Definition 2.1 (Log pair).

Let X and S be quasi projective normal varieties such that there exists a projective surjective morphism $f\colon X\to S$ . Let $\Delta $ be a $\mathbb {Q}$ -Weil divisor on X such that $ K_X+\Delta $ is $\mathbb {Q}$ -Cartier. Then we call $(X,\Delta )$ a log subpair over S. If $\Delta $ is further effective, then we call $(X,\Delta )$ a log pair. For any subpair $(X,\Delta )$ , we take a log resolution $\sigma \colon Y\to X$ such that $\sigma $ is a projective birational morphism from a smooth variety Y such that $\mathrm {Ex}(\sigma )+\sigma ^{-1}_{*}\Delta $ is snc, where $\mathrm {Ex}(\sigma )$ is the exceptional locus. If an algebraic group G acts on X, then there exists a log resolution $\sigma $ such that $\sigma $ is G-equivariant (cf. [Reference Kollár53, 3.9.1]) and we call such $\sigma $ a G-equivariant log resolution. We call F a prime divisor over X if there exists a log resolution $\sigma :Y\to X$ of $(X,\Delta )$ such that F is a prime divisor on Y. A valuation v is called divisorial if there exist a prime divisor F over X and $c\in \mathbb {Q}_{>0}$ such that $v=c\,\mathrm {ord}_F$ . Then we set the log discrepancy of $(X,\Delta )$ with respect to F as

$$\begin{align*}A_{(X,\Delta)}(v)=v(K_Y+F-\sigma^{*}(K_X+\Delta)). \end{align*}$$

In particular, if $v=\mathrm {ord}_F$ , then we denote $A_{(X,\Delta )}(F)$ . A log subpair $(X,\Delta )$ is called subklt (resp. sublc; $\delta $ -sublc, where $\delta>0$ ) if $A_{(X,\Delta )}(v)>0$ (resp. $\ge 0$ ; $\ge \delta $ ) for any divisorial valuation v. If $(X,\Delta )$ is further a pair, we call this klt (resp. lc). If a (sub)lc pair $(X,\Delta )$ has a log resolution $\pi \colon Y\to X$ such that $A_{(X,\Delta )}(E)>0$ for any $\pi $ -exceptional divisor E, we say that $(X,\Delta )$ is (sub)dlt. Furthermore, if $(X,\Delta )$ is lc and $A_{(X,\Delta )}(E)>0$ for any prime divisor E over X whose center is not a prime divisor, then we say that $(X,\Delta )$ is a plt pair. A subvariety $Z\subset X$ is called a lc center of an lc pair $(X,\Delta )$ if there exists a prime divisor E over X such that $A_{(X,\Delta )}(E)=0$ and the center of E is the generic point of Z. It is well known that if $(X,\Delta )$ is a dlt pair, then a lc center of $(X,\Delta )$ is a stratum of $\lfloor \Delta \rfloor $ (cf. [Reference Kollár54, Theorem 4.16]). On the other hand, if for any Weil divisor D on X, there exists $m\in \mathbb {Z}_{>0}$ such that $mD$ is Cartier, then we say that X is $\mathbb {Q}$ -factorial.

Definition 2.3 (Models).

Let $(X,\Delta )$ be a log pair over S. Let $\phi :X\dashrightarrow Y$ be a birational map such that Y is a normal variety projective over S. Let $\mathrm {Ex}(\phi )$ be the sum of all $\phi $ -exceptional prime divisors. $\phi :X\dashrightarrow Y$ is called a birational contraction if there exists no $\phi ^{-1}$ -exceptional divisor. Here, suppose that $\phi $ is a birational contraction preserving the structure morphisms to S. Take a resolution $\Gamma $ of indeterminacy of $\phi $ , and let $p:\Gamma \to X$ and $q:\Gamma \to Y$ be the canonical projections. Suppose that $K_Y+\phi _{*}\Delta $ is $\mathbb {Q}$ -Cartier, and let E be the q-exceptional $\mathbb {Q}$ -divisor on $\Gamma $ defined by

$$\begin{align*}E=p^{*}(K_X+\Delta)-q^{*}(K_Y+\phi_{*}\Delta). \end{align*}$$

We say that $\phi $ is $K_X+\Delta $ -nonpositive (resp. $K_X+\Delta $ -negative) if E is effective (resp. and further $\mathrm {Supp}\,E$ contains all $\phi $ -exceptional prime divisors). We call $(Y,\phi _{*}\Delta )$ (or simply call Y)

