Uniform K-stability of 𝐺-varieties of complexity 1

Yan Li; Zhenye Li

doi:10.1515/crelle-2025-0087

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Uniform K-stability of 𝐺-varieties of complexity 1

Yan Li and Zhenye Li

Published/Copyright: November 20, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

Abstract

Let k be an algebraically closed field of characteristic 0 and 𝐺 a connected, reductive, linear algebraic group of simply connected type over k . Let 𝑋 be a projective 𝐺-variety of complexity 1. We classify 𝐺-equivariant normal test configurations of 𝑋 with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on 𝑋. Based on this formula, when 𝑋 is ℚ-Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12571058

Award Identifier / Grant number: 12571016

Funding statement: Yan Li was partially supported by NSFC Grant 12571058 and the Beijing Institute of Technology Research Fund Program for Young Scholars. Zhenye Li was partially supported by NSFC Grant 12571016.

A Appendix

A.1 Lemmas on colours in spherical homogeneous spaces

The following lemma is a combination of several known results (cf. [52, Section 30.10] and [20, Remark 11.8]); we include it here for readers’ convenience.

Lemma A.1

Let G / H be a spherical homogeneous space and σ ∈ Aut G ⁢ ( G / H ) a 𝐺-equivariant isomorphism. Suppose that two different colours 𝐷 and D ′ satisfies D = σ ⁢ ( D ′ ) . Then both 𝐷 and D ′ are of type 𝑎, and there is a simple root α ∈ Π G so that

D B ⁢ ( G / H ; α ) = { D , D ′ } .

Moreover, v D = v D ′ in the hyperspace of G / H .

Proof

It is obvious that if D ∈ D B ⁢ ( G / H ; α ) , then so is D ′ . Thus # ⁢ D B ⁢ ( G / H ; α ) ≥ 2 . This is possible only if both 𝐷 and D ′ are of type 𝑎 (cf. [52, Section 30.10]), and

D B ⁢ ( G / H ; α ) = { D , D ′ } .

For the last point, it suffices to show that, for any e λ ∈ k ⁢ ( G / H ) λ ( B ) , ord D ⁢ ( e λ ) = ord D ′ ⁢ ( e λ ) . Since 𝜎 commutes with the 𝐺-action, σ ⋅ e λ ∈ k ⁢ ( G / H ) λ ( B ) . Hence σ ⋅ e λ = c ⁢ e λ for some constant c ≠ 0 , and

ord D ⁢ ( e λ ) = ord σ ⁢ ( D ′ ) ⁢ ( e λ ) = ord D ′ ⁢ ( σ − 1 ⋅ e λ ) = ord D ′ ⁢ ( e λ ) ,

which concludes the lemma. ∎

Lemma A.2

Let ( X , L ) be a polarized 𝐺-spherical variety and d = ∑ D ∈ B ⁢ ( X ) m D ⁢ D a divisor of 𝐿, which is the divisor of some 𝐵-semiinvariant rational section 𝑠 of 𝐿 with 𝐵-weight λ 0 . Suppose that D ∈ D B ⁢ ( X ; α ) is a colour of type a ′ or 𝑏 that corresponds to a simple root α ∈ Π G . Then α ∨ ⁢ ( λ 0 ) = m D , and for any λ ∈ Γ Q so that v D ⁢ ( λ ) + m D = 0 , it holds

⟨ α ∨ , λ + λ 0 ⟩ = 0 .

Proof

Let G / H be the spherical homogeneous space that is embedded in 𝑋. Then, for each D ∈ D B , there is a line bundle L D on G / H so that

D = div ⁡ ( s D ) for some ⁢ s D ∈ H 0 ⁢ ( G / H , L D ) λ D ( B ) ,

where λ D is the 𝐵-weight of s D . Since ( d − ∑ D ∈ D B m D ⁢ D ) | G / H = 0 , we have

f : = s | G / H ∏ D ∈ D B s D m D ∈ O ( G / H ) × .

By [31, Proposition 1.3], 𝐺 acts on 𝑓 through a character λ G ∈ X ⁢ ( G ) . Consequently,

λ 0 = ∑ D ∈ D B m D ⁢ λ D + λ G .

Let 𝛼 be a simple root in Π G . It holds

⟨ α ∨ , λ 0 ⟩ = ∑ D ∈ D B m D ⁢ ⟨ α ∨ , λ D ⟩ = ∑ D ∈ D B ⁢ ( X ; α ) m D ⁢ ⟨ α ∨ , λ D ⟩ .

When 𝛼 is of type a ′ , D B ⁢ ( X ; α ) contains precisely one colour (denoted by 𝐷), and by [17, Section 2.2, Theorem 2.2] (see also [19, Proposition 2.2] or [52, Lemma 30.24]),

⟨ α ∨ , λ 0 ⟩ = m D ⁢ ⟨ α ∨ , λ D ⟩ = 2 ⁢ m D .

Combining with the fact that v D = 1 2 ⁢ α ∨ | Γ for the type a ′ colour 𝐷, we get the lemma.

