Delta invariants of projective bundles and projective cones of Fano type

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 β(V ) of Tian [Tia92] measures how far V is away from a KE manifold. As shown in [BBJ18,CRZ19], β(V ) is equal to the algebraic δ-invariant introduced in [FO18,BJ20], which serves as the right threshold for V to be Ding-stable (cf. [BJ18,BBJ18,BLZ20]).
So an important problem in algebraic geometry is to compute the δ-invariants of Fano varieties. Although some efforts to tackle this problem have been made in low dimensions (see e.g. [PW18,CRZ19,CZ19]), 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 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 δinvariant. This generalizes the example considered in [Szé11, Section 3.1]. More generally we will also consider the (singular) log Fano setting and derive formulas for the corresponding δ-invariants. In the log case, one can still carry out the Calabi trick as in [LL19]. But we will take a purely algebraic approach, again making use of the C * -action. Indeed, by [BJ20,Gol20], to compute δ, it is enough to consider C * -invariant divisorial valuations, which can greatly simplify the computation.
1.2. Main results. We work over complex number 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 ) ≥ 2. So we can find an ample line bundle L such that L = − 1 r K V for some rational number r > 1. (1) We putỸ which is the P 1 -bundle over V with respect to L −1 ⊕ O V . Let V 0 denote the zero section and V ∞ the infinity section ofỸ . Then is ample and henceỸ is an (n + 1)-dimensional Fano manifold. We put β 0 := n + 1 n + 2 · (r + 1) n+2 − (r − 1) n+2 (r + 1) n+1 − (r − 1) n+1 − (r − 1) −1 . (2) Using binomial formula, one can easily verify the following elementary fact: Our first main result is stated as follows.
In particular,Ỹ cannot admit KE metrics. But as we shall see in Section A,Ỹ does admit a family of twisted conical KE metrics. When δ(V ) ≥ 1/r + β 0 (1 − 1/r) (this holds for example when V is K-semistable), we deduce that δ(V ) = β 0 .
In this case, there always exists a prime divisor F over V computing δ(V ) (see [BLZ20,Theorem 6.7]). This divisor naturally induces a divisor F overỸ , and we will show that δ(Ỹ ) is computed by F .
So to some extent, Theorem 1.1 generalizes this product formula.
The smoothness assumption ofỸ 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 X := Spec ⊕ k∈N H 0 (V, kL), and the projective cone over X associated to L to be Y := Proj ⊕ k,j∈N H 0 (V, kL)s j .
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 ⊕ k∈N H 0 (V, kmL) where m is sufficiently divisible such that mL is very ample, then ⊕ k∈N H 0 (V, kL) is integral over ⊕ k∈N H 0 (V, kmL)), and Y is obtained by adding an infinite divisor V ∞ to X, where V ∞ is the infinite divisor on Y defined by s = 0.
We provide another viewpoint of X and Y . Let p :Ỹ := P V (L −1 O V ) → V be the projective P 1 -bundle over V associated to L −1 O V . Let V ∞ and V 0 be the infinite and zero sections of p respectively. Then we know there is a contraction φ : If the base V is a Q-Fano variety and L ∼ Q − 1 r K V is an ample line bundle on V for some positive rational number r, then it is not hard to see thatỸ (for r > 1) and Y (for r > 0) are Q-Fano varieties. Moreover, both (Ỹ , aV 0 + bV ∞ ) and (Y, cV ∞ ) are log Fano varieties for any rational 0 ≤ a < 1, 0 ≤ b < 1, 0 ≤ c < 1 if r > 1, and 1 − r < a < 1, 0 ≤ b < 1, 0 ≤ c < 1 if 0 < r ≤ 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 Q-Fano variety of dimension n and L ∼ Q − 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 ≤ a < 1, 0 ≤ b < 1 if r > 1, and 1 − r < a < 1, 0 ≤ b < 1 if 0 < r ≤ 1. Then we have the following formula computing delta invariants, 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 .
Note that when V is a Fano manifold, we naturally have r ≤ n + 1, e.g [Kol96, Chapter 5], thus one can easily have the comparison 1 n+1 n+2 This is precisely what we get in Theorem 1.
