2 Model metrics, curvature forms, and psh envelopes

Non-Archimedean pluripotential theory in higher dimensions was introduced by Boucksom, Favre, and Jonsson [Reference Boucksom, Favre and Jonsson6]. We recall basic notions of non-Archimedean pluripotential theory following [Reference Gubler, Jell, Künnemann and Martin10, Subsections 2.1–2.8].

Let X be a proper variety over a non-Archimedean field K. Recall that the topological space underlying the Berkovich analytification $X^{\mathrm {an}}$ of X consists of pairs $(p,|\phantom {a}|_p)$ with $p\in X$ and $|\phantom {a}|_p$ an absolute value on $\kappa (p)=\mathcal O_{X,p}/\mathfrak m_{X,p}$ that extends $|\phantom {a}|_K$ and is equipped with coarsest topology such that the map $\pi \colon X^{\mathrm {an}}\rightarrow X$ , $(p,|\phantom {a}|_p) \mapsto p$ is continuous and for all Zariski open subsets U of X and all regular functions $f \in \mathcal O_X(U)$ the map $|f|\colon U^{\mathrm {an}}=\pi ^{-1}(U)\rightarrow {\mathbb R},\,(p,| \phantom {a}|_p) \mapsto |f(p)| := |f+ \mathfrak m_{X,p}|_p$ is continuous as well.

A model of X is a proper flat scheme ${\mathscr X}$ over together with an isomorphism h from X to the generic fiber ${\mathscr X}_{\eta }$ of the S-scheme ${\mathscr X}$ . The special fiber of ${\mathscr X}$ is denoted by ${\mathscr X}_s$ . Given a model, the valuative criterion of properness yields a natural reduction map $\mathrm {red}_{\mathscr X}\colon X^{\mathrm {an}}\to {\mathscr X}_s$ . Let L be a line bundle over X. A model of $(X,L)$ is given by a model ${\mathscr X}$ of X and a line bundle ${\mathscr L}$ over ${\mathscr X}$ with an isomorphism of L to $h^*({\mathscr L}|_{{\mathscr X}_{\eta }})$ . A continuous metric $\| \ \|$ of L associates with each section $s\in \Gamma (U,L)$ on some Zariski open subset U of the variety X a continuous function $\|s\|\colon U^{\mathrm {an}}\to [0,\infty )$ such that one has $\|f\cdot s\|=|f|\cdot \|s\|$ for each regular function f in ${\mathcal O}_X(U)$ . It is furthermore required that $\|s\|>0$ , if s is a nowhere vanishing section of L.

Let $({\mathscr X},{\mathscr L})$ be a model of $(X,L^{\otimes m})$ for some $m\in {\mathbb N}_{>0}$ . The model metric $\| \ \|_{{\mathscr L}}$ of L over $X^{\mathrm {an}}$ is determined by $\|s\|_{{\mathscr L}}:= \sqrt [m]{|g|}$ on $U^{\mathrm { an}}\cap \mathrm {red}^{-1}({\mathscr U}_s)$ , where ${\mathscr U}$ is open in ${\mathscr X}$ , s is a section of L over , b is a nowhere vanishing section of ${\mathscr L}$ over ${\mathscr U}$ , $g\in {\mathcal O}_X(U)$ is a regular function such that $s^{\otimes m}=gb$ over U, and $\mathrm {red}_{\mathscr X}\colon X^{\mathrm {an}}\to {\mathscr X}_s$ is the reduction map. A model metric $\| \ \|$ on ${\mathcal O}_X^{\mathrm {an}}$ induces a so-called model function $-\log \|1\|\colon X^{\mathrm {an}}\to {\mathbb R}$ . The ${\mathbb Q}$ -vector space of all model functions on X is denoted by ${\mathscr D}(X)$ .

Let $\mathfrak a$ be an ideal sheaf on a model ${\mathscr X}$ of X which is supported in the special fibre. The exceptional divisor E of the blowup ${\mathscr Y}$ of ${\mathscr X}$ in $\mathfrak a$ defines a model $({\mathscr Y},{\mathcal O}_{{\mathscr Y}}(E))$ of $(X,{\mathcal O}_X)$ . The associated model function in ${\mathscr D}(X)$ is denoted by $\log |\mathfrak a|$ .

