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Geodesic rays and stabilities in the cscK problem

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson. We then show that twisted Monge-Amp\`{e}re slopes of maximal geodesic rays are continuous under decreasing approximations by smooth positive non-Archimedean metrics. Combined with previous works, these two results reduce the proof of Yau-Tian-Donaldson conjecture for constant scalar curvature K\"{a}hler metrics to a convergence conjecture for a newly formulated non-Archimedean entropy functional. Moreover we propose a strengthening of K-stability condition to study the YTD conjecture. This also makes connection with positivity in birational algebraic geometry and produces some existence result from algebraic stability condition. Another result we prove is that the Mabuchi slope of a geodesic ray associated to a test configuration is equal to the non-Archimedean Mabuchi invariant.

In this paper, we are interested in destabilizing geodesic rays in the variational approach. We first recall the following important quantities associated to geodesic rays. See section 2.2 for definition of geodesic rays and (47) for the expression of Mabuchi energy.
The existence of the above limit, which may be +∞, follows from the convexity of M along Φ as proved in [5] based on the convexity along C 1,1 -geodesics proved in [1]. By the beautiful work [4] of Berman-Boucksom-Jonsson, to any psh ray of linear growth we can associate a finite energy non-Archimedean metric, and to any finite energy non-Archimedean metric we can associate a maximal geodesic ray (see Theorem 2.11).
Our first result is (see section 2.2 for the notations in the statement): Theorem 1.2. Let Φ : R ≥0 → E 1 (L) be a geodesic ray. Assume that M ′∞ (Φ) < ∞. Then Φ is maximal. As a consequence, we have the identity: We then show that the twisted Monge-Ampère slopes of maximal geodesic rays are non-Archimedean quantities. In this paper H NA (L) denotes the set of smooth positive non-Archimedean metrics that are associated to test configurations of (X, L). Theorem 1.3. Assume that Φ : R ≥0 → E 1 (L) is a maximal geodesic ray. Let {φ m } ⊂ H NA (L) be any decreasing sequence converging to Φ NA , and Φ m be the (maximal) geodesic ray associated to φ m . Then for any smooth closed (1, 1)-form χ, we have the convergence: Remark 1.4. Boucksom told me that this result was indeed expected to be true. However see Remark 2.16. Actually, by working harder, one can prove that the convergence (3) holds true for any sequence {φ m } that converges to φ = Φ NA ∈ E 1,NA in the strong topology (in the sense defined in [15,Section 6.6]). See Remark 4.1 In fact, a more general result is true (see Theorem 4.3), which will be used in our study of YTD conjecture in section 5. The above two results resolve some difficulty in Berman-Boucksom-Jonsson's variational program (see [9]), and, when combined with the recent progresses mentioned above, essentially reduce the proof of (the G-uniform version of) YTD conjecture for cscK metrics to proving that the entropy slope is a non-Archimedean quantity. Indeed, by the proof of Theorem 5.12, it now suffices to prove the following result: Conjecture 1.5. Let Φ be a maximal geodesic ray. Then there is a sequence φ m ∈ H NA (L) converging to Φ NA in the strong topology such that In fact, we expect the equality can be taken in (4). To apply this to YTD conjecture, one may even assume that Φ is C 1,1 by [27]. In exploring possible routes to (4), it seems to us that one difficulty in proving (4) is that there is not yet an explicit formula for H ′∞ (see Conjecture 5.6). In section 5.1, we will define a non-Archimdean entropy using the non-Archimedean analogue of supremum characterization of the usual entropy. This turns to be related to a non-Archimedean formula in [13] for H NA (φ) for φ ∈ H NA (see Proposition 5.4). A comparison with H ′∞ (Φ) allows us to state a refined Conjecture 5.7 which might be more doable. Both conjectures (which are conjectured to be equivalent), if true, would be the (refined) non-Archimedean analogue of Berman-Darvas-Lu's result on approximation of entropies (see [5,Lemma 3.1]).
Moreover we will propose a strengthening of K-stability (Definition 5.11) and show that it is a sufficient condition for the existence of cscK metrics (Theorem 5.12) . Interestingly, in this study we naturally single out models with at worst log canonical singularities and some positive condition, which suggests some connection with the birational algebraic geometry. In particular the new stability condition may well be studied by the tools of Minimal Model Program. For details, see section 5.2 and Remark 5. 13.
This stronger stability condition also leads us to the following sufficient algebraic criterion for the existence of cscK metric, which is related to the study of the so-called J-equation in the literature (see e.g. [19,48]). The proof of this result is however independent, depending on Theorem 1.2, Theorem 1.3 and the variational argument. See Definition 2.4, Theorem 5.15 and the discussion there. Theorem 1.6 (Theorem 5.15). If (X, L) is J K X -semistable, then X admits a cscK metric.
