde{Y}}}})\}. \end{aligned}$$

For T-invariant divisors whose centers lie in $V_\infty$ but not equal to $V_\infty$, the analysis is similar. One can write

$$\begin{aligned}{\mathrm{ord}}_{{\tilde{F}}}={\mathrm{ord}}_{{\tilde{F}}}(V_\infty )\cdot {\mathrm{ord}}_{V_\infty }+c\cdot {\mathrm{ord}}_{F_{{\tilde{Y}}}}\end{aligned}$$

for some positive rational c, so by (10) and [23] we have

$$\begin{aligned} A_{({\tilde{Y}},aV_0+bV_\infty )}({\tilde{F}})&={\mathrm{ord}}_{{\tilde{F}}}(V_\infty )\cdot A_{({\tilde{Y}},aV_0+bV_\infty )}(V_\infty )+c\cdot A_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}})\\&={\mathrm{ord}}_{{\tilde{F}}}(V_\infty )(1-b)+c\cdot A_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}}), \end{aligned}$$

and

$$\begin{aligned} S_{({\tilde{Y}},aV_0+bV_\infty )}({\tilde{F}})&={\mathrm{ord}}_{{\tilde{F}}}(V_\infty )\lim _{m\rightarrow \infty }\frac{\sum _{j=0}^{m(B-A)}(m(B-A)-j){\tilde{r}}_{m,j}}{\sum _{j=0}^{m(B-A)}m{\tilde{r}}_{m,j}}+c\cdot S_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}})\\&= {\mathrm{ord}}_{{\tilde{F}}}(V_\infty )\bigg \{B-\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}} \bigg \}+ c\cdot S_{({\tilde{Y}},aV_0+bV_\infty )}(F_{{\tilde{Y}}}), \end{aligned}$$

so by Lemma 3.1 the following holds,

$$\begin{aligned} \delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{{\tilde{F}}})\ge \min \{\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_\infty }),\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{F_{{\tilde{Y}}}})\}. \end{aligned}$$

Combine inequality (18), we have the following

$$\begin{aligned}\delta ({\tilde{Y}},aV_0+bV_\infty )\ge \min \bigg \{\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_0}),\delta _{({\tilde{Y}},aV_0+bV_\infty )}({\mathrm{ord}}_{V_\infty }), \frac{r\delta (V)}{\frac{n+1}{n+2}\frac{B^{n+2}-A^{n+2}}{B^{n+1}-A^{n+1}}}\bigg \},\end{aligned}$$

which finishes the proof. $\square$

4 Delta invariants of projective cones of Fano type

In this section we prove Theorem 1.4. Similarly as bundle case in previous section, we start by the computation of $\delta _{Y,cV_\infty }({\mathrm{ord}}_{V_0})$ and $\delta _{Y,cV_\infty }({\mathrm{ord}}_{V_\infty })$.

Lemma 4.1

Notation as in Theorem 1.4, we have

$$\begin{aligned}\delta _{Y,cV_\infty }({\mathrm{ord}}_{V_0})= \frac{(n+2)r}{(n+1)(r+1-c)}, \delta _{Y,cV_\infty }({\mathrm{ord}}_{V_\infty })=\frac{(n+2)(1-c)}{r+1-c}\end{aligned}$$

Proof

By Lemma 2.2 we know that

$$\begin{aligned}K_{{\tilde{Y}}}+V_0+cV_\infty =p^*(K_Y+cV_\infty )+rV_0,\end{aligned}$$

hence $A_{Y,cV_\infty }(V_0)=r$ and $p^*(K_Y+cV_\infty )\sim _\mathbb {Q}K_{{\tilde{Y}}}+(1-r)V_0+cV_\infty$. Go back to the proof of Lemma 3.1, we just let $a=1-r$ and $b=c$, although the values of a, b may not satisfy our requirement on them anymore, it doesn’t effect the computation of volumes. A direct computation concludes the lemma. $\square$

By the same method used for the bundle case, we next work out another upper bound for $\delta (Y,cV_\infty )$, that is $\delta (Y,cV_\infty )\le \frac{(n+2)r}{(n+1)(r+1-c)}\delta (V)$.

Lemma 4.2

Notation as in Theorem 1.4, we have $\delta (Y,cV_\infty )\le \frac{(n+2)r}{(n+1)(r+1-c)}\delta (V)$

Proof

We choose a sufficiently divisible natural number m such that $-m(K_Y+cV_\infty )$ is an ample line bundle, we have

$$\begin{aligned} \phi ^*(-m(K_Y+cV_\infty ))\sim m(r+1-c)V_\infty \sim jp^*L+(m(r+1-c)-j)V_\infty +jV_0, \end{aligned}$$

(19)

Let $R_m$ denote $H^0(Y,-m(K_Y+cV_\infty ))$, we define

$$\begin{aligned}R_{m,j}:=\{s\in R_m|{{\mathrm{ord}}_{V_0}(s)=j}\}.\end{aligned}$$

One directly sees that

$$\begin{aligned}R_m=\bigoplus ^{m(r+1-c)}_{j=0} R_{m,j},\end{aligned}$$

and by (19),

$$\begin{aligned} R_{m,j}\cong H^0(V,jL). \end{aligned}$$

(20)

Write $r_{m,j}=\dim R_{m,j}=\dim H^0(V,jL)$, we choose a j-basis type divisor $D_{m,j}$ of L such that $D_{m,j}$ computes $S_{j,L}(E)$. Then we lift $D_{m,j}$ to be a divisor $\mathcal {D}_{m,j}$ over Y. By (19), it is not hard to see that

$$\begin{aligned} \mathcal {D}_m:=\frac{\sum _{j=0}^{m(r+1-c)}r_{m,j}(j\mathcal {D}_{m,j}+jV_0+(m(r+1-c)-j)V_\infty )}{m\sum _{j=0}^{m(r+1-c)} r_{m,j}} \end{aligned}$$

(21)

is an m-basis type divisor of $\phi ^*(-(K_Y+cV_\infty ))$. Let E be a prime divisor over V and $E_Y$ the divisor over Y which is the natural extension of E, then we have

$$\begin{aligned} S_{m,(Y,cV_\infty )}(E_Y)\ge {\mathrm{ord}}_{E_Y}(\mathcal {D}_m)=\frac{\sum _{j=0}^{m(r+1-c)}jr_{m,j}S_{j,L}(E)}{m\sum _{j=0}^{m(r+1-c)}r_{m,j}}. \end{aligned}$$

(22)

Let m tends to infinity, by the same computation as in Lemma 3.3, one gets

$$\begin{aligned} S_{Y,cV_\infty }(E_Y)\ge \frac{(n+1)(r+1-c)}{n+2}\cdot S_L(E), \end{aligned}$$

(23)

hence we have

$$\begin{aligned} \delta _{Y,cV_\infty }({\mathrm{ord}}_{E_Y})\le \frac{n+2}{(n+1)(r+1-c)}\delta _L({\mathrm{ord}}_E). \end{aligned}$$

(24)

As E is arbitrarily chosen over V, we know

$$\begin{aligned}\delta (Y,cV_\infty )\le \frac{n+2}{(n+1)(r+1-c)}\delta _V(L)=\frac{(n+2)r}{(n+1)(r+1-c)}\delta (V).\end{aligned}$$

$\square$

Above all, one obtains the upper bound in Theorem 1.4.

Theorem 4.3

Notation as in Theorem 1.4, we have

$$\begin{aligned} \delta (Y,cV_\infty )\le \min \bigg \{\frac{(n+2)r}{(n+1)(r+1-c)}\delta (V), \frac{(n+2)r}{(n+1)(r+1-c)}, \frac{(n+2)(1-c)}{r+1-c}\bigg \}. \end{aligned}$$

(25)

Proof of theorem 1.4

For the converse direction, one can use totally the same way as in the final proof of Theorem 1.3 for bundle case. As there is no need to repeat it, we just leave it out. $\square$

We will take another more elegant way below for the lower bound, although we can only work out the case where $0<r\le n+1$. However, this upper bound of r is usually satisfied, at least for K-semistable log Fano pairs of dimension n, see [19, 28]. First we recall the following lemma [35, Proposition 2.11] which establishes the relationship between K-stability of the base and that of projective cone over it, see also [29].

