2.1 Valuative stability

Although we are ultimately interested in divisorial stability, which is a notion of stability defined through convex combinations of divisorial valuations on X, the general theory is quite intricate and simplifies considerably for a single divisorial valuation. Thus, we begin by explaining the theory for a single valuation, as introduced by the first author and Legendre [Reference Dervan and Legendre15], generalising the work of Fujita and Li in the Fano setting [Reference Fujita20, Reference Li25]. A reference explaining the background to the material presented here is [Reference Lazarsfeld24].

A prime divisor F over X equivalently induces a valuation $\operatorname {\mathrm {ord}} F$ on the function field $k(X)$ of X.

By passing to a log resolution of singularities if necessary, we assume that the pair $(Y,F)$ is log smooth. To such a divisorial valuation, we associate a numerical invariant called the beta invariant of F, defined through the following standard invariants in birational geometry.

Definition 2.3. Suppose L is a line bundle on X. We define the volume of L to be

$$\begin{align*}\operatorname{\mathrm{Vol}}(L) = \limsup_{r\to\infty} \frac{\dim H^0(X,rL)}{r^n/n!},\end{align*}$$

and we say that L is big if $\operatorname {\mathrm {Vol}}(L)>0$ .

The volume satisfies the homogeneity property $\operatorname {\mathrm {Vol}}(lL) = l^n\operatorname {\mathrm {Vol}}(L)$ , and hence, the definition extends to $\mathbb {Q}$ -line bundles; it further extends to $\mathbb {R}$ -line bundles by a continuity argument. We use two foundational results concerning the volume. First, the $\limsup $ involved in the definition is actually a limit. Second, the volume is actually a continuously differentiable function on the cone of big ( $\mathbb {R}$ -)line bundles on X [Reference Boucksom, Favre and Jonsson7]. We extend this definition to take F into account by defining

$$\begin{align*}\operatorname{\mathrm{Vol}}(L-xF) = \operatorname{\mathrm{Vol}}(\pi^*L-xF),\end{align*}$$

where the latter is calculated on Y.

We also require a measure of singularities, for which we use our hypothesis that $K_X+B$ is $\mathbb {Q}$ -Cartier.

Definition 2.4. We define the log discrepancy of F to be

$$\begin{align*}A_{(X,B)}(F) = \operatorname{\mathrm{ord}}_F(K_Y - \pi^*(K_X+B)) + 1.\end{align*}$$

Here, we use that $(Y,F)$ is log smooth; if not, one should work on a log resolution of singularities of $(Y,F)$ . The beta invariant is simply a combination of these invariants. Denote

$$\begin{align*}S_L(F) = \frac{ \int_0^{\infty} \operatorname{\mathrm{Vol}}(L-xF) dx}{L^n}.\end{align*}$$

Definition 2.5. The beta invariant of F is defined to be

$$\begin{align*}\beta(F) = A_{(X,B)}(F) + \nabla_{K_X+B} S_L(F),\end{align*}$$

where

$$\begin{align*}\nabla_{K_X+B} S_L(F) = \frac{d}{dt}\bigg|_{t=0} S_{L+t(K_X+B)}(F).\end{align*}$$

Example 2.6. If $L=-K_X$ and $B=0$ , so that X is a Fano variety, this invariant reduces to

$$\begin{align*}\beta(F) = A_X(F)- S_{-K_X}(F),\end{align*}$$

which is precisely the invariant introduced by Fujita and Li [Reference Fujita20, Reference Li25]. In general, our presentation of $\beta (F)$ agrees with that of Boucksom–Jonsson; see [Reference Boucksom and Jonsson10, Theorem 2.18] for the equivalence with the original presentation [Reference Dervan and Legendre15].

Strictly speaking, in [Reference Dervan and Legendre15], valuative stability required the divisors to be dreamy, which is a finite-generation hypothesis. As this plays no role in the present work – and in light of the work of Boucksom–Jonsson on divisorial stability [Reference Boucksom and Jonsson10], appears generally less relevant – we choose to omit this hypothesis (as in [Reference Liu27]).

2.2 Test configurations and K-stability

We next define test configurations and the associated Monge–Ampère energy (which will be used in the subsequent sections) and uniform K-stability (which will play a secondary role to divisorial stability).

The divisor $\mathcal {B} \subset \mathcal {X}$ is canonically defined by taking the $\mathbb G_m$ -closure of $B \subset X \cong \mathcal {X}_1$ . A test configuration admits a canonical compactification to a family over $\mathbb {P}^1$ by compactifying trivially at infinity, and we will use the same notation for the resulting family $((\mathcal {X},\mathcal {B});\mathcal {L}) \to \mathbb {P}^1$ .

Definition 2.10. The Monge–Ampère energy of a nef test configuration $(\mathcal {X},\mathcal {L})$ is defined to be

$$\begin{align*}E(\mathcal{X},\mathcal{L}) = \frac{\mathcal{L}^{n+1}}{(n+1)L^n},\end{align*}$$

where this intersection number is calculated on the compactified test configuration over $\mathbb {P}^1$ . When we wish to emphasise the dependence on L, we denote this by $E_L(\mathcal {X},\mathcal {L})$ .

Note that this quantity is independent of $\mathcal {B}$ , and hence, we omit $\mathcal {B}$ from the notation. To define further numerical invariants, it is useful to pass to a resolution of indeterminacy of the natural rational map $\mathcal {X} \dashrightarrow X\times \mathbb {P}^1;$ we thus obtain a new test configuration with an equivariant morphism to $X\times \mathbb {P}^1$ . We replace $\mathcal {X}$ by the associated resolution of indeterminacy, which we then say dominates the trivial test configuration $(X\times \mathbb {P}^1, L)$ .

Definition 2.11. The minimum norm of a nef test configuration $(\mathcal {X},\mathcal {L})$ is defined to be

$$\begin{align*}\|(\mathcal{X},\mathcal{L})\|_{\min} = \frac{\mathcal{L}^{n+1}}{(n+1)L^n} - \frac{\mathcal{L}^n.(\mathcal{L}-L)}{L^n},\end{align*}$$

where L is pulled back to $\mathcal {X}$ through the morphism $\mathcal {X} \to X$ .

The minimum norm is called the ‘non-Archimedean $I-J$ -functional’ in [Reference Boucksom, Hisamoto and Jonsson9]; we follow the terminology of [Reference Dervan14]. This quantity is again independent of $\mathcal {B}$ . The Mabuchi functional, however, does actually depend on $\mathcal {B}$ . In order to define this, for a test configuration dominating the trivial one, define the entropy

$$\begin{align*}\operatorname{\mathrm{Ent}}(\mathcal{X},\mathcal{L}) = \frac{1}{L^n}\left(\mathcal{L}^n.K_{(\mathcal{X},\mathcal{B})/(X,B)\times\mathbb{P}^1} + \mathcal{L}^n.(\mathcal{X}_0 - \mathcal{X}_{0,\operatorname{\mathrm{red}}})\right),\end{align*}$$

where

$$\begin{align*}K_{(\mathcal{X},\mathcal{B})/(X,B)\times\mathbb{P}^1} = K_{\mathcal{X}}+\mathcal{B} - \pi^*(K_{X\times\mathbb{P}^1} + B)\end{align*}$$

is the relative canonical class and $\mathcal {X}_0, \mathcal {X}_{0,\operatorname {\mathrm {red}}}$ denote the central fibre of $\mathcal {X}$ and the reduced the central fibre, respectively. To make sense of this definition, one uses that $\mathcal {X}$ is normal to ensure that $K_{\mathcal {X}}$ is a Weil divisor.

Definition 2.12. We define the Mabuchi functional on the set of normal, nef test configurations to take the value

$$\begin{align*}M((\mathcal{X},\mathcal{B});\mathcal{L}) =\operatorname{\mathrm{Ent}}(\mathcal{X},\mathcal{L}) + \nabla_{K_X+B}E_L(\mathcal{X},\mathcal{L}),\end{align*}$$

where $((\mathcal {X},\mathcal {B});\mathcal {L})$ is such a test configuration.

This is often called the non-Archimedean Mabuchi functional; as its Archimedean counterpart plays no role in the present work, we simplify the terminology. It agrees with the more traditional Donaldson–Futaki invariant of a normal, nef test configuration provided $\mathcal {X}_0$ is reduced; the associated notions of stability – which we next define – can be seen to be equivalent by a base-change argument [Reference Boucksom, Hisamoto and Jonsson9, Proposition 7.15]. The directional derivative involved is defined by

$$\begin{align*}\nabla_{K_X+B}E_L (\mathcal{X},\mathcal{L}) = \frac{d}{dt}\bigg|_{t=0} \frac{(\mathcal{L}+t(K_X+B))^{n+1}}{(n+1)(L+t(K_X+B))^n},\end{align*}$$

where we assume (as we may) that the test configuration dominates the trivial one and $K_X+B$ is also then used to denote its pullback to $\mathcal {X}$ ; this derivative can be computed explicitly to produce a version of the Mabuchi functional more commonly used in the literature.

The relationship between uniform K-stability and K-stability is as follows: uniform K-stability with respect to test configurations with irreducible central fibre is equivalent to uniform valuative stability with respect to dreamy prime divisors [Reference Dervan and Legendre15, Theorem 1.1]. Roughly speaking, the central fibre of a test configuration with irreducible central fibre induces a prime divisor over X, and conversely, under a finite generation hypothesis, the reverse of this construction also succeeds; the beta invariant is defined in such a way that it equals the value of the Mabuchi functional at the associated test configuration. To obtain stronger results – giving a fuller valuative interpretation of K-stability, in particular allowing test configurations with reducible central fibres – one needs further tools from non-Archimedean geometry, which we now turn to.

2.3 Berkovich analytification

The appropriate way of viewing convex combinations of divisorial valuations is as a certain type of measure on the Berkovich analytification $X^{\mathrm {an}}$ of X, which we now define. As throughout, we assume that X is a normal projective variety defined over an algebraically closed field of characteristic zero; the boundary divisor B will be irrelevant in the present section. We refer to Reboulet [Reference Reboulet37, Section 2] or Boucksom–Jonsson [Reference Boucksom and Jonsson11] for further details and proofs of the results stated below.