Given a model ${\mathscr X}$ of X one defines $N^1({\mathscr X}/S)_{{\mathbb Q}}$ as the quotient of the ${\mathbb Q}$ -vector space $\mathrm {Pic}({\mathscr X})_{{\mathbb Q}}:=\mathrm {Pic}({\mathscr X})\otimes _{\mathbb Z}{\mathbb Q}$ by the subspace generated by line bundles whose restriction to every closed curve C in the special fiber ${\mathscr X}_s$ has degree zero. Call nef if $\alpha \cdot C\geq 0$ for all such curves.

The space of closed (1,1)-forms on X is defined as

where the direct limit is taken over all isomorphism classes of models of X and the transition maps are induced by pullback between dominating models. For the model metric induced by a model ${\mathscr L}$ of $L^{\otimes m}$ , its curvature form $c_1(L,\| \ \|_{{\mathscr L}})$ is the image of ${\mathscr L}^{\otimes \frac {1}{m}}\in N^1({\mathscr X}/S)_{{\mathbb Q}}$ in ${\mathcal Z}^{1,1}(X)$ . By construction we have a map $dd^c\colon {\mathscr D}(X)\to {\mathcal Z}^{1,1}(X)$ given by $g\mapsto c_1({\mathcal O}_X,\| \ \|\cdot \mathrm {e}^{-g})$ . Call a closed $(1,1)$ -form $\theta \in {\mathcal Z}^{1,1}(X)$ semipositive if it can be represented by a nef element in $N^1({\mathscr X}/S)$ for some model ${\mathscr X}$ . Denote by $N^1(X)$ the quotient of $\mathrm {Pic}(X)_{\mathbb R}:=\mathrm {Pic}(X)\otimes _{\mathbb Z}{\mathbb R}$ by numerical equivalence. The map $\{\phantom {a}\}\colon {\mathcal Z}^{1,1}(X)\to N^1(X)$ induced by the restriction maps $N^1({\mathscr X}/S)\to N^1(X)$ sends a closed $(1,1)$ -form $\theta $ to its de Rham class $\{\theta \}$ . A class in $N^1(X)$ is called ample if it is an ${\mathbb R}_{>0}$ -linear combination of classes induced by ample line bundles on X.

We fix $\theta \in {\mathcal Z}^{1,1}(X)$ . A model function $\varphi \in {\mathscr D}(X)$ is called $\theta $ -plurisubharmonic ( $\theta $ -psh for short) if the class $\theta +dd^c\varphi \in {\mathcal Z}^{1,1}(X)$ is semipositive. The space of all psh model functions is denoted by $\mathrm {PSH}_{{\mathscr D}}(X,\theta )$ . For a continuous function $u\in {\mathscr C}^0(X^{\mathrm {an}})$ , its $\theta $ -psh envelope $P_{\theta }(u)\colon X^{\mathrm {an}}\to {\mathbb R}\cup \{-\infty \}$ is the function defined by

For $u\in C^0(X^{\mathrm {an}})$ , $v\in {\mathscr D}(X)$ and $t\in {\mathbb R}_{>0}$ we have

(2.1) $$ \begin{align} P_\theta(u)-v&= P_{\theta+dd^cv}(u-v), \end{align} $$

(2.2) $$ \begin{align} P_{t\theta}(t\theta)&= t P_\theta(u). \qquad\qquad \end{align} $$

If the de Rham class $\{\theta \}\in N^1(X)$ is ample, then $\mathrm {PSH}_{\mathscr D}(X,\theta )$ is non-empty, $P_{\theta }(u)$ takes value in ${\mathbb R}$ , and we have

(2.3) $$ \begin{align} \sup_{X^{\mathrm{an}}}|P_{\theta}(u)-P_{\theta}(u')|\leq \sup_{X^{\mathrm{an}}}|u-u'|. \end{align} $$

for all $u,u'\in C^0(X^{\mathrm {an}})$ . For further properties of $\theta $ -psh model functions and the $\theta $ -psh envelope we refer to [Reference Boucksom, Favre and Jonsson6] and [Reference Gubler, Jell, Künnemann and Martin10, Section 2]