Lemma 4.4

Let $(V,\Delta )$ be an n-dimensional log Fano variety, and L an ample line bundle on V such that $L\sim _\mathbb {Q}-\frac{1}{r}(K_V+\Delta )$ for some $0<r\le n+1$. As before, Y is the projective cone over V associated to L, then $(V,\Delta )$ is K-semistable if and only if $(Y,\Delta _Y+(1-\frac{r}{n+1})V_\infty )$ is K-semistable, where $\Delta _Y$ is the divisor on Y naturally extended by $\Delta$.

By above lemma, we can obtain the converse direction for projective cones.

Theorem 4.5

Notation as before, assume $0<r\le n+1$, then

$$\begin{aligned} \delta (Y)\ge \min \bigg \{\frac{(n+2)r}{(n+1)(r+1)}\delta (V), \frac{(n+2)r}{(n+1)(r+1)}\bigg \}. \end{aligned}$$

(26)

Proof

We first consider the case $\delta (V)\ge 1$, i.e. V is K-semistable. By above Lemma 4.4, one knows $\delta (Y,(1-\frac{r}{n+1})V_\infty )\ge 1$, which directly implies $\delta (Y)\ge \frac{(n+2)r}{(n+1)(r+1)}$. So in this case we conclude the result. We next deal with the case $\delta (V)<1$. Choose a sequence positive rational numbers $\delta _i$ which tends to $\delta (V)$ and $\delta _i<\delta (V)$ for each i. For each $\delta _i$ one can choose an effective $\mathbb {Q}$-divisor $\Delta _i\sim _\mathbb {Q}-K_V$ such that $(V,(1-\delta _i)\Delta _i)$ is a K-semistable log Fano pair (even uniformly K-stable), see [7, 14]. Then we know $L\sim _\mathbb {Q}-\frac{1}{r_i}(K_V+(1-\delta _i)\Delta _i)$, where $r_i=\delta _ir\le n+1$. By Lemma 4.4, one sees $(Y,(1-\delta _i)\Delta _{i,Y}+(1-\frac{r_i}{n+1})V_\infty )$ is K-semistable, where $\Delta _{i,Y}$ is a divisor on Y naturally extended by $\Delta _i$. Then we have $\delta (Y,(1-\delta _i)\Delta _{i,Y}+(1-\frac{r_i}{n+1})V_\infty )\ge 1$. It is clear that

$$\begin{aligned}-(K_Y+(1-\delta _i)\Delta _{i,Y}+(1-\frac{r_i}{n+1})V_\infty )\sim _\mathbb {Q}\frac{(n+2)r_i}{n+1}V_\infty \sim _\mathbb {Q}\frac{(n+2)r_i}{(n+1)(r+1)}(-K_Y),\end{aligned}$$

hence $\delta (Y)\ge \frac{(n+2)r_i}{(n+1)(r+1)}$. Recall $r_i=\delta _ir$, and let i tends to infinity, one obtains

$$\begin{aligned}\delta (Y)\ge \frac{(n+2)r}{(n+1)(r+1)}\delta (V) ,\end{aligned}$$

concluded. $\square$

Proof of Theorem 1.4for $0<r\le n+1$ and $c=0$. The proof is a combination of Theorem 4.3 and Theorem 4.5. $\square$

Proof of Corollary 1.6

If V is K-semistable, then the Fano index cannot be larger than $n+1$, hence $r\le n+1$. So by Theorem 1.4, $\delta (Y)= \frac{(n+2)r}{(n+1)(r+1)}$. By Lemma 4.1, the value is achieved by $V_0$. The proof is finished. If V is not K-semistable, then by [6, Theorem D], we have

$$\begin{aligned}\delta (V)^n\cdot {\mathrm{vol}}(-K_V)\le (n+1)^n .\end{aligned}$$

As $-K_V\sim _\mathbb {Q}rL$ and L is an ample line bundle, one has $\delta (V)^nr^n\le (n+1)^n$, which implies $r\delta (V)\le n+1$. Hence

$$\begin{aligned}\frac{(n+2)r}{(n+1)(r+1)}\delta (V)\le \frac{n+2}{r+1}.\end{aligned}$$

Apply Theorem 1.4 to the case $c=0$ one directly concludes. $\square$

5 K-stability with optimal angle

In this section, as in the setting of Theorem 1.7, we fix V to be a projective Fano manifold of dimension n, and S is a smooth divisor on V such that $S\sim _\mathbb {Q}-\lambda K_V$ for some positive rational number $\lambda$. Write

$$\begin{aligned}E(V,S):=\{a\in [0,1)| (V,aS)\textit{ is K-semistable} \},\end{aligned}$$

we prove the following theorem on optimal angle.

Theorem 5.1

(= Theorem 1.7) Notation as above, suppose V and S are both K-semistable and $0<\lambda <1$, then $E(V,S)= [0,1-\frac{r}{n}]$, where $r=\frac{1}{\lambda }-1$. In particular, if V and S are both K-polystable, then the pair (V, aS) is K-polystable for any $a\in [0, 1-\frac{r}{n})$. We also note that $1-\frac{r}{n}\ge 0$, since $\lambda \ge \frac{1}{n+1}$ if V is K-semistable.

In the above theorem, we do not consider the case $\lambda \ge 1$, since in this case the K-stability of the pair (V, aS) is well known to experts. We state it here and provide a proof for the readers’ convenience.

Theorem 5.2

Suppose V is a smooth K-semistable Fano manifold, and $S\sim _\mathbb {Q}-\lambda K_V$ is a smooth divisor on V for some positive rational number $\lambda \ge 1$. Then (V, aS) is K-semistable for any $a\in [0,\frac{1}{\lambda })$ and K-stable for $a\in (0,\frac{1}{\lambda })$.

Proof

We first deal with the case $\lambda =1$, i.e. $S\sim _\mathbb {Q}-K_V$, then (V, S) is a log smooth Calabi-Yau pair. We show that $\alpha (V,(1-\beta )S)=1$ for sufficiently small rational $0<\beta \ll 1$ (see also [3]). For the definition of alpha invariant of a log Fano pair please refer to [6, 13, 15, 42] etc. It is clear that $\alpha (V,(1-\beta )S)\le 1$. Suppose the inequality is strict, then we can find a divisor $D\sim _\mathbb {Q}-K_V$ such that the pair $(V,(1-\beta )S+\beta D)$ is not log canonical. After subtracting certain amount of S from D we may assume that $S\nsubseteq {\text {Supp}}(D)$, then by inversion of adjunction $(S,\beta D|_S)$ is not log canonical, contradicting to the choice of $\beta$ (note that $\beta$ is sufficiently small). So $(V,(1-\beta )S)$ is K-stable for sufficiently small $0<\beta \ll 1$. By interpolation of K-stability, see [1, Proposition 2.13], we conclude the result for $\lambda =1$.

For the case $\lambda >1$, it is clear that we should require $a<\frac{1}{\lambda }$ to make sure that (V, aS) is a log Fano pair. Then for any prime divisor E over V, we have

$$\begin{aligned}\frac{A_{V,aS}(E)}{S_{V,aS}(E)}\ge \frac{(1-a)A_V(E)}{(1-\lambda a)S_V(E)}\ge \frac{1-a}{1-\lambda a}>1\end{aligned}$$

for any $a\in (0,\frac{1}{\lambda })$. Thus $\delta (V,aS)>1$ for any $a\in (0,\frac{1}{\lambda })$. Concluded. $\square$

We turn to prove Theorem 1.7. Let us first begin with the following lemma, see [35, Lemma 2.12] or [36, Theorem 5.1].