For our purposes, it will be sufficient to define $X^{\mathrm {an}}$ as a topological space; we note that it naturally carries the richer structure of a locally ringed space. We endow the field k with the trivial absolute value.

As a topological space, we first consider an affine chart $U \subset X$ , where we define a topology on $U^{\mathrm {an}}$ by requiring that for all $f\in \mathcal {O}_X(U)$ , the evaluation map

$$\begin{align*}f: U^{\mathrm{an}}\to \mathbb{R}\end{align*}$$

defined by $ f(V, |\cdot |_V) = |f|_V$ be continuous. These topologies agree on intersections of affine charts of X, and hence glue to a topology on $X^{\mathrm {an}}$ which is compact and Hausdorff. The association $X\to X^{\mathrm {an}}$ is functorial, in the sense that a morphism $Y\to X$ of projective varieties induces a morphism of analytic spaces $Y^{\mathrm {an}}\to X^{\mathrm {an}}$ .

If we take $V=X$ , then we simply obtain the function field of X. The space $X^{\operatorname {\mathrm {val}}} \subset X^{\mathrm {an}}$ of valuations on X is then an open dense subset of $X^{\mathrm {an}}$ , and $X^{\mathrm {an}}$ is thus a compactification of $X^{\operatorname {\mathrm {val}}}$ . We further denote $X^{\operatorname {\mathrm {div}}}$ the space of divisorial valuations on X, so that $X^{\operatorname {\mathrm {div}} }\subset X^{\operatorname {\mathrm {val}}} \subset X^{\mathrm {an}}.$

The $\mathbb {Q}$ -line L bundle on X induces a $\mathbb {Q}$ -line bundle $L^{\mathrm {an}}$ on $X^{\mathrm {an}}$ ; rather than $L^{\mathrm {an}}$ itself, what will be important is the space of non-Archimedean metrics on $L^{\mathrm {an}}$ . We will give a shallow treatment of non-Archimedean metrics, omitting how to view Fubini–Study metrics as genuine metrics (in the sense of assigning a nonnegative number to a section at a point) and instead viewing them as certain functions on $X^{\mathrm {an}}$ . In Kähler geometry, the quotient of two Hermitian metrics can be viewed as a function, and our treatment is reasonable due to the presence of the trivial metric in the non-Archimedean setting of interest here. Here, the trivial metric $\varphi _{{\mathrm {triv}}}$ is induced by the trivial test configuration $(X\times \mathbb A^1,L)$ , with X given the trivial $\mathbb G_m$ -action.

To define a function on $X^{\mathrm {an}}$ associated to a test configuration, we may first assume that $(\mathcal {X},\mathcal {L})$ dominates the trivial test configuration by passing to a $\mathbb G_m$ -equivariant resolution of indeterminacy of the rational map $X\times \mathbb {A}^1 \dashrightarrow \mathcal {X}$ if necessary. We can thus write $\mathcal {L} - L = D$ (with L the pullback of L from $X\times \mathbb {A}^1$ to $\mathcal {X}$ ), where D is a $\mathbb {Q}$ -Cartier divisor supported on $\mathcal {X}_0$ . Writing $\mathcal {X}_0 = \sum _{j} b_j E_j$ as a cycle, so that the $E_j \subset \mathcal {X}_0$ are reduced and irreducible, this function is given as

(2.1) $$ \begin{align}\varphi_{(\mathcal{X},\mathcal{L})}(v_{a\operatorname{\mathrm{ord}} E_j}) = b_{j}^{-1}a\operatorname{\mathrm{ord}}_{E_j}(D),\end{align} $$

with the function vanishing elsewhere. One checks that if $p: \mathcal {X}' \to \mathcal {X}$ is a $\mathbb G_m$ -equivariant morphism with $(\mathcal {X}',p^*\mathcal {L})$ and $(\mathcal {X},\mathcal {L})$ test configurations for $(X,L)$ , then $\varphi _{(\mathcal {X}',p^*\mathcal {L})} = \varphi _{(\mathcal {X},\mathcal {L})}$ . We always identify $\varphi _{(\mathcal {X},\mathcal {L})}$ and $\varphi _{(\mathcal {X}',\mathcal {L}')}$ if the associated functions on $X^{\mathrm {an}}$ are equal. Finally, note that that $\varphi _{(\mathcal {X},\mathcal {L})}$ is supported on $X^{\operatorname {\mathrm {val}}}$ .

When L is clear from context, we denote this simply by $\mathcal {H}^{\operatorname {\mathrm {NA}}}$ . We next define flag ideals which allow us to obtain a more concrete picture on the relationship between Fubini–Study metrics and test configurations. Studied in [Reference Boucksom, Hisamoto and Jonsson9, Reference Odaka35], we refer to [Reference Boucksom and Jonsson11, Section 2.1] for further details.

Here, a vertical ideal sheaf by definition satisfies the condition that $\mathcal {O}_{X\times \mathbb A^1}/\mathfrak {a}$ is supported on $X\times \{0\}$ . Fractional ideal sheaves are included as in the definition of a test configuration we allow $\mathcal {L}$ to be a $\mathbb {Q}$ -line bundle. Any flag ideal admits a decomposition

$$\begin{align*}\mathfrak{a} = \sum_{\lambda\in \mathbb{Z}}t^{-\lambda} \mathfrak{a}_{\lambda},\end{align*}$$

where t is the coordinate of $\mathbb {A}^1$ and $\mathfrak {a}_{\lambda }\subset \mathcal {O}_X$ is a non-increasing sequence of integrally closed ideals on X with $\mathfrak {a}_{\lambda } = 0$ for $\lambda \gg 0$ and $\mathfrak {a}_{\lambda } = \mathcal {O}_X$ for $\lambda \ll 0$ . In addition, every test configuration $\mathcal {X}$ dominating the trivial one is given as the blowup

$$\begin{align*}\mathcal{X} = \operatorname{Bl}_{\mathfrak{a}}X\times\mathbb{P}^1,\end{align*}$$

where with D the exceptional divisor, we define Under this correspondence, the function $\varphi _{(\mathcal {X},\mathcal {L})} = \varphi _{\mathfrak {a}}$ satisfies the property [Reference Boucksom and Jonsson11, Equation 2.4]

(2.2) $$ \begin{align}\varphi_{\mathfrak{a}\cdot\mathfrak{a}'} = \varphi_{\mathfrak{a}} + \varphi_{\mathfrak{a}'},\end{align} $$

where we use that the product of two flag ideals is a flag ideal.

Much as in Kähler geometry, it is also helpful to consider singular metrics.

Example 2.18. For two Fubini–Study metrics $\varphi _{(\mathcal {X},\mathcal {L})}, \varphi _{(\mathcal {X}',\mathcal {L}')}$ , the condition

$$\begin{align*}\varphi_{(\mathcal{X}',\mathcal{L}')}\geq \varphi_{(\mathcal{X},\mathcal{L})}\end{align*}$$

means that we can find a $\mathbb G_m$ -equivariant birational model $\mathcal {Y}$ with morphisms to both $\mathcal {X},\mathcal {X}'$ such that on $\mathcal {Y}$ , the difference $\mathcal {L}' - \mathcal {L}$ of pullbacks to $\mathcal {Y}$ is effective.

One should think that the theory of non-Archimedean geometry allows a language for discussing sequences of test configurations and, in particular, for discussing compactness properties for sequences of test configurations.

We will use the following extension of the Monge–Ampère energy of a test configuration.

Definition 2.20 [Reference Boucksom and Jonsson11, Sections 3, 7].

We define the Monge–Ampère energy of a Fubini–Study metric $\varphi _{(\mathcal {X},\mathcal {L})}$ to be $E(\varphi _{(\mathcal {X},\mathcal {L})})= E(\mathcal {X},\mathcal {L})$ . We extend this definition to arbitrary psh metrics $\psi $ by setting

$$\begin{align*}E(\psi) = \inf \{E(\varphi_{(\mathcal{X},\mathcal{L})}): \varphi_{(\mathcal{X},\mathcal{L})}\geq \psi\}\end{align*}$$

and define a psh metric to be of finite energy if $E(\psi )>-\infty $ . We denote by $\mathcal {E}^1(L^{\mathrm {an}})$ the space of finite energy psh metrics on $L^{\mathrm {an}}$ , or simply $\mathcal {E}^1$ when the polarisation L is clear from context.

We endow $\mathcal {E}^1(L^{\mathrm {an}})$ with the strong topology: the coarsest refinement of the weak topology (which requires convergence $\varphi _j \to \varphi $ if this holds pointwise) such that the Monge–Ampère energy $E: \mathcal {E}^1(L^{\mathrm {an}})\to \mathbb {R}$ is continuous. With this topology, Fubini–Study metrics are dense in $\mathcal {E}^1(L^{\mathrm {an}})$ .

2.4 Measures on $X^{\mathrm {an}}$

As we have seen, test configurations are analogous to Fubini–Study metrics in non-Archimedean geometry. In Kähler geometry, it is beneficial to consider volume forms and more general measures. The non-Archimedean construction of a measure associated to a metric is the following. Throughout, if $v = c\operatorname {\mathrm {ord}}_F$ is a divisorial valuation, viewed as an element of $X^{\operatorname {\mathrm {div}}} \subset X^{\mathrm {an}}$ , we will denote by $\delta _{c\operatorname {\mathrm {ord}}_F}$ the Dirac mass (or Dirac measure) supported at $ v = c\operatorname {\mathrm {ord}}_F$ .

Definition 2.22 [Reference Boucksom and Jonsson11, Section 3.2].

Denote by $\mathcal {X}_0 = \sum _j b_j E_j$ the central fibre of a test configuration $(\mathcal {X},\mathcal {L})$ as a cycle, so that the $E_j$ are reduced and irreducible. We define the Monge–Ampère measure $\mathrm {MA}(\varphi _{(\mathcal {X},\mathcal {L})})$ of $\varphi _{(\mathcal {X},\mathcal {L})}$ to be

$$\begin{align*}\mathrm{MA}(\varphi_{(\mathcal{X},\mathcal{L})}) = \frac{1}{L^n}\sum_j b_j(\mathcal{L}^n.E_j) \delta_{b_j^{-1}\operatorname{\mathrm{ord}}_{E_j}}.\end{align*}$$

In the following, we denote by $\mathcal {M}$ the set of Radon probability measures on $X^{\mathrm {an}}$ (i.e., the dual space $C^0(X^{\mathrm {an}})^{\vee }$ , which we endow with the weak topology).