Another result we will prove is (see section 2.1 for the notations): Theorem 1.7. Given an ample test configuration π : (X , L) → C, let Φ = {ϕ(s)} be a geodesic ray associated to (X , L). Then we have the following slope formula: To prove this we will use the slope formula by Boucksom-Hisamoto-Jonsson ( [13]) and Tian ([50, 52]) for M ′∞ when Φ is replaced by a smooth S 1 -invariant subgeodesic ray induced by a smooth positively curved metric on L (see also [49]). We note questions related to above results were also asked in [54,Conjecture 5.1] and [22,Remark 3.1].
Acknowledgement: The author is partially supported by NSF (Grant No. DMS-1810867) and an Alfred P. Sloan research fellowship. I would like to thank Sebastien Boucksom for helpful comments on a preliminary work.

Non-Archimedean metrics
Definition 2.1. A model of X is a flat family of projective varieties π X : X → C satisfying: (i) There is a C * -action on X such that π X is C * -equivariant; (ii) There is a C * -equivariant isomorphism X \ X 0 ∼ = X × C * . This is also called a test configuration of X in [16]. A model Y is called dominating if there exists a C * -equivariant birational morphism µ Y : Y → X × C. We will denote by MO (resp. DMO) the set of all models (resp. all dominating models).
For any two models X i , i = 1, 2 of X, a common refinement of X i , i = 1, 2 is a model X 3 with two C * -equivariant birational morphisms µ i : X 3 → X i , i = 1, 2.
A (normal) semi-ample (resp. ample) test configuration of (X, L) is an model π X : X → C together with a C * -equivariant Q-line bundle L → X satisfying: (iii) X is normal and L is π X -semi-ample (resp. ample).
Two test configurations (X i , L i ), i = 1, 2 are called equivalent if there exists a test configuration (X 3 , L 3 ) and two C * -equivariant birational morphisms µ i : In this paper, for any C * -equivariant data • over C, we will use• to denote its natural C * -equivariant compactification over P 1 .
Following [4], PSH NA (L) denotes the set of non-Archimedean metrics (on (X NA , L NA )) which are always represented by L NA -psh functions on X div Q . Any semi-ample dominating test configuration (X , L) µ X → X C defines a smooth positive non-Archimedean metric φ (X ,L) on (X NA , L NA ): for any v ∈ X div Q , let G(v) be the Gauss extension and set: Note that any semi-ample test configuration is equivalent to a dominating one. Two equivalent test configurations define the same non-Archimedean metric (by definition). We always denote by H NA = H NA (L) the set of (smooth and positive) non-Archimedean metrics coming from test configurations. By abuse of notations, we will interchangeably use the notation of test configurations and smooth positive non-Archimedean metrics. Moreover we will follow the convenient notations in [12,13,15]. First through this paper we use: For any φ ∈ H NA (L), define: In the above formula, φ triv is the trivial non-Archimedean metric associated to the trivial test configuration, φ n+1 denotesL ·n+1 and similar interpretation applies for other products.
The following stability condition was introduced in [36] based Darvas-Rubinstein's principle for proving Tian's properness conjecture. It refines the (uniform) K-stability (see [50,33,12,32]). We will modify this stability in Definition 5.11. Definition 2.2 ([36, Definition 3.1], see also [39]). Let G be any connected and reductive group of Aut(X, L) 0 . Let T ∼ = (C * ) r be the center of G. Set N Z = Hom(C * , T) and N R = N Z ⊗ Z R. Then we say that (X, L) is G-uniformly K-stable if there exists γ > 0 such that for any G-equivariant test configuration φ = (X , L), According to [52,Lemma 2.2] and [13], by using base change to make the central fibre reduced, this definition is equivalent to the definition of G-uniform K-stability via the CMweights. We state the G-uniform version of Yau-Tian-Donaldson conjecture. Based on Darvas-Rubinstein's principle and his slope formula for J T (ϕ) = inf σ∈T J(σ * ϕ) (see (25)), Hisamoto in [36,Theorem 3.3] proved the "only if" part.
In this paper, we will also consider the following stronger stability condition.
Definition 2.4. (X, L) is called uniformly J K X -stable (resp. J K X -semistable) if there exists γ > 0 such that for any test configuration φ = (X , L), we have: An advantage of Boucksom-Jonsson's non-Archimedean approach to K-stability is that it is a natural set-up for studying and compactifying the space of (equivalent classes of) test configurations. For example [15], the non-Archimedean analogue of finite energy spaces, which is a natural compactification of H NA , were introduced: As in the Archimedean case, set We will see in the following sections (Definition 4.4 and Definition 5.2) that all the non-Archimedean invariants in (8)- (16) can be defined for any φ ∈ E 1,NA (L).

Finite energy rays
We refer to the papers [2,4,24,34] for precise meanings of the notations in the following discussion. Denote by H(L) the space of smooth Hermitian metrics on L with Kähler curvature forms. Fix a smooth reference metric ψ ∈ H(L). In this paper, we always set V = L ·n For any ϕ ∈ H(L), define For any ϕ ∈ PSH(L), define: and set: . For any ϕ i ∈ E 1 , i = 1, 2, we have the following important functionals: The inequalities in the following lemma will be useful to us.