When 𝛼 is of type 𝑏, again D B ⁢ ( X ; α ) contains only one colour (denoted by 𝐷); similarly to above,

⟨ α ∨ , λ 0 ⟩ = m D ⁢ ⟨ α ∨ , λ D ⟩ = m D .

Combining with the fact that v D = α ∨ | Γ for the type 𝑏 colour 𝐷, we get the lemma. ∎

A.1.1 A combinatorial property of 𝐵-stable ample divisors

Let ( X , L ) be a polarized 𝐺-variety of complexity 1, and d = ∑ D ∈ B ⁢ ( X ) m D ⁢ D a divisor of 𝐿, which is the divisor of some 𝐵-semiinvariant rational section 𝑠 of 𝐿 with 𝐵-weight λ 0 . We have the following lemma.

Lemma A.3

Suppose that

𝑋 is a one-parameter 𝐺-variety and D ∈ D B ⁢ ( α ) a colour of type a ′ or 𝑏, or
𝑋 is a quasihomogeneous 𝐺-variety and 𝐷 a central colour that descends to a type a ′ or 𝑏 colour in D B ⁢ ( α ) on Z ′ in the proof of Theorem 3.8.

Then ⟨ α ∨ , λ + λ 0 ⟩ = 0 whenever λ ∈ Γ Q satisfies v D ⁢ ( λ ) + m D = 0 .

Proof

We start with case (1) when 𝑋 is a one-parameter 𝐺-variety. In this case, a general 𝐺-orbit in 𝑋 is isomorphic to some spherical homogeneous space 𝑂. For simplicity, we denote by 𝑂 any fixed such an orbit. Recall that Γ ≅ Γ ⁢ ( O ) . Let D ∈ D B be a colour of 𝑋 of type a ′ or 𝑏. Then, from the construction in Section 3.1.1 and Lemma A.1, D ∩ O = D ̂ is a single colour of 𝑂 which has the same type with 𝐷, and satisfies v D = v D ̂ . By restricting 𝐿 and 𝑠 on 𝑂, we get s | O a 𝐵-semiinvariant section of L | O with the same weight λ 0 and d | O its divisor. The statement then follows from Lemma A.2.

Now we turn to case (2). Assume that 𝑋 is quasihomogeneous which contains a homogeneous space G / H . Then, for any colour 𝐷, D ∩ ( G / H ) is the divisor of a 𝐵-semiinvariant section s D ∈ H 0 ⁢ ( G / H , L D ) λ D ( B ) for some λ D ∈ X ⁢ ( B ) . Recall the set C o constructed in Section 3.2 above. We claim that λ X z ≡ λ o (modulo a 𝐺-character) for some λ o ∈ X ⁢ ( B ) for any X z with z ∈ C o . Note that, for any z 1 , z 2 ∈ C ̊ , the difference of the colours X z 1 − X z 2 coincides with the divisor of a 𝐵-invariant function f 0 : = z − z 1 z − z 2 ∈ k ( P 1 ) ≅ k ( X ) B . This implies

f ′ : = s X z 1 f 0 ⁢ s X z 2 ∈ O ( G / H ) × .

By [31, Proposition 1.3], 𝐺 acts on f ′ through some character 𝜇, whence λ X z 1 = λ X z 2 + μ , and we get the claim.

By restricting 𝑠 on G / H , we get

f = s | G / H ∏ D ∈ D B s D m D ∈ O ⁢ ( G / H ) × .

Again by [31, Proposition 1.3], we can decompose

λ 0 = m ⁢ λ o + λ 0 ′ + λ 0 ′′ + λ G ,

where m = ∑ z ∈ C o m X z ∈ N + , λ 0 ′ = ∑ D ∈ D B ∖ B ⁢ ( X ) P ′ m D ⁢ λ D (every 𝐷 appears in this sum descends to a colour of Z ′ ), λ 0 ′′ = ∑ D ∈ ( D B ) P ′ , x D ∉ C o m D ⁢ λ D (every 𝐷 in this sum descends to an L P ′ -stable divisor in Z ′ ), and λ G ∈ X ⁢ ( G ) . In particular, λ o and λ 0 ′′ are L P ′ -characters, since P ′ stabilizes X z for z ∈ C o and any D ∈ ( D B ) P ′ . Thus ⟨ α ∨ , λ 0 ⟩ = ⟨ α ∨ , λ 0 ′ ⟩ for all α ∈ Π L P ′ . On the other hand, for a colour 𝐷 that descends to a colour D ′ of type a ′ or 𝑏 in Z ′ , v D = 1 2 ⁢ α ∨ | Γ or α ∨ | Γ for α ∈ Π L P ′ . The lemma then follows from the previous case. ∎

A.2 Lemmas on one-parameter 𝐺-varieties of type I

Lemma A.4

Let 𝑋 be a one-parameter 𝐺-variety of type I and 𝒪 any 𝐺-orbit in it. Then the coloured cone ( C , R ) of 𝒪 cannot be totally contained in 𝒬.