• • a good minimal model of $(X,\Delta )$ if $K_Y+\phi _{*}\Delta $ is relatively semiample over S and $\phi $ is $K_X+\Delta $ -negative,

• • the log canonical (lc) model of $(X,\Delta )$ if $K_Y+\phi _{*}\Delta $ is relatively ample over S and $\phi $ is $K_X+\Delta $ -nonpositive.

For details, see [Reference Kollár and Mori55, §2,3] and [Reference Birkar, Cascini, Hacon and McKernan9].

Definition 2.4 (G-equivariant models).

Let G be a connected algebraic group and suppose that G acts on S. If $(X,\Delta )$ is a log pair over S, there exists $f\colon X\to S$ and G acts on X in the way that f is G-equivariant and $\Delta $ is G-stable, then we say that $(X,\Delta )$ admits a G-action. Suppose that G acts on $(X,\Delta )$ and let $\phi \colon X\dashrightarrow Y$ be a birational map. If Y admits a G-action and the graph $\Gamma $ of $\phi $ is a G-invariant subvariety of $X\times Y$ , then we say that $\phi $ is G-equivariant. If $(Y,\phi _{*}\Delta )$ is a good minimal model of $(X,\Delta )$ and $\phi $ is G-equivariant, we call Y a G-equivariant good minimal model. By the argument of the proof of [Reference Li and Xu61, Theorem 7] (see also [Reference Andreatta5, 1.5]), if Y is a good minimal model obtained by a certain minimal model program of $(X,\Delta )$ with scaling, then Y is a G-equivariant good minimal model. Note also that if Y is the lc model of $(X,\Delta )$ , then Y has the natural G-action and the canonical birational map $\phi \colon X\dashrightarrow Y$ is G-equivariant since $Y=\mathbf {Proj}_{S}(\oplus _{l\ge 0}f_{*}\mathcal {O}_{X}(lr_0(K_{X}+\Delta )))$ , where $r_0(K_{X}+\Delta )$ is Cartier.

Definition 2.5 (Test configuration).

Let $\pi :\mathcal {X}\to \mathbb {P}^1$ be a projective and flat morphism such that $\pi $ is $\mathbb {G}_m$ -equivariant, where $\mathbb {P}^1$ admits the canonical $\mathbb {G}_m$ -action by multiplication. If there exists a $\mathbb {G}_m$ -equivariant isomorphism $\pi ^{-1}(\mathbb {P}^1\setminus \{0\})\cong X\times (\mathbb {P}^1\setminus \{0\})$ over $\mathbb {P}^1\setminus \{0\}$ , where $\mathbb {G}_m$ trivially acts on the first component of $X\times (\mathbb {P}^1\setminus \{0\})$ , $\mathcal {X}$ is called a test configuration for X. Assume further that $\mathcal {X}$ admits a relatively (semi)ample $\mathbb {G}_m$ -linearized $\mathbb {Q}$ -line bundle $\mathcal {L}$ over $\mathbb {P}^1$ such that $\mathcal {L}|_{\pi ^{-1}(\mathbb {P}^1\setminus \{0\})}$ coincides with $L\otimes \mathcal {O}_{\mathbb {P}^1\setminus \{0\}}$ with the trivial $\mathbb {G}_m$ -action on $\mathcal {L}|_{\pi ^{-1}(\infty )}$ . Here, we say that a $\mathbb {Q}$ -line bundle $\mathcal {L}$ is $\mathbb {G}_m$ -linearized if there exists a positive integer m such that $m\mathcal {L}$ is a $\mathbb {G}_m$ -linearlized line bundle in the sense of [Reference Mumford, Fogarty and Kirwan66, Definition 1.6]. Then, $(\mathcal {X},\mathcal {L})$ is called a (semi)ample test configuration for $(X,L)$ .