, then the above formula can be simplified as We just note here that β a,b is the delta invariant computed by the divisor V 0 , while β a,b is computed by the divisor V ∞ , and rδ(V ) 1−a+Aβ a,b β 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 Q-Fano variety of dimension n and L ∼ Q − 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 δ(Y, cV ∞ ) = min (n + 2)r (n + 1)(r + 1 − c) δ(V ), (n + 2)r (n + 1)(r + 1 − c) , where 0 ≤ c < 1 is a rational number.
Corollary 1.6. Let V be a Q-Fano variety of dimension n and L ∼ Q − 1 r K V an ample line bundle on V for some positive rational number r. ( 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 ∼ Q −λK V for some positive rational number λ. Write 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 < λ < 1, In particular, if V and S are both K-polystable, then (V, aS) is K-polystable for any a ∈ [0, 1 − r n ). This result has been essentially known to experts (cf. [LS14, LZ19, LL19] 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 [Der16, Remark 3.6] The paper is organized as follows. In Section 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 Section 3 and Section 4, we prove Theorem 1.3 and Theorem 1.4 respectively, by purely algebraic method. In Section 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.

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, ∆) is a log pair if V is a projective normal variety and ∆ is an effective Q-divisor on V such that K V + ∆ is Q-Cartier. Suppose f : W → V is a proper birational mophism between normal varieties and E is a prime divisor on W , then we define to be the log discrepancy of the divisor E associated to the pair (V, ∆). The log pair (V, ∆) is called a log Fano variety if it admits klt singularities and −(K V + ∆) is ample. If ∆ = 0, we just say V is a Q-Fano variety. For the concepts of klt singularities, please refer to [KM98,Kol13].
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, where V ∞ is the infinite section ofỸ . There is a natural contraction φ :Ỹ → Y (resp. φ :X → X) which contracts V 0 , where V 0 is the zero section ofỸ . We just list the properties ofỸ and Y in the following lemma.
Lemma 2.1. Notation as above, let p :Ỹ : Proof. By the construction we see that is the natural blowup of the vertex o ∈ Y , thus OỸ (1) is defined by s = 0, which is the infinite section V ∞ , and V 0 | V 0 ∼ L −1 . It's clear that V ∞ − V 0 is a relative trivial line bundle for the projection map p :Ỹ → V , thus there is a line bundle M on V such that p * M ∼ V ∞ − V 0 . Restrict it to V 0 we see M ∼ L, and restrict it to V ∞ we the see that V ∞ | V∞ ∼ 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 ∞ ) are indeed log Fano varieties, specially with klt singularities. This follows from the following Theorem 2.3.
Let ∆ be an effective Q-divisor on V such that K V + ∆ is Q-Cartier, and ∆ X and ∆X be the corresponding extending divisors on X andX respectively. We have following results [Kol13].
(5) We assume more that K X + ∆ X is Q-Cartier, then there is some rational number r such that L ∼ Q − 1 r (K V + ∆).
(6) Conversely, if K V + ∆ is proportional to L, then K X + ∆ X is Q-Cartier.
Proof. For (1), one only needs to note thatX is an A 1 -bundle over V . For (2), we consider V ∼ = V 0 →X → X, where the last arrow denoted by φ means the blowup of the cone vertex.
For any line bundle M on X, it is pulled back toX to be a trivial line bundle, thus M is also a trivial line bundle. For (3), consider the exact sequence then the facts Cl(X \ o) ∼ = Cl(X) and V 0 | V 0 ∼ L −1 conclude the result. For (4), it is directly implied if we write down the differential form. For (5), as m(K X + ∆ X ) is a trivial line bundle for a divisible m due to (2), m(K X\o + ∆ X\o ) is also trivial, then by the exact sequence above, one sees there is a rational number r such that p * (K V + ∆) ∼ Q rV 0 . Hence (5) is finished by restriction. For (6), one only needs to note that m(K X + ∆ X ) is Cartier if and only if m(K X + ∆ X ) is a trivial line bundle if and only if m(K X\o + ∆ X\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 ∼ Q − 1 r (K V + ∆), so KX + ∆X + V 0 ∼ Q,φ rV 0 by (4).
