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.


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 [43] measures how far V is away from a KE manifold. As shown in [2,14], β(V ) is equal to the algebraic δ-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 δ-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 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 δ-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 δ-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 C * -action. Indeed, by [6,21], to compute δ, it is enough to consider C * -invariant divisorial valuations, which can greatly simplify the computation.

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.
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 Using binomial formula, one can easily verify the following elementary fact: Our first main result is stated as follows.
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 , k L), and the projective cone over X associated to L to be Y := Proj ⊕ k, j∈N H 0 (V , k L)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 , km L) where m is sufficiently divisible such that m L is very ample, then ⊕ k∈N H 0 (V , k L) is integral over ⊕ k∈N H 0 (V , km L)), 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 :Ỹ → Y which only contracts divisor V 0 , andỸ \V 0 ∼ = Y \o, where o is the cone vertex of X .
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, 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 Then we have the following formula computing delta invariants, 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 [26,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 r δ(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 where 0 ≤ c < 1 is a rational number.
Assume c = 0. If δ(V ) ≥ 1, we naturally have r ≤ n + 1, see [19,28]. If δ(V ) < 1 then we have r δ(V ) ≤ n + 1 by [6,Theorem D]. Hence the following result is deduced. 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, 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 [ 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.

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 [24,25].

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
If 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 both (Ỹ , aV 0 + bV ∞ ) and (Y , cV ∞ ) are log Fano pairs, where Proof By the construction we see that 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 [25].

Lemma 2.2 Notation as above, we have
where we also use p to denote the projectionX → V . (5) We assume more that K X + X is Q-Cartier, then there is some rational number r such where r is as above and r = A X , X (V 0 ).
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 is a trivial line bundle, then apply the above exact sequence. For (7), we apply (5) to get a rational number r such The above lemma directly implies the following result on cone singularities, also see [25].

Theorem 2.3 Let (V , ) be a log pair of dimension n, then
is an ample line bundle on V for some rational r > 0, then (X , X ) admits klt singularities.

) is a log canonical Calabi-Yau pair and L is an ample line bundle on V, then
(X , X ) is log canonical.

Corollary 2.4 Let V be a projective normal variety and L an ample line bundle on V. If
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 [18] 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 [6,8,10,20] etc.. We first define m-basis type divisors of (V , ).

Divisors of this form are called m-basis type divisors. It is clear that
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 [6,18].

(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 , ) (E) = ord E (D m ). (2) Let m tend to infinity, then the limits in above definition indeed exist, denoted by S (V , ) (E)
and δ (V , ) (ord E ).

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 [6,18], we just state it here.

The greatest Ricci lower bound
Let V be a Fano manifold. Then the greatest Ricci lower bound β(V ) of V is defined to be 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 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 δand β-invariants (see [2] for more information).

Delta invariants of projective bundles of Fano type
In this section, we will prove Theorem 1.3. We start by the following lemma computing

Lemma 3.1 Notation as in Theorem 1.3, we have
and Note that the above lemma is essentially a Futaki invariant computation under a fancy δinvariant guise for the natural C * -action onỸ (see the proof of Lemma A.4), and it tells us that

Lemma 3.2 Notation as in Theorem
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) 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 ∞ ).
, we can choose an mr − m + ma + j-basis 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

Lemma 3.3 Notation as in the proof of Lemma 3.2, we have
Proof The lemma is concluded by the following two computations Combine Lemmas 3.1 and 3.2 , we have the following result on upper bound.

Theorem 3.4 Notation as in Theorem
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.

Lemma 3.5 Notation as in the proof of Lemma 3.2, the m-basis divisorD m we construct is compatible with EỸ , that is
Proof Recall in the proof of Lemma 3.2, we obtained As EỸ induces a C * -invariant divisorial valuation overỸ , then the filtration induced by EỸ onR m is compatible with the filtration induced by E on H 0 (V , (mr − m + ma + j)L), concluded.
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 [21,Section 4] or [6, 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 overỸ extended by F, then by [4] 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 [23] 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 for some positive rational c, so by (10) and [23] 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 ∞ ). By the same method used for the bundle case, we next work out another upper bound for

Lemma 4.2 Notation as in Theorem
Proof We choose a sufficiently divisible natural number m such that −m(K Y + cV ∞ ) is an ample line bundle, we have One directly sees that and by (19), Write r m, j = dim R m, j = dim H 0 (V , j L), 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 Above all, one obtains the upper bound in Theorem 1.4.

Theorem 4.3 Notation as in Theorem
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 [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 , ) be an n-dimensional log Fano variety, and L an ample line bundle on
where Y is the divisor on Y naturally extended by .
By above lemma, we can obtain the converse direction for projective cones.
Theorem 4.5 Notation as before, assume 0 < r ≤ n + 1, then Proof We first consider the case δ(V ) ≥ 1, i.e. V is K-semistable. By above Lemma 4.4, one knows δ(Y , ( . So in this case we conclude the result. We next deal with the case δ(V ) < 1. Choose a sequence positive rational numbers δ i which tends to δ(V ) and δ i < δ(V ) for each i. For each δ i one can choose an effective Q-divisor i ∼ Q −K V such that (V , (1 − δ i ) i ) is a K-semistable log Fano pair (even uniformly K-stable), see [7,14]. Then we know L ∼ Q − 1 . Recall r i = δ i r , and let i tends to infinity, one obtains concluded.
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. 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 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 [3]). For the definition of alpha invariant of a log Fano pair please refer to [6,13,15,42] 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 + β D) 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, β D| 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 [1, 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 [35,Lemma 2.12]  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 − 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 ∞ ), 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 [33], the generalised Futaki invariant (or Donaldson-Futaki invariant) of the test configuration is as follows, 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.
In particular, when a > 1 − r n , we have

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.