If $\mathcal {X}$ is $\mathbb {G}_m$ -equivariantly isomorphic to $X\times \mathbb {P}^1$ with the trivial action, then we call $\mathcal {X}$ trivial. We denote $(X\times \mathbb {P}^1,L\otimes \mathcal {O}_{\mathbb {P}^1})$ with the trivial action by $(X_{\mathbb {P}^1},L_{\mathbb {P}^1})$ . Similarly, we denote $\Delta \times \mathbb {P}^1\subset X_{\mathbb {P}^1}$ by $\Delta _{\mathbb {P}^1}$ for any $\mathbb {Q}$ -divisor $\Delta $ on X.

Let Y be another projective variety and $f\colon X\to Y$ be a morphism. If $\mathcal {X}$ (resp. $\mathcal {Y}$ ) is a test configuration for X (resp. Y), we note that there exists a unique $\mathbb {G}_m$ -equivariant rational map $F\colon \mathcal {X}\dashrightarrow \mathcal {Y}$ induced by $f\times \mathrm {id}_{\mathbb {P}^1\setminus \{0\}}$ . We call such F the canonical rational map. If F is further a morphism, then we say that there exists a canonical morphism $F\colon \mathcal {X}\to \mathcal {Y}$ .

Definition 2.8 [Reference Boucksom, Hisamoto and Jonsson13, §6, §7].

Let $(X,\Delta ,L)$ be a polarized log subpair of dimension n. Take an arbitrary normal semiample test configuration $(\mathcal {X},\mathcal {L})$ for $(X,L)$ . Let $\rho \colon \mathcal {X}\dashrightarrow X_{\mathbb {P}^1}$ be the canonical birational map. It is well known that there exist a normal test configuration $\mathcal {Y}$ and a canonical morphism $\sigma \colon \mathcal {Y}\to \mathcal {X}$ such that the canonical birational map $\rho _{\mathcal {Y}}=\rho \circ \sigma $ is a morphism. Furthermore, let $\mathcal {Y}_{0,\mathrm {red}}$ be the reduced structure of $\mathcal {Y}_0$ , $\Delta _{\mathcal {Y}}$ be the closure of $\Delta \times (\mathbb {P}^1\setminus \{0\})$ in $\mathcal {Y}$ and $K^{\mathrm {log}}_{(\mathcal {Y},\Delta _{\mathcal {Y}})/\mathbb {P}^1}:=K_{(\mathcal {Y},\Delta _{\mathcal {Y}})/\mathbb {P}^1}+(\mathcal {Y}_{0,\mathrm {red}}-\mathcal {Y}_0)$ , where $\pi \colon \mathcal {Y}\to \mathbb {P}^1$ is the canonical morphism and $K_{(\mathcal {Y},\Delta _{\mathcal {Y}})/\mathbb {P}^1}:=K_{\mathcal {Y}}+\Delta _{\mathcal {Y}}-\pi ^{*}K_{\mathbb {P}^1}$ . Then we define

• • $I^{\mathrm {NA}}(\mathcal {X},\mathcal {L})=(L^n)^{-1}(\sigma ^{*}\mathcal {L}\cdot \rho _{\mathcal {Y}}^{*}L_{\mathbb {P}^1}^n)-(L^n)^{-1}(\sigma ^{*}\mathcal {L}- \rho _{\mathcal {Y}}^{*}L_{\mathbb {P}^1})\cdot (\sigma ^{*}\mathcal {L})^n,$

• • $E^{\mathrm {NA}}(\mathcal {X},\mathcal {L})=\frac {\mathcal {L}^{n+1}}{(n+1)(L^n)},$

• • $J^{\mathrm {NA}}(\mathcal {X},\mathcal {L})=(L^n)^{-1}(\sigma ^{*}\mathcal {L}\cdot \rho _{\mathcal {Y}}^{*}L_{\mathbb {P}^1}^n)- E^{\mathrm {NA}}(\mathcal {X},\mathcal {L}),$