The inverse problem – associating a non-Archimedean metric to a measure – is the content of the non-Archimedean Calabi–Yau theorem (originating in [Reference Boucksom, Favre and Jonsson8]). We will require a general version of this, which involves finite norm measures.

Definition 2.24 [Reference Boucksom and Jonsson11, Definition 9.1].

The norm of a measure $\mu \in \mathcal {M}$ is defined as

The space $\mathcal {M}^1 \subset \mathcal {M}$ of measures of finite norm is defined as the set

The following then allows us to pass freely between measures and non-Archimedean metrics.

Theorem 2.25 [Reference Boucksom and Jonsson11, Theorem A].

The Monge–Ampère operator defines a bijection

$$\begin{align*}\mathcal{E}^1(L^{\mathrm{an}})/\mathbb{R} \to \mathcal{M}^1,\end{align*}$$

where $\mathcal {E}^1(L^{\mathrm {an}})/\mathbb {R}$ denotes finite energy metrics modulo the addition of constants. Furthermore, given a measure $\mu $ , if $\mathrm {MA}(\varphi ) = \mu \in \mathcal {M}^1$ and $\int _{X^{\mathrm {an}}}\varphi \operatorname {\mathrm {d\mu }} = 0$ , then

$$\begin{align*}\|\mu\|_L = E(\varphi).\end{align*}$$

The bijection $\mathcal {E}^1(L^{\mathrm {an}})/\mathbb {R} \to \mathcal {M}^1$ – induced by the non-Archimedean Calabi–Yau theorem – can be upgraded to a homeomorphism if $\mathcal {M}^1$ is given the strong topology, though this will not be used in the present work. We will, however, use a further differentiability result in associating numerical invariants to measures:

Theorem 2.27 [Reference Boucksom and Jonsson10, Theorem A].

Fix a measure $\mu \in \mathcal {M}^1$ , and denote $\operatorname {\mathrm {Amp}}_{\mathbb {Q}}(X)$ the space of ample $\mathbb {Q}$ -divisors modulo numerical equivalence. Then the function $\operatorname {\mathrm {Amp}}_{\mathbb {Q}}(X) \to \mathbb {R}$ defined by

$$\begin{align*}L \to \|\mu\|_L\end{align*}$$

extends by continuity to a function on $\operatorname {\mathrm {Amp}}_{\mathbb {R}}(X)$ which is continuously differentiable.

For an $\mathbb {R}$ -divisor H, we denote

the resulting directional derivative.

A more well-behaved subspace of $\mathcal {M}^1$ will be sufficient for our purposes.

Definition 2.28. We define a divisorial measure on $X^{\mathrm {an}}$ to be a probability measure of the form

$$\begin{align*}\mu = \sum_j a_j\delta_{v_{j}}\end{align*}$$

for some finite collection $v_{j}$ of divisorial valuations on X, so in particular, $\sum _{j=0}^m a_j = 1$ . We denote by $\mathcal {M}^{\operatorname {\mathrm {div}}}$ the set of divisorial measures.

2.5 Uniform K-stability on $\mathcal {E}^1$

With the construction of the Monge–Ampère measure of a finite energy metric $\varphi $ in hand, we may extend the uniform K-stability from test configurations to $\mathcal {E}^1$ . First, it is easily checked that the value taken by the Mabuchi functional at a test configuration depends only on the associated Fubini–Study metric (and similarly the minimum norm has the same property); we denote the resulting functional

$$\begin{align*}M: \mathcal{H}^{\operatorname{\mathrm{NA}}} \to \mathbb{R}.\end{align*}$$

We extend the Mabuchi functional in the following manner. For this, we first recall that there is a natural way to extend the log discrepancy function $A_{(X,B)}: X^{\operatorname {\mathrm {div}}} \to \mathbb {R}$ (which is nonnegative by definition if $(X,B)$ has at worst log canonical singularities) to a function

$$\begin{align*}A_{(X,B)}: X^{\operatorname{\mathrm{NA}}} \to \mathbb{R},\end{align*}$$

using semicontinuity of $A_X$ on $X^{\operatorname {\mathrm {div}}}$ [Reference Boucksom and Jonsson10, Definition A.2].

Integration against the Monge–Ampère measure associated to $((\mathcal {X},\mathcal {B});\mathcal {L})$ produces [Reference Boucksom, Hisamoto and Jonsson9, Corollary 7.18]

$$\begin{align*}\int_{X^{\mathrm{an}}} A_{(X,B)} \mathrm{MA}(\varphi_{(\mathcal{X},\mathcal{L})}) =\operatorname{\mathrm{Ent}}(\mathcal{X},\mathcal{L});\end{align*}$$

we denote

$$\begin{align*}\operatorname{\mathrm{Ent}}(\varphi_{(\mathcal{X},\mathcal{L})}) = \operatorname{\mathrm{Ent}}(\mathcal{X},\mathcal{L}).\end{align*}$$

Second, the functional defined on $\mathcal {H}^{\operatorname {\mathrm {NA}}}$ by

$$\begin{align*}\varphi \to \nabla_{K_X+B} E_L(\varphi)\end{align*}$$

extends in a continuous manner to $\mathcal {E}^1$ , essentially because it contains only Monge–Ampère energy (and ‘mixed Monge–Ampère energy’) terms [Reference Boucksom and Jonsson11, Theorem 7.14], producing a natural extension to a functional

$$\begin{align*}M: \mathcal{E}^1 \to \mathbb{R}\end{align*}$$

taking the form

$$\begin{align*}M(\varphi) = \operatorname{\mathrm{Ent}}(\varphi) + \nabla_{K_X+B} E_{L_X}(\varphi).\end{align*}$$

The minimum norm similarly extends by continuity to a functional on $\mathcal {E}^1$ .

Definition 2.30. We say that $((X,B);L)$ is uniformly K-stable on $\mathcal {E}^1$ if there is an $\varepsilon>0$ such that for all $\varphi \in \mathcal {E}^1$ , we have

$$\begin{align*}M(\varphi) \geq \varepsilon \|\varphi\|_{\min}.\end{align*}$$

This condition is equivalent to Boucksom–Jonsson’s notion of uniform K-stability with respect to filtrations and Li’s notion of uniform K-stability with respect to models; see Li [Reference Li26, Section 2.1.3] for a discussion and further details.

2.6 Divisorial stability

We are now in a position to associate numerical invariants to divisorial measures (rather than metrics), and hence to define divisorial stability, following Boucksom–Jonsson [Reference Boucksom and Jonsson10]. We begin with the entropy of $((X,B);L)$ , which extends the log discrepancy of a single divisorial valuation to a general divisorial measure.

Definition 2.31. We define the entropy $\operatorname {Ent}_{(X,B)}: \mathcal {M}^{\operatorname {\mathrm {div}}} \rightarrow \mathbb {R}$ to be

$$\begin{align*}\operatorname{Ent}_{(X,B)}(\mu) = \int_{X^{\mathrm{an}}} A_{(X,B)}\operatorname{\mathrm{d\mu}},\end{align*}$$

where $\mu $ is a divisorial measure.

Writing $\mu = \sum _j a_j\delta _{v_{j}}$ , the entropy is given explicitly as

$$\begin{align*}\operatorname{Ent}(\mu) = \sum_j a_j A_{(X,B)}(v_j).\end{align*}$$

Note that the entropy is independent of the ample line bundle L. This allows us to define the beta invariant of a divisorial measure on $((X,B);L)$ .

Definition 2.32 [Reference Boucksom and Jonsson10, Definition 4.1].

The beta invariant of a divisorial measure $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}}}$ is defined to be

This allows us to define divisorial stability.

We may extend the beta invariant of a divisorial measure to a general finite norm measure in a way analogous to the extension of the Mabuchi functional to $\mathcal {E}^1$ : the entropy is defined in the same way for divisorial and finite norm measures, whereas the norm itself remains differentiable (in the polarisation) for a general finite energy measure [Reference Boucksom and Jonsson10, Theorem 2.15], meaning we can define

$$\begin{align*}\beta(\mu) = \operatorname{Ent}_{(X,B)}(\mu) + \nabla_{K_{X}+B}\lVert\mu \rVert_{L}\end{align*}$$

for $\mu \in \mathcal {E}^1$ . The resulting notion of stability is equivalent to divisorial stability by continuity of the various quantities in the measure.

Example 2.34. If $\varphi \in \mathcal {E}^1$ , then Boucksom–Jonsson prove the key equality [Reference Boucksom and Jonsson10, Equation (4.5)]

$$\begin{align*}M(\varphi) = \beta(\mathrm{MA}(\varphi)),\end{align*}$$

which implies that divisorial stability is equivalent to uniform K-stability on $\mathcal {E}^1$ (using also the non-Archimedean Calabi–Yau theorem); in particular, divisorial stability implies uniform K-stability. If instead $\mu = \delta _{v_F}$ for a divisorial valuation $v_F$ on X associated to a prime divisor F over X, then $\beta (\mu ) = \beta (F)$ with $\beta (F)$ the $\beta $ -invariant of Definition 2.5, and relatedly, $S_L(F) = \|\mu \|_L$ [Reference Boucksom and Jonsson10, Theorem 2.18]. Thus, the $\beta $ -invariant of finite norm measures (in particular, divisorial measures) can be seen as a simultaneous generalisation of the $\beta $ -invariant of divisorial valuations and the Mabuchi functional on the set of Fubini–Study metrics (in particular, test configurations).

2.7 Equivariant divisorial stability

Consider now a finite group G acting on the projective variety X. Since G acts on X, it acts on the function field of X and hence on $X^{\operatorname {\mathrm {div}}}$ by setting $(g(v))(f) = v(g^*f)$ .

Definition 2.35. We say that a divisorial measure $\mu = \sum _{j} a_j\delta _{v_{j}}$ is G-invariant if for all $g \in G$ ,

$$\begin{align*}\mu = \sum_{j} a_j\delta_{g(v_{j})}.\end{align*}$$

We denote the space of G-invariant divisorial measures by $\mathcal {M}^{\operatorname {\mathrm {div}}, G}_Y$ .