Lemma 5.3

Notation as in Theorem 1.7. Suppose V and S are both K-semistable, then $(V,(1-\frac{r}{n})S)$ is K-semistable and $[0, 1-\frac{r}{n}]\subset E(V,S)$.

Proof

Write $M:= S|_S\sim _\mathbb {Q}-\frac{1}{r}K_S$ to be an ample line bundle on S, then by [35, Lemma 2.12], $(V,(1-\frac{r}{n})S)$ can be specially degenerated to $(C_p(S, M), (1-\frac{r}{n})S_\infty )$, where $(C_p(S, M)$ is the projective cone over S associated to M, and $S_\infty$ is the infinite section. Since S is K-semistable, by [35, Proposition 2.11], $(C_p(S, M), (1-\frac{r}{n})S_\infty )$ is K-semistable, hence $(V,(1-\frac{r}{n})S)$ is K-semistable by lower semicontinuity of delta invariants, see [7]. By interpolation of K-stability ([1, Proposition 2.13]) we know $[0, 1-\frac{r}{n}]\subset E(V,S)$. $\square$

In the above lemma, we have proved one direction of inclusion, the converse inclusion is implied by following lemma. This has been proved in [31, Theorem 1.4] in analytic setting, we give a purely algebraic proof below.

Lemma 5.4

Notation as above, if $1-\frac{r}{n}<a<1$, then (V, aS) is K-unstable.

Proof

As we have seen in the proof of Lemma 5.3, by [35, Lemma 2.12], the pair (V, aS) can be specially degenerated to $(C_p(S, M), aS_\infty )$, where $C_p(S, M)$ is the projective cone over S associated to M, and $S_\infty$ is the infinite section. We denote this special test configuration by $\phi : ({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})\rightarrow \mathbb {P}^1$, where ${\bar{\mathcal {L}}}:=-(K_{{\bar{\mathcal {V}}}/\mathbb {P}^1}+a{\bar{\mathcal {S}}})$ is the polarization of this test configuration. Note that $-K_V\sim _\mathbb {Q}(1+r)S$, so there is a $k\in \mathbb {Q}$ such that $-K_{{\bar{\mathcal {V}}}/\mathbb {P}^1}\sim _\mathbb {Q}(1+r){\bar{\mathcal {S}}}-\phi ^*\mathcal {O}_{\mathbb {P}^1}(k)$, thus ${\bar{\mathcal {L}}}\sim _\mathbb {Q}(1+r-a){\bar{\mathcal {S}}}-\phi ^*\mathcal {O}_{\mathbb {P}^1}(k)$. By [33], the generalised Futaki invariant (or Donaldson-Futaki invariant) of the test configuration is as follows,

$$\begin{aligned} {\mathrm{Fut}}({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})&=-\frac{1}{n+1}\frac{{\bar{\mathcal {L}}}^{n+1}}{(-K_V-aS)^n}\\&=-\frac{1}{n+1}\frac{\{(1+r-a){\bar{\mathcal {S}}}-\phi ^*\mathcal {O}_{\mathbb {P}^1}(k)\}^{n+1}}{(-K_V-aS)^n}\\&= k-\frac{1+r-a}{n+1}\frac{{\bar{\mathcal {S}}}^{n+1}}{S^n}. \end{aligned}$$

By [35, Lemma 2.12], ${\mathrm{Fut}}({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})=0$ if $a=1-\frac{r}{n}$. By Lemma 5.3, we know ${\mathrm{Fut}}({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})\ge 0$ when $a\le 1-\frac{r}{n}$, so in the case $a>1-\frac{r}{n}$ we have ${\mathrm{Fut}}({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})\le 0$. Suppose $a>1-\frac{r}{n}$, if ${\mathrm{Fut}}({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})< 0$ then we are done; if ${\mathrm{Fut}}({\bar{\mathcal {V}}},a{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})= 0$, then ${\mathrm{Fut}}({\bar{\mathcal {V}}},t{\bar{\mathcal {S}}}; {\bar{\mathcal {L}}})=0$ for any $t\in [0,1)$. Suppose (V, aS) is K-semistable, then by [32, Section 3], the pair $(C_p(S,M),aS_\infty )$ is also K-semistable. By the next lemma, we know $\delta (C_p(S,M),aS_\infty )<1$ when $a>1-\frac{r}{n}$. The contradiction implies that (V, aS) is K-unstable for $a>1-\frac{r}{n}$. The proof is finished. $\square$

The following lemma is in fact a rephrase of Theorem 1.4, we still state it here to fit our setting for the convenience of readers.

Lemma 5.5

Let S be a $\mathbb {Q}$-Fano variety of dimension $n-1$ and M an ample line bundle on S such that $M\sim _\mathbb {Q}-\frac{1}{r}K_S$. Write $Y:=C_p(S,M)$ to be the projective cone over S associated to M, and $S_\infty$ the infinite section, then we have following formula computing delta invariant of $(C_p(S,M), aS_\infty )$,

$$\begin{aligned}\delta (C_p(S,M), aS_\infty )=\min \bigg \{\frac{(n+1)r}{n(r+1-a)}\delta (S), \frac{(n+1)r}{n(r+1-a)}, \frac{(n+1)(1-a)}{r+1-a}\bigg \}.\end{aligned}$$

In particular, when $a>1-\frac{r}{n}$, we have

$$\begin{aligned}\delta (C_p(S,M), aS_\infty )\le \frac{(n+1)(1-a)}{r+1-a} <1.\end{aligned}$$

Proof of Theorem 1.7

The proof is a combination of Lemmas 5.3, 5.4, 5.5, and following lemma on interpolation for K-polystability. $\square$

Lemma 5.6

Let V be an n-dimensional Fano manifold and S a smooth prime divisor on V such that $-K_V\sim _\mathbb {Q}(1+r)S$ for some positive rational number r. Assume the following two conditions,

• (1) V and S are both K-polystable,

• (2) there is a positive rational number $0<a<1$ such that (V, aS) is K-semistable,

then we have that $(V,a'S)$ is K-polystable for any rational $0<a'<a$.

Proof

It suffices to assume a to be the optimal angle for K-stability ($a=1-\frac{r}{n}$), i.e. (V, bS) is K-unstable for $a<b<1$. Fix a rational number $a'\in (0,a)$. We aim to show the pair $(V,a'S)$ is K-polystable. Since it’s K-semistable, by [32, Section 3] there is a test configuration $(\mathcal {V}, a'\mathcal {S}; \mathcal {L})\rightarrow \mathbb {A}^1$ such that

• (1) ${\mathrm{Fut}}(\mathcal {V},a'\mathcal {S}; \mathcal {L})=0$,

• (2) $(\mathcal {V}_0,a'\mathcal {S}_0)$ is K-polystable.

By interpolation of Futaki invariants (see [1, Proposition 2.13]) we know ${\mathrm{Fut}}(\mathcal {V},t\mathcal {S};\mathcal {L})=0$ for any $t\in [0,1)$, which implies $\mathcal {V}_0\cong V$ since V is K-polystable. As ${\mathrm{Fut}}(\mathcal {V},a\mathcal {S};\mathcal {L})=0$, by [32, Section 3] we know $(\mathcal {V}_0,a\mathcal {S}_0)$ is K-semistable. By the openness property of K-polystability, cf. [1, Proporsition 3.18], (V, aS) is strictly K-semistable (i.e. K-semistable but not K-polystable), then there is a test configuration $({\tilde{\mathcal {V}}},a{\tilde{\mathcal {S}}};{\tilde{\mathcal {L}}})$ of $(V,aS;-(K_V+aS))$ such that

• (1) ${\mathrm{Fut}}({\tilde{\mathcal {V}}},a{\tilde{\mathcal {S}}};{\tilde{\mathcal {L}}})=0$,

• (2) $({\tilde{\mathcal {V}}}_0, a{\tilde{\mathcal {S}}}_0)$ is K-polystable.