• • the non-Archimedean J $^H$ -energy is defined to be

  $$\begin{align*}(\mathcal{J}^H)^{\mathrm{NA}}(\mathcal{X},\mathcal{L})=(L^n)^{-1}(\rho_{\mathcal{Y}}^{*}H_{\mathbb{P}^1}\cdot (\sigma^{*}\mathcal{L})^{n})- (L^n)^{-1}(nH\cdot L^{n-1})E^{\mathrm{NA}}(\mathcal{X},\mathcal{L}), \end{align*}$$
  where H is a $\mathbb {Q}$ -line bundle on X,

• • the non-Archimedean entropy is defined to be

  $$ \begin{align*} H_\Delta^{\mathrm{NA}}(\mathcal{X},\mathcal{L})&=(L^n)^{-1}\sum_Eb_EA_{(X,B)}(v_E)(E\cdot(\sigma^{*}\mathcal{L})^n)\\ &=(L^n)^{-1}(K^{\mathrm{log}}_{(\mathcal{Y},\Delta_{\mathcal{Y}})/\mathbb{P}^1}-\rho_{\mathcal{Y}}^{*}K_{(X_{\mathbb{P}^1},\Delta_{\mathbb{P}^1})/\mathbb{P}^1})\cdot(\sigma^{*}\mathcal{L})^n, \end{align*} $$
  where E runs over all irreducible components of $\mathcal {Y}_0$ that is not the strict transform of $X\times \{0\}$ ,

• • the log Donaldson–Futaki invariant is defined to be

  $$ \begin{align*}\mathrm{DF}_{\Delta}(\mathcal{X},\mathcal{L})=(L^n)^{-1}(K_{(\mathcal{Y},\Delta_{\mathcal{Y}})/\mathbb{P}^1}-\rho_{\mathcal{Y}}^{*}K_{(X_{\mathbb{P}^1},\Delta_{\mathbb{P}^1})/\mathbb{P}^1})\cdot\mathcal{L}^n+(\mathcal{J}^{K_{X}+\Delta})^{\mathrm{NA}}(\mathcal{X},\mathcal{L}),\end{align*} $$

• • the non-Archimedean Mabuchi energy

  $$ \begin{align*}M_\Delta^{\mathrm{NA}}(\mathcal{X},\mathcal{L})=H_\Delta^{\mathrm{NA}}(\mathcal{X},\mathcal{L})+(\mathcal{J}^{K_{X}+\Delta})^{\mathrm{NA}}(\mathcal{X},\mathcal{L}).\end{align*} $$

It is well known that these invariants are independent of the choice of $\sigma \colon \mathcal {Y}\to \mathcal {X}$ . We say that a semiample test configuration $(\tilde {\mathcal {X}},\tilde {\mathcal {L}})$ dominates $(\mathcal {X},\mathcal {L})$ via $\mu $ if there exists a canonical morphism $\mu :\tilde {\mathcal {X}}\to \mathcal {X}$ such that $\tilde {\mathcal {L}}=\mu ^{*}\mathcal {L}$ . It is proved in [Reference Boucksom, Hisamoto and Jonsson13, §§7.5] that those functionals are pullback invariant as defined in [Reference Boucksom, Hisamoto and Jonsson13], that is, if $(\tilde {\mathcal {X}},\tilde {\mathcal {L}})$ dominates $(\mathcal {X},\mathcal {L})$ , then $J^{\mathrm {NA}}(\tilde {\mathcal {X}},\tilde {\mathcal {L}})=J^{\mathrm {NA}}(\mathcal {X},\mathcal {L})$ cf. [Reference Boucksom, Hisamoto and Jonsson13, §6, §7] for example.

Definition 2.10 (K-stability and J-stability).

Let $(X,\Delta ,L)$ be a polarized log pair. $(X,\Delta ,L)$ is called K-stable if

$$\begin{align*}\mathrm{DF}_{\Delta}(\mathcal{X},\mathcal{L})>0 \end{align*}$$

for any nontrivial normal ample test configuration. On the other hand, $(X,\Delta ,L)$ is called uniformly K-stable (resp, K-semistable) if there exists a constant $c>0$ (resp., $c\ge 0$ ) such that

$$\begin{align*}\mathrm{DF}_{\Delta}(\mathcal{X},\mathcal{L})\ge c J^{\mathrm{NA}}(\mathcal{X},\mathcal{L}) \end{align*}$$

for any semiample test configuration $(\mathcal {X},\mathcal {L})$ for $(X,L)$ . The above condition is known by [Reference Boucksom, Hisamoto and Jonsson13, §8] to be equivalent to that for any semiample normal test configuration $(\mathcal {X},\mathcal {L})$

$$\begin{align*}M^{\mathrm{NA}}_{\Delta}(\mathcal{X},\mathcal{L})\ge c J^{\mathrm{NA}}(\mathcal{X},\mathcal{L}). \end{align*}$$