We will consider pushforwards of measures in Section 3.3, where we will show that this condition asks $g_*\mu = \mu $ for all $g \in G$ .

We are now in a position to introduce the notion of G-equivariant divisorial stability.

To compare with equivariant notions of uniform K-stability, we first make the following definition.

Remark 2.37. In Section 3.2, we define pullbacks of metrics under morphisms, which for $g: X \to X$ , we denote $g^*\varphi $ . Furthermore, in Corollary 3.12, we show that for a G-invariant divisorial measure $\mu $ , the sup defining the norm of $\mu $ can be taken over G-invariant psh metrics – namely,

$$\begin{align*}\lVert \mu \rVert_{L}= \sup_{\varphi\in \mathcal{E}^{1,G}} \bigg\{ E(\varphi) - \int_{X^{\mathrm{an}}}\varphi\operatorname{\mathrm{d\mu}}\bigg\}\end{align*}$$

with $\mathcal {E}^{1,G}(L_X^{\mathrm {an}})$ the space of G-invariant finite energy metrics, giving some justification for the definition.

3.1 Finite maps between analytifications

The map $\pi : Y \to X$ induces a map $\pi ^{\mathrm {an}}: Y^{\mathrm {an}}\to X^{\mathrm {an}}$ defined by

$$\begin{align*}\pi^{\mathrm{an}}(V, |\cdot|_V) = (\pi(V), |\cdot|_{\pi(V)});\end{align*}$$

we begin by giving a more explicit geometric description of this map on divisorial valuations. For a divisorial valuation, V is simply taken to be Y, so since $\pi $ is surjective, we obtain a valuation on X from one on Y. Recall in general that the image of a valuation v on Y is defined for a rational function $f\in k(X)$ by setting

$$\begin{align*}\pi(v)(f) = v(\pi^*f).\end{align*}$$

Proof. We begin by replacing an arbitrary birational model $Y'$ of Y with a birational model $Y" \to Y$ , which admits a lift of the G-action, in such a way that the morphism $Y" \to Y$ is G-equivariant. It suffices to construct $Y"$ as a blowup of Y along a G-invariant subscheme of Y, since, in this case, the G-action lifts automatically by the universal property of blowups.

Since $Y'$ is birational to Y, we may write $Y' = \operatorname {\mathrm {Bl}}_{\mathcal {I}}Y$ , where $\mathcal {I}$ is an ideal sheaf. We consider the orbit

$$\begin{align*}\mathcal{I}\cdot g^{-1}\mathcal{I} \cdot\ldots \cdot (g^{m-1})^{-1}\cdot \mathcal{I} \subset Y\end{align*}$$

of $\mathcal I$ , which is a G-invariant ideal sheaf (and where $g^{-1}\mathcal I$ denotes the inverse image of $\mathcal I$ under g). Letting , by [Reference Moody33, Corollary 1] (namely, we use that the blowup of a product of ideal sheaves is the successive blowup of one factor and then the total transform of the other factor), we obtain birational morphisms $Y"\rightarrow Y' \rightarrow Y$ , and by construction, $Y"$ admits a G-action. Thus, we replace $F\subset Y'$ with its proper transform on $Y"$ , which does not modify the divisorial valuation itself.

As we now assume $Y'$ admits a G-action making the morphism $Y'\to Y$ a G-equivariant morphism, we may take the quotient $Y'/G$ of $Y'$ by G; we define $X'=Y'/G$ . We then have a commutative diagram

since $Y' \to X$ is G-invariant.

Setting $D = \pi '(F)$ , it follows, for example, from [Reference Silverman38, Exercise 2.2] that

$$\begin{align*}\operatorname{\mathrm{ord}}_F((\pi')^* f) =e_F \operatorname{\mathrm{ord}}_D(f),\end{align*}$$

where $e_F$ is the ramification index of $Y' \to X'$ along F and $f \in k(X)$ . This completes the proof by the definition of the image of a valuation.

3.2 Pullbacks and pushforwards of metrics under finite covers

Our next goal is define pushforwards and pullbacks of metrics in order to relate G-invariant Fubini–Study metrics on Y to Fubini–Study metrics on X.

We recall the definition of the pullback of a psh metric, as defined by Boucksom–Jonsson [Reference Boucksom and Jonsson11, Proposition 3.6]. We begin with a Fubini–Study metric $\varphi $ on $L^{\mathrm {an}}$ induced by a test configuration $(\mathcal {X},\mathcal {L}_{\mathcal {X}})$ for $(X,L_X)$ . The $\mathbb {G}_m$ -equivariant rational map $Y\times \mathbb {A}^1 \dashrightarrow \mathcal {X}$ induced by $\pi $ admits a $\mathbb {G}_m$ -equivariant resolution of indeterminacies, inducing a test configuration $(\mathcal {Y}, \mathcal {L}_{\mathcal {Y}})$ for $(Y,L_Y)$ , where $\mathcal {L}_{\mathcal {Y}}$ is the pullback of $\mathcal {L}_{\mathcal {X}}$ through the morphism $\mathcal {Y} \to \mathcal {X}$ ; this test configuration dominates $Y\times \mathbb A ^1$ by construction.

The pullback extends to arbitrary psh metrics by an approximation argument.

We next define pushforwards of G-invariant Fubini–Study metrics in a similar spirit. Let $\varphi $ be a G-invariant Fubini–Study metric $\varphi $ on $L_Y^{\mathrm {an}}$ , which hence corresponds to a test configuration $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ , which can be taken to dominate the trivial test configuration. We will show in Proposition 3.5 that $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ can be taken to be G-invariant, in the sense that it admits a G-action inducing the fixed action on Y and commuting with the $\mathbb G_m$ -action. Then by [Reference Dervan13, Lemma 3.1], taking the quotient of $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ by the G-action induces a test configuration $(\mathcal {X},\mathcal {L}_{\mathcal {X}})$ for $(X,L_X)$ such that $\pi ^*\mathcal {L}_{\mathcal {X}} = \mathcal {L}_{\mathcal {Y}}$ , where $\pi $ is the quotient map; by construction, $(\mathcal {X},\mathcal {L}_{\mathcal {X}})$ dominates the trivial test configuration provided $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ dominates the trivial test configuration.

We next prove that a G-invariant psh metric $\varphi $ on $L_Y^{\mathrm {an}}$ (which is a decreasing limit of Fubini–Study metrics) is a decreasing limit of G-invariant Fubini–Study metrics $\varphi _k$ and that these G-invariant Fubini–Study metrics can further be taken to be associated to G-invariant test configurations.

Proof. Let $\varphi $ be a G-invariant psh metric on $Y^{\mathrm {an}}$ with finite energy. By Remark 2.19, we may realise $\varphi $ as a decreasing limit of Fubini–Study metrics $\varphi _k$ associated to test configurations $(\mathcal {Y}_k,\mathcal {L}_k)$ for $(Y,L_Y)$ , which we may assume dominate the trivial test configuration. We define

$$\begin{align*}\varphi_k^G = \sum_{g\in G}\frac{1}{|G|}g^*\varphi_k,\end{align*}$$

which is, by definition, a G-invariant function. This is a convex combination of psh metrics, and is hence itself psh [Reference Boucksom and Jonsson11, Theorem 4.7 (ii)]; we will see positivity more explicitly in what follows.

We will first show that $\varphi _k^G\to \varphi $ . Since $\varphi $ is the decreasing limit of the Fubini–Study metrics $\varphi _k$ , for a point $y\in Y^{\mathrm {an}}$ , $\varphi _k(g(y))$ decreases to $\varphi (g(y)) = \varphi (y)$ , where we have used that $\varphi $ is G-invariant. Hence, $\varphi _k(y)^G=\varphi _k^G(g(y))$ decreases to

$$\begin{align*}\sum_{g\in G}\frac{1}{|G|}g^*\varphi = \varphi.\end{align*}$$

We next give an explicit description of $\varphi _k^G$ as a Fubini–Study metric, for which we employ flag ideals. As $\varphi _k$ is Fubini–Study, it corresponds to a nef test configuration $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ , and in addition, $\mathcal {Y} = \operatorname {\mathrm {Bl}}_{\mathfrak {a}}X\times \mathbb {P}^1$ for a flag ideal $\mathfrak {a}$ . Using a similar idea to the proof of Proposition 3.1, we define

Notice that $\mathfrak {a}^G$ is a flag ideal on $Y\times \mathbb {P}^1$ , and set

$$\begin{align*}\mathcal{Y}^G = \operatorname{\mathrm{Bl}}_{\mathfrak{a}^G} Y\times\mathbb{P}^1,\end{align*}$$

which admits a G-action by construction. By [Reference Boucksom and Jonsson11, Theorem 2.7, Proposition 3.6], each $g^*\varphi _k$ is Fubini–Study with associated flag ideal $g^*\varphi _k =\varphi _{g^{-1}\mathfrak {a}} $ , where $g^{-1}\mathfrak {a}$ denotes the inverse image of $\mathfrak {a}$ under g. By Equation (2.2), the product of ideal sheaves defining $\mathfrak {a}^G$ corresponds precisely to the sum

$$\begin{align*}\varphi_{\mathfrak{a}^G} = \sum_{g\in G}g^*\varphi_k.\end{align*}$$

Thus, $\varphi _k^G$ is associated to the flag ideal $\mathfrak {a}^G$ .

To understand the line bundle $\mathcal {L}^G$ on $\mathcal {Y}^G$ associated to the Fubini–Study metric $\varphi _k^G$ , note first that $\mathcal {Y}^G$ admits a morphism to the test configuration associated to $g^{j*}\varphi _k$ for each j in the same way as the proof of Proposition 3.1. The pullback metric $g^*\varphi _{\mathcal {L}_k}$ can then be represented on $\mathcal {Y}^G$ itself through the pullback line bundle $g^*\mathcal {L}$ on $\mathcal {Y}^G$ , since pullback of Fubini–Study metrics is defined through pulling back line bundles. Thus, $\varphi _k^G$ corresponds to the test configuration $(\mathcal {Y}^G, \mathcal {L}^G)$ , where

$$\begin{align*}\mathcal{L}^G = \sum_{g\in G}\frac{1}{|G|}g^*\mathcal{L};\end{align*}$$

note $\mathcal {L}^G$ is relatively nef as $\mathcal {L}$ is so, and similarly, relatively semiample provided $\mathcal {L}$ is so (this can alternatively be obtained from [Reference Boucksom and Jonsson11, Proposition 2.25]). Since $\mathcal {L}$ can be viewed as a G-invariant $\mathbb {Q}$ -Cartier divisor, it admits a lift of the G-action, meaning we have represented $\varphi _k^G$ by a G-invariant test configuration $(\mathcal {Y}^G, \mathcal {L}^G)$ .