The above lemma directly implies the following result on cone singularities, also see [Kol13]. (1) If (V, ∆) is a log Fano pair and L = − 1 r (K V + ∆) is an ample line bundle on V for some rational r > 0, then (X, ∆ X ) admits klt singularities.
(2) If (V, ∆) is a general type pair and L = − 1 r (K V + ∆) is an ample line bundle on V for some rational r < 0, then (X, ∆ X ) is not log canonical. Corollary 2.4. Let V be a projective normal variety and L an ample line bundle on V. If X := Spec ⊕ k∈bN H 0 (V, kL) admits klt singularities, specially, K X is Q-Cartier, then there is a positive rational number r > 0 such that L ∼ Q − 1 r K V . In particular, V is a Q-Fano variety.

Delta invariant.
In this section we assume (V, ∆) is a log Fano variety of dimension n. Delta invariant is introduced in [FO18] to measure the singularities of the anti-canonical divisor of a 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 [Fuj19, BJ20, BX19, BLX20] etc.. We first define m-basis type divisors of (V, ∆).
Definition 2.5. For a sufficiently divisible fixed natural number m such that −m(K V + ∆) is an ample line bundle, one chooses any complete basis of H 0 (V, −m(K V + ∆)), say . Divisors of this form are called m-basis type divisors. It is clear that D m ∼ Q −(K V + ∆). For a prime divisor E over V (i.e. there exists a proper birational morphism from a normal variety W → V such that E is a prime divisor on W ), we define the following S m -invariant and m-th delta invariant of (V, ∆), The following lemma is due to [BJ20,FO18].
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 (2) Let m tend to infinity, then the limits in above definition indeed exist, denoted by S-invariant can be computed by the following integration, Definition 2.7. Delta invariant of the log Fano pair (V, ∆) is defined to be where E runs through all prime divisors over V .
We sometimes leave out (V, ∆) in the subscript if there is no confusion. It is clear that the above definition applies to any Q-ample line bundle L on V , and we respectively get m-basis type divisors associated to L just by replacing −(K V + ∆) by L. In this case, we use S m,L (E) and S L (E) (resp. δ m,L (ord E ) and δ L (ord E )) to denote the S m -invariant and S-invariant (resp. m-th delta invariant and delta invariant), and δ(L) The following result is well known by works [BJ20,FO18], we just state it here.
This invariant was first implicitly studied by Tian [Tia92] and then explicitly introduced in [Rub08, Rub09]. Recently it is shown independently by [CRZ19] and [BBJ18] that Finally we remark that, suppose in addition that there is a semipositive (1, 1)-current θ on X, then there are analogous results for the θ-twisted δ-and β-invariants (see [BBJ18] for more information).
Lemma 3.1. Notation as in Theorem 1.3, we have Note that the above lemma is essentially a Futaki invariant computation under a fancy δ-invariant guise for the natural C * -action onỸ (see the proof of Lemma A.4), and it tells us . Lemma 3.2. Notation as in Theorem 1.3, we have Proof. We choose sufficiently divisible m such that ma, mb and mr are all integers, and by Lemma 2.1 we have By (9), one sees thatR and by (10)R Let E be a prime divisor over V and EỸ the natural extended divisor overỸ , then we want to explore the relationship between δ L (ord E ) and δ (Ỹ ,aV 0 +bV∞) (ord EỸ ). As it is clear that , it suffices to explore the relationship between S L (E) and S (Ỹ ,aV 0 +bV∞) (EỸ ). We first construct a special m-basis type divisor of −(KỸ + aV 0 + bV ∞ ).
Writer m,j = dimR m,j = dim H 0 (V, (mr−m+ma+j)L), we can choose an mr−m+ma+jbasis type divisorD m,j of L such thatD m,j computes S mr−m+ma+j,L (E). In fact,D m,j is created by the filtration induced by ord E on H 0 (V, (mr − m + ma + j)L). Then we liftD m,j to be a divisorD m,j overỸ . By (10), it is not hard to see that is an m-basis type divisor of −(KỸ + aV 0 + bV ∞ ). Then we have As m tends to infinity, by the computation of mr m,j in the next lemma, one directly obtains that where As the prime divisor E is arbitrarily chosen, we have Proof. The lemma is concluded by the following two computations Combine Lemma 3.1 and 3.2, we have the following result on upper bound.
