K-stability for varieties with a big anticanonical class

We extend the algebraic K-stability theory to projective klt pairs with a big anticanonical class. While in general such a pair could behave pathologically, it is observed in this note that K-semistability condition will force them to have a klt anticanonical model, whose stability property is the same as the original pair.


Introduction
There has been tremendous progress in algebraic K-stability theory of log Fano pairs (see [Xu21] for a survey of the topic).In the recent works [DZ22] and [DR22], the Kähler-Einstein problem is considered for a Kähler manifold (X, ω) such that −K X is big.More precisely, in [DZ22] the authors prove a transcendental Yau-Tian-Donaldson theorem for twisted big Kähler-Einstein metrics.As a consequence of their result, in the algebraic setting, uniform K-stability of X with a big anticanonical class implies the existence of a Kähler-Einstein metric.In this note we want to show that the K-stability theory in this setting, i.e. a projective klt pair with a big anticanonical class, essentially follows from the original (log) Fano case.
In general, there could be pathological examples in projective varieties X with a big anticanonical class −K X ; e.g. the anticanonical ring R(X, −K X ) = m∈N H 0 (X, −mK X ) is not necessarily finitely generated (see Example 3.8).However, we will show that the K-stability condition implies that X is of log Fano type.
Here δ(X, ∆) is defined in the exactly same fashion as in the case when −K X −∆ is ample (see [FO18,BJ20] .
For the stronger and more precise statement, see Theorem 3.4.We note that the above finite generation is asked in [DR22].
The above observation makes it possible to use existing birational geometry techniques to study K-stability questions for X with a big anticanonical class.In fact, without too much difficulty, it reduces K-stability questions for (X, ∆) to K-stability questions for its anticanonical model (Z, ∆ Z ), as we can see from the following statement.
Remark 1.3.In [DR22], Ding stability notions for a projective klt pair (X, ∆) with big −K X − ∆ are developed.If one assumes R = m∈r•N H 0 (−m(K X + ∆)) is finitely generated and denotes by (Z, ∆ Z ) the anticanonical model, then one can show a similar statement to Theorem 1.2; i.e. the Ding stability notions for (X, ∆) are equivalent to the notions for (Z, ∆ Z ).

Notation and Convention.
Throughout this paper, we work over an algebraically closed field k of characteristic 0. We follow the standard terminology from [KM98,Kol13].
For a normal log pair (X, ∆) such that K X + ∆ is Q-Cartier and a divisor E over X, we denote by A X,∆ (E) the log discrepancy of E with respect to (X, ∆).
We say a klt projective pair (X, ∆) is log Fano if (X, ∆) is klt and −K X − ∆ is ample, and a klt projective pair (X, ∆) is of log Fano type if there exists an effective Q-divisor D such that (X, ∆ + D) is a log Fano pair.
We say an effective Q-divisor Γ on a projective log pair (X, ∆) is an N -complement for a positive integer N if N (K X + ∆ + Γ ) ∼ 0 and (X, ∆ + Γ ) is log canonical.A Q-complement is an N -complement for some N .

S-invariants
Let (X, ∆) be an n-dimensional projective normal pair such that −K X − ∆ is big.For any prime divisor E which appears on a birational model µ : Y → X, the S-invariant is defined as , where E runs through all valuations over (X, ∆).We say (X, ∆) Remark 2.2.When (X, ∆) is log Fano, the equivalence between this way of defining K-stability notions using valuations and the original one using test configurations, called the Fujita-Li criterion, is proved in [Fuj19], [Li17] and [BX19].For (X, ∆) with a big anticanonical class, the current definition is formulated in [DZ22].
Remark 2.3.Theorem 1.2 says K-stability is indeed the same as uniform K-stability.For a log Fano pair, this is proved in [LXZ22] (see [XZ22] for a different proof).
Here the basis {s 1 , . . ., s N m } is compatible with where .
The following are basic properties proved in [BJ20].
Theorem 2.4.Keep the notation as above.
(2) For any ε > 0, there exists an m 0 such that for any E over X and m ≥ m 0 with m ∈ r • N, (3) We have δ m (X, ∆) = inf D lct(X, ∆; D), where D runs through all m-basis type divisors.
Proof.Statement (1) follows from [BJ20, Lemma 2.9] and ( 2 We can consider more general filtrations. Definition 2.5.By a (linearly bounded) filtration (3) there exist e − , e + ∈ R such that F me − R m = R m and F me + R m = 0 for any m; (4) For any filtration F on R, we can define S m (F ) and S(F ) as in [BJ20, Sections 2.5 and 2.6, pp.15-16], and we have Thus . □