Thus, any G-invariant psh metric is a decreasing limit of G-invariant Fubini–Study metrics induced by G-invariant test configurations, as claimed.

We next relate pushforwards and pullbacks.

Proof. Consider by Proposition 3.5 a G-invariant Fubini–Study metric $\varphi $ on $L_Y^{\mathrm {an}}$ which has an associated G-invariant test configuration $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ , so that by definition of the pushforward, $\pi _*\varphi $ is associated to the quotient test configuration $(\mathcal {X},\mathcal {L}_{\mathcal {X}}$ ). To prove that the pushforward and pullback are mutual inverses, it thus suffices to prove that

$$\begin{align*}\pi^*\pi_*\varphi = \varphi.\end{align*}$$

By definition, the pullback $\pi ^*\pi _*\varphi $ is the G-invariant Fubini–Study metric on $Y^{\mathrm {an}}$ corresponding to the G-invariant test configuration $(\mathcal {Y}',\mathcal {L}_{\mathcal {Y}'})$ , where $\mathcal {Y}' = \mathcal {X} \times _{X\times \mathbb {P}^1} Y \times \mathbb {P}^1$ is the fibre product

we set $\mathcal {L}_{\mathcal {Y}'} = p_{\mathcal {X}}^*\mathcal {L}_{\mathcal {X}}$ . By the universal property of the fibre product $\mathcal {Y}'$ , there is an induced morphism $\rho \colon \mathcal {Y}\rightarrow \mathcal {Y}'$ , which then satisfies $\rho ^*\mathcal {L}_{\mathcal {Y}'} = \mathcal {L}_{\mathcal {Y}}$ . It follows that the non-Archimedean metrics associated to $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ and $(\mathcal {Y}',\mathcal {L}_{\mathcal {Y}'})$ are equal, proving that pullback and pushforward induce a bijection between G-invariant Fubini–Study metrics on $L_Y^{\mathrm {an}}$ and Fubini–Study metrics on $L_X^{\mathrm {an}}$ .

We finally prove that this bijection is energy-preserving. Denote by $(\mathcal {X},\mathcal {L}_{\mathcal {X}})$ a test configuration associated to $\varphi _{(\mathcal {X}, \mathcal {L}_{\mathcal {X}})}$ , and denote $(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})$ the test configuration associated to the pullback $\pi ^*\varphi _{(\mathcal {X}, \mathcal {L}_{\mathcal {X}})}$ defined through constructing an equivariant resolution of indeterminacy of $Y\times \mathbb A^1 \dashrightarrow \mathcal {X}$ . Let $p: \mathcal {Y} \to \mathcal {X}$ be the resulting morphism. We calculate

$$ \begin{align*} E(\varphi_{(\mathcal{X}, \mathcal{L}_{\mathcal{X}})}) &= \frac{(\mathcal{L}_{\mathcal{X}})^{n+1}}{(n+1)(L_X)^n}, \\ &= \frac{(p^*\mathcal{L}_{\mathcal{X}})^{n+1}}{(n+1)(\pi^*L_X)^n}, \\ &= \frac{(\mathcal{L}_{\mathcal{Y}})^{n+1}}{(n+1)(L_Y)^n},\\ &= E(\varphi_{(\mathcal{Y}, \mathcal{L}_{\mathcal{Y}})}), \end{align*} $$

which shows that $E(\varphi _{(\mathcal {X}, \mathcal {L}_{\mathcal {X}})}) = E(\varphi _{(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})})$ . A similar calculation shows for a G-invariant Fubini–Study metric $\varphi _{(\mathcal {Y},\mathcal {L}_{\mathcal {Y}})}$ on $L_Y^{\mathrm {an}}$ that

$$\begin{align*}E(\varphi_{(\mathcal{Y},\mathcal{L}_{\mathcal{Y}})}) = E(\pi_*\varphi_{(\mathcal{Y},\mathcal{L}_{\mathcal{Y}})}),\end{align*}$$

proving the result.

3.3 Pullbacks and pushforwards of measures under finite covers

We next relate divisorial measures on Y to those on X. We begin by recalling the explicit construction of divisorial valuations on X from those on Y. Given a prime divisor $F\subset Y' \to Y$ over Y, by Proposition 3.1, we may assume that $Y'$ admits a G-action, meaning we may form the quotient $X' = Y'/G$ . We denote by $\pi (F)$ the prime divisor over X given by the image of F under the morphism $Y' \to X'$ . Proposition 3.1 then shows that the image of the divisorial valuation $c\operatorname {\mathrm {ord}}_F$ under the map $Y^{\operatorname {\mathrm {val}}} \to X^{\operatorname {\mathrm {val}}}$ is the divisorial valuation $e_F c\operatorname {\mathrm {ord}}_{\pi (F)}$ , where $e_F$ is the ramification index of $Y' \to X'$ along F.

Let $D = \pi (F)$ be a prime divisor over Y. Then $\pi ^*D$ , the pullback cycle, takes the form

$$\begin{align*}\pi^*D = \sum_{F_j\in \pi^{-1}(D)}e_{F_j}F_j,\end{align*}$$

with $e_{F_j}$ the ramification index along $F_j$ . The ramification indices are equal for all such $F_j$ , so in this expression, $e_{F_j} = e_{F_l}$ for all $j,l$ . In addition, the divisors $F_j$ and $F_l$ belong to the same G-orbit, in the sense that for all $j,l$ , there exists a $g\in G$ such that $F_j = g(F_l)$ .

Restating Definition 2.35 through the explicit interpretation of the image of a divisorial valuation, a divisorial measure

$$\begin{align*}\mu = \sum_i a_i \delta_{c_i\operatorname{ord}_{F_i}}\end{align*}$$

is G-invariant if $a_i=a_l$ and $c_i=c_l$ for all $i, l$ such that $F_i$ and $F_l$ lie in the same G-orbit, or equivalently, such that $\pi (F_i) = \pi (F_l) = D$ for a prime divisor D over X. We will use the following notation for G-invariant divisorial measures on $Y^{\mathrm {an}}$ :

(3.1) $$ \begin{align}\mathcal{M}^{\operatorname{\mathrm{div}}, G}_Y \ni \mu = \sum_{D/X}a_D \left( \sum_{F_j\in \pi^{-1}(D)}\delta_{c_D\operatorname{ord}_{F_j}}\right).\end{align} $$

Here, the first sum $\sum _{D/X}$ is taken over all prime divisors D over X and is finite since there is a finite number of nonzero $a_D$ , while the second sum is taken over all divisors $F_j$ over Y in the preimage of D. The coefficients $a_D$ are arbitrary nonnegative coefficients such that $\int _{Y^{\mathrm {an}}} \operatorname {\mathrm {d\mu }} = 1$ , so that the measure is a probability measure.

We next consider pushforwards and pullbacks of measures. For a divsiorial measure, the pushforward is given explicitly by the following expression.

Lemma 3.8. If

$$\begin{align*}\mu = \sum_i a_i\delta_{c_i\operatorname{\mathrm{ord}}_{F_i}}\in \mathcal{M}^{\operatorname{\mathrm{div}}}_Y\end{align*}$$

is a divisorial measure on $Y^{\mathrm {an}}$ , then

$$\begin{align*}\pi_*\mu = \sum_i a_i\delta_{e_{F_i}c_i\operatorname{\mathrm{ord}}_{\pi(F_i)}}\in \mathcal{M}^{\operatorname{\mathrm{div}}}_X\end{align*}$$

is a divisorial measure on $X^{\mathrm {an}}$ , where $e_{F_i}$ is the ramification index along $F_i$ .

Proof. For a single Dirac mass $ \delta _u$ supported at a point $u\in Y^{\operatorname {\mathrm {div}}}$ , for $U\subset X^{\mathrm {an}}$ , we have

$$\begin{align*}(\pi_*\delta_u)(U) = \delta_{\pi(u)}\end{align*}$$

by definition of the pushfoward. Thus, $\pi _* \delta _u \in \mathcal {M}^{\operatorname {\mathrm {div}}}_X$ is a divisorial measure since Proposition 3.1 implies that $\pi (u)$ is itself a divisorial valuation. Writing $u = c\operatorname {\mathrm {ord}}_F$ , by Proposition 3.1, its image is given explicitly by

$$\begin{align*}\pi(c\operatorname{\mathrm{ord}}_F) = e_{F}c\operatorname{\mathrm{ord}}_{\pi(F)}.\end{align*}$$

The general case is identical.

Although there is a canonical pushforward, we must define pullbacks explicitly. For a single divisorial valuation $a\operatorname {\mathrm {ord}}_D$ , a divisorial valuation has image $a\operatorname {\mathrm {ord}}_D$ in $X^{\mathrm {an}}$ if and only if it takes the form $e_F^{-1}a\operatorname {\mathrm {ord}}(F),$ where $\pi (F) = D$ and $e_F$ is the ramification index along F, since

$$\begin{align*}\pi(e_F^{-1}c\operatorname{\mathrm{ord}}(F)) = c\operatorname{\mathrm{ord}}_D\end{align*}$$

by Proposition 3.1. Writing

$$\begin{align*}\pi^*D = \sum_{F_j\in \pi^{-1}(D)}e_{F} F_j,\end{align*}$$

where, as before, the ramification indices are equal for each j, we define

$$\begin{align*}\pi^*\delta_{a\operatorname{\mathrm{ord}}_D} = \sum_{F_j\in \pi^{-1}(D)}\frac{e_F}{|G|}\delta_{e_F^{-1}c\operatorname{\mathrm{ord}}(F_j)}.\end{align*}$$

Note that $\pi ^*\delta _{a\operatorname {\mathrm {ord}}_D}$ is still a probability measure since by the orbit-stabiliser theorem,

$$\begin{align*}\sum_{F_j\in \pi^{-1}(D)} \frac{e_F}{|G|} = 1.\end{align*}$$

We now define pullbacks of general divisorial measures in a similar way, essentially extending linearly.