In particular, when a = b = 0 (in this case, r > 1 automatically), we have .
Now we turn to the converse direction. First recall that in the proof of Lemma 3.2, we construct a special m-basis type divisorD m for −(KỸ + aV 0 + bV ∞ ), however, we don't know whether it is compatible with EỸ , so we only have S m,(Ỹ ,aV 0 +bV∞) (EỸ ) ≥ ord EỸ (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.
Proof. Recall in the proof of Lemma 3.2, we obtained 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 Let T := C * . As δ(Ỹ , aV 0 +bV ∞ ) can be approximated by T -equivariant divisorial valuations overỸ , see [Gol20, Section 4] or [BJ20, Section 7], we need to deal with the T -invariant valuations overỸ whose centers lie in V 0 or V ∞ . AssumeF = V 0 is a T -invariant prime divisor overỸ such that cỸ (F ) ⊂ V 0 , which means the center of ordF lies in V 0 , then there is a positive rational number c and a prime divisor F over V such that r(ordF ) = c · ord F , where r(ordF ) is the restriction of ordF to K(V ). Let FỸ be the induced prime divisor over Y extended by F , then by [BHJ17] ordF is a quasimonomial valuation along V 0 and FỸ with weights (ordF (V 0 ), c), that is ordF = ordF (V 0 ) · ord V 0 + c · ord FỸ , so by (10) and [JM12] we have , so by Lemma 3.1 the following holds, For T -invariant divisors whose centers lie in V ∞ but not equal to V ∞ , the analysis is similar. One can write ordF = ordF (V ∞ ) · ord V∞ + c · ord FỸ for some positive rational c, so by (10) and [JM12] we have so by Lemma 3.1 the following holds, Combine inequality (18), we have the following , which finishes the proof.

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 δ Y,cV∞ (ord V 0 ) and δ Y,cV∞ (ord V∞ ).
Lemma 4.1. Notation as in Theorem 1.4, we have Proof. By Lemma 2.2 we know that 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.
By the same method used for the bundle case, we next work out another upper bound for δ(Y, cV ∞ ), that is δ(Y, cV ∞ ) ≤ One directly sees that and by (19), 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 D m,j over Y . By (19), it is not hard to see that is an m-basis type divisor of φ * (−(K Y + cV ∞ )). 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 Let m tends to infinity, by the same computation as in Lemma 3.3, one gets hence we have As E is arbitrarily chosen over V , we know δ(Y, cV ∞ ) ≤ n + 2 (n + 1)(r + 1 − c) δ V (L) = (n + 2)r (n + 1)(r + 1 − c) δ(V ).
Above all, one obtains the upper bound in Theorem 1.4.
Theorem 4.3. Notation as in Theorem 1.4, we have δ(Y, cV ∞ ) ≤ min (n + 2)r (n + 1)(r + 1 − c) δ(V ), (n + 2)r (n + 1)(r + 1 − c) , 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.
We will take another more elegant way below for the lower bound, although we can only work out the case where 0 < r ≤ n + 1. However, this upper bound of r is usually satisfied, at least for K-semistable log Fano pairs of dimension n, see [Liu18,Fuj18]. First we recall the following lemma [LZ19, Proposition 2.11] which establishes the relationship between K-stability of the base and that of projective cone over it, see also [LL19].
Lemma 4.4. Let (V, ∆) be an n-dimensional log Fano variety, and L an ample line bundle on V such that L ∼ Q − 1 r (K V + ∆) for some 0 < r ≤ n + 1. As before, Y is the projective cone over V associated to L, then (V, ∆) is K-semistable if and only if (Y, ∆ Y +(1− r n+1 )V ∞ ) is K-semistable, where ∆ Y is the divisor on Y naturally extended by ∆.
By above lemma, we can obtain the converse direction for projective cones.
Proof of Theorem 1.4 for 0 < r ≤ n + 1 and c = 0. The proof is a combination of Theorem 4.3 and Theorem 4.5.