In this paper, C n will denote any constant depending only on the dimension n. By [2,2], I satisfies a quasi-triangle inequality: for any ϕ i ∈ E 1 , i = 1, 2, 3, we have: By general theory of metric topological spaces, I defines a metrizable structure on E 1 0 (L) (note that I is translation invariant). Darvas [24] defined a Finsler-type d 1 -distance on E 1 (L) and proved that (E 1 (L), d 1 ) is the metric completion of (H(L), d 1 ) whose metric topology coincides with the strong topology introduced in [2]. Moreover, he proved Theorem 2.7 ([24, Theorem 3]). There exists a universal constant C > 0, such that for any ϕ 1 , ϕ 2 ∈ E 1 (L), where Note that in general we then have: In this note, as in [4], a psh ray (resp. psh path) is a map such that the S 1 -invariant Hermitian metric Φ = {ϕ(− log |t| 2 )} on p * 1 L over X×{t ∈ C * ; |t| ≤ 1} (resp. X × {t ∈ C * ; |t| ∈ [e −b/2 , e −a/2 ]}) has positive curvature current. A (finite energy) geodesic joining ϕ i , i = 1, 2 ∈ E 1 (L) is by definition the largest psh path dominated by ϕ 0 and ϕ 1 . Any (finite energy) geodesic ray in this paper is a finite energy geodesic ray emanating from the fixed reference metric ψ. We say that Φ is sup-normalized if sup(ϕ(s) − ψ) = 0 for any s ∈ R ≥0 .
By the work of Berman-Darvas-Lu and Chen-Cheng, we have the important: be two geodesic rays emanating from ψ. Then the function s → d 1 (ϕ 1 (s), ϕ 2 (s)) is convex on [0, ∞). As a consequence, the following limit exists, which may be +∞: Moreover exactly one of the two alternatives holds: either d c It is known that E = E ψ is affine along any geodesic. In particular, E is linear along any geodesic ray (emanating from ψ). For any geodesic ray Φ = {φ(s)}, we will set: Proof. Because E(ϕ i (s)) is linear in s, this follows easily from the following domination principle in [5,Proposition 4 (30), the latter domination principle is reduced to S.Dinew's domination principle in [8,Proposition 5.9] which in turn depends on his uniqueness result.

and the equality holds if and only
. So the above lemma is a corollary of Theorem 2.8. However we state it separately since its proof is in some sense simpler and it is enough for proving Theorem 1.2.
For the details of the following definition, we refer to [4,6].
Any psh ray Ψ of linear growth defines a non-Archimedean metric Ψ NA ∈ PSH NA (L) which is represented by the following function on X div Q : for any v ∈ X div Q , let G(v) be its Gauss extension and set

A geodesic ray
Any geodesic ray is of linear growth and hence defines a finite energy non-Archimedean metric.
2. For any φ ∈ E 1,NA , there exists a unique maximal geodesic ray Φ : R ≥0 → E 1 emanating from ψ and Φ NA = φ. We will also denote this maximal geodesic ray by ρ φ . In particular, for any test configuration there is a unique maximal geodesic ray emanating from ψ.

A geodesic ray
is maximal if and only if equality holds in (39), or equivalently E(ϕ(s)) = s · E NA (Φ NA ). Moreover, in this case, let {φ m } ⊂ H NA (L) be any decreasing sequence converging to Φ NA , the following limit identity holds true: The proof of Theorem 2.11 hinges on the following important construction by Berman-Boucksom-Jonsson, which in particular shows that any φ ∈ E 1,NA (L) can be approximated by a decreasing sequence {φ m } ⊂ H NA . Similar construction has also appeared in the earlier study of non-Archimedean Monge-Ampère equations by Boucksom-Favre-Jonsson.
BBJ's construction: Let Φ : R ≥0 → E 1 be a finite energy geodesic ray such that sup(ϕ(t)− ψ) = 0. We extend Φ to be a singular positively curved Hermitian metric on p * Then by Castelnuovo-Mumford criterion and Nadel vanishing, for m 0 sufficiently large and any m ≫ 1, (X m , L m ) is a semiample test configuration of (X, L). Let Φ m be the geodesic ray associated to (X m , L m ) (as constructed in [45] or in [4, Theorem 6.6]). Then Moreover, we can get a decreasing approximating sequence by settingΦ m = Φ 2 m . Theň Φ m,NA decreases to Φ NA andΦ m decreases to the maximal geodesic ray ρ Φ NA . Because of this minor change of subscripts, by abuse of notations, we will also write Φ m (resp. φ m := Φ m,NA ) forΦ m (resp.Φ m,NA ).
Lemma 2.12. Let Φ : R ≥0 → E 1 be a geodesic ray. Then the limit exists and is finite.
Proof. Any smooth (1, 1)-form χ can written as χ = χ 1 − χ 2 with χ i , i = 1, 2 being smooth Kähler forms. So we can assume χ is smooth and Kähler. Then we know that E χ (ϕ(s)) is convex with respect to s. Indeed, this is well known if Φ is C 1,1 (see [17,4.1,Proposition 2]). In general, we can approximate any segment of Φ by C 1,1 -geodesic segments and use the continuity of E χ under d 1 -convergence (see [28,Lemma 5.23]) to get the convexity. Moreover because Φ has linear growth, E χ (ϕ(s)) also grows at most linearly with respect to s. So the slope (E χ ) ′∞ (Φ) indeed exists and is finite.