Q-complements and finite generation
For a Q-divisor D with |rD| 0 and any m ∈ N, we denote by Bs(|mrD|) the base ideal.We can define the log canonical threshold of the asymptotic linear series as follows: We can define a sequence of multiplier ideals By the ascending chain condition of ideals, this sequence will stabilize.We denote the maximal element by I (X, ∆; ∥ − K X − ∆∥) and call it the asymptotic multiplier ideal sheaf of D. For more background, see [Laz04, Section 11.1].Recall that for any ideal a ⊆ O X , we have lct(X, ∆; a) > 1 if and only if I (X, ∆; a) > 1.

Now we can show the following.
Theorem 3.4.Let (X, ∆) satisfy Assumption 3.3; then (X, ∆) is of log Fano type.In particular, any Cartier divisor E on X satisfies that R(X, E) := m∈N H 0 (X, mE) is finitely generated.Proof.Let us first prove this when δ(X, ∆) > 1 as it is quite straightforward.By Theorem 2.4, we know that for a sufficiently large m and any m-basis type divisor D, lct(X, ∆; D) ≥ δ m (X, ∆) > 1.
Thus we can apply Lemma 3.1.
In the general case, we may assume δ(X, ∆) ≤ 1, and we need some perturbation argument.By our definition of a(X, ∆), for any t ∈ (0, a(X, ∆)), there exists an ample where E 1 is an effective Q-divisor and A 0 is an ample Q-divisor.Moreover, by (3.2) we may assume Fix m 0 ∈ N such that |m 0 A| is base-point-free.Then for any prime divisor H ∈ |m 0 A|, by Lemma 2.6, .
We can choose an m-basis type Q-divisor D m compatible with H, so we can write .
Proof.We know f : X Z is a birational contraction; i.e.Ex(f −1 ) does not contain any divisor, and It follows from Theorem 3.4 that there exists a

K-stability of the anticanonical model
Let (X, ∆) be a projective log pair with big −K X − ∆.Let (Z, ∆ Z ) be its anticanonical model; i.e.Z = Proj R(X, −r(K X + ∆)), and ∆ Z is the birational transform of ∆ on Z.Let Y be a common resolution.
Proof.For the log discrepancy function, this follows directly from the definition.Since we have S X,∆,m (E) = S Z,∆ Z ,m (E) + ord E (B).Therefore, the same is true for the S-function.□ Lemma 3.7.If (Z, ∆ Z ) is klt, there exists a t > 0 depending on Z (but not E) such that for any divisor E over Proof.Since (Z, ∆ Z ) is klt, we know that there exists a t > 0 such that if we write π where t is the constant from Lemma 3.7.□ Example 3.8.This example has appeared in several works to present pathological phenomena, see e.g.
[Gon12]: Let S be the blowup of P 2 at nine very general points.Then −K S is known to be nef but not semiample.In fact, there will be a unique cubic curve passing through these nine points, and if we denote by E its birational transform on S, then for any m ∈ N, | − mK S | has one element mE.
Let H be an ample Cartier divisor on S and X = P S (E), where E := O S + O S (H).Denote by π : X → S the natural morphism.We claim −K X is big.In fact, since we have is not surjective for any m.Thus we need generators from H 0 (O S (H − mK S )) for every m.By Theorem 1.1, we know δ(X) < 1.Here we give a direct verification of this.We denote by Y ⊆ X the section given by Then similarly to before, we have , where we follow the convention that if m 0 > 2m, then the direct sum is 0. Hence a direct calculation implies By an elementary calculation, which implies δ(X) < 3 5 .