Apply Theorem 1.4 to the case c = 0 one directly concludes.

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 ∼ Q −λK V for some positive rational number λ. Write 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 < λ < 1, then E(V, S) = [0, 1 − r n ], where r = 1 λ − 1. In particular, if V and S are both K-polystable, then the pair (V, aS) is K-polystable for any a ∈ [0, 1 − r n ). We also note that 1 − r n ≥ 0, since λ ≥ 1 n+1 if V is K-semistable. In the above theorem, we do not consider the case λ ≥ 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 ∼ Q −λK V is a smooth divisor on V for some positive rational number λ ≥ 1. Then (V, aS) is K-semistable for any a ∈ [0, 1 λ ) and K-stable for a ∈ (0, 1 λ ). Proof. We first deal with the case λ = 1, i.e. S ∼ Q −K V , then (V, S) is a log smooth Calabi-Yau pair. We show that α(V, (1 − β)S) = 1 for sufficiently small rational 0 < β 1 (see also [Ber13]). For the definition of alpha invariant of a log Fano pair please refer to [Tia87, Che01, CS08, BJ20] etc. It is clear that α(V, (1 − β)S) ≤ 1. Suppose the inequality is strict, then we can find a divisor D ∼ Q −K V such that the pair (V, (1 − β)S + βD) is not log canonical. After subtracting certain amount of S from D we may assume that S Supp(D), then by inversion of adjunction (S, βD| S ) is not log canonical, contradicting to the choice of β (note that β is sufficiently small). So (V, (1 − β)S) is K-stable for sufficiently small 0 < β 1. By interpolation of K-stability, see [ADL19, Proposition 2.13], we conclude the result for λ = 1.
For the case λ > 1, it is clear that we should require a < 1 λ to make sure that (V, aS) is a log Fano pair. Then for any prime divisor E over V , we have for any a ∈ (0, 1 λ ). Thus δ(V, aS) > 1 for any a ∈ (0, 1 λ ). Concluded.
We turn to prove Theorem 1.7. Let us first begin with the following lemma, see [LZ19, Lemma 2.12] or [LZ20, Theorem 5.1]. In the above lemma, we have proved one direction of inclusion, the converse inclusion is implied by following lemma. This has been proved in [LS14,Theorem 1.4] in analytic setting, we give a purely algebraic proof below.
Proof. As we have seen in the proof of Lemma 5.3, by [LZ19, Lemma 2.12], the pair (V, aS) can be specially degenerated to (C p (S, M ), aS ∞ ), where C p (S, M ) is the projective cone over S associated to M , and S ∞ is the infinite section. We denote this special test configuration by φ : (V, aS;L) → P 1 , whereL := −(KV /P 1 + aS) is the polarization of this test configuration.
. By [LX14], the generalised Futaki invariant (or Donaldson-Futaki invariant) of the test configuration is as follows,  (S, M ), aS ∞ ) is also K-semistable. By the next lemma, we know δ(C p (S, M ), aS ∞ ) < 1 when a > 1 − r n . The contradiction implies that (V, aS) is K-unstable for a > 1 − r n . The proof is finished.
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 Q-Fano variety of dimension n − 1 and M an ample line bundle on S such that M ∼ Q − 1 r K S . Write Y := C p (S, M ) to be the projective cone over S associated to M, and S ∞ the infinite section, then we have following formula computing delta invariant of (C p (S, M ), aS ∞ ), δ(C p (S, M ), aS ∞ ) = min (n + 1)r n(r + 1 − a) δ(S), (n + 1)r n(r + 1 − a) , (n + 1)(1 − a) r + 1 − a .
In particular, when a > 1 − r n , we have Proof of Theorem 1.7. The proof is a combination of Lemma 5.3, Lemma 5.4, Lemma 5.5, and following lemma on interpolation for K-polystability.
Lemma 5.6. Let V be an n-dimensional Fano manifold and S a smooth prime divisor on V such that −K V ∼ 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.