For $\theta \in {\mathcal Z}^{1,1}(X)$ that has an ample de Rham class $\{\theta \}$ , we consider $\varphi \in \mathrm {PSH}_{\mathscr D}(X,\theta )$ and we assume that there exists a normal model ${\mathscr X}$ of X such that $\theta $ and $\varphi $ are induced by line bundles ${\mathscr M}$ and ${\mathscr L}$ on ${\mathscr X}$ . The Monge–Ampère measure $\mathrm {MA}_\theta (\varphi )$ is the discrete measure

on $X^{\mathrm {an}}$ where V runs through the irreducible components of ${\mathscr X}_s$ , $x_V\in X^{\mathrm {an}}$ is the unique preimage of the generic point $\xi _V$ of V under the reduction map $\mathrm {red}_{\mathscr X}\colon X^{\mathrm {an}}\to {\mathscr X}_s$ , and $\delta _{x_V}$ denotes the Dirac probability measure supported in $x_V$ . For generalization of the Monge–Ampère measure to all $\theta \in {\mathcal Z}^{1,1}(X)$ with ample de Rham class $\{\theta \}$ and to more general classes of $\theta $ -psh functions, we refer to [Reference Boucksom, Favre and Jonsson5], [Reference Gubler, Jell, Künnemann and Martin10] and Section 8.

3 Global test ideals in mixed characteristic

In this section, we introduce the global $+$ -test ideals defined and studied by Hacon, Lamarche, and Schwede in [Reference Hacon, Lamarche and Schwede13]. The $+$ -test ideals form a mixed characteristic analog of the theories of multiplier ideals in characteristic zero and test ideals in positive characteristic. The theory is based on the notion of $+$ -stable sections introduced in [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek3], [Reference Takamatsu and Yoshikawa21].

Let $(R,{\mathfrak m},k)$ be a complete noetherian local ring of mixed characteristic. Let $p>0$ denote the characteristic of the residue field k. Let ${\mathscr X}$ be a normal integral scheme which is proper over . The canonical sheaf $\omega _{\mathscr X}$ is reflexive ([20, Tag 0AWK] and [Reference Hartshorne12, Theorem 1.9]) and we fix a canonical divisor $K_{\mathscr X}$ with $\omega _{\mathscr X}={\mathcal O}_{\mathscr X}(K_{\mathscr X})$ [20, Tag 0EBM].

Definition 3.1. For a reflexive sheaf ${\mathscr M}={\mathcal O}_{\mathscr X}(M)$ associated with a divisor M and an effective ${\mathbb Q}$ -divisor B on ${\mathscr X}$ , the subspace of $+$ -stable sections of the adjoint line bundle $\omega _{\mathscr X}\otimes {\mathscr M}$ relative to B

$$\begin{align*}\mathbf{B}^0\bigl({\mathscr X},B,{\mathcal O}_{\mathscr X}(K_{\mathscr X}+M)\bigr) \subset H^0\bigl({\mathscr X},{\mathcal O}_{\mathscr X}(K_{\mathscr X}+M)\bigr) \end{align*}$$

is defined by

$$\begin{align*}\bigcap_{f\colon {\mathscr Y}\to {\mathscr X}}\mathrm{Im} \Bigl(H^0\bigl({\mathscr Y},{\mathcal O}_{\mathscr Y}(K_{\mathscr Y}+\lceil f^*(M-B)\rceil\bigr) \longrightarrow H^0\bigl({\mathscr X},{\mathcal O}_{\mathscr X}(K_{\mathscr X}+M)\bigr)\Bigr), \end{align*}$$

where an algebraic closure $\overline {\kappa ({\mathscr X})}$ of the function field $\kappa ({\mathscr X})$ is fixed, and f runs through the finite surjective morphisms $f\colon {\mathscr Y}\to {\mathscr X}$ from a normal integral scheme ${\mathscr Y}$ together with an embedding $\kappa ({\mathscr Y})\hookrightarrow \overline {\kappa ({\mathscr X})}$ [Reference Hacon, Lamarche and Schwede13, Definition 3.2 and Lemma 3.8].

For the rest of this section, we assume that the scheme ${\mathscr X}$ is regular and projective over S.