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.
Proof It suffices to assume a to be the optimal angle for K-stability (a = 1 − r n ), i.e. (V , bS) is K-unstable for a < b < 1. Fix a rational number a ∈ (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 (V, a S; L) → A 1 such that By interpolation of Futaki invariants (see [1, Proposition 2.13]) we know Fut(V, tS; L) = 0 for any t ∈ [0, 1), which implies V 0 ∼ = V since V is K-polystable. As Fut(V, aS; L) = 0, by [32, Section 3] we know (V 0 , aS 0 ) is K-semistable. By the openness property of Kpolystability, 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 (Ṽ, aS;L) of (V , aS; −(K V + aS)) such that Since S is K-polystable, by [35,Proporsition 2.11], (C p (S, M), aS ∞ ) 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 ∞ denotes the infinite divisor. By the uniqueness of K-polystable degeneration [10] we know (Ṽ 0 , aS 0 ) ∼ = (C p (S, M), aS ∞ ). By [32,Section 3], there is a test configuration which has vanishing generalized Futaki invariant and degenerates (V 0 , aS 0 ) to (Ṽ 0 , aS 0 ). That is, we first degenerate (V , aS) to (V 0 ∼ = V , aS 0 ), then we degenerate (V 0 , aS 0 ) to (Ṽ 0 , aS 0 ) ∼ = (C p (S, M), aS ∞ ). Focusing on the boundary, we first degenerate S to S 0 , then degenerate S 0 to S ∞ ∼ = S. By K-polystability of S we at once see S 0 ∼ = S. Hence, the pair (V 0 , a 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. 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.

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 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 Continue the process, we get hypersurfaces 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 , k L)s l , then by Theorem 1.4 we know, Thus we have following formula We have the following results.
Corollary 6.2 Let V be a smooth hypersurface of degree 2 ≤ d ≤ n + 1 in P n+1 , which is cut out by the form f d (x 0 , x 1 , . . . , x n+1 ), then for any positive natural number l, the variety Y ⊂ P n+1+l defined by f d (x 0 , x 1 , . . . , x n+1 ) is K-unstable. Moreover,

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 [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.

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 oñ V . We consider the following cone overṼ associated to f * L. i.e.

Corollary 6.6 Let V be a Q-Gorenstein projective normal variety of dimension n and L an ample
, n + 2 r + 1 .

Example 6.7
This example is taken from [35]. Let V = P n and L = n+1 n+2 S n+1 − n H, 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 We choose k, d to satisfy that (n + 1)k − (k − 1)d > 0. By Corollary 6.6, we have We have the following lemma about δ(V , L ). Above all, we have following result. Corollary 6.9 Let Y ⊂ P n+2 be a hypersurface determined by . , x n ), where g d determines a smooth hypersurface in P n of degree n +1 ≤ d ≤ 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 δ(Y ) = min (n + 2)r (n + 1)(r + 1) , where r = (n + 1)k − (k − 1)d ≤ n + 1.

Example 6.10
Let k = 2, d = n + 1, and n is even. Then the hypersurface Y ⊂ P n+2 determined by x 2 n+1 x n−1 n+2 = g n+1 (x 0 , x 1 , . . . , x n ) is K-semistable, where g n+1 determines a general hypersurface of degree n + 1 in P n . For example, if we take n = 2, then the 3dimensional hypersurface x 2 where g 3 cuts out a smooth elliptic curve in P 2 . Example 6.11 Let k = 3, d = n + 1, and 3|n or 3|2n + 1. Then the hypersurface Y ⊂ P n+2 determined by x 3 n+1 x n−2 n+2 = g n+1 (x 0 , x 1 , . . . , x n ) is K-semistable, where g n+1 determines a general hypersurface of degree n + 1 in P n . For example, take n = 1, then the surface where g 2 cuts out two different points in P 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 , 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 on L −1 \{V 0 } of the form 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 Moreover we may assume that L −1 is trivialized over U by a nowhere vanishing holomorphic section σ ∈ (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 the result follows from a tedious computation using V S(ω)ω n = (2π) n n(−K V ) n , (39) and (40).
As a consequence, we have the following Lemma A. 4 One has Proof By considering the slope of the twisted Mabuchi energy along the geodesic ray generated by v, the result follows from [40,Proposition 7]. Now we derive a lower bound for the greatest Ricci lower bound β(Ỹ ). We will follow the approach in [40, 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 [29,Section 3.2].
Recall that V is also a Fano manifold. We fix and choose Kähler forms ω, α ∈ 2πc 1 (V ) such that Take an ample line bundle L with L = − 1 r K V , for some r > 1.
So Proposition A.1 is proved.
As we have noted in the above two cases, the limit space is stillỸ itself. So in the view of [9], the optimal destabilization ofỸ 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.