Corollary 5.7. (=Corollary 1.8) For the pair (P n , S d ) where S d is a smooth hypersurface of degree 1 ≤ d ≤ n. If S d is K-polystable (this is expected to be true), then we have In particular, (P n , aS d ) is K-polystable for any a ∈ [0, 1 − r n ). Proof. Just replace (P n , S d ) by (V, S), then apply Theorem 1.7.

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 ⊂ P n+1 be a smooth hypersuface of degree d defined by a homogeneous polynomial f d , where 2 ≤ d ≤ n + 1.
It is clear that V 1 d is still a degree d hypersuface but lies in 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 2 d ∼ r 1 H 1 , where r 1 = n + 3 − d = r 0 + 1. Continue the process, we get hypersurfaces (V i d , H i ) of degree d in P n+1+i , i ≥ 1, which is still cut out by f d = 0, and H i is the hyperplane section with −K We want to compute delta invariants of these Fano varieties. Let us first recall following result.
Let V be a Q-Fano variety of dimension n, and L an ample line bundle on V such that L ∼ Q − 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 := Proj ⊕ k∈N ⊕ l∈N H 0 (V, kL)s l , then by Theorem 1.4 we know, δ(Y ) = min (n + 2)r (n + 1)(r + 1) δ(V ), (n + 2)r (n + 1)(r + 1) , n + 2 r + 1 .
Thus we have following formula We have the following results.
Example 6.3. Let V be a smooth conic curve in P 2 , then δ(V ) = 1. Let Y be the projective cone over V , then Y is isomorphic to a quadratic surface defined by x 2 0 + x 2 1 + x 2 2 = 0 in P 3 .
In this case, n = 1, d = 2, so δ(Y ) = 3 4 . We know that a smooth quadratic surface in P 3 is isomorphic to P 1 × P 1 , which is K-polystable and the delta invariant is 1.
Example 6.4. Let V be a smooth cubic surface in P 3 , then δ(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 P 4 . In this case, n = 2, d = 3, so δ(Y ) = 2 3 . We know that a smooth cubic 3-fold in P 4 is K-stable, see [LX19].
Question 6.5. The delta invariants of smooth cubic surfaces has been estimated in [PW18,CZ19]. 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 Q-line bundles. In the whole previous contents, the cones are associated to ample line bundles. We now deal with the case of ample Q-line bundles. Let V be a Q-Gorenstein projective normal variety of dimension n and L an ample Q-line bundle on V . We will view L to be a Q-Cartier divisor and write ∆ L to be the branched divisor associated to L, that is, we construct a finite morphism f :Ṽ → V such that KṼ = f * (K V + ∆ L ) and f * L is an ample line bundle onṼ . We consider the following cone overṼ associated to f * L. i.e.
Example 6.7. This example is taken from [LZ19]. Let V = P n and L = n+1 n+2 S n+1 − nH, where S n+1 is a general hypersurface of degree n + 1 in P n and H is a hyperplane. Then n+2 H. Thus r = n + 1, δ(V, ∆ L ) ≥ 1. So by Theorem 6.6, δ(Y ) = 1. In fact, Y is isomorphic to a hypersurface in P n+2 determined by x n+2 n+1 = g n+1 x n+2 , where g n+1 (x 0 , x 1 , ..., x n ) is the equation of S n+1 ⊂ P n . We will deal with such kind of hypersurfaces in detail.
From now on, we fix V := P n and S d ⊂ P n is a smooth hypersurface of degree d determined by a homogeneous form g d (x 0 , x 1 , ..., x n ). Then we can construct a cover morphism f :Ṽ → V , which is ramified along S d with multiplicity k. It is clear that we can chooseṼ to be a hypersurface in the weighted projective space P n+1 (k, k, ..., k, d) determined by x k n+1 = g d (x 0 , x 1 , ..., x n ). Suppose H is the hyperplane class in P n and denote L : 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 overṼ associated to f * L is exactly the hypersurface in C n+2 determined by x k n+1 = g d (x 0 , x 1 , ..., x n ), thus the corresponding projective cone is exactly the hypersurface in P n+2 determined by x k n+1 = g d · x k−d n+2 , denoted by Y . In this case, We choose k, d to satisfy that (n + 1)k − (k − 1)d > 0. By Corollary 6.6, we have δ(Y ) = min (n + 2)r (n + 1)(r + 1) δ(V, ∆ L ), (n + 2)r (n + 1)(r + 1) , n + 2 r + 1 , where r = (n + 1)k − (k − 1)d. We have the following lemma about δ(V, ∆ L ).