Proof

Note that any 𝐺-orbit in general position is spherical. By [3, Proposition 1], any 𝐺-orbit is spherical. Also, X O : = O ̄ is normal by [52, Theorem 16.25], whence a spherical variety. In particular, X O contains only finitely many 𝐺-orbits by Akhiezer’s theorem [1]. Thus there are only finitely many coloured cones in the fan F X of 𝑋 whose relative interior intersects 𝒱 and contains ( C , R ) as a face.

Denote by 𝐶 the smooth projective curve so that k ⁢ ( X ) G = k ⁢ ( C ) . If ( C , R ) ⊂ Q , then for almost every x ∈ C , ( Cone ⁡ ( q x , C ) , R ) is a coloured cone in F X . Also, RelInt ⁡ ( C ) ∩ V ≠ ∅ since ( C , R ) is the coloured cone of a 𝐺-subvariety, which implies

RelInt ⁡ ( Cone ⁡ ( q x , C ) ) ∩ V ≠ ∅

for those x ∈ C above. Thus X O contains infinitely many 𝐺-orbits, a contradiction. ∎

The following lemma is a special case of a general result [52, Proposition 12.12]. We include an elementary proof below in our case only for readers’ convenience.

Lemma A.5

Let 𝑋 be a one-parameter 𝐺-variety of type I and 𝐶 the smooth projective curve so that k ⁢ ( X ) G = k ⁢ ( C ) . Then the rational quotient pr B : X ⇢ C is a morphism pr B : X → C separating general 𝐺-orbits.

Proof

Suppose that X ̊ ⊂ X is any 𝐵-chart given by the coloured data ( W , R ) . Then there is an open subset X o ⊂ X ̊ where pr B is defined. By definition, for any x ∈ C , we have that pr B − 1 ⁢ ( x ) ∩ X o is the union of all D ∩ X o , where D ∈ W ⊔ R so that x D = x and h D > 0 . We hope to show that, under our assumptions, pr B in fact can be extended to the whole 𝑋.

By Lemma A.4, for any 𝐺-orbit O ⊂ X , there is a unique x ∈ C so that the coloured cone of X O ⊂ Q x , + . Thus there is a 𝐺-invariant map Pr : X → C that maps any point in 𝒪 to x ∈ C . This is a map globally defined on 𝑋, and Pr | X o = pr B | X o . It suffices to show that Pr is regular on the whole 𝑋.

Suppose that ( C , R ) is a hypercone of type I so that each ( C x , R x ) , x ∈ C , is a coloured cone in F X . As in [42, 34], we denote the locus of ( C , R ) by

Loc ( C , R ) : = { x ∈ C ∣ ( C x , R x ) ⊄ Q } .

Denote by U ⁢ ( C , R ) the corresponding 𝐵-chart defined by ( C , R ) . Note that h D ≥ 0 for any D ∈ B ⁢ ( X ) . For any affine subset C ̊ ⊂ C , k ⁢ [ C ̊ ] ⊂ k ⁢ ( C ) = k ⁢ ( X ) G is contained in k ⁢ [ U ⁢ ( C , R ) ] if and only if Loc ⁡ ( C , R ) ⊂ C ̊ . On the other hand, since every f ∈ k ⁢ [ C ̊ ] is 𝐺-invariant, 𝑓 is regular on the 𝐺-span of U ⁢ ( C , R ) if and only if f ∈ k ⁢ [ U ⁢ ( C , R ) ] . Then Pr is regular on the 𝐺-span of every U ⁢ ( C , R ) so that Loc ⁡ ( C , R ) ⊂ C ̊ , and is a globally defined morphism. ∎

A.3 Lemmas on counting integral points

Lemma A.6

Suppose there are 𝑚 numbers α 1 , … , α m ∈ R so that

(A.1) ∑ i = 1 m α i ≥ 0 > ∑ i = 1 m [ α i ] .

Then

(A.2) m > ∑ i = 1 m α i ≥ 0 > ∑ i = 1 m [ α i ] > − m .

Proof

Note that

α i = [ α i ] + { α i } , i = 1 , … , m .

From the right-hand side inequality of (A.1), we have

m > ∑ i = 1 m { α i } > ∑ i = 1 m α i ≥ 0 .

At the same time, the left-hand side inequality of (A.1) yields

∑ i = 1 m [ α i ] ≥ − ∑ i = 1 m { α i } > − m .

Hence we get the lemma. ∎

From the above lemma, we get the following estimates.

Lemma A.7

Let d 0 be an ample divisor given by (2.7). Set

T k : = { λ ∈ Δ Z ( d 0 ) ∩ 1 k Γ | ∑ x ∈ C [ k A x ( d 0 , λ ) ] < 0 } , k ∈ N + .

Then there are a k 0 ∈ N + , depending only on d 0 , and constants c , c ′ > 0 so that the cardinal number # ⁢ T k ≤ c ⁢ k r − 1 for any k ∈ N ≥ k 0 , and

− c ′ < ∑ x ∈ C [ k ⁢ A x ⁢ ( d 0 , λ ) ] < k ⁢ A ⁢ ( d 0 , λ ) < c ′ for all ⁢ k ∈ N ≥ k 0 ⁢ and ⁢ λ ∈ T k .