The constant scalar curvature Kähler metrics are global minimizers of Mabuchi energy. We will use the following Chen-Tian's formula for Mabuchi energy and other related functionals.
Definition 2.13. For any smooth volume forms Ω and ν, and any ϕ ∈ E 1 (L), define: The following result connects the Archimedean and non-Archimedean functionals : Theorem 2.14 (see [12,13] and references therein). For any φ = (X , L) ∈ H NA (L), let Φ be any locally bounded S 1 -invariant Hermitian metric on L. Then for any F ∈ {E, Λ, J, I, J , R, H, M}, we have the identity: In particular, the identity is true if Φ is the maximal geodesic ray associated to φ.
As an application, we can use Jensen's inequality and Tian's α-invariant to get: We will prove in Theorem 1.7 that the same is true for F = M and Φ is the maximal geodesic ray. Remark 2.16. Most non-Archimdedean quantities can be defined for φ ∈ E 1,NA (L) by using the slopes of corresponding Archimedean functionals along the associated maximal geodesic ray at infinity (see Definition 4.4). However we want the quantities to be continuous when φ is approximated by some decreasing (or even strongly converging) sequence from H NA . 1 We achieve this for E χ , in particular for R = E −Ric(Ω) , in Theorem 1.3, while this is not yet clear for M NA (or equivalently for H or S) (see Conjecture 5.7).
In the above formula, by replacing ψ, ϕ by ϕ 1 , ϕ 2 respectively, we define S ϕ 1 (ϕ 2 ) and M ϕ 1 (ϕ 2 ) and so on. It is easy to verify that the functionals S ϕ 1 (ϕ 2 ) and M ϕ 1 (ϕ 2 ) depend only on ϕ i , i = 1, 2 but not on the volume form Ω. We will use the following simple but important co-cycle property: for any ϕ i ∈ E 1 , i = 1, 2, 3, and for F ∈ {E, E χ , S, M}, This is well-known if ϕ i are smooth. For general ϕ i , it can be verified directly using integration by parts.

Geodesic rays with finite Mabuchi slopes
Let Ω be a fixed smooth volume form on X satisfying X Ω = X ( √ −1∂∂ϕ) n = (2π) n V . Denote by B = {t ∈ C; |t| ≤ 1} the unit disc in C. We will interchangeably use the variable t to denote coordinate on C and the variable s = − log |t| 2 ∈ [0, +∞) for any t ∈ B \ {0}.
Lemma 3.1. Let Φ : R ≥0 → E 1 be a finite energy geodesic ray. LetΦ = ρ Φ NA be the associated maximal geodesic ray. Then for any α > 0, we have: Proof. By [4, Proposition 1.10], s → sup(ϕ(s) − ψ) is affine. So by subtracting an affine function, we can assume sup(ϕ(s) − ψ) = 0. We then carry out BBJ's construction (see the end of section 2.2) to get test configurations (X m , L m ). Let Φ m be the geodesic ray associated to (X m , L m ). Locally the singularity of Φ m is comparable to . By Demailly's regularization theorem which depends on the Ohsawa-Takegoshi extension theorem, locally (m + m 0 )Φ m is less singular than mΦ. Let Ψ triv = p * 1 ψ be the trivial geodesic ray. Then there exists C = C m such that: Because Φ NA =Φ NA , we have J (mΦ) = J (mΦ) by the valuative description of multiplier ideal sheaves ([10, Theorem 5.5]). So similarly we can assume that Then we have: When α = m, we get: This is integrable by the definition of multiplier ideal sheaf J (mΦ) (see [30, Proof of Lemma 2]). Since m can be arbitrarily big, we get the conclusion.
Proof of Theorem 1.2. Recall that we have the formula: By Lemma 2.12, we know that the slope (E −Ric(Ω) ) ′∞ (Φ) exists and is finite. Moreover E ′∞ (Φ) is also finite (since Φ has a linear growth). So we know that H ′∞ exists and is finite. So we just need to show that if Φ is not maximal, then H ′∞ Ω (Φ) is arbitrary large, which would contradict the assumption of finite Mabuchi slope. In By the S 1 -invariance, we use the variable s = − log |t| 2 to re-write the integral as +∞ 0 X e αu(s) Ω e −s ds < +∞.
So there exist s j → +∞ such that So we can assume that Re-write the above inequality as: By Jensen's inequality for the probability measure ((2π) n V ) −1 ( √ −1∂∂ϕ(s j )) n , we get: The above inequality is valid when √ −1∂∂ϕ(s j ) is a smooth Kähler form. We claim that it is still true when ϕ(s j ) ∈ E 1 . To see this, we first use the fact that the entropy of √ −1∂∂ϕ(s j ) can be approximated by entropies of smooth Kähler forms ([5, Lemma 3.1]). More precisely, there exist ϕ k ∈ H(L) such that Then the same application of Jensen's inequality gives: As k → +∞, the left hand side converges to H Ω (ϕ). By [7, Proposition 5.6], X u( It is then easy to show that the right-hand-side of (60) indeed converges to the right-hand-side of (59) (for fixed s j ). So the claim follows.