Since S is K-polystable, by [35, Proporsition 2.11], $(C_p(S,M), aS_\infty )$ is a K-polystable degeneration of (V, aS), where $C_p(S,M)$ is the projective cone over S associated to $M:=S|_S$ and $S_\infty$ denotes the infinite divisor. By the uniqueness of K-polystable degeneration [10] we know $({\tilde{\mathcal {V}}}_0, a{\tilde{\mathcal {S}}}_0)\cong (C_p(S,M),aS_\infty )$. By [32, Section 3], there is a test configuration which has vanishing generalized Futaki invariant and degenerates $(\mathcal {V}_0,a\mathcal {S}_0)$ to $({\tilde{\mathcal {V}}}_0,a{\tilde{\mathcal {S}}}_0)$. That is, we first degenerate (V, aS) to $(\mathcal {V}_0\cong V, a\mathcal {S}_0)$, then we degenerate $(\mathcal {V}_0, a\mathcal {S}_0)$ to $({\tilde{\mathcal {V}}}_0,a{\tilde{\mathcal {S}}}_0)\cong (C_p(S,M), aS_\infty )$. Focusing on the boundary, we first degenerate S to $\mathcal {S}_0$, then degenerate $\mathcal {S}_0$ to $S_\infty \cong S$. By K-polystability of S we at once see $\mathcal {S}_0\cong S$. Hence, the pair $(\mathcal {V}_0, a'\mathcal {S}_0)$, which is the K-polystable degeneration of $(V,a'S)$, is induced by a test configuration of product type. Thus the pair $(V,a'S)$ is K-polystable. The proof is finished. $\square$

Corollary 5.7

(=Corollary 1.8) For the pair $(\mathbb {P}^n, S_d)$ where $S_d$ is a smooth hypersurface of degree $1\le d\le n$. If $S_d$ is K-polystable (this is expected to be true), then we have $E(\mathbb {P}^n, S_d)=[0, 1-\frac{r}{n}]$, where $r=\frac{n+1-d}{d}$. In particular, $(\mathbb {P}^n, aS_d)$ is K-polystable for any $a\in [0, 1-\frac{r}{n})$.

Proof

Just replace $(\mathbb {P}^n, S_d)$ by (V, S), then apply Theorem 1.7. $\square$

Example 5.8

The pair $(\mathbb {P}^2, aC)$ is K-polystable for $a\in [0, \frac{3}{4})$, where C is a smooth conic curve, see [31, Theorem 1.5].

6 Examples

In this section, we will compute delta invariants of some special hypersurfaces in projective space, which can be realized as the projective cones over some lower dimensional log Fano pairs.

6.1 Projective cones over smooth Fano hypersurfaces

Let $V^0_d\subset \mathbb {P}^{n+1}$ be a smooth hypersuface of degree d defined by a homogeneous polynomial $f_d$, where $2\le d\le n+1$. Let $H^0$ denote the hyperplane section of $V^0_d$, then $-K_{V_d^0}\sim r_0 H^0$, where $r_0=n+2-d$. Consider the projective cone over $V^0_d$ associated to $H^0$, denoted by $V^1_d$, that is $V^1_d:={\mathrm{Proj}}\oplus _{k\in \mathbb {N}}\oplus _{l\in \mathbb {N}} H^0(V_d^0, kH^0) s^l$. It is clear that $V_d^1$ is still a degree d hypersuface but lies in $\mathbb {P}^{n+2}$, and it is still cut out by equation $f_d=0$. Write $H^1$ the hyperplane section of $V^1_d$, then we see $-K_{V_d^2}\sim r_1 H^1$, where $r_1=n+3-d=r_0+1$. Continue the process, we get hypersurfaces $(V_d^i, H^i)$ of degree d in $\mathbb {P}^{n+1+i}, i\ge 1$, which is still cut out by $f_d=0$, and $H^i$ is the hyperplane section with $-K_{V^i_d}\sim r_iH^i$, where $r_i=n+2+i-d=r_0+i$. Note that $V_d^i\subset \mathbb {P}^{n+1+i}, i\in \mathbb {N}$ are all $\mathbb {Q}$-Fano varieties. We want to compute delta invariants of these Fano varieties. Let us first recall following result.

Let V be a $\mathbb {Q}$-Fano variety of dimension n, and L an ample line bundle on V such that $L\sim _\mathbb {Q}-\frac{1}{r}K_V$ for some positive rational number $r>0$. Let Y be the projective cone over V associated to L, that is, $Y:={\mathrm{Proj}}\oplus _{k\in \mathbb {N}}\oplus _{l\in \mathbb {N}} H^0(V, kL) s^l$, then by Theorem 1.4 we know,

$$\begin{aligned}\delta (Y)=\min \bigg \{\frac{(n+2)r}{(n+1)(r+1)}\delta (V), \frac{(n+2)r}{(n+1)(r+1)}, \frac{n+2}{r+1}\bigg \}.\end{aligned}$$

So if $\delta (V)\le 1$ and $r\le n+1$, then $\delta (Y)=\frac{(n+2)r}{(n+1)(r+1)}\delta (V)$. Assume $\delta (V_d^0)\le 1$, then

$$\begin{aligned}&\delta (V_d^1)=\frac{(n+2)r_0}{(n+1)(r_0+1)}\delta (V_d^0)< 1,\\&\delta (V_d^2)=\frac{(n+3)r_1}{(n+2)(r_1+1)}\delta (V_d^1)< 1,\\&\dots \\&\delta (V_d^i)=\frac{(n+1+i)r_{i-1}}{(n+i)(r_{i-1}+1)}\delta (V_d^{i-1})< 1. \end{aligned}$$

Thus we have following formula

$$\begin{aligned} \delta (V_d^i)=\frac{(n+1+i)r_0}{(n+1)r_i}\delta (V_d^0)=\frac{(n+2-d)(n+1+i)}{(n+1)(n+2+i-d)}\delta (V_d^0)<1. \end{aligned}$$

Similarly, if $\delta (V_d^0)\ge 1$, then $\delta (V_d^i)=\frac{(n+2-d)(n+1+i)}{(n+1)(n+2+i-d)}<1$. We have the following results.

Corollary 6.1

Let V be a smooth hypersurface of degree d in $\mathbb {P}^{n+1}$, where $2\le d\le n+1$, and H the hyperplane section. Then the projective cone Y over V associated to H is K-unstable. In other words, $\mathbb {Q}$-Fano variety cut out by the form $f_d(x_0,x_1, \dots , x_{n+1})=0$ in $\mathbb {P}^{n+2}$ is K-unstable for $2\le d\le n+1$, where $f_d$ is a homogeneous form of degree d which determines a smooth Fano manifold in $\mathbb {P}^{n+1}$. Moreover,

$$\begin{aligned}\delta (Y)=\min \bigg \{ \frac{(n+2)(n+2-d)}{(n+1)(n+3-d)}\delta (V),\frac{(n+2)(n+2-d)}{(n+1)(n+3-d)}\bigg \}<1.\end{aligned}$$

Corollary 6.2

Let V be a smooth hypersurface of degree $2\le d\le n+1$ in $\mathbb {P}^{n+1}$, which is cut out by the form $f_d(x_0,x_1, \dots ,x_{n+1})$, then for any positive natural number l, the variety $Y\subset \mathbb {P}^{n+1+l}$ defined by $f_d(x_0,x_1, \dots ,x_{n+1})$ is K-unstable. Moreover,

$$\begin{aligned}\delta (Y)=\min \bigg \{\frac{(n+2-d)(n+1+l)}{(n+1)(n+2+l-d)}\delta (V), \frac{(n+2-d)(n+1+l)}{(n+1)(n+2+l-d)} \bigg \} <1.\end{aligned}$$

Example 6.3

Let V be a smooth conic curve in $\mathbb {P}^2$, then $\delta (V)=1$. Let Y be the projective cone over V, then Y is isomorphic to a quadratic surface defined by $x_0^2+x_1^2+x_2^2=0$ in $\mathbb {P}^3$. In this case, $n=1, d=2$, so $\delta (Y)=\frac{3}{4}$. We know that a smooth quadratic surface in $\mathbb {P}^3$ is isomorphic to $\mathbb {P}^1\times \mathbb {P}^1$, which is K-polystable and the delta invariant is 1.