Proof

Note that there are only finitely many points x 1 , … , x m ∈ C so that

A x i ⁢ ( d 0 , λ ) ≢ 0 for ⁢ i = 1 , … , m .

Take α i = k ⁢ A x i ⁢ ( d 0 , λ ) . By Lemma A.6,

(A.3) T k ⊂ { λ ∈ Δ Z ⁢ ( d 0 ) ∩ 1 k ⁢ Γ | A ⁢ ( d 0 , λ ) < m k } .

On the other hand, the piecewise linear function A ⁢ ( d 0 , λ ) > 0 on Δ Z ⁢ ( d 0 ) . Thus there is a k 0 ∈ N + that depends only on d 0 (more precisely, the integer 𝑚, the function A ⁢ ( d 0 , λ ) , and the shape of Δ Z ⁢ ( d 0 ) which are completely determined by d 0 ) such that the right-hand side of (A.3) is contained in a strip near the boundary of Δ Z ⁢ ( d 0 ) with facets parallel to that of the boundary. It is also direct to check that, for k ≥ k 0 , this strip can be chosen so that its width is at most c 0 ⁢ m / k , where c 0 > 0 is again a constant that depends only on the function A ⁢ ( d 0 , λ ) and the shape of Δ Z ⁢ ( d 0 ) . In particular, c 0 is independent of k ∈ N + . Thus # ⁢ T k ≤ c ⁢ k r − 1 for some uniform c > 0 . The last point follows from (A.2) by taking c ′ = m . ∎

The following lemma can be derived from a general result of Pukhlikov–Khovanskij [44]. We include it here for readers’ convenience.

Lemma A.8

Let M ≅ Z r be a lattice and Δ ⊂ M R a solid, integral convex polytope in it. Let π : Δ → R be a monomial of degree 𝑑 on M R and f : Δ → R a concave, piecewise linear function whose domains of linearity { Ω a } a = 1 N f consist of integral polytopes in Δ. Suppose that, on each domain of linearity Ω a ,

f ⁢ ( λ ) = 1 p a ⁢ ( q a ⁢ ( λ ) + r a ) ,

where ( p a , − q a ) ∈ Z ⊕ Z is a primitive vector, r a ∈ Q , and 𝑓 takes integral value at every vertex f its domains of linearity. Define

S k ( f ; π ) : = ∑ λ ∈ k ⁢ Δ ∩ M [ k f ( λ k ) ] π ( λ ) , k ∈ N + .

Then

S k ⁢ ( f ; π ) = k r + d + 1 ⁢ ∫ Δ f ⁢ ( λ ) ⁢ π ⁢ ( λ ) ⁢ d λ + 1 2 ⁢ k r + d ⁢ ∫ ∂ Δ f ⁢ ( λ ) ⁢ π ⁢ ( λ ) ⁢ d σ − 1 2 ⁢ k r + d ⁢ ∑ a = 1 N f ∫ Ω a ( 1 − 1 | p a | ) ⁢ π ⁢ ( λ ) ⁢ d λ + O ⁢ ( k r + d − 1 ) , k → + ∞ ,

where d ⁢ σ is the induced lattice measure on ∂ Δ .

Proof

Since 𝑓 is concave, min Δ ⁡ f is attained at some vertex of Δ. Hence min Δ ⁡ f ∈ Z . Consider a convex polytope

Δ m : = { ( t , λ ) | λ ∈ Δ , t ∈ R , min Δ f ≤ t ≤ f ( λ ) } .

Then Δ m is an ( r + 1 ) -dimensional convex integral polytope in M R ⊕ R . Clearly,

[ k ⁢ f ⁢ ( λ k ) ] = [ k ⁢ ( f ⁢ ( λ k ) − min Δ ⁡ f ) ] + k ⁢ min Δ ⁡ f = # ⁢ { ( { 1 k ⁢ λ } × 1 k ⁢ Z ) ∩ Δ m } + k ⁢ min Δ ⁡ f − 1 .

Thus

(A.4) ∑ λ ∈ k ⁢ Δ ∩ M [ k ⁢ f ⁢ ( λ k ) ] ⁢ π ⁢ ( λ ) = ∑ ( λ , t ) ∈ Δ m ∩ 1 k ⁢ ( M ⊕ Z ) π ⁢ ( k ⁢ λ ) + ( k ⁢ min Δ ⁡ f − 1 ) ⁢ ∑ λ ∈ Δ ∩ 1 k ⁢ M π ⁢ ( k ⁢ λ ) = k d ⁢ ∑ ( λ , t ) ∈ Δ m ∩ 1 k ⁢ ( M ⊕ Z ) π ⁢ ( λ ) + k d ⁢ ( k ⁢ min Δ ⁡ f − 1 ) ⁢ ∑ λ ∈ Δ ∩ 1 k ⁢ M π ⁢ ( λ ) .