To continue the estimate of the right-hand-side of (59), we use (28) to get: Now dividing both sides by s j and letting s j → +∞, we get: If Φ is not maximal, then Φ =Φ butΦ ≥ Φ. Hence by Lemma 2.9, E ′∞ (Φ) − E ′∞ (Φ) is positive. But α can be arbitrary large. This is impossible if H ′∞ (Φ) is finite.

Proof of Theorem 1.3
With the notations in the statement of Theorem 1.3, our goal is to prove the convergence: By writing χ as χ 2 − χ 1 , we can assume that χ is Kähler so that E χ is monotone increasing.
On the other hand, there exists a constant C > 0 such that χ ≤ C( √ −1∂∂ψ). For simplicity of notation, we denote the Kähler form √ −1∂∂ψ by ω. By using the co-cycle condition (50) and Φ m ≥ Φ, we easily get: If the right-hand-side converges to 0 as m → +∞, so does the left hand side. So from now on we just assume that χ = √ −1∂∂ψ = ω. To prove the convergence of slopes, it suffices to show the following estimate: there exists a sequence δ m → 0 as m → +∞ such that for any In order to estimate the quantity: we will adapt the idea of proof of d 1 -continuity for E ω in [28,Lemma 5.23] which in turn depends on [7,Lemma 5.8].
The key is to make the estimate from [7, Proof of Lemma 5.8] more effective in our case with the extra information that Φ is maximal. Moreover, we will use the simpler I distance to carry out our argument (although one could also use Darvas's d 1 -distance via (34)). First note that we have the identity: Because Φ is maximal, the linear functions E(ϕ m (s)) decrease to the linear function E(ϕ(s)) as m → +∞. So we get: lim m→+∞ ǫ m = 0.
Since we will not use the strong topology on E 1,NA and this convergence in this paper, we will just sketch its proof. First one can show that if a sequence {φ ′ m } decreases to φ, then φ ′ m converges to φ in the strong topology. Applying the same proof as [5, Proposition 2.6] to the non-Archimedean setting, one can find a decreasing sequence {φ ′ m } ⊂ E 1,NA such that φ m ≤ φ ′ m , {φ ′ m } is decreasing and converges to φ as m → +∞. Then one can use the co-cycle condition to estimate: It is easy to see that there still exists ǫ m → 0, such that: Now we can use the same method as in the above proof to show that the right-hand-side of (78) is bounded by ǫ (1/2) n m s whose slope converges to 0 as m → +∞. The same remark applies to the following Theorem 4.3.
The following technical estimate from [7] plays a crucial role in the above argument. . For ϕ, ϕ ′ ,φ ∈ E 1 , then we have the following estimate: where b p (defined in the following proof ) satisfies the following inequalities: For completeness, we sketch the proof following [7].
and define the quantities: As in [7, (5.4)], by using integration by parts, Cauchy-Schwarz inequality and √ −1∂∂ϕ ≤ 2 √ −1∂∂v, we get, for any p = 0, . . . , n − 1, which gives: So we just need to estimate b p . Using integration by parts, Cauchy-Schwarz inequality and √ −1∂∂φ ≤ 2 √ −1∂∂v as before, we get (see [7, (5.5) Finally we have the inequality: By refining the method in the above proof, one can prove the following result which will be used in the next section. (1) Φ = {ϕ(s)} is a maximal geodesic ray. Let {φ m } ⊂ H NA (L) be a decreasing sequence converging to to Φ NA and Φ m = {ϕ m (s)} be the associated maximal geodesic rays.
(2) π Y : Y → C is a model of X (see Definition 2.1). Let Q be a C * -equivariant Q-line bundle over Y, and Ψ Q = {ψ Q (s)} be a smooth S 1 -invariant Hermitian metric on Q.
Then the functions f (s) = E where f ′∞ and f ′∞ m are the slopes of f (s) and f m (s) at infinity: Proof. Since the idea of proof is the same as the proof of Theorem 1.3, we just sketch the proof. Using the functorial embedded resolution of singularities by a sequence of blowing-ups along smooth subvarieties, we can find a model π Y ′ : Y ′ → C with C * -equivariant birational morphisms p : Y ′ → X C and q : Y ′ → Y, and moreover there is a π Y ′ -ample line bundle L ′ such that (Y ′ , L ′ ) becomes an ample test configuration for (X, L). There exists ℓ ≫ 1 such that q * Q + ℓL ′ is π Y ′ -ample. As a consequence, there is a smooth Hermitian metric Φ L ′ such that both Φ L ′ = {ϕ L ′ (s)} and Ψ Q + ℓ · Φ L ′ have positive curvature forms. We just need to prove the convergence when (Q, Ψ Q ) is replaced by the π Y ′ -positive Hermitian Q-line bundles Moreover when ℓ ≫ 1, 2ℓL ′ − (q * Q + ℓL ′ ) is also π Y ′ -ample. By using the fact Φ m ≥ Φ and co-cycle property as before, it suffices to prove the convergence for the π Y ′ -ample Hermitian Q-line bundle (L ′ , Φ L ′ ). Now we can carry out the same arguments as the proof of Theorem 1.3. We just need to observe that, to get (72)-(73), we use the following version of inequality (30): I(ψ L ′ (s), ϕ(s)) ≤ C(I(ψ, ψ L ′ (s)) + I(ψ, ϕ(s))) ≤ Cs, which follows from the asymptotic expansion of the I functional stated in Theorem 2.14.