Example 6.4

Let V be a smooth cubic surface in $\mathbb {P}^3$, then $\delta (V)>1$. Let Y be the projective cone over V, then Y is isomorphic to a cubic 3-fold defined by the same $f_3(x_0,x_1,x_2,x_3)=0$ in $\mathbb {P}^4$. In this case, $n=2, d=3$, so $\delta (Y)=\frac{2}{3}$. We know that a smooth cubic 3-fold in $\mathbb {P}^4$ is K-stable, see [34].

Question 6.5

The delta invariants of smooth cubic surfaces has been estimated in [16, 37]. It is an interesting question to estimate delta invariants of smooth cubic 3-folds and quartic 3-folds.

6.2 Projective cones associated to ample $\mathbb {Q}$-line bundles

In the whole previous contents, the cones are associated to ample line bundles. We now deal with the case of ample $\mathbb {Q}$-line bundles. Let V be a $\mathbb {Q}$-Gorenstein projective normal variety of dimension n and L an ample $\mathbb {Q}$-line bundle on V. We will view L to be a $\mathbb {Q}$-Cartier divisor and write $\Delta _L$ to be the branched divisor associated to L, that is, we construct a finite morphism $f: {\tilde{V}}\rightarrow V$ such that $K_{{\tilde{V}}}=f^*(K_V+\Delta _L)$ and $f^*L$ is an ample line bundle on ${\tilde{V}}$. We consider the following cone over ${\tilde{V}}$ associated to $f^*L$. i.e.

$$\begin{aligned}Y:= {\mathrm{Proj}}\oplus _{k\in \mathbb {N}}\oplus _{l\in \mathbb {N}}H^0({\tilde{V}},kf^*L)s^l.\end{aligned}$$

We assume Y admits klt singularities, then $f^*L$ is proportional to $-K_{{\tilde{V}}}$, see [25] or results in Section 2, that is ${\tilde{V}}$ is a $\mathbb {Q}$-Fano variety. Hence $(V,\Delta _L)$ is also a log Fano pair and L is proportional to $-(K_V+\Delta _L)$. We assume $f^*L\sim _\mathbb {Q}-\frac{1}{r}K_{{\tilde{V}}}$ for some positive rational r, then we have $L\sim _\mathbb {Q}-\frac{1}{r}(K_V+\Delta _L)$. By Theorem 1.4,

$$\begin{aligned}\delta (Y)=\min \bigg \{\frac{(n+2)r}{(n+1)(r+1)}\delta ({\tilde{V}}), \frac{(n+2)r}{(n+1)(r+1)}, \frac{n+2}{r+1}\bigg \} .\end{aligned}$$

Combine the recent work [36, Theorem 1.2] (See also [17]), we have $\delta ({\tilde{V}})\ge 1$ if and only if $\delta (V,\Delta _L)\ge 1$, and $\delta ({\tilde{V}})=\delta (V,\Delta _L)$ if either one is K-unstable. Thus we conclude the following result.

Corollary 6.6

Let V be a $\mathbb {Q}$-Gorenstein projective normal variety of dimension n and L an ample $\mathbb {Q}$-line bundle. Suppose $L\sim _\mathbb {Q}-\frac{1}{r}(K_V+\Delta _L)$. Write $Y:= {\mathrm{Proj}}\oplus _{k\in \mathbb {N}}\oplus _{l\in \mathbb {N}}H^0(V,\lfloor kL\rfloor )s^l$, then we have

$$\begin{aligned}\delta (Y)=\min \bigg \{\frac{(n+2)r}{(n+1)(r+1)}\delta (V,\Delta _L), \frac{(n+2)r}{(n+1)(r+1)}, \frac{n+2}{r+1}\bigg \}.\end{aligned}$$

Example 6.7

This example is taken from [35]. Let $V=\mathbb {P}^n$ and $L=\frac{n+1}{n+2}S_{n+1}-nH$, where $S_{n+1}$ is a general hypersurface of degree $n+1$ in $\mathbb {P}^n$ and H is a hyperplane. Then $\Delta _L=\frac{n+1}{n+2}S_{n+1}$, $L\sim _\mathbb {Q}\frac{1}{n+2}H$, $-(K_V+\Delta _L)\sim _\mathbb {Q}\frac{n+1}{n+2}H$. Thus $r=n+1$, $\delta (V,\Delta _L)\ge 1$. So by Theorem 6.6, $\delta (Y)=1$. In fact, Y is isomorphic to a hypersurface in $\mathbb {P}^{n+2}$ determined by $x_{n+1}^{n+2}=g_{n+1}x_{n+2}$, where $g_{n+1}(x_0,x_1, \dots ,x_n)$ is the equation of $S_{n+1}\subset \mathbb {P}^n$. We will deal with such kind of hypersurfaces in detail.

From now on, we fix $V:=\mathbb {P}^n$ and $S_d\subset \mathbb {P}^n$ is a smooth hypersurface of degree d determined by a homogeneous form $g_d(x_0,x_1, \dots ,x_n)$. Then we can construct a cover morphism $f: {\tilde{V}}\rightarrow V$, which is ramified along $S_d$ with multiplicity k. It is clear that we can choose ${\tilde{V}}$ to be a hypersurface in the weighted projective space $\mathbb {P}^{n+1}(k,k, \dots ,k,d)$ determined by $x_{n+1}^k=g_d(x_0,x_1, \dots ,x_n)$. Suppose H is the hyperplane class in $\mathbb {P}^n$ and denote $L:=\frac{l}{k}S_d-\frac{dl-1}{k}H\sim _\mathbb {Q}\frac{1}{k}H$, where l is a positive natural number such that $l<k, (k,l)=1$ and $k| dl-1$. It is not hard to see the affine cone over ${\tilde{V}}$ associated to $f^*L$ is exactly the hypersurface in $\mathbb {C}^{n+2}$ determined by $x_{n+1}^k=g_d(x_0,x_1, \dots ,x_n)$, thus the corresponding projective cone is exactly the hypersurface in $\mathbb {P}^{n+2}$ determined by $x_{n+1}^k=g_d\cdot x_{n+2}^{k-d}$, denoted by Y. In this case, $\Delta _L=\frac{k-1}{k}S_d$, so $-(K_V+\Delta _L)\sim _\mathbb {Q}(n+1-\frac{(k-1)d}{k})H\sim _\mathbb {Q}((n+1)k-(k-1)d)L$. We choose k, d to satisfy that $(n+1)k-(k-1)d>0$. By Corollary 6.6, we have

$$\begin{aligned}\delta (Y)=\min \bigg \{\frac{(n+2)r}{(n+1)(r+1)}\delta (V,\Delta _L),\frac{(n+2)r}{(n+1)(r+1)}, \frac{n+2}{r+1} \bigg \},\end{aligned}$$

where $r= (n+1)k-(k-1)d$. We have the following lemma about $\delta (V,\Delta _L)$.

Lemma 6.8

Let $V:=\mathbb {P}^n$ and $S_d\subset \mathbb {P}^n$ be a smooth hypersurface of degree $d\ge n+1$, then $(V, aS_d)$ is K-semistable for any rational $0\le a<\frac{n+1}{d}$, i.e. $\delta (V,aS_d)\ge 1$.