Lemma 6.8. Let V := P n and S d ⊂ P n be a smooth hypersurface of degree d ≥ n + 1, then (V, aS d ) is K-semistable for any rational 0 ≤ a < n+1 d , i.e. δ(V, aS d ) ≥ 1. Proof. Apply Theorem 5.2.
Above all, we have following result.

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 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.
A.1. Calabi ansatz. We review a well-studied and powerful construction, pioneered by Calabi [Cal79,Cal82], 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., [HS02] 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 [Szé14, Section 4.4].
Let (V, ω) be an n-dimensional compact Kähler manifold, where ω is a Kähler form on V . Let L → V be a holomorphic line bundle equipped with a smooth Hermitian metric h such that its curvature form R h satisfies for some constant λ > 0. Let 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 \{V 0 }. The idea is to make use of the fiberwise norm on L −1 induced by h −1 . We put So s is a globally defined function on L −1 \{V 0 }. The goal is to construct a Kähler metric η where f is a function to be determined. We will carry out the computation locally. Choose p ∈ V and let U, z = (z 1 , ..., z n ) be a local coordinate system around p such that ω can be expressed by a Kähler potential: where P (z) = |z| 2 + O(|z| 4 ).
Moreover we may assume that L −1 is trivialized over U by a nowhere vanishing holomorphic section σ ∈ Γ(U, L −1 ) such that . Under this trivialization, we have an identification: Let w be the holomorphic coordinate function in the fiber direction. So we have s = log(|w| 2 e λP (z) ) on U × C * .
Such a choice of coordinates has the advantage that, on the fiber π −1 (p) over p, one has So direct computation gives over p. Thus we get Now observe that this expression of volume form is true not just over p. Indeed, if we choose a different trivialization w = q(z)w, the expression (34) remains the same. So (34) holds everywhere on U × C * . Expression (33) indicates that, to make η positively definite, f should be a strictly convex function with f > 0. So let us introduce Then, over p, the Ricci form is given by Taking trace, we also get the scalar curvature: Note that there is a natural holomorphic vector field on L −1 , which can be formulated as v := ∇ 1,0 η τ.
In our chosen local coordinates, simply v = w ∂ ∂w , which is in fact the vector field generated by the obvious C * -action on the fiber.
Remark A.2. It is worth mentioning that, Calabi ansatz also applies to projective bundles of higher ranks (see [HS02] for more general discussions).
A.2. Apply to our setting. Assume further now we are in the setting of Theorem 1.1. with ω ∈ 2πc 1 (V ). Then we have In what follows, we always assume that ϕ(r − 1) = ϕ(r + 1) = 0.
Lemma A.3. Assume that ϕ satisfies (39) and (40), then the Futaki invariant F ut(v) satisfies 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 iθ . Then it is easy to deduce that Thus by (37) and (34), Then the result follows from a tedious computation using V S(ω)ω n = (2π) n n(−K V ) n , (39) and (40).
Proof. By considering the slope of the twisted Mabuchi energy along the geodesic ray generated by v, the result follows from [Szé11, Proposition 7]. Now we derive a lower bound for the greatest Ricci lower bound β(Ỹ ). We will follow the approach in [Szé11, Section 3.1] to construct a family of Kähler metrics η ∈ 2πc 1 (Ỹ ) with Ricci curvature as positive as possible. Similar treatment also appears in [LL19, Section 3.2].
Take an ample line bundle L with L = − 1 r K V , for some r > 1.
So Proposition A.1 is proved.
So as in the previous case, the limit space is stillỸ itself. Note that the limit metric η is not unique (as µ, ω and α are allowed to vary).
As we have noted in the above two cases, the limit space is stillỸ itself. So in the view of [BLZ20], the optimal destabilization ofỸ should be a product (but non-trivial) test configuration. In the toric case, similar phenomena also appeared in [Li11]. We actually expect that the optimal destabilization of a toric Fano variety is always itself.