Definition 3.9. We define the pullback of a divisorial measure

$$\begin{align*}\nu =\sum_{D/X} a_D\delta_{c_D\operatorname{\mathrm{ord}}_{D}} \in \mathcal{M}^{\operatorname{\mathrm{div}}}_X\end{align*}$$

by

$$\begin{align*}\pi^*\nu = \sum_{D/X}a_D \left(\sum_{F_j \in \pi^{-1}(D)}\frac{e_{F_j}}{|G|} \delta_{e_{F_j}^{-1}c_D\operatorname{\mathrm{ord}}(F_j)}\right).\end{align*}$$

As before, the sum $D/X$ denotes a finite sum of prime divisors over X. As in the case when $\mu $ is a Dirac mass at a single divisorial valuation, it follows from the orbit-stabiliser theorem that $\pi _*\mu $ is a probability measure.

In particular, pushforward and pullback induce an isomorphism

$$\begin{align*}\mathcal{M}^{\operatorname{\mathrm{div}}, G}_Y \cong \mathcal{M}^{\operatorname{\mathrm{div}}}_X.\end{align*}$$

Proof. We consider a G-invariant divisorial measure $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}}, G}_Y$ and begin by showing that

$$\begin{align*}\pi^*\pi_*\mu = \mu.\end{align*}$$

Continuing the notation used in Equation (3.1), we denote

$$\begin{align*}\mu = \sum_{D/X}a_D\left( \sum_{F_j\in \pi^{-1}(D)}\delta_{c_D\operatorname{ord}_{F_j}}\right),\end{align*}$$

where the first sum is taken over a finite sum of prime divisors D over X and the $a_D$ are coefficients such that $\sum _{D/X} a_D = 1$ , and the second sum is taken over all divisors $F_j$ in the preimage of D.

We calculate

$$ \begin{align*} \pi_*\mu &= \sum_{D/X} a_D \left( \sum_{F_j\in \pi^{-1}(D)} \delta_{e_{F_j}c_D\operatorname{ord}_D}\right) \\ &= \sum_{D/X} a_D \delta_{e_{F_j}c_D\operatorname{ord}_D}\left(\sum_{F_j\in \pi^{-1}(D)} 1\right), \end{align*} $$

where $e_{F_j}$ is the common ramification index of the $F_j$ , so $\pi ^*D = e_{F_j} \sum _j F_j$ . Then,

$$ \begin{align*} \pi^*\pi_*\mu &= \sum_{D/X} a_D \left( \sum_{F_l\in \pi^{-1}(D)}\delta_{c_D\operatorname{\mathrm{ord}}_{F_l}}\frac{e_{F_l}}{|G|}\left(\sum_{F_j\in \pi^{-1} (D)}1\right)\right), \\ &= \sum_{D/X} a_D \sum_{F_l\in \pi^{-1}(D)}\delta_{c_D\operatorname{\mathrm{ord}}_{F_l}}, \end{align*} $$

where we use that

$$\begin{align*}\frac{e_{F_l}}{|G|}\sum_{F_j\in \pi^{-1}(D)} 1 =1\end{align*}$$

by the orbit-stabiliser theorem and the fact that $e_{F_l}=e_{F_j}$ for all $l,j$ . Thus,

$$\begin{align*}\pi^*\pi_*\mu = \mu,\end{align*}$$

as claimed.

In the reverse direction, let $\nu \in \mathcal {M}^{\operatorname {\mathrm {div}}}_X$ take the form

$$\begin{align*}\nu =\sum_{D/X} a_D\delta_{c_D\operatorname{\mathrm{ord}}_{D}},\end{align*}$$

so that by definition,

$$\begin{align*}\pi^*\nu = \sum_{D/X}a_D \left(\sum_{F_j \in \pi^{-1}(D)}\frac{e_{F_j}}{|G|} \delta_{e_{F_j}^{-1}c_D\operatorname{\mathrm{ord}}_{F_j}}\right).\end{align*}$$

The pushforward of this measure is then given by

$$ \begin{align*} \pi_*\pi^*\nu &= \sum_{D/X}a_D \left(\sum_{F_j \in \pi^{-1}(D)} \frac{e_{F_j}}{|G|}\delta_{c_D\operatorname{\mathrm{ord}}_D}\right),\\ & = \sum_{D/X}a_D \delta_{c_D\operatorname{\mathrm{ord}}_D}\sum_{F_j \in \pi^{-1}(D)} \frac{e_{F_j}}{|G|},\\ & = \sum_{D/X}a_D \delta_{c_D\operatorname{\mathrm{ord}}_D}, \\ &=\nu, \end{align*} $$

where we again use the orbit-stabiliser theorem. This completes the proof.

We end this section by showing that G-invariant measures of finite norm are given as Monge–Ampère measures of G-invariant psh metrics when G is a finite group.

Proposition 3.11. Let $G \subset \operatorname {\mathrm {Aut}}(Y,L_Y)$ be a finite group, and suppose $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}},G}_Y$ is a G-invariant divisorial measure. Then the solution $\varphi \in \mathcal {E}^1(L_Y^{\mathrm {an}})$ of the Monge–Ampère equation

$$\begin{align*}\mathrm{MA}(\varphi) = \mu\end{align*}$$

is a G-invariant metric.

Proof. Letting $g \in G \subset \operatorname {\mathrm {Aut}}(Y,L_Y)$ be such that $g_*\mu =\mu $ , we must show that $g^*\varphi = \varphi $ . We first claim that it is enough to show that

(3.2) $$ \begin{align}\mathrm{MA}(\varphi) = g_*\mathrm{MA}(g^*\varphi).\end{align} $$

Indeed, by Proposition 3.10,

$$\begin{align*}g^*g_*\mathrm{MA}(g^*\varphi) = \mathrm{MA}(g^*\varphi),\end{align*}$$

so by Equation (3.2),

$$ \begin{align*} \mathrm{MA}(g^*\varphi) &= g^* \mathrm{MA}(\varphi), \\ &= g^*\mu, \\ &=\mu,\end{align*} $$

where we use that $\mu $ is G-invariant. Thus, $\varphi $ and $g^*\varphi $ both solve the Monge–Ampère equation for the measure $\mu $ , meaning they must be equal up to the addition of a constant, by uniqueness of solutions of the Monge–Ampère equation – namely, Theorem 2.25. Since, for example, $\varphi $ and $g^*\varphi $ have the same supremum, they must genuinely be equal.

It therefore suffices to prove that for $g \in G \subset \operatorname {\mathrm {Aut}}(Y,L_Y)$ ,

$$\begin{align*}\mathrm{MA}(\varphi) = g_*\mathrm{MA}(g^*\varphi).\end{align*}$$

By Remark 2.19, we may realise $\varphi \in \mathcal {E}^1(L_Y^{\mathrm {an}})$ as a decreasing limit of Fubini–Study metrics $\varphi _k$ , associated to test configurations $(\mathcal {Y}_k, \mathcal {L}_{\mathcal {Y}_k})$ , which we may assume dominate the trivial test configuration. As such, we will prove the equality of Equation (3.2) for Fubini–Study metrics, and the result will follow in general by continuity of the Monge–Ampère operator.

Thus, let $(\mathcal {Y}, \mathcal {L}_{\mathcal {Y}})$ be a test configuration on $(Y, L_Y)$ dominating the trivial test configuration, associated to a Fubini–Study metric $ \psi _{(\mathcal {Y}, \mathcal {L}_{\mathcal {Y}})}$ , and denote the central fibre of $\mathcal {Y}$ by

$$\begin{align*}\mathcal{Y}_0 = \sum_jb_jE_j,\end{align*}$$

where the $E_j$ are reduced and irreducible. By definition of the Monge–Ampère operator,

$$\begin{align*}\mathrm{MA}(\psi_{(\mathcal{Y}, \mathcal{L}_{\mathcal{Y}})}) = \frac{1}{L_Y^n}\sum_j b_j(\mathcal{L}^n_{\mathcal{Y}}.E_j) \delta_{b_j^{-1}\operatorname{\mathrm{ord}}_{E_j}},\end{align*}$$

where $\delta _{b_j^{-1}\operatorname {\mathrm {ord}}_{E_j}}$ is the Dirac mass at the divisorial valuation $b_j^{-1}\operatorname {\mathrm {ord}}_{E_j} \in Y^{\operatorname {\mathrm {div}}}\subset Y^{\mathrm {an}}$ .

The morphism g induces by pullback a test configuration $(\mathcal {Y}', \mathcal {L}_{\mathcal {Y}'})$ for $(Y,L_Y)$ , where by Proposition 3.5, we may (and do) assume that $\mathcal {Y}'\cong \mathcal {Y}$ , but where the line bundle $\mathcal {L}_{\mathcal {Y}'} = g^* \mathcal {L}_{\mathcal {Y}}$ may not agree with $\mathcal {L}_{\mathcal {Y}}$ (indeed, we wish to show $g^* \psi _{(\mathcal {Y}, \mathcal {L}_{\mathcal {Y}})} = \psi _{(\mathcal {Y}, \mathcal {L}_{\mathcal {Y}})}$ , which precisely asks that $g^* \mathcal {L}_{\mathcal {Y}} = \mathcal {L}_{\mathcal {Y}})$ . Since $g: \mathcal {Y} \to \mathcal {Y}$ is an isomorphism, for each j, the pullback $g^{*}E_j$ (which is simply the preimage $g^{-1}(E_j)$ ) is a single irreducible component of $\mathcal {Y}_0$ , and further, if $g^{*}E_j = E_l$ , then $b_j=b_l$ ; we use these properties frequently in what follows.