Proof

Apply Theorem 5.2. $\square$

Above all, we have following result.

Corollary 6.9

Let $Y\subset \mathbb {P}^{n+2}$ be a hypersurface determined by

$$\begin{aligned}x_{n+1}^k\cdot x_{n+2}^{d-k}=g_d(x_0,x_1, \dots ,x_n),\end{aligned}$$

where $g_d$ determines a smooth hypersurface in $\mathbb {P}^n$ of degree $n+1\le d\le n+2$. Suppose there is a positive natural number l such that $l<k, (k,l)=1, k|dl-1$ and $(n+1)k-(k-1)d>0$, then

$$\begin{aligned}\delta (Y)=\min \bigg \{\frac{(n+2)r}{(n+1)(r+1)}, \frac{n+2}{r+1} \bigg \}=\frac{(n+2)r}{(n+1)(r+1)}\le 1 ,\end{aligned}$$

where $r= (n+1)k-(k-1)d\le n+1$.

Example 6.10

Let $k=2, d=n+1$, and n is even. Then the hypersurface $Y\subset \mathbb {P}^{n+2}$ determined by $x_{n+1}^2x_{n+2}^{n-1}=g_{n+1}(x_0,x_1, \dots ,x_n)$ is K-semistable, where $g_{n+1}$ determines a general hypersurface of degree $n+1$ in $\mathbb {P}^n$. For example, if we take $n=2$, then the 3-dimensional hypersurface $x_3^2x_4=g_3(x_0,x_1,x_2)$ is K-semistable, where $g_3$ cuts out a smooth elliptic curve in $\mathbb {P}^2$.

Example 6.11

Let $k=3, d=n+1$, and 3|n or $3| 2n+1$. Then the hypersurface $Y\subset \mathbb {P}^{n+2}$ determined by $x_{n+1}^3x_{n+2}^{n-2}=g_{n+1}(x_0,x_1, \dots ,x_n)$ is K-semistable, where $g_{n+1}$ determines a general hypersurface of degree $n+1$ in $\mathbb {P}^n$. For example, take $n=1$, then the surface $x_2^3=g_2(x_0,x_1)x_3$ is K-semistable, where $g_2$ cuts out two different points in $\mathbb {P}^1$.

Example 6.12

Let $k=2, d=n+2$, and n is odd. Then the hypersurface $Y\subset \mathbb {P}^{n+2}$ determined by $x_{n+1}^2x_{n+2}^n=g_{n+2}(x_0,x_1, \dots ,x_n)$ is K-unstable with $\delta (Y)=\frac{(n+2)n}{(n+1)^2}<1$, where $g_{n+2}$ determines a general hypersurface of degree $n+2$ in $\mathbb {P}^n$. For example, take $n=1$, then the surface $Y\subset \mathbb {P}^3$ determined by $x_2^2x_3=g_3(x_0,x_1)$ is K-unstable with $\delta (Y)=\frac{3}{4}$, where $g_3$ cuts out 3 different points in $\mathbb {P}^1$.

Appendix A.: Calabi symmetry and twisted Kähler–Einstein edge metrics

In this appendix we prove a weaker version of Theorem 1.1 using purely analytic methods. More precisely, we will show

Proposition A.1

In the setting of Theorem 1.1, we have

$$\begin{aligned}\min \bigg \{\frac{\delta (V)r\beta _0}{1+\beta _0(r-1)},\beta _0\bigg \}\le \delta ({\tilde{Y}})\le \beta _0.\end{aligned}$$

We will use Calabi ansatz to carry out the proof. It is interesting to note that this approach also yields the bound obtained from the algebraic approach. Before proceeding, we first give some necessary backgrounds for the reader’s convenience.

1.1 A.1 Calabi ansatz

We review a well-studied and powerful construction, pioneered by Calabi [11, 12], which can effectively produce various explicit examples of canonical metrics in Kähler geometry. The idea is to work on complex manifolds with certain symmetries so that one can reduce geometric PDEs to simple ODEs. This approach is often referred to as the Calabi ansatz in the literature, which has been studied and generalized to different extent by many authors; see e.g., [22] for some general discussions and historical overviews.

For our purpose, we will work on the total space of line bundles over Kähler manifolds. The goal is to construct canonical metrics on this space. Our computation will follow the exposition in [41, Section 4.4].

Let $(V,\omega )$ be an n-dimensional compact Kähler manifold, where $\omega$ is a Kähler form on V. Let $L\rightarrow V$ be a holomorphic line bundle equipped with a smooth Hermitian metric h such that its curvature form $R_h$ satisfies

$$\begin{aligned} R_h:={\sqrt{-1}}\partial {\bar{\partial }}\log h^{-1}=\lambda \omega \end{aligned}$$

(27)

for some constant $\lambda >0.$ Let

$$\begin{aligned}L^{-1}\xrightarrow {\pi }V\end{aligned}$$

be the dual bundle of L. whose zero section will be denoted by $V_0$ (so $V_0$ is a copy of V sitting inside the total space $L^{-1}$). In the following we will construct a Kähler metric on $L^{-1}\backslash \{V_0\}$.

The idea is to make use of the fiberwise norm on $L^{-1}$ induced by $h^{-1}$. We put

$$\begin{aligned}s(t):=\log ||t||^2=\log h^{-1}(t,t),\ \text {for }t\in L^{-1}\backslash \{V_0\}.\end{aligned}$$

So s is a globally defined function on $L^{-1}\backslash \{V_0\}$. The goal is to construct a Kähler metric $\eta$ on $L^{-1}\backslash \{V_0\}$ of the form

$$\begin{aligned} \eta ={\sqrt{-1}}\partial {\bar{\partial }}f(s), \end{aligned}$$

(28)

where f is a function to be determined.

We will carry out the computation locally. Choose $p\in V$ and let $\big (U,z=(z_1, \dots ,z_{n})\big )$ be a local coordinate system around p such that $\omega$ can be expressed by a Kähler potential:

$$\begin{aligned} \omega ={\sqrt{-1}}\partial {\bar{\partial }}(P(z)), \end{aligned}$$

(29)

where

$$\begin{aligned}P(z)=|z|^2+O(|z|^4).\end{aligned}$$

Moreover we may assume that $L^{-1}$ is trivialized over U by a nowhere vanishing holomorphic section $\sigma \in \Gamma (U,L^{-1})$ such that

$$\begin{aligned}||\sigma ||_{h^{-1}}^2=h^{-1}(\sigma ,\sigma )=e^{\lambda P(z)}.\end{aligned}$$

Under this trivialization, we have an identification:

$$\begin{aligned} \pi ^{-1}(U)\cong U\times {\mathbb {C}}. \end{aligned}$$

(30)

Let w be the holomorphic coordinate function in the fiber direction. So we have

$$\begin{aligned} s=\log (|w|^2e^{\lambda P(z)})\ \text {on }U\times {\mathbb {C}}^*. \end{aligned}$$

(31)

Such a choice of coordinates has the advantage that, on the fiber $\pi ^{-1}(p)$ over p, one has

$$\begin{aligned} \partial P(z)={\overline{\partial }} P(z)=0. \end{aligned}$$

(32)

So direct computation gives

$$\begin{aligned} \eta ={\sqrt{-1}}\partial {\bar{\partial }}f(s)=\lambda f^\prime \pi ^*\omega +f^{\prime \prime }\frac{\sqrt{-1} dw\wedge d{\overline{w}}}{|w|^2}. \end{aligned}$$

(33)

over p. Thus we get

$$\begin{aligned} \eta ^{n+1}=\frac{(n+1) \lambda ^n (f^\prime )^nf^{\prime \prime }}{|w|^2}(\pi ^*\omega )^n\wedge \sqrt{-1} dw\wedge d{\overline{w}}. \end{aligned}$$

(34)

Now observe that this expression of volume form is true not just over p. Indeed, if we choose a different trivialization $w^\prime =q(z)w$, the expression (34) remains the same. So (34) holds everywhere on $U\times {\mathbb {C}}^*$.