By [44], the first term is

(A.5) ∑ ( λ , t ) ∈ Δ m ∩ 1 k ⁢ ( M ⊕ Z ) π ⁢ ( λ ) = k r + 1 ⁢ ∫ Δ m π ⁢ ( λ ) ⁢ d t ∧ d ⁢ λ + 1 2 ⁢ k r ⁢ ∫ ∂ Δ m π ⁢ ( λ ) ⁢ d σ ̄ + O ⁢ ( k r − 1 ) , k → + ∞ ,

where d ⁢ σ ̄ is the induced lattice measure on ∂ Δ m . We have

(A.6) ∫ Δ m π ⁢ ( λ ) ⁢ d t ∧ d ⁢ λ = ∫ Δ ( f ⁢ ( λ ) − min Δ ⁡ f ) ⁢ π ⁢ ( λ ) ⁢ d λ .

The boundary ∂ Δ m consists of three parts:

on F 1 : = ( R × ∂ Δ ) ∩ Δ m ,
(A.7) ∫ F 1 π ⁢ ( λ ) ⁢ d σ ̄ = ∫ ∂ Δ ( f ⁢ ( λ ) − min Δ ⁡ f ) ⁢ π ⁢ ( λ ) ⁢ d σ ,
where d ⁢ σ is the induced lattice measure on ∂ Δ ;
on F 2 = { min Δ ⁡ f } × Δ ,
(A.8) ∫ F 2 π ⁢ ( λ ) ⁢ d σ ̄ = ∫ Δ π ⁢ ( λ ) ⁢ d σ ̄ ;
on F 3 = { graph of ⁢ f } , since 𝑓 is rational, on each domain of linearity Ω where
f ⁢ ( λ ) = 1 p ⁢ ( q ⁢ ( λ ) + r ′ )
for primitive vector ( p , − q ) ∈ Z ⊕ M ∗ and r ′ ∈ Q ,
(A.9) ∫ graph ⁢ f ⁢ on ⁢ Ω π ⁢ ( λ ) ⁢ d σ ̄ = ∫ Ω π ⁢ ( λ ) ⁢ 1 | ( p , − q ) | ⁢ d σ 0 = 1 | p | ⁢ ∫ Ω π ⁢ ( λ ) ⁢ d λ .

Here

d ⁢ σ 0 = 1 + | q | 2 p 2 ⁢ d ⁢ λ

is the standard induced Lebesgue measure. Plugging (A.6)–(A.9) into (A.5), we get

∑ ( λ , t ) ∈ Δ m ∩ 1 k ⁢ ( M ⊕ Z ) π ⁢ ( λ ) = k r + 1 ⁢ ∫ Δ ( f ⁢ ( λ ) − min Δ ⁡ f ) ⁢ π ⁢ ( λ ) ⁢ d λ + 1 2 ⁢ k r ⁢ ∫ ∂ Δ ( f ⁢ ( λ ) − min Δ ⁡ f ) ⁢ π ⁢ ( λ ) ⁢ d σ + 1 2 ⁢ k r ⁢ ∑ a = 1 N f ∫ Ω a ( 1 − 1 | p a | ) ⁢ π ⁢ ( λ ) ⁢ d λ + O ⁢ ( k r − 1 ) , k → + ∞ .

Similarly,

∑ λ ∈ Δ ∩ 1 k ⁢ M π ⁢ ( λ ) = k r ⁢ ∫ Δ π ⁢ ( λ ) ⁢ d λ + 1 2 ⁢ k r − 1 ⁢ ∫ ∂ Δ π ⁢ ( λ ) ⁢ d σ + O ⁢ ( k r − 2 ) , k → + ∞ .

Plugging the above two relations into (A.4), we get the lemma. ∎

A.4 An alternative proof to Theorem 5.5

In the following, we give an alternative proof via an intersection formula of the Futaki invariant (cf. [38, 5]). Given a normal test configuration ( X , L ) of ( X , L ) , the Futaki invariant can also be interpreted as intersection numbers

(A.10) Fut ⁡ ( X , L ) = 1 V ⁢ K X / P 1 ⋅ L ⋅ n + S ̄ V ⁢ ( n + 1 ) ⁢ L ⋅ ( n + 1 ) ,

where V = L n is the volume of ( X , L ) , S ̄ the mean value of the scalar curvature of ( X , L ) , and

K X / P 1 = K X − pr ∗ ⁢ K P 1

a Weil divisor on 𝒳.

Assume that ( X , L ) is a polarized spherical variety with 𝐿 having a divisor d 0 given by (2.7), and d K = ∑ D ∈ B ⁢ ( X ) m ̄ D ⁢ D a 𝐵-stable anti-canonical divisor of 𝑋 corresponds to weight κ P . Let ( X , L ) be a normal test configuration associated to v 0 ∈ Q x 0 , + and m = − 1 . Without loss of generality, we may also assume that ℒ has a divisor 𝔇 given by (4.1) with r 0 = 1 and m ∞ = 0 (otherwise, one needs to divide (A.10) by r 0 n + 1 ). Then, by Theorems 3.5 (in the one-parameter case) and 3.8 (in the quasihomogeneous case), it is direct to see

d X / P 1 = − ∑ D ∈ B ⁢ ( X ) m ̄ D ⁢ D ̄ − h 0 ⁢ ( a x 0 − 1 ) ⁢ X 0 .