The push-pull formula gives

$$\begin{align*}\mathcal{L}_{\mathcal{Y}'}^n.g^{*}E_j = \mathcal{L}_{\mathcal{Y}}^n.E_j,\end{align*}$$

which in turn produces

$$ \begin{align*} \mathrm{MA}(g^*\psi_{(\mathcal{Y}, \mathcal{L}_{\mathcal{Y}})}) &= \frac{1}{L_Y^n}\sum_j b_j(\mathcal{L}^n_{\mathcal{Y}'}.g^{*}E_j) \delta_{b_j^{-1} \operatorname{\mathrm{ord}}_{g^*E_j}}, \\ &= \frac{1}{L_Y^n}\sum_j b_j(\mathcal{L}^n_{\mathcal{Y}}.E_j) \delta_{b_j^{-1} \operatorname{\mathrm{ord}}_{g^{*}E_j}}. \end{align*} $$

By Lemma 3.8, using that g is an $g: \mathcal {Y} \to \mathcal {Y}$ is an isomorphism and hence is unramified,

$$ \begin{align*} g_*\mathrm{MA}(g^*\psi_{(\mathcal{Y}, \mathcal{L}_{\mathcal{Y}})})&=\frac{1}{L_Y^n}\sum_j b_j(\mathcal{L}^n_{\mathcal{Y}}.E_j) \delta_{b_j^{-1}\operatorname{\mathrm{ord}}_{g(g^{*}(E_j))}},\\ &= \frac{1}{L_Y^n}\sum_j b_j(\mathcal{L}^n_{\mathcal{Y}}.E_j) \delta_{b_j^{-1}\operatorname{\mathrm{ord}}_{E_j}},\\ &=\mathrm{MA}(\psi_{(\mathcal{Y}, \mathcal{L}_{\mathcal{Y}})}), \end{align*} $$

as required.

A consequence, using the fact that the norm of a measure is computed as the energy of its Monge–Ampère-inverse by Theorem 2.25, is the following:

Corollary 3.12. Let $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}},G}_Y$ be a G-invariant divisorial measure on $Y^{\mathrm {an}}$ . Then

$$\begin{align*}\lVert \mu \rVert_{L}= \sup_{\varphi\in \mathcal{E}^{1,G}} \bigg\{ E(\varphi) - \int_{Y^{\mathrm{an}}}\varphi\operatorname{\mathrm{d\mu}}\bigg\}.\end{align*}$$

3.4 Proof of the main theorem

We next turn to comparing numerical invariants of divisorial measures under finite covers and, in particular, to the proof of Theorem 1.1. We begin with the energies of the measures, and throughout this section, we include subscripts in various numerical invariants to emphasise whether we are calculating quantities on X or on Y.

Proposition 3.13. Let $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}}, G}_Y$ be a divisorial measure on X. Then

$$ \begin{align*} \lVert \mu \rVert_{L_Y} = \lVert \pi_*\mu \rVert_{L_X}. \end{align*} $$

Proof. By Corollary 3.12, we must show that

$$\begin{align*}\sup_{\varphi\in \mathcal{E}_Y^{1,G}} \bigg\{ E(\varphi) - \int_{Y^{\mathrm{an}}}\varphi\operatorname{\mathrm{d\mu}}\bigg\} = \sup_{\psi\in \mathcal{E}_X^{1}} \bigg\{ E(\psi) - \int_{X^{\mathrm{an}}}\psi\pi_*(\operatorname{\mathrm{d\mu}})\bigg\}.\end{align*}$$

We may take the supremum defining $\lVert \mu \rVert _{L_Y} $ with respect to G-invariant Fubini–Study metrics on $L_Y^{\mathrm {an}}$ by Proposition 3.5 and the continuity property of the Monge–Ampère energy stated in Proposition 2.21. Similarly, the norm $\lVert \pi _*\mu \rVert _{L_X}$ can be computed as a supremum over Fubini–Study metrics on $L_X^{\mathrm {an}}$ .

Taking an arbitrary $\psi \in \mathcal H^{\operatorname {\mathrm {NA}}}(L_X^{\mathrm {an}})$ , by Proposition 3.6, its pullback $\pi ^*\psi $ is a G-invariant Fubini–Study metric satisfying $E(\psi ) = E(\pi ^*\psi )$ ; Proposition 3.6 also implies any G-invariant Fubini–Study metric $\varphi \in \mathcal H^{\operatorname {\mathrm {NA}}}(L_Y^{\mathrm {an}})$ can be realised in this way. Then

$$\begin{align*}\int_{Y^{\mathrm{an}}}\pi^*{\psi}\operatorname{\mathrm{d\mu}} = \int_{X^{\mathrm{an}}} \psi \pi_*(\operatorname{\mathrm{d\mu}})\end{align*}$$

by definition of the pushforward measure, completing the proof.

Corollary 3.14. Let $\nu \in \mathcal {M}^{\operatorname {\mathrm {div}}}_X$ be a divisorial measure on Y. Then

$$ \begin{align*} \lVert \nu \rVert_{L_X} = \lVert \pi^*\nu \rVert_{L_Y}. \end{align*} $$

Proof. From Proposition 3.10, the pullback $ \pi ^*\nu \in \mathcal {M}^{\operatorname {\mathrm {div}},G}_Y$ satisfies $\pi _*\pi ^*\nu = \nu $ , so by Proposition 3.13,

$$ \begin{align*} \lVert \pi^*\nu \rVert_{L_Y} &= \lVert \pi_*\pi^*\nu \rVert_{L_X}, \\ &= \lVert\nu \rVert_{L_X}, \end{align*} $$

as required.

The following is a trivial consequence.

Corollary 3.15. Let H be an $\mathbb {R}$ -divisor on X. Then for $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}}}_X$ ,

$$\begin{align*}\nabla_{H}\lVert\mu \rVert_{L_X} = \nabla_{\pi^*H}\lVert\pi^*\mu \rVert_{\pi^*L_X} .\end{align*}$$

Similarly, for $\nu \in \mathcal {M}^{\operatorname {\mathrm {div}},G}_Y$ ,

$$\begin{align*}\nabla_{H}\lVert \pi_*\nu \rVert_{L_X} = \nabla_{\pi^*H}\lVert\nu \rVert_{\pi^*L_X} .\end{align*}$$

We are now in a position to prove our main result. We recall the setup, which is a cyclic Galois cover $\pi \colon (Y,L_Y) \rightarrow (X, L_X)$ of normal projective varieties of degree m with branch divisor $B \subset X$ . We assume $\pi ^*L_X = L_Y$ and that $K_X+(1-1/m)B$ is $\mathbb {Q}$ -Cartier, so that by Riemann–Hurwitz,

$$\begin{align*}K_Y = \pi^*(K_X+(1-1/m)B).\end{align*}$$

Proof. We prove the result for semistability; the proof for stability is identical. First, supposing that $((X,B);L_X)$ is divisorially semistable, we aim to show that $\beta (\mu )\geq 0$ for any $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}}, G}_Y$ . As in the proof of Proposition 3.1, we may assume that each of the divisors comprising $\mu $ lives on a model $Y'$ of Y, which admits a G-action, making $Y' \to Y$ a G-equivariant morphism, so that taking quotients produces a commutative diagram

Denote

$$\begin{align*}\mu = \sum_i a_i \delta_{c_i\operatorname{ord}_{F_i}}\in \mathcal{M}^{\operatorname{\mathrm{div}}, G}_Y,\end{align*}$$

where the $F_i$ are each prime divisors on $Y'$ . The prime divisors $F_i$ then have image which we denote $\pi '(F_i) = D_i$ in $X'$ ; we let $e_{F_i}$ denote the ramification index along $F_i$ .

We next calculate the pushforward measure, which by Proposition 3.1 and Lemma 3.8 is given by

$$ \begin{align*}\pi_* \mu &= \sum_i a_i \delta_{\pi(c_i\operatorname{ord}_{F_i})}, \\ &= \sum_i a_i \delta_{e_{F_i}c_i\operatorname{ord}_{D_i}}.\end{align*} $$

We can now use [Reference Kollár and Mori22, Proof of Proposition 5.20] to conclude that the discrepancies satisfy

$$ \begin{align*} A_Y(F_i) = e_{F_i}A_{(X,B)}(D_i), \end{align*} $$

implying that the entropies satisfy

$$ \begin{align*} \operatorname{Ent}_{(X,B)}(\pi_*\mu)&= \sum_i a_i e_{F_i}c_iA_{(X,B)}(D_i),\\ &= \sum_i a_i c_iA_Y(F_i), \\ &= \operatorname{Ent}_{Y}(\mu), \end{align*} $$

where we use the definition of the entropy of a general divisorial valuation given in Remark 2.8.

Corollary 3.15 proves a similar inequality for the derivatives of the energies of measures involved in the definitions of $\beta (\mu )$ and $\beta (\pi _*\mu )$ , where we use the Riemann–Hurwitz formula $K_Y = \pi ^*\left (K_X +\left (1-1/m\right )B \right )$ , proving equality of the two beta invariants.

Since any divisorial measure on $X^{\mathrm {an}}$ is of the form $\pi _*\mu $ for $\mu \in \mathcal {M}^{\operatorname {\mathrm {div}}, G}_Y$ by Proposition 3.10, the other direction also follows.

4.1 Divisorial stability in the asymptotic regime

We consider a normal projective variety X endowed with an ample $\mathbb {Q}$ -line bundle $L_X$ . Recall that given an effective $\mathbb {Q}$ -divisor B on X, we say that $(X,B)$ is log canonical if $K_X+B$ is $\mathbb {Q}$ -Cartier and $A_{(X,B)} \geq 0$ on $X^{\operatorname {\mathrm {div}}}$ .