Definition 3.3. Take a very ample line bundle ${\mathscr H}$ on ${\mathscr X}$ . For an effective ${\mathbb Q}$ -divisor B on ${\mathscr X}$ and for each $i\in {\mathbb N}$ the subspace

(3.1) $$ \begin{align} {\mathbf B}^0\bigl({\mathscr X},B,\omega_{\mathscr X}\otimes {\mathscr H}^i\bigr) \subset H^0\bigl({\mathscr X},\omega_{\mathscr X}\otimes {\mathscr H}^i\bigr) \end{align} $$

generates the subsheaf ${\mathscr N}_i\subset \omega _{\mathscr X}\otimes {\mathscr H}^i$ which defines

(3.2)

The sequence $({\mathscr J}_i)_{i\in {\mathbb N}}$ is increasing and becomes stationary. Define the $+$ -test submodule $\tau _+(\omega _{\mathscr X},B)$ to be ${\mathscr J}_i$ for $i\gg 0$ [Reference Hacon, Lamarche and Schwede13, Definition 4.3].

Remark 3.4. (i) Definition 3.3 does not depend on the choice of ${\mathscr H}$ [Reference Hacon, Lamarche and Schwede13, Proposition 4.5].

(ii) For $i\gg 0$ we have $H^0({\mathscr X},\tau _+(\omega _{\mathscr X},B)\otimes {\mathscr H}^i)= {\mathbf B}^0({\mathscr X},B,\omega _{\mathscr X}\otimes {\mathscr H}^i))$ [Reference Hacon, Lamarche and Schwede13, Proposition 4.7]. If $B'\geq B$ , then $\tau _+(\omega _{\mathscr X},B')\subset \tau _+(\omega _{\mathscr X},B)$ and if F is an effective Cartier divisor, then [Reference Hacon, Lamarche and Schwede13, Lemma 4.8]

(3.3) $$ \begin{align} \tau_+(\omega_{\mathscr X},B+F)=\tau_+(\omega_{\mathscr X},B)\otimes {\mathcal O}_{\mathscr X}(-F). \end{align} $$

(iii) Equality (3.3) allows one to define $\tau _+(\omega _{\mathscr X},B)$ for a not necessarily effective ${\mathbb Q}$ -divisor B as $\tau _+(\omega _{\mathscr X},B+F)\otimes {\mathcal O}_{\mathscr X}(F)$ where F is a Cartier divisor such that $B+F$ is effective [Reference Hacon, Lamarche and Schwede13, Definition 4.14].

Definition 3.5. For an effective ${\mathbb Q}$ -divisor B on ${\mathscr X}$ , the ideal sheaf

(3.4)

is the called the +-test ideal associated with B [Reference Hacon, Lamarche and Schwede13, Definition 4.15, Lemma 4.18]. Using $F=-K_{\mathscr X}$ in Remark 3.4(iii), this is indeed a coherent ideal sheaf.

We recall the subadditivity conjecture of Hacon, Lamarche, and Schwede [Reference Hacon, Lamarche and Schwede13, Conjecture 8.3].

Conjecture 3.6. Given effective ${\mathbb Q}$ -divisors D and E on ${\mathscr X}$ , we have

$$\begin{align*}\tau_+({\mathcal O}_{\mathscr X},D+E)\subset \tau_+({\mathcal O}_{\mathscr X},D)\cdot \tau_+({\mathcal O}_{\mathscr X},E). \end{align*}$$

The above conjecture is still open, but Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek prove subadditivity for their new perturbation-friendly test ideals which we will introduce in the next section.

4 Perturbation friendly global test ideals

In this section, we present the perturbation friendly global test ideals after Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4]. They are suitable for our purpose as subadditivity is known for them in contrast to the test ideals considered in the previous section.

Let $(R,{\mathfrak m},k)$ be a complete discrete valuation ring of mixed characteristic with field of fractions K. We fix a flat projective integral regular scheme ${\mathscr X}$ over $S=\mathrm {Spec}(R)$ . We fix a canonical divisor $K_{\mathscr X}$ with $\omega _{\mathscr X}={\mathcal O}_{\mathscr X}(K_{\mathscr X})$ .