The definition of J $^H$ -stability is of similar form (cf. [Reference Hattori45]). That is, we say that $(X,L)$ is uniformly J $^H$ -stable (resp. J $^H$ -semistable) if there exists $c>0$ (resp. $c\ge 0$ ) such that

$$\begin{align*}(\mathcal{J}^H)^{\mathrm{NA}}(\mathcal{X},\mathcal{L})\ge c J^{\mathrm{NA}}(\mathcal{X},\mathcal{L}). \end{align*}$$

Definition 2.12. Let $(X,B)$ be a subpair and D be an effective $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor. Suppose that there exists $l\in \mathbb {Q}$ such that $(X,B+lD)$ is sublc. Then we set the log canonical threshold of $(X,B)$ with respect to D as

$$\begin{align*}\mathrm{lct}_{(X,B)}(D)=\inf\{r\in\mathbb{Q}|(X,B+rD) \ \mathrm{is}\ \mathrm{sublc}\}=\min_v\frac{A_{(X,B)}(v)}{v(D)}, \end{align*}$$

where v runs over all divisorial valuations such that $v(D)\ne 0$ .

Definition 2.13. Let $(X,B,L)$ be a polarized klt pair. Then the alpha invariant of $(X,B)$ with respect to L is

$$\begin{align*}\alpha_{(X,B)}(L)=\inf_D \mathrm{lct}_{(X,B)}(D), \end{align*}$$

where D runs over all elements of $|L|_{\mathbb {Q}}$ , which is the set of effective $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisors D such that $D\sim _{\mathbb {Q}}L$ . If we set $T_L(v)=\sup _{D\in |L|_{\mathbb {Q}}}v(D)$ , then we have

$$\begin{align*}\alpha_{(X,B)}(L)=\inf_v\frac{A_{(X,B)}(v)}{T_L(v)}, \end{align*}$$

where v runs over all divisorial valuations.

Definition 2.14 ([Reference Fujita and Odaka39], [Reference Blum and Jonsson10]).

Let $(X,B,L)$ be a polarized klt pair. Take $r\in \mathbb {Z}_{>0}$ such that $rL$ is a line bundle. For any $m\in \mathbb {Z}_{>0}$ , $D\in |L|_{\mathbb {Q}}$ is called a divisor of $mr$ -basis type if there exists a basis $\{f_j\}_{j=1}^{h^0(X,mrL)}\subset H^0(X,mrL)$ such that

$$\begin{align*}D=(mrh^0(X,mrL))^{-1}\sum_{j=1}^{h^0(X,mrL)}\mathrm{div}_{mrL}(f_j). \end{align*}$$

Here, $\mathrm {div}_{mrL}(f_j)$ is an effective Cartier divisor defined by $f_j$ as a section of $H^0(X,mrL)$ . We denote by $|L|_{mr\text {-basis}}$ the set of all $mr$ -basis type divisors D with respect to L. We set $S^{mr}_{L}(v):=\sup _{D\in |L|_{mr\text {-basis}}}v(D)$ for any divisorial valuation v of X. Then, we set

$$\begin{align*}\delta^{mr}_{(X,B)}(L):=\inf_{D\in|L|_{mr\text{-basis}}}\mathrm{lct}_{(X,B)}(D)=\inf_v\frac{A_{(X,B)}(v)}{S^{mr}_{L}(v)}, \end{align*}$$

where v runs over all divisorial valuations on X. By [Reference Blum and Jonsson10, Theorem A], $\{\delta ^{mr}_{(X,B)}(L)\}_{m=1}^{\infty }$ converges and set the delta invariant of $(X,B)$ with respect to L as

$$\begin{align*}\delta_{(X,B)}(L):=\lim_{m\to\infty}\delta^{mr}_{(X,B)}(L). \end{align*}$$