Expression (33) indicates that, to make $\eta$ positively definite, f should be a strictly convex function with $f^\prime >0$. So let us introduce

$$\begin{aligned} \tau =f^\prime (s),\ \varphi (\tau )=f^{\prime \prime }(s). \end{aligned}$$

(35)

Then, over p, the Ricci form is given by

$$\begin{aligned} \begin{aligned} {{\text {Ric}}}(\eta )&=-{\sqrt{-1}}\partial {\bar{\partial }}\log {\text {det}}(\eta )\\&=\pi ^*{{\text {Ric}}}(\omega )-\bigg (n\lambda \frac{\varphi }{\tau }+\lambda \varphi ^\prime \bigg )\pi ^*\omega \\&\ \ \ \ -\varphi \bigg (n\frac{\varphi }{\tau }+\varphi ^\prime \bigg )^\prime \frac{\sqrt{-1} dw\wedge d{\overline{w}}}{|w|^2}. \end{aligned} \end{aligned}$$

(36)

Taking trace, we also get the scalar curvature:

$$\begin{aligned} S(\eta )=\frac{r\pi ^*S(\omega )}{\tau }-\frac{(\tau ^n\varphi )^{\prime \prime }}{\tau ^n}. \end{aligned}$$

(37)

Note that there is a natural holomorphic vector field on $L^{-1}$, which can be formulated as

$$\begin{aligned} v:=\nabla ^{1,0}_\eta \tau . \end{aligned}$$

(38)

In our chosen local coordinates, simply $v=w\frac{\partial }{\partial w}$, which is in fact the vector field generated by the obvious $\mathbb {C}^*$ -action on the fiber.

Remark A.2

It is worth mentioning that, Calabi ansatz also applies to projective bundles of higher ranks (see [22] for more general discussions).

1.2 A.2 Apply to our setting

Assume further now we are in the setting of Theorem 1.1. with $\omega \in 2\pi c_1(V)$. Then we have

$$\begin{aligned} r=\frac{1}{\lambda }. \end{aligned}$$

In what follows, we always assume that

$$\begin{aligned} \varphi (r-1)=\varphi (r+1)=0. \end{aligned}$$

(39)

For simple cohomological reason, this guarantees that $\eta$ extends to ${\tilde{Y}}$ and lies in $2\pi c_1({\tilde{Y}})$ (but possibly singular along $V_0$ or $V_\infty$). If moreover

$$\begin{aligned} \varphi ^\prime (r-1)=1\text { and }\varphi ^\prime (r+1)=-1, \end{aligned}$$

(40)

then $\eta$ will indeed be a smooth Kähler form in $2\pi c_1({\tilde{Y}})$.

Lemma A.3

Assume that $\varphi$ satisfies (39) and (40), then the Futaki invariant Fut(v) satisfies

$$\begin{aligned} Fut(v)=\int _{{\tilde{Y}}}\tau (S(\eta )-n-1)\eta ^{n+1}=(2\pi )^{n+1}(\beta _0^{-1}-1)(-K_{{\tilde{Y}}})^{n+1}. \end{aligned}$$

Proof

The first equality is exactly the definition of Futaki’s invariant. While the second equality follows from a direct computation. Indeed, using the local coordinates in the previous subsection, we write $w=re^{i\theta }$. Then it is easy to deduce that

$$\begin{aligned} \frac{\sqrt{-1}dw\wedge d{\bar{w}}}{|w|^2}=2\frac{dr}{r}\wedge d\theta =\frac{d\tau }{\varphi }\wedge d\theta . \end{aligned}$$

Thus by (37) and (34),

$$\begin{aligned}\int _{{\tilde{Y}}}\tau (S(\eta )-n-1)\eta ^{n+1}=2\pi (n+1)\int _V\int _{r-1}^{r+1}\tau \bigg (\frac{rS(\omega )}{\tau }-\frac{(\tau ^n\varphi )^{\prime \prime }}{\tau ^n}-n-1\bigg )\frac{\tau ^n}{r^n}d\tau \wedge \omega ^n.\\\end{aligned}$$

Then the result follows from a tedious computation using $\int _VS(\omega )\omega ^n=(2\pi )^nn(-K_V)^n$, (39) and (40). $\square$

As a consequence, we have the following

Lemma A.4

One has

$$\begin{aligned} \beta ({\tilde{Y}})\le \beta _0. \end{aligned}$$

Proof

By considering the slope of the twisted Mabuchi energy along the geodesic ray generated by v, the result follows from [40, Proposition 7]. $\square$

Now we derive a lower bound for the greatest Ricci lower bound $\beta ({\tilde{Y}})$. We will follow the approach in [40, Section 3.1] to construct a family of Kähler metrics $\eta \in 2\pi c_1({\tilde{Y}})$ with Ricci curvature as positive as possible. Similar treatment also appears in [29, Section 3.2].

Recall that V is also a Fano manifold. We fix

$$\begin{aligned}\mu \in (0,\beta (V))\end{aligned}$$

and choose Kähler forms $\omega ,\alpha \in 2\pi c_1(V)$ such that

$$\begin{aligned} {{\text {Ric}}}(\omega )=\mu \omega +(1-\mu )\alpha . \end{aligned}$$

(41)

Take an ample line bundle L with

$$\begin{aligned} L=-\frac{1}{r} K_V,\ \text {for some }r>1. \end{aligned}$$

Then using Calabi ansatz, we can consider Kähler metrics $\eta$ on ${\tilde{Y}}$ of the form (in special local coordinates)

$$\begin{aligned} \eta =\frac{\tau }{r}\pi ^*\omega +\varphi \frac{\sqrt{-1} dw\wedge d{\overline{w}}}{|w|^2}, \end{aligned}$$

whose Ricci forms are given by

$$\begin{aligned} \begin{aligned} {{\text {Ric}}}(\eta )&=\bigg (\mu -\frac{n\varphi }{r\tau }-\frac{\varphi ^\prime }{r}\bigg )\pi ^*\omega +(1-\mu )\pi ^*\alpha \\&\ \ \ \ -\varphi \bigg (\frac{n\varphi }{\tau }+\varphi ^\prime \bigg )^\prime \frac{\sqrt{-1} dw\wedge d{\overline{w}}}{|w|^2}. \end{aligned} \end{aligned}$$

(42)

Here $\varphi =\varphi (\tau )$ with $\tau \in (r-1,r+1)$ is a one-variable positive function to be determined and w denotes the fiberwise coordinate. To cook up $\eta \in 2\pi c_1({\tilde{Y}})$ with ${{\text {Ric}}}(\eta )\ge \beta \eta$ (possibly in the current sense), we will impose the following conditions for $\varphi$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \varphi (r-1)=\varphi (r+1)=0,\\ \varphi ^\prime (r-1)\in (0,1],\\ \varphi ^\prime (r+1)\in [-1,0),\\ \end{array}\right. } \end{aligned}$$

(43)

and

$$\begin{aligned} -\bigg (n\frac{\varphi }{\tau }+\varphi ^\prime \bigg )^\prime =\beta \text { for }\tau \in (r-1,r+1), \end{aligned}$$

(44)

where $\beta$ is any constant that satisfies

$$\begin{aligned} 0<\beta \le \min \bigg \{\frac{\mu \beta _0}{1/r+\beta _0(1-1/r)},\beta _0\bigg \} \end{aligned}$$

(45)

(recall the definition of $\beta _0$ in (2)).