Proof. Since $(X,B)$ is log canonical, for any divisorial measure $\mu $ on $X^{\mathrm {an}}$ , the entropy satisfies

$$\begin{align*}\operatorname{Ent}_{(X,B)}(\mu) \geq 0.\end{align*}$$

Thus, to prove the result, it suffices to prove that there is a $k>0$ and an $\varepsilon>0$ such that for all divisorial measures $\mu $ , we have

$$\begin{align*}\nabla_{K_{X}+B}\lVert\mu \rVert_{L_X} \geq \varepsilon \|\mu\|_{L_X}.\end{align*}$$

We next reduce to an analogous claim regarding metrics instead of measures. Recall from Example 2.34 that if $\varphi \in \mathcal {E}^1$ , then

$$\begin{align*}M(\varphi) = \beta(\mathrm{MA}(\varphi)).\end{align*}$$

Boucksom–Jonsson’s proof of this statement, in fact, proves that

$$\begin{align*}\nabla_{K_X+B} E_{L_X}(\varphi) = \nabla_{K_X+B} \|\mathrm{MA}(\varphi)\|_{L_X}.\end{align*}$$

Thus, by Theorem 2.25 – namely, the non-Archimedean Calabi–Yau theorem – if we can prove that there is a $k>0$ and an $\varepsilon>0$ such that for all $\varphi \in \mathcal {E}^1$

$$\begin{align*}\nabla_{K_X+B} E_{L_X}(\varphi) \geq \varepsilon \|\varphi\|_{\min},\end{align*}$$

the proof is concluded. But this inequality on $\mathcal {H}^{\operatorname {\mathrm {NA}}}$ is (modulo our non-Archimedean language) proven as part of [Reference Dervan and Ross17, Proof of Theorem 3.7], where we use that $B \in |kL_X|$ . The corresponding inequality on $\mathcal {E}^1$ follows by continuity of the extension of mixed Monge–Ampère energies from $\mathcal {H}^{\operatorname {\mathrm {NA}}}$ to $\mathcal {E}^1$ .

Remark 4.2. As explained in the introduction, this result is the (divisorial) ‘log’ version of its (metric) ‘twisted’ counterpart [Reference Dervan and Ross17, Theorem 3.7], which in turn is the algebro-geometric counterpart of analytic results of Hashimoto [Reference Hashimoto21] and Zeng [Reference Zeng45] regarding twisted constant scalar curvature Kähler metrics and Aoi–Hashimoto–Zheng regarding conical constant scalar curvature Kähler metrics [Reference Aoi, Hashimoto and Zheng4, Theorem 1.10] when working over $\mathbb {C}$ . Our proof is completely algebro-geometric, and the k needed is explicit: for a $\mathbb {Q}$ -line bundle H, setting

$$\begin{align*}\mu(X,L_X)= \frac{-K_X.L_X^{n-1}}{L_X^n},\end{align*}$$

a sufficient condition is that

$$\begin{align*}\frac{n}{n+1} \mu(X,L_X)L_X - K_X + \frac{k}{2n(n+1)}L_X\end{align*}$$

be nef (which is certainly true for $k \gg 0$ ). This requirement can be sharpened using analytic techniques when X is smooth and X and Y are defined over $\mathbb {C}$ ; see Remark 4.9.

We next note the following interpolation result for divisorial stability, analogous to (for example) [Reference Dervan13, Lemma 2.6], which concerns log K-stability. In what follows, we assume that both $K_X$ and B are $\mathbb {Q}$ -Cartier, so that $K_X+cB$ is $\mathbb {Q}$ -Cartier for all $c \in \mathbb {Q}$ .

Proof. We denote the beta invariant of a divisorial measure $\mu $ defined with respect to $((X,cB);L_X)$ and $(X,L_X)$ by $\beta _{((X,cB);L_X)}$ and $\beta _{(X,L_X)}$ , respectively, for clarity. For a divisorial measure $\mu $ , we then wish to compare $\beta _{((X,cB);L_X)}(\mu )$ and $\beta _{(X,L_X)}(\mu )$ , and we begin with their respective entropy terms $\operatorname {\mathrm {Ent}}_{(X,cB)}(\mu )$ and $\operatorname {\mathrm {Ent}}_{X}(\mu )$ .

Write $\mu = \sum _j a_j \delta _{b_j\operatorname {\mathrm {ord}} F_j}$ for a finite collection of divisorial valuations $b_j\operatorname {\mathrm {ord}} F_j$ on X. For a single divisorial valuation $b_j\operatorname {\mathrm {ord}} F_j$ with $F_j \subset Y_j$ and $\pi _j: Y_j\to X$ the associated birational morphism, we note

$$ \begin{align*}A_{(X,cB)}(b_j\operatorname{\mathrm{ord}} F_j) &= b_jA_{(X,cB)}(F_j), \\ &= b_j(\operatorname{\mathrm{ord}}_{F_j}(K_{Y_j} - \pi^*(K_X + cB)+1), \\ &= b_j(\operatorname{\mathrm{ord}}_{F_j}(K_{Y_j} - \pi^*K_X+1)) - b_jc\operatorname{\mathrm{ord}}_{F_j}(\pi^*B),\\ &= A_X(b_j\operatorname{\mathrm{ord}}_{F_j})- c\left(b_j\operatorname{\mathrm{ord}}_{F_j}(\pi^*B)\right).\end{align*} $$

By linearity, we thus obtain

$$\begin{align*}\operatorname{\mathrm{Ent}}_{(X,cB)}(\mu) - \operatorname{\mathrm{Ent}}_{X}(\mu) = -c\left(\sum_j b_j\operatorname{\mathrm{ord}}_{F_j}(\pi^*B)\right).\end{align*}$$

We next consider the dependence on c of the remaining term $ \nabla _{K_{X}+cB}\lVert \mu \rVert _{L_X}$ comprising $\beta _{((X,cB);L_X)}(\mu )$ , which by linearity satisfies (using that B is $\mathbb {Q}$ -Cartier)

$$\begin{align*}\nabla_{K_{X}+cB}\lVert\mu \rVert_{L_X} = \nabla_{K_{X}}\lVert\mu \rVert_{L_X} +c \nabla_{B}\lVert\mu \rVert_{L_X}.\end{align*}$$

Thus, if we define, for a divisorial measure $\mu = \sum _j a_j \delta _{b_j\operatorname {\mathrm {ord}} F_j}$ , a functional

$$\begin{align*}\tilde \beta(\mu): \mathcal{M}^{\operatorname{\mathrm{div}}} \to \mathbb{R}\end{align*}$$

by

$$\begin{align*}\tilde \beta(\mu) = \nabla_{B}\lVert\mu \rVert_{L_X} - \sum_jb_j\operatorname{\mathrm{ord}}_{F_j}(\pi^*B),\end{align*}$$

then

$$\begin{align*}\beta_{((X,cB);L_X)}(\mu) = \beta_{(X,L_X)}(\mu) + c\tilde \beta(\mu).\end{align*}$$

By hypothesis, there is an $\varepsilon>0$ such that for all divisorial measures $\mu $ ,

$$\begin{align*}\beta_{(X,L_X)}(\mu) \geq 0 \textrm{ and } \beta_{((X,B);L_X)} \geq \varepsilon \|\mu\|_{L_X}.\end{align*}$$

Writing

$$\begin{align*}\beta_{((X,cB);L_X)}(\mu) = c\left( \beta_{(X,L_X)}(\mu) + \tilde \beta(\mu)\right) + (1-c) \beta_{(X,L_X)}(\mu),\end{align*}$$

we next recall that by hypothesis,

$$\begin{align*}\beta_{(X,L_X)}(\mu) + \tilde \beta(\mu) \geq \varepsilon \lVert\mu \rVert_{L_X}.\end{align*}$$

Thus, since $0 < c \leq 1$ , we obtain

$$\begin{align*}\beta_{((X,cB);L_X)}(\mu) \geq c\varepsilon \lVert\mu \rVert_{L_X},\end{align*}$$

proving log divisorial stability.

The following is then an automatic consequence of these two results.

In particular, Theorem 1.1 produces the following, which proves Theorem 1.3. We continue the notation of Corollary 4.4 and let G be the cyclic group of order m.

We refer to Kollár for the construction of m-fold branched coverings over Cartier divisors [Reference Kollár23, Section 2.11].

4.2 G-equivariant divisorial and uniform K-stability

We end by using Corollary 4.5 to produce constant scalar curvature Kähler metrics, for which we need Y to be smooth. We need the following equivariant analogue of Boucksom–Jonsson’s work relating divisorial and uniform K-stability, proved using their results.

Proof. Let $\mu \in \mathcal {M}^{1,G}$ , and let $\varphi \in \mathcal {E}^{1,G}$ be such that

$$\begin{align*}\mathrm{MA}(\varphi) = \mu,\end{align*}$$

where G-invariance of such a $\varphi $ follows from Proposition 3.11. By Proposition 3.5, there is a sequence $\varphi _k \in \mathcal {H}^{\operatorname {\mathrm {NA}},G}$ of G-invariant Fubini–Study metrics decreasing to $\varphi $ , and in particular, $\mathrm {MA}(\varphi _k)$ converges to $\mathrm {MA}(\varphi ) = \mu $ by continuity of the Monge–Ampère operator. Thus, we have produced a sequence of G-invariant divisorial measures converging to $\mu $ . It follows that G-equivariant divisorial stability is equivalent to the existence of an $\varepsilon>0$ such that for all $\mu \in \mathcal {M}^{1,G}$ ,

$$\begin{align*}\beta(\mu) \geq \varepsilon \|\mu\|_{L_Y},\end{align*}$$

by continuity of the quantities involved.

We next show that the latter condition is equivalent to uniform K-stability on $\mathcal {E}^{1,G}$ . As the content of their proof that divisorial stability is equivalent to uniform K-stability on $\mathcal {E}^1$ (along with the non-Archimedean Calabi–Yau theorem), Boucksom–Jonsson prove that for any $\varphi \in \mathcal {E}^1$ , we have equality

(4.1) $$ \begin{align} M(\varphi) = \beta(\mathrm{MA}(\varphi));\end{align} $$

see Example 2.34. But Theorem 3.11 implies that the Monge–Ampère operator induces a bijection between finite norm G-invariant measures and finite energy G-invariant metrics (modulo addition of constants), meaning that Equation (4.1) proves the equivalence of the conditions

$$\begin{align*}\beta(\mu) \geq \varepsilon \|\mu\|_{L_Y} \textrm{ for all } \mu \in \mathcal{M}^{1,G}\end{align*}$$

and

$$\begin{align*}M(\varphi) \geq \varepsilon \|\varphi\|_{\min} \textrm { for all } \varphi \in \mathcal{E}^{1,G}\end{align*}$$

and hence proves the result.

We now take the field k over which the varieties X and Y are defined to be $\mathbb {C}$ . We use the following important result of Li [Reference Li26, Theorem 1.10].

From this, we obtain the following consequence of Corollary 4.5, which proves Corollary 1.4.