In particular, if 𝑋 is ℚ-Fano and K = K X − 1 , we can take d K = d 0 (in particular, m ̄ D = m D for all D ∈ B ⁢ ( X ) ) and

d X / P 1 = − ( D + ( h 0 ⁢ ( a x 0 − 1 ) − m 0 ) ⁢ pr ∗ ⁢ ( [ 0 ] ) )

is also a ℚ-Cartier divisor that corresponds to the weight − κ P .

The line bundle

L ϵ : = L + ϵ K X / P 1

is a ℚ-line bundle on 𝒳, and is ample when 0 < ϵ ≪ 1 . The divisor

D ϵ : = D + ϵ d X / P 1 = ∑ D ∈ B ⁢ ( X ) ( m D − ϵ m D ) D ̄ + ( m 0 − ϵ ( a x 0 − 1 ) ) X 0

is a divisor of L ϵ that corresponds to weight ( 1 − ϵ ) ⁢ κ P , and the associated function is

A ̃ x ϵ ( λ , t ) : = { ( 1 − ϵ ) ⁢ min ⁡ { A x 0 ⁢ ( d , λ 1 − ϵ ) , − t + m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) + ℓ 0 ⁢ ( λ ) h 0 ⁢ ( 1 − ϵ ) } when ⁢ x = x 0 , ( 1 − ϵ ) ⁢ A x ⁢ ( d , λ 1 − ϵ ) when ⁢ x ≠ x 0 ,

for λ ∈ Γ R and A ̃ ϵ ⁢ ( λ , t ) = ∑ x ∈ C A ̃ x ϵ ⁢ ( λ , t ) . The polytope is

(A.11) Δ ̃ Z ϵ ⁢ ( D ϵ ) = { ( λ , t ) ∣ A ̃ ϵ ⁢ ( λ , t ) ≥ 0 } = { ( λ , t ) ∣ λ ∈ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) , 0 ≤ t ≤ τ ϵ 0 ⁢ ( λ ) } ,

where

τ ϵ 0 ⁢ ( λ ) = m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) + ℓ 0 ⁢ ( λ ) + h 0 ⁢ ( 1 − ϵ ) ⁢ ( A ⁢ ( λ 1 − ϵ ) − A x 0 ⁢ ( λ 1 − ϵ ) ) .

Set

τ ̃ ϵ 0 ⁢ ( λ ) = m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) + ℓ 0 ⁢ ( λ ) − h 0 ⁢ ( 1 − ϵ ) ⁢ A x 0 ⁢ ( λ 1 − ϵ ) .

Then, for λ ∈ Γ R ,

(A.12) A ̃ ϵ ( λ , t ) : = { ( 1 − ϵ ) ⁢ A ⁢ ( d 0 , λ 1 − ϵ ) when ⁢ 0 ≤ t ≤ τ ̃ 0 ϵ ⁢ ( λ ) , ( 1 − ϵ ) ⁢ ∑ x ≠ x 0 A x ⁢ ( d 0 , λ 1 − ϵ ) + m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) − t + ℓ 0 ⁢ ( λ ) h 0 when ⁢ τ ̃ ϵ 0 ⁢ ( λ ) ≤ t ≤ τ ϵ 0 ⁢ ( λ ) .

Note that, when ϵ ∈ Q ,

L ϵ ⋅ ( n + 1 ) = L ⋅ ( n + 1 ) + ( n + 1 ) ⁢ ϵ ⁢ K X / P 1 ⋅ L ⋅ n + O ⁢ ( ϵ 2 ) , ϵ → 0 + .

Applying the intersection formula [50, Theorem 8] to L ϵ , we get

(A.13) L ⋅ ( n + 1 ) = ( n + 1 ) ! ⁢ ∫ Δ ̃ Z ⁢ ( D ) A ̃ ⁢ ( λ , τ ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ τ = ( n + 1 ) ! ⁢ ∫ Δ Z ⁢ ( d 0 ) ∫ 0 τ 0 ⁢ ( λ ) A ̃ ⁢ ( λ , τ ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ τ ,

(A.14) K X / P 1 ⋅ L ⋅ n = 1 n + 1 ⁢ d d ⁢ ϵ | ϵ = 0 ⁢ L ϵ ⋅ ( n + 1 ) = n ! ⁢ d d ⁢ ϵ | ϵ = 0 ⁢ ∫ Δ ̃ Z ϵ ⁢ ( D ϵ ) A ̃ ϵ ⁢ ( λ , τ ) ⁢ π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d λ ∧ d ⁢ τ .