Moreover, thanks to [Reference Blum and Jonsson10, §2], we have that $\lim _{m\to \infty }S^{mr}_{L}(v)=S_L(v)$ for any divisorial valuation v on X and

$$\begin{align*}\delta_{(X,B)}(L)=\inf_v\frac{A_{(X,B)}(v)}{S_{L}(v)}, \end{align*}$$

where v runs over all divisorial valuations on X and $S_{L}(v):=\mathrm {vol}(L)^{-1}\int _0^{\infty }\mathrm {vol}(L-xv)dx$ . Here, $\mathrm {vol}(L-xv)$ means $\mathrm {vol}(\pi ^{*}L-xcE)$ , where $c\in \mathbb {Q}_{>0}$ , E is a prime divisor over X and $\pi \colon Y\to X$ is a log resolution such that $v=c\,\mathrm {ord}_E$ and E is a prime divisor on Y. See [Reference Blum and Jonsson10] for details.

Definition 2.18 [Reference Hattori48].

$f\colon (X,\Delta ,H)\to (C,L)$ is called a polarized algebraic fiber space (sub)pair if $(X,\Delta , H)$ is a polarized (sub)pair, $(C,L)$ is a polarized normal variety and $f\colon X\to C$ is a contraction, that is, f is projective and $f_{*}\mathcal {O}_X\cong \mathcal {O}_C$ . Let $f:(X,\Delta ,H)\to (C,L)$ be a polarized algebraic fiber space pair. Then $(X,\Delta ,H)$ is called adiabatically K-(semi)stable over $(C,L)$ if $(X,\Delta ,\epsilon H+f^{*}L)$ is K-(semi)stable for sufficiently small $\epsilon \in \mathbb {Q}_{>0}$ . Furthermore, $(X,\Delta ,H)$ is called

• • uniformly adiabatically K-stable over $(C,L)$ if there exist positive constants $\delta>0$ and $\epsilon _0>0$ such that $\mathrm {DF}_{\Delta }(\mathcal {X},\mathcal {M})\ge \delta J^{\mathrm {NA}}(\mathcal {X},\mathcal {M})$ for any $\epsilon \in (0,\epsilon _0)\cap \mathbb {Q}$ and semiample test configuration $(\mathcal {X},\mathcal {M})$ for $(X,\epsilon H+f^{*}L)$ ,

• • adiabatically K-unstable over $(C,L)$ if for any sufficiently small rational number $\epsilon>0$ , there exists a semiample test configuration $(\mathcal {X},\mathcal {M})$ for $(X,\epsilon H+f^{*}L)$ such that $\mathrm {DF}_{\Delta }(\mathcal {X},\mathcal {M})<0$ .

We say that $(X,H)$ has cscK metrics adiabatically if X is smooth, $\Delta =0$ and there exists $\delta _0>0$ such that $(X,\delta H+f^{*}L)$ has a cscK metric for any $0<\delta <\delta _0$ . Here, we will simply denote $H+f^{*}L$ by $H+L$ .

Definition 2.19. A quadruple $(X,B,T,L)$ is called a polarized log-twisted pair if $(X,B,L)$ is a polarized log pair and T is a $\mathbb {Q}$ -line bundle on X.

We call $(X,B,T,L)$ a log-twisted Fano pair if $L\sim _{\mathbb {Q}}-(K_X+B+T)$ and $(X,B)$ is klt.