Definition 4.4. For any φ ∈ E 1,NA (L), let Φ be the maximal geodesic ray associated to φ. Let χ be any closed smooth (1, 1)-form, set: In particular, we define: More generally, let (Q, Ψ Q ) → Y be like in Theorem 4.3, we define: Then Theorem 1.3 and Theorem 4.3 can be restated as: The following result can be proved by using the method as in [13,Lemma 3.9] by using the formalism of Deligne pairing: Proposition 4.6. With the same notations as in the above definition, if φ ∈ H NA (L), then we have: where the intersections are calculated by pulling back the data to a common refinement of X and Y.
5 Towards the YTD conjecture for cscK metrics

Modified M NA invariants
To explain our modified approach to the YTD conjecture, we first generalize the S NA invariant in (14): has log canonical singularities. Recall that this means that K Y + Y red 0 is Q-Cartier and for any divisor E over (Y, Y red 0 ), we have: We will denote by LC the set of all log canonical models of X.
2. We say a model Y ∈ MO is a SNC (for simple normal crossing) model of X if (Y, Y red 0 ) is a simple normal crossing pair. We denote by SN the set of all SNC models. We will denote by DSN the set of dominating SNC models of X.
Definition 5.2. Assume (X , L) is a semi-ample test configuration and let φ (X ,L) ∈ H NA be the associated non-Archimedean metric (see (6)). For any Y ∈ MO (see Definition 2.1), let Z be a common refinement (see Definition 2.1) of X and Y which dominates X and Y via birational morphisms p and q respectively, and define: More generally, let φ ∈ E 1,NA (L) and {φ m } ⊂ H NA (L) be a decreasing sequence converging to φ. Define: For any E 1,NA (L), we also write this quantity as Moreover, for any φ ∈ E 1,NA (L), we define (note that in the supremum in (94), Y ranges only over log canonical models): The existence of limit in (92) follows from Theorem 4.3. Indeed, if Φ is the maximal geodesic ray associated to φ and Ψ Y = {ψ Y (s)} be an S 1 -invariant smooth Hermitian metric on K log Y/C , then and the identity (92) becomes (83). The definitions in (94)-(95) are justified by the following lemma. See [47] and [44] for other attempts to define Futaki-type invariant for filtrations and sequence of test configurations.
2. For any φ ∈ E 1,NA (L), let Φ = {ϕ(s)} be the associated maximal geodesic ray. We have the inequality: Moreover if φ = (X , L) ∈ H NA (L), then S NA (φ) coincides with (14) and satisfies: The above statements also hold for M NA and H NA .
Proof. Choose any Y ∈ LC. Let ν be a smooth Hermitian metric on K log Y/P 1 . We write the entropy part as: By Jensen's inequality, the first term is bigger than − log Xt ν + log((2π) n V ). We claim that because (Y, Y red 0 ) is log canonical, we have Assuming this claim, we can take slopes on both sides of (101) and use Definition 4.4 and (89) to get: Because R ′∞ (Φ) = R NA (φ) = 1 V K log X P 1 /P 1 · φ n , we indeed get (99): for any Y ∈ LC, Next we verify (102). First by using C * -equivariant resolution of singularities, we can find an a model Z that dominates Y via µ : Z → Y such that Z 0 = (π Y • µ) −1 (0) is snc. Then we can write: . Choose a smooth reference metric ν on −K log Y/P 1 . Then ν ∼ |s| 2 where s is any local frame of K log Y/P 1 considered as an n-form. To find such a frame, choosing local holomorphic coordinates such that the projection Z → C is given by t = p j=0 z b j j with 0 ≤ p ≤ n. Then a local frame of K log Z/C is given by: For (106) (105), the pull back of a n-form s of K log Y/C is then given by Since A i ≥ 0, we still have B∩Zt |µ * s| 2 ≤ B∩Zt |θ| 2 = O(s d ) which indeed implies (102). The identity (105) also shows that Moreover we can choose Z to dominate X × C. So we get the identities in (98). Finally, for (100), the first identity will be proved by Theorem 1.7. The second identity is by definition in (14) and (91): For the last identity in (100), by (99) we just need to prove the " ≤ " direction. Let µ : Y → X to be a log resolution of singularities of X such that Y 0 has simple normal crossings. Then because µ * L has zero self-intersection number when restricted to exceptional divisors, by the projection formula, we indeed get: Note that the definition of H NA in (115) is a non-Archimedean analogue of the well-known supremum characterization of Archimedean entropy: In particular, the following lower semi-continuity is always true: for any {φ m } ⊂ H NA decreasing to (or just converging strongly to) φ ∈ E 1,NA , we have: It turns out that there is a formula for H NA (φ) that can be derived from theory of non-Archimedean metrics (as developed in [16,38]) which generalizes the formula in [13].