By (A.11) and (A.12),

∫ Δ ̃ Z ϵ ⁢ ( D ϵ ) A ̃ ϵ ⁢ ( λ , τ ) ⁢ π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d λ ∧ d ⁢ τ = ∫ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) ∫ 0 τ ̃ ϵ 0 ⁢ ( λ ) ( 1 − ϵ ) ⁢ A ⁢ ( d 0 , λ 1 − ϵ ) ⁢ π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d λ ∧ d ⁢ t + ∫ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) ∫ τ ̃ ϵ 0 ⁢ ( λ ) τ ϵ 0 ⁢ ( λ ) ( 1 − ϵ ) ⁢ ∑ x ≠ x 0 A x ⁢ ( d 0 , λ 1 − ϵ ) × π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d ⁢ λ ∧ d ⁢ t + ∫ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) ∫ τ ̃ ϵ 0 ⁢ ( λ ) τ ϵ 0 ⁢ ( λ ) m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) − t + ℓ 0 ⁢ ( λ ) h 0 × π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d ⁢ λ ∧ d ⁢ t .

Using change of variables λ → ( 1 − ϵ ) ⁢ λ , t → ( 1 − ϵ ) ⁢ t , and relation (2.12), we can rewrite the first two terms as

∫ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) ∫ 0 τ ̃ ϵ 0 ⁢ ( λ ) ( 1 − ϵ ) ⁢ A ⁢ ( d 0 , λ 1 − ϵ ) ⁢ π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d λ ∧ d ⁢ t + ∫ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) ∫ τ ̃ ϵ 0 ⁢ ( λ ) τ ϵ 0 ⁢ ( λ ) ( 1 − ϵ ) ⁢ ∑ x ≠ x 0 A x ⁢ ( d 0 , λ 1 − ϵ ) × π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d ⁢ λ ∧ d ⁢ t = ( 1 − ϵ ) n + 1 ⁢ ∫ Δ Z ⁢ ( d 0 ) ∫ 0 1 1 − ϵ ⁢ τ ̃ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) A ⁢ ( d 0 , λ ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ t + ( 1 − ϵ ) n + 1 ⁢ ∫ Δ Z ⁢ ( d 0 ) ∫ 1 1 − ϵ ⁢ τ ̃ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) 1 1 − ϵ ⁢ τ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) ∑ x ≠ x 0 A x ⁢ ( d 0 , λ ) ⁢ π ⁢ ( λ + κ P ) ⁢ d ⁢ λ ∧ d ⁢ t ,

and the last term can be rewritten as

∫ ( 1 − ϵ ) ⁢ Δ Z ⁢ ( d 0 ) ∫ τ ̃ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) τ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) − t + ℓ 0 ⁢ ( λ ) h 0 ⁢ π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d λ ∧ d ⁢ t = ( 1 − ϵ ) n + 1 ⁢ ∫ Δ Z ⁢ ( d 0 ) ∫ 1 1 − ϵ ⁢ τ ̃ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) 1 1 − ϵ ⁢ τ ϵ 0 ⁢ ( ( 1 − ϵ ) ⁢ λ ) 1 1 − ϵ ⁢ ( m 0 − ϵ ⁢ h 0 ⁢ ( a x 0 − 1 ) ) − t + ℓ 0 ⁢ ( λ ) h 0 × π ⁢ ( λ + κ P ) ⁢ d ⁢ λ ∧ d ⁢ t

Taking variation and using Lemma 5.2, we get

d d ⁢ ϵ | ϵ = 0 ⁢ ∫ Δ ̃ Z ϵ ⁢ ( D ϵ ) A ̃ ϵ ⁢ ( λ , τ ) ⁢ π ⁢ ( λ + ( 1 − ϵ ) ⁢ κ P ) ⁢ d λ ∧ d ⁢ τ = − ( n + 1 ) ⁢ ∫ Δ ̃ Z ⁢ ( D ) A ̃ ⁢ ( D , λ , τ ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ τ + ∫ Δ Z ⁢ ( d 0 ) ∫ τ ̃ 0 ⁢ ( λ ) τ 0 ⁢ ( λ ) ( m 0 h 0 − a x 0 + 1 ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ t .

Combining with (A.10), (A.13)–(A.14), we get

Fut ⁡ ( X , L ) = 1 V ⁢ ∫ Δ Z ⁢ ( d 0 ) ∫ τ ̃ 0 ⁢ ( λ ) τ 0 ⁢ ( λ ) m 0 − h 0 ⁢ ( a x 0 − 1 ) h 0 ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ t − 1 V ⁢ ∫ Δ Z ⁢ ( d 0 ) ∫ 0 τ 0 ⁢ ( λ ) A ̃ ⁢ ( D , λ , t ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ ∧ d ⁢ t = 1 V ⁢ ∫ Δ x 0 O ⁢ ( K X − 1 ) ⟨ ( κ P − λ , − t ) , v 0 ⟩ ⁢ π ⁢ ( λ ) ⁢ d λ ∧ d ⁢ t ,

where in the last line we used (5.25) and the fact that

V = ( K X − 1 ) ⋅ n n ! = ∫ Δ X ⁢ ( d 0 ) A ⁢ ( d 0 , λ ) ⁢ π ⁢ ( λ + κ P ) ⁢ d λ = ∫ Δ x 0 O ⁢ ( K X − 1 ) π ⁢ ( λ ) ⁢ d λ ∧ d ⁢ t .