Definition 2.20. Let $(X,B,T,L)$ be a polarized log-twisted pair and $(\mathcal {X},\mathcal {L})$ be a semiample test configuration for $(X,L)$ . Then the log-twisted Donaldson–Futaki invariant is

$$\begin{align*}\mathrm{DF}_{(B,T)}(\mathcal{X},\mathcal{L})=\mathrm{DF}_{B}(\mathcal{X},\mathcal{L})+(\mathcal{J}^T)^{\mathrm{NA}}(\mathcal{X},\mathcal{L}). \end{align*}$$

We can also define the log-twisted non-Archimedean Mabuchi functional as

$$\begin{align*}M^{\mathrm{NA}}_{(B,T)}(\mathcal{X},\mathcal{L})=M^{\mathrm{NA}}_{B}(\mathcal{X},\mathcal{L})+(\mathcal{J}^T)^{\mathrm{NA}}(\mathcal{X},\mathcal{L}). \end{align*}$$

Then, $(X,B,T,L)$ is called (log-twisted)

• • K-semistable if

  $$\begin{align*}M^{\mathrm{NA}}_{(B,T)}(\mathcal{X},\mathcal{L})\ge0 \end{align*}$$
  for any normal semiample test configuration,

• • K-stable if

  $$\begin{align*}M^{\mathrm{NA}}_{(B,T)}(\mathcal{X},\mathcal{L})>0 \end{align*}$$
  for any nontrivial normal ample test configuration,

• • uniformly K-stable if there exists a positive constant $\epsilon>0$ such that

  $$\begin{align*}M^{\mathrm{NA}}_{(B,T)}(\mathcal{X},\mathcal{L})\ge \epsilon J^{\mathrm{NA}}(\mathcal{X},\mathcal{L}) \end{align*}$$
  for any normal semiample test configuration.

Definition 2.21. Let T be a $\mathbb {Q}$ -line bundle on X. If T is semiample, we say that $A\sim _{\mathbb {Q}}T$ is a general effective $\mathbb {Q}$ -divisor if there exists $m\in \mathbb {Z}_{>0}$ such that $mT$ is globally generated and $mA\in |mT|$ . When T is not semiample, $A\sim _{\mathbb {Q}}T$ is called general if there exist two general semiample $\mathbb {Q}$ -divisors $A_1$ and $A_2$ such that $T\sim _{\mathbb {Q}}A_1-A_2$ .

Definition 2.25. Let $(X,B,T,L=-K_X-B-T)$ be a log-twisted Fano pair, T is semiample and $(\mathcal {X}^{\mathrm {s}},\mathcal {L}^{\mathrm {s}})$ be a normal ample test configuration. Let $\rho ^{\mathrm {s}}\colon \mathcal {X}^{\mathrm {s}}\dashrightarrow X_{\mathbb {P}^1}$ be the canonical birational morphism and $\mathcal {B}^{\mathrm {s}}:=(\rho ^{\mathrm {s}})^{-1}_{*}B_{\mathbb {P}^1}$ . Then $(\mathcal {X}^{\mathrm {s}},\mathcal {L}^{\mathrm {s}})$ is a log-twisted special test configuration for $(X,B,T,L)$ if $(\mathcal {X}^{\mathrm {s}},\mathcal {B}^{\mathrm {s}}+(\rho ^{\mathrm {s}})^{-1}_{*}T_{1,\mathbb {P}^1}+\mathcal {X}_0)$ is plt (cf., [Reference Kollár and Mori55, 2.34]) and $\mathcal {L}^{\mathrm {s}}\sim _{\mathbb {Q},\mathbb {P}^1}-(K_{\mathcal {X}^{\mathrm {s}}}+\mathcal {B}^{\mathrm {s}}+(\rho ^{\mathrm {s}})^{-1}_{*}T_{1,\mathbb {P}^1})$ for any general effective $\mathbb {Q}$ -divisor $T_1\sim _{\mathbb {Q}} T$ .

Definition 2.26 (Klt-trivial fibrations).

Let S be a quasi-projective normal variety. Let $f:X\to C$ be a proper surjective morphism of normal varieties projective over S such that $f_{*}\mathcal {O}_X\cong \mathcal {O}_C$ . $K(X)$ denotes the sheaf of quotient fields of X. For any $\mathbb {Q}$ -divisor $\Delta $ on X, $f\colon (X,\Delta )\to C$ is called a sublc-trivial fibration (resp. subklt-trivial fibration) if $(X,\Delta )$ is a sublc (resp. subklt) pair satisfying the following properties:

• • $ K_{X}+\Delta \sim _{C,\mathbb {Q}}0$ ,

• • rank $f_{*}\mathcal {O}_X(\lceil \mathbf {A}^{*}(X,\Delta )\rceil )=1$ ,

• • $(X,\Delta )$ is sublc (resp. subklt) over the generic point of C,

where we define $\mathcal {O}_X(\lceil \mathbf {A}^{*}(X,\Delta )\rceil )$ as follows (cf. [Reference Fujino32, Lemma 3.22]). Let $\pi :Y\to X$ be a log resolution of $(X,\Delta )$ , and suppose that $K_Y=\pi ^{*}(K_X+\Delta )+\sum _ia_iE_i$ and $\pi _{*}(\sum _ia_iE_i)=-\Delta $ . Then we set

$$\begin{align*}\mathcal{O}_X(\lceil \mathbf{A}^{*}(X,\Delta)\rceil)=\pi_{*}\mathcal{O}_Y(\sum_{a_i\ne-1}\lceil a_i\rceil E_i). \end{align*}$$

The definition of $\mathcal {O}_X(\lceil \mathbf {A}^{*}(X,\Delta )\rceil )$ is independent of the choice of $\pi $ and hence $\mathcal {O}_X(\lceil \mathbf {A}^{*}(X,\Delta )\rceil )$ is coherent. We say that f is a lc (resp., klt)-trivial fibration if $(X,\Delta )$ is a lc (resp., klt) pair and $K_{X}+\Delta \sim _{C,\mathbb {Q}}0$ . It is easy to see that if f is a lc (resp., klt)-trivial fibration, then f is a sublc (resp., subklt)-trivial fibration. We remark that sublc-trivial fibrations are usually called lc-trivial fibrations (e.g., in [Reference Fujino and Gongyo34]). Furthermore, if $f\colon (X,\Delta ,H)\to (C,L)$ is a polarized algebraic fiber space pair, then we say that $f\colon (X,\Delta ,H)\to (C,L)$ is a polarized lc (resp. klt)-trivial fibration.

Definition 2.27 (Ambro model).

We say that $M_C$ is stable under birational base change if $M_C$ is $\mathbb {Q}$ -Cartier and

$$\begin{align*}M_{C'}=h^{*}M_C \end{align*}$$

for any birational morphism $h:C'\to C$ , where $M_{C'}$ is the moduli divisor defined by $M_C$ and f. Then, we call C an Ambro model for f. If moreover, $M_C$ is semiample, we call C a good Ambro model in this paper. Thanks to [Reference Fujino and Gongyo34, Theorem 3.6], there exists an Ambro model $C'$ dominating C for any sublc-trivial fibration and then the moduli divisor is nef. For subklt-trivial fibrations, this fact was originally proved in [Reference Ambro4].

It immediately follows from the same argument of [Reference Ambro3, §5] that if $f:(X,\Delta )\to C$ is a sublc-trivial fibration over a curve C, then C is an Ambro model.

Let $f\colon (X,\Delta )\to C$ be an lc (resp. klt)-trivial fibration with ample $\mathbb {Q}$ -divisors H and L on X and C respectively. Then we call $f\colon (X,\Delta ,H)\to (C,L)$ a polarized lc (resp. klt)-trivial fibration.

Definition 2.31. Let $f\colon X\to C$ be a contraction such that $K_X\sim _{\mathbb {Q},C}0$ . Suppose that C is a proper smooth curve and X is a normal surface with only canonical singularities and let $\pi \colon \tilde {X}\to X$ be the minimal resolution of singularities. Then we call $f\colon X\to C$ (or X) an elliptic surface. If $X=\tilde {X}$ , we call X a smooth elliptic surface. On the other hand, if $\tilde {X}$ is rational, then we call X a rational elliptic surface.

If X is a rational elliptic surface and has a section S over C such that S is f-ample, then the pair $(X,S)$ or $f\colon X\to C$ is called a rational Weierstrass fibration over C.

Definition 2.33. Let $f\colon X\to \mathbb {P}^1$ be a rational elliptic surface. We set the multiplicity $m(X)$ of X as follows. Let $\tilde {X}$ be the minimal resolution of X. If $\tilde {X}$ has a $_mI_N$ -type fiber for some $m\ge 2$ , we set $m(X)=m$ . Otherwise, we set $m(X)=1$ .