Proposition 5.4. For any φ ∈ E 1,NA (L), let Φ be the associated maximal geodesic ray (see Theorem 2.11). Assume that H ′∞ (Φ) < +∞. Then we have the formula: =: Proof. We only sketch the proof of the identity (113), since it depends on more advanced set-up of non-Archimedean geometry (as developed in [11,16]). First by (98), we get: Let {φ m } ⊂ H NA (L) be a sequence decreasing to φ. For any Y ∈ DSN, by Theorem 4.5 we have: On the other hand, by using the definition of the non-Archimedean Monge-Ampère measure (see [16,3]) one can easily show the identity where X qm is the set of quasi-monomial points and ev Y : X NA → ∆ Y is the retraction map (see [16,4] Together with (115), this proves the identity (113).
Remark 5.5. Note that we have defined the right-hand-side of the identity (114) using its left-hand-side, although we believe it to be true without this definition. Indeed, because X NA is an inverse limit of the dual complexes {∆ Y ; Y ∈ DSN} (see [11,Corollary 3.2]) and there is an identity A X (v) = sup Y∈DSN A X (ev Y (v)) (see [14, (5.3)] or [38, 5.2]), the identity in (114) is formally a consequence of the monotone convergence theorem. However, since the supremum is taken as an inverse limit (instead of the usual limit) over a net (instead of a sequence) of functions, one needs to be more careful in making this argument precise.
We in fact the inequality in (99) is an equality.
Conjecture 5.6. For any φ ∈ E 1,NA (L), let Φ be the associated maximal geodesic ray (see Theorem 2.11). Then: Now we can state a conjecture which refines Conjecture 1.5 and may be more amenable to algebraic approach: Conjecture 5.7. For any φ ∈ E 1,NA (L), there exists a sequence {φ m } ⊂ H NA (L) converging to φ in the strong topology (or even a decreasing sequence converging to φ) such that: or equivalently (see Proposition 5.4 and Remark 5.5): If Conjecture 5.6 is true, then Conjecture 5.7 is equivalent to Conjecture 1.5. The point is that Conjecture 5.7 will complete the proof of (G-uniform version of) YTD conjecture. See Proposition 5.14.
Next we consider a modified version of S NA .
Definition 5.8. 1. A Ricci configuration π Y : Y → C is a log canonical model of of X (see Definition 5.1) and satisfies a positivity condition: there exists a common refinement Z of Y and X × C such that is π Z -nef over C, where p Z , q Z , π Z denote the natural maps in the following commutative: Denote by RC the set of all Ricci configurations. In particular, RC ⊂ LC ⊂ MO.
2. For any φ ∈ E 1,NA (L), we define (using (92)): Remark 5.9. For any φ ∈ E 1,NA (L), by the defining identities (94) and (122), resp. (95) and (123), we always have: Moreover, by choosing Y = X C , we get the inequality (see (12)): and hence (see (15))M The main reason for introducingS NA is that the corresponding version for (118) holds true for decreasing sequences: Proposition 5.10. Assume φ ∈ E 1,NA (L) and let Φ be the associated maximal geodesic ray. Let {φ m } ⊂ H NA be a decreasing sequence converging to φ. Then we have: Proof. Because M NA − S NA =M NA −S NA = S n+1 E NA , the inequality (128) follows from (127) and the convergence of E slopes for maximal geodesic rays. So we just need to prove (127). Note that the first inequality already follows from (99) and (124).
Fix anyφ ∈ H NA satisfyingφ ≥ φ and letΦ be the maximal geodesic ray associated tõ φ. Then we haveΦ ≥ Φ. Let ν be any smooth Hermitian metric on q * (−K loḡ Y/P 1 ). For any t = 0, ν| Xt is a smooth volume form and we can decompose: where the term S(φ; ν) was defined in (48) and The g 0 part is not a problem by the proof of Lemma 5.3. For g, by integration by parts We now use the assumption that Y ∈ RC so that −K log Y/X C is π Z -nef (see diagram (121)). We first assume that it is semi-ample. Then there exists ℓ ∈ Z >0 such that is semi-ample. So we can find a smooth Hermitian metric ν on q * (−K loḡ Y/P 1 ) such that the following induced metric on −K has smooth semi-positive curvature form: where e −ϕ P 1 is the standard Fubini-Study metric on O P 1 (1). In particular, for any t = 0, Ric(ν) − Ric(Ω) ≥ 0. SinceΦ ≥ Φ, by (129) the quantity g ≥ 0 . So we get: Taking slopes on both sides and using Theorem 4.3 we get: Now we apply the above discussion toφ = φ m and let Y ∈ RC vary to get: Letting m → +∞ and using Theorem 1.3 to get the wanted inequality: Now we assume that −K loḡ Y/X C is just nef. Fix any ample line bundleĀ overȲ. Then for any ǫ > 0, −K loḡ Y/X C + ǫĀ is ample over P 1 . Fix any smooth metric e −ϕĀ onĀ. Then there exists a smooth Hermitian metric ν on K loḡ Y/P 1 and ℓ ∈ Z >0 such that has a positive smooth curvature form. Then the same argument as above gives the estimate: where We can now use the same argument as before to get the slope inequality: Since ǫ can be arbitrarily small, we can let ǫ → 0 to get Then the rest of proof is same as before.