A.5 On the polytope Δ x 0 O ⁢ ( K X − 1 )

The polytope Δ x 0 O ⁢ ( K X − 1 ) has another geometric meaning

Proposition A.9

Suppose that ( X , L ) is a 𝐺-equivariant special test configuration of ( X , K X − 1 ) associated to ( v 0 , − 1 ) . Assume that v 0 = ℓ 0 + h 0 ⁢ q x 0 with h 0 ≠ 0 . Then ( X 0 , L 0 ) is a polarized G × k × -spherical variety, and the associated moment polytope is

Δ x 0 O ⁢ ( K X − 1 ) − ( 0 , a x 0 − 1 ) .

Proof

Recall that, for any k ∈ N ,

H 0 ⁢ ( X 0 , L 0 k ) ( λ , τ ) ( B × k × ) ≅ ( F ( X , L ) τ ⁢ R k / F ( X , L ) > τ ⁢ R k ) λ ( B ) ⊂ Gr ⁡ ( F ( X , L ) ) .

We decompose H 0 ⁢ ( X 0 , L 0 k ) into irreducible G × k × -modules using Proposition 4.4. From Proposition 4.4 (2) and (4.7), we see that

H 0 ⁢ ( X 0 , L 0 k ) ( λ , τ ) ( B × k × ) ≠ 0

if and only if

deg ⁡ ( δ k ⁢ ( λ , τ ) ) ≥ 0 , where ⁢ δ k ⁢ ( λ , τ ) ⁢ is defined by ( 4.4 ) , deg ⁡ ( δ k ⁢ ( λ , τ ) ) − deg ⁡ ( δ k ⁢ ( λ , τ + 1 ) ) > 0 .

The above conditions require that ( λ , τ ) satisfies

{ τ ≤ k ⁢ τ 0 ⁢ ( λ k − κ P ) , where ⁢ τ 0 ⁢ is defined by ( 4.11 ) , A x 0 ⁢ ( d , λ k − κ P ) ≥ − τ k + m 0 + ℓ 0 ⁢ ( λ k − κ P ) h 0 ,

or equivalently ( λ , τ ) ∈ k ⁢ Δ Z o ⁢ ( L ) (see (5.2) for the definition). Conversely, by Lemma A.7, for any k ∈ N + , all points at which the first condition may fail lie in a strip of uniform (to 𝑘) width along k ⁢ ∂ Δ Z o ⁢ ( L ) . Thus the closure of

⋃ k = 0 + ∞ 1 k ⁢ { ( λ , τ ) ∈ ( Γ + k ⁢ κ P ) × Z ∣ H 0 ⁢ ( X 0 , L 0 k ) ( λ , τ ) ( B × k × ) ≠ 0 }

in ( Γ R + κ P ) × R is Δ Z o ⁢ ( L ) .

However, to get the moment polytope, we need some normalization. Take a (unimodular and integral) transformation of 𝜎 on Γ ⊕ Z , σ : ( λ , τ ) → ( λ , τ − m 0 − ℓ 0 ⁢ ( λ ) ) , which induces a transformation

( λ + κ P , τ ) → ( λ + κ P , τ − m 0 − ℓ ⁢ ( λ ) )

on ( Γ + κ P ) × Z . The polytope Δ Z o ⁢ ( L ) ⊂ ( Γ + κ 0 ) R × R is then transformed to

{ ( λ , t ) ∣ − h 0 ⁢ A x 0 ⁢ ( d , λ − κ P ) ≤ t ≤ h 0 ⁢ A ⁢ ( d , λ − κ P ) − h 0 ⁢ A x 0 ⁢ ( d , λ − κ P ) } ⊂ ( Γ + κ 0 ) R × R .

On the other hand, from (4.7), we see the second condition also requires

− τ + k ⁢ m 0 + ℓ 0 ⁢ ( λ − k ⁢ λ 0 ) ∈ h 0 ⁢ Z .

Hence the group generated by k × -weights on ⋃ k = 0 + ∞ H 0 ⁢ ( X 0 , L 0 k ) is h 0 ⁢ Z . By rescaling it to the standard ℤ, we get the moment polytope of ( X 0 , L 0 ) is Δ x 0 O ⁢ ( K X − 1 ) − ( 0 , a x 0 − 1 ) . ∎

We remark that Proposition A.9 for 𝑇-varieties of complexity 1 has been proved in [24, Corollary 4.6]. Also, the above method applies to 𝐺-equivariant special test configurations of a general polarized ( X , L ) (not necessarily L = K X − 1 ).

Acknowledgements

We sincerely thank Prof. D. A. Timashëv for kindly introducing us to his book [52] and many helpful discussions. We also thank the referees for careful reading and valuable comments, which improve this paper a lot.

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Received: 2024-09-06

Revised: 2025-06-06

Published Online: 2025-11-20

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https://doi.org/10.1515/crelle-2025-0087