Let us explain the exact meanings of these conditions. The boundary condition (43) makes sure that $\eta \in 2\pi c_1({\tilde{Y}})$ and $\eta$ possibly possesses certain amount of edge singularities along $V_0$ and $V_\infty$. Solving the ODE (44), we obtain that

$$\begin{aligned} \tau ^n\varphi =-\frac{\beta }{n+2}\tau ^{n+2}+A\tau ^{n+1}+B \end{aligned}$$

(46)

where

$$\begin{aligned}{\left\{ \begin{array}{ll} A=\frac{\beta }{n+2}\cdot \frac{(r+1)^{n+2}-(r-1)^{n+2}}{(r+1)^{n+1}-(r-1)^{n+1}},\\ B=\frac{-2\beta }{n+2}\cdot \frac{(r^2-1)^{n+1}}{(r+1)^{n+1}-(r-1)^{n+1}}.\\ \end{array}\right. }\end{aligned}$$

From this, we easily derive that

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta _1:=\varphi ^\prime (r-1)=\frac{\beta }{\beta _0},\\ \beta _2:=-\varphi ^\prime (r+1)=\frac{\beta (2\beta _0-1)}{\beta _0}.\\ \end{array}\right. } \end{aligned}$$

(47)

Then (3) and (45) simply imply that

$$\begin{aligned} 0<\beta _2<\beta _1\le 1. \end{aligned}$$

So $\eta$ has edge singularities with angles $\beta _1$ and $\beta _2$ along $V_0$ and $V_\infty$ respectively. Moreover (45) also guarantees that

$$\begin{aligned} \begin{aligned} \mu -\frac{n\varphi }{r\tau }-\frac{\varphi ^\prime }{r}&=\mu -\beta _1/r-\beta (1-1/r-\tau )\\&=(\mu -\beta /(r\beta _0)-\beta (1-1/r))+\tau \beta \\&\ge \tau \beta .\\ \end{aligned} \end{aligned}$$

(48)

Therefore $\eta$ satisfies ${{\text {Ric}}}(\eta )\ge \beta \eta$ in the current sense. More precisely, $\eta$ solves the following twisted Kähler–Einstein edge equation:

$$\begin{aligned} \begin{aligned} {{\text {Ric}}}(\eta )&=\beta \eta +\big (\mu -\beta /(r\beta _0)-\beta (1-1/r)\big )\pi ^*\omega +(1-\mu )\pi ^*\alpha \\&\ \ \ \ \ \ \ \ +2\pi (1-\beta /\beta _0)[V_0]+2\pi \big (1-\beta (2\beta _0-1)/\beta _0\big )[V_\infty ]. \end{aligned} \end{aligned}$$

(49)

Then by Proposition A.1, (3), (8) and [2, Theorem C],

$$\begin{aligned} \beta ({\tilde{Y}})=\delta ({\tilde{Y}})\ge \delta _\theta ({\tilde{Y}})\ge \beta _\theta ({\tilde{Y}})\ge \beta , \end{aligned}$$

(50)

where

$$\begin{aligned} \theta =\frac{(\mu -\beta /(r\beta _0)-\beta (1-1/r))}{2\pi }\pi ^*\omega +\frac{1-\mu }{2\pi }\pi ^*\alpha +(1-\beta _1)[V_0]+(1-\beta _2)[V_\infty ] \end{aligned}$$

is a semi-positive current in $(1-\beta )c_1({\tilde{Y}})$. Using (45) and letting $\mu \rightarrow \beta ({\tilde{Y}})$, we obtain

$$\begin{aligned} \beta ({\tilde{Y}})\ge \min \bigg \{\frac{\beta (V)\beta _0}{1/r+\beta _0(1-1/r)},\beta _0\bigg \}. \end{aligned}$$

Finally, using (8) again, we get the following

Proposition A.5

One has

$$\begin{aligned} \delta ({\tilde{Y}})\ge \min \bigg \{\frac{\delta (V)\beta _0}{1/r+\beta _0(1-1/r)},\beta _0\bigg \}. \end{aligned}$$

So Proposition A.1 is proved.

1.3 A.3 Limiting behavior

Now we briefly study the degeneration of metrics on ${\tilde{Y}}$ with positive Ricci curvature as they approach the threshold.

Suppose that V admits a KE metric $\omega _{KE}\in 2\pi c_1(V)$. (In this case $\beta ({\tilde{Y}})=\beta _0$ by Theorem 1.1). Then as in [40, Section 3.1], for any $\beta \in (0,\beta _0)$ we can construct a smooth Kähler form $\omega _\beta \in 2\pi c_1({\tilde{Y}})$ with ${{\text {Ric}}}(\omega _\beta )>\beta \omega _\beta$ such that, as $\beta \rightarrow \beta _0$, one has $({\tilde{Y}},\omega _\beta )\xrightarrow {G.H.}({\tilde{Y}},\eta ),$ with $\eta$ solving

$$\begin{aligned} {{\text {Ric}}}(\eta )=\beta _0\eta +(1-1/r-\beta _0(1-1/r))\pi ^*\omega _{KE}+2\pi (2-2\beta _0)[V_\infty ]. \end{aligned}$$

In particular the limit space is still ${\tilde{Y}}$. This generalizes [40], where an $\eta$ satisfying

$$\begin{aligned} {{\text {Ric}}}(\eta )=\frac{6}{7}\eta +\frac{1}{7}\pi ^*\omega _{FS}+2\pi (1-\frac{5}{7})[V_\infty ] \end{aligned}$$

was constructed on $Bl_1\mathbb {P}^2$.

Suppose in general that V does not necessarily admit KE, but $\beta (V)>1/r+\beta _0(1-1/r)$. (In this case again $\beta ({\tilde{Y}})=\beta _0$ by Theorem 1.1). We choose $\mu \in (1/r+\beta _0(1-1/r),\beta (V))$ and hence there are Kähler forms $\omega , \alpha \in 2\pi c_1(V)$ satisfying (41). Then the same construction as in [40, Section 3.1] shows that, for any $\beta \in (0,\beta _0)$ there is a smooth Kähler form $\omega _\beta \in 2\pi c_1({\tilde{Y}})$ with ${{\text {Ric}}}(\omega _\beta )>\beta \omega _\beta$ such that, as $\beta \rightarrow \beta _0$, one has $({\tilde{Y}},\omega _\beta )\xrightarrow {G.H.}({\tilde{Y}},\eta ),$ with $\eta$ solving

$$\begin{aligned} {{\text {Ric}}}(\eta )=\beta _0\eta +(\mu -1/r-\beta _0(1-1/r))\pi ^*\omega +(1-\mu )\pi ^*\alpha +2\pi (2-2\beta _0)[V_\infty ]. \end{aligned}$$

So as in the previous case, the limit space is still ${\tilde{Y}}$ itself. Note that the limit metric $\eta$ is not unique (as $\mu$, $\omega$ and $\alpha$ are allowed to vary).

As we have noted in the above two cases, the limit space is still ${\tilde{Y}}$ itself. So in the view of [9], the optimal destabilization of ${\tilde{Y}}$ should be a product (but non-trivial) test configuration. In the toric case, similar phenomena also appeared in [27]. We actually expect that the optimal destabilization of a toric Fano variety is always itself.

Finally, suppose that $\beta (V)\le 1/r+\beta _0(1-1/r)$. Then by Theorem 1.1, $\beta ({\tilde{Y}})=\frac{\delta (V)\beta _0)}{1/r+\beta _0(1-1/r)}$. This case turns out to be more subtle. Firstly, it seems that the Calabi ansatz does not easily provide smooth Kähler forms $\omega _\beta$ such that ${{\text {Ric}}}(\omega )\ge \beta \omega _\beta$ as $\beta \rightarrow \beta ({\tilde{Y}})$. Secondly, as $\mu \rightarrow \beta (X)$, the Kähler form $\omega$ we chose from the base V (recall (41)) is supposed to develop certain singularities, which suggests that V itself would degenerate in the Gromov–Hausdorff topology to some other $\mathbb {Q}$-Fano variety. So at this stage it is unclear how ${\tilde{Y}}$ would degenerate. We leave this case to future studies.