Existence result for cscK metrics
In the following discussion, we will use the following notations. Let G be any connected and reductive group of Aut(X, L) 0 . Denote by T ∼ = (C * ) r = ((S 1 ) r ) C the center of G. Set N Z = Hom(C * , T) and N R = N Z ⊗ Z R. Let K be a maximal compact subgroup containing (S 1 ) r .
Definition 5.11. Let G be any connected group of Aut(X, L) 0 . Then we say that (X, L) is G-uniformlyK-stable if G is reductive and there exists γ > 0 such that for any G-equivariant test configuration (X , L),M A main goal in this section is to prove: Theorem 5.12. If (X, L) is G-uniformlyK-stable, then (X, L) admits a cscK metric.
Before giving the proof of this result, we make some discussion in the following remark.
Remark 5.13. By the inequality (124), G-uniformK-stability is indeed stronger than Guniform K-stability. It is then natural to ask: Question: Is G-uniformK-stability equivalent to G-uniform K-stability? In particular, does cscK imply G-uniformK-stability?
Writing out the relevant definitions, it is easy to see that it suffices to answer: Question: Is it true that, for all φ ∈ H NA (L), we have the identity In other words, for any semi-ample test configuration (X , L), which can be assumed to be dominating and satisfy that the pair (X , X red 0 ) is simple normal crossing, can we find {Y k } ⊂ RC such that K loḡ X /P 1 ·L n = lim sup The intersection on the right-hand-side is on a common refinement of Y k and X . If the answer is yes (maybe over-optimistic), then by Theorem 5.12 we would solve the YTD conjecture 2.3.
The proof of the identity (113) also suggests another approach to the above question. First recall that it has been known by the work of de-Fernex-Kollár-Xu that dual complexes can be defined also for dlt models, which are indeed log canonical. We ask: Since the Minimal Model Program (MMP) is a powerful tool to achieve nefness of divisors, it seems interesting to study these questions using the MMP, maybe in a similar fashon as in [41,42].
Next we give the proof of Theorem 5.12, which is along the similar line as the proof in [39]  To see this, we use the fact that there exists a universal constant C = C(ψ) such that for any ϕ ∈ E 1 : In our case, sup(ϕ(s) − ψ) − E(ϕ(s)) = E(ϕ(s)) = −s is linear. So for any s 1 , s 2 ∈ R >0 , we have: J(ϕ(s 1 )) ≥ s 1 − C = s 1 s 2 (s 2 ) − C ≥ s 1 s 2 J(ϕ(s 2 )) − C.
Step 2: We denote by φ = Φ NA the non-Archimedean metric associated to Φ and carry out BBJ's construction (see the end of section 2.2) to get G-equivariant test configurations (X 2 m , L 2 m ) whose associated K-invariant geodesic ray and non-Archimedean metric are denoted by Φ m and φ m := Φ m,NA .
For any compact interval of r, {c s (r)} converges uniformly in strong topology to Φ = {ϕ(r)} which is the geodesic ray associated to (X , L). Indeed by the convexity in Theorem 2.8, we have: d 1 (c s (r), ϕ(r)) ≤ r s d 1 (φ(s), ϕ(s)) ≤ C r s .
The last identity used the slope formula from [12]. Dividing both sides by r and letting r → +∞, we get the inequality. For the converse direction of the inequality, we first construct a (special) S 1 -invariant smooth subgeodesic rayΦ as follows. Note that by [20,Theorem 1.4], we are free to choose the initial point of Φ without changing M ′∞ (Φ). Since L is assumed to be ample, it is wellknown that the test configuration (X , L) is associated to a one-parameter C * -action on P Np−1 where N p = dim H 0 (X, pL) for p ≫ 1. In other words, we can assume that there is a C *equivariant embedding ι : X → P Np−1 × C such that X = {(σ η (t)(X), t); t ∈ C} ⊂ P Np−1 × C where η ∈ gl(N p , C) generates a C * -action and σ η (t) = exp(−(log t)η) ∈ GL(N p , C). Let ψ FS be the standard Fubini-Study metric on the hyperplane bundle over P Np−1 . Setφ(t) = σ η (t) * ϕ 1/p FS | ση(t)(X) . Now let Φ be the geodesic ray associated to (X , L). It is known that Φ is C 1,1 on X × C * ( [22,46]). Moreover Φ andΦ is L ∞ comparable: there exists C > 0 such that |Φ −Φ| ≤ C over X × (B \ {0}).