Definition 4.1. Let B be a ${\mathbb Q}$ -divisor on ${\mathscr X}$ . By [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Proposition 7.14(f)], there is a unique perturbation friendly test module $\tau (\omega _{\mathscr X},B)$ such that there exists an effective Cartier divisor $G^0$ with the property that

(4.1) $$ \begin{align} \tau(\omega_{\mathscr X},B)= \tau_+(\omega_{\mathscr X},B+ \varepsilon G) \end{align} $$

for all divisors $G \geq G^0$ on ${\mathscr X}$ and all $0<\varepsilon \ll 1$ (depending on G).

We define the perturbation friendly test ideal

(4.2)

Both the test module $\tau (\omega _{\mathscr X},B)$ and the test ideal $\tau ({\mathcal O}_{\mathscr X},B)$ are coherent [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Definition 7.12, Theorem 7.13]. If B is effective, then the fractional ideal sheaf $\tau ({\mathcal O}_{\mathscr X},B)$ is indeed an ideal sheaf in ${\mathcal O}_{{\mathscr X}}$ [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Definition 7.18].

Here are some properties of perturbation friendly test ideals $\tau ({\mathcal O}_{\mathscr X},B)$ including subadditivity which holds only conjecturally [Reference Hacon, Lamarche and Schwede13, Conjecture 8.3] for the test ideals $\tau _+({\mathcal O}_{\mathscr X},B)$ .

Theorem 4.2 (Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, Witaszek).

Let $B, B_1,B_2$ be ${\mathbb Q}$ -divisors on ${\mathscr X}$ .

(i) There exists a Cartier divisor $G^0$ on ${\mathscr X}$ with the property that

$$\begin{align*}\tau({\mathcal O}_{\mathscr X},B)= \tau_+({\mathcal O}_{\mathscr X},B+ \varepsilon G)\end{align*}$$

for all divisors $G \geq G^0$ on ${\mathscr X}$ and all $0<\varepsilon \ll 1$ (depending on G).

(ii) If B is a ${\mathbb Q}$ -divisor and F is a divisor on ${\mathscr X}$ , then

(4.3) $$ \begin{align} \tau(\omega_{\mathscr X},B+F)=\tau(\omega_{\mathscr X},B)\otimes {\mathcal O}_{\mathscr X}(-F), \end{align} $$

(4.4) $$ \begin{align} \tau({\mathcal O}_{\mathscr X},B+F)=\tau({\mathcal O}_{\mathscr X},B)\otimes {\mathcal O}_{\mathscr X}(-F). \end{align} $$

(iii) (Subadditivity) The perturbation friendly test ideals satisfy the subadditivity property, that is, we have

$$\begin{align*}\tau({\mathcal O}_{\mathscr X},B_1+B_2)\subset \tau({\mathcal O}_{\mathscr X},B_1)\cdot \tau({\mathcal O}_{\mathscr X},B_2) \end{align*}$$

for effective ${\mathbb Q}$ -divisors $B_1,B_2$ on ${\mathscr X}$ .

(iv) (Effective global generation) Let B be effective and let D be a divisor such that the ${\mathbb Q}$ -divisor $D-K_{\mathscr X}-B$ is big and nef, and let H be a globally generated ample divisor. Then $\tau ({\mathcal O}_{\mathscr X},B)\otimes {\mathcal O}_{\mathscr X}(nH+D)$ is globally generated by $\mathbf B^0({\mathscr X},B, {\mathcal O}_{\mathscr X}(nH+D))$ for all $n\geq \mathrm {dim}({\mathscr X}\otimes _Rk)$ .

(v) ( $\tau ({\mathcal O}_{\mathscr X},0)$ is vertical) The ideal sheaf $\tau ({\mathcal O}_{\mathscr X},0)$ is vertical, that is, its support is contained in the special fiber.

Proof. Everything is shown in [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4]. We give precise references. Property (i) holds by [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Corollary 7.19(b)] and (ii) follows from (3.3), (4.1), and (4.2). Statement (iii) is the subadditivity property [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Theorem 7.20(e)] and (iv) is the effective global generation property [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Corollary 7.22]. Recall for (v) that a coherent ideal of ${\mathscr X}$ is called vertical if its support is contained in the special fiber. Let denote the generic fibre of ${\mathscr X}$ , put , and recall that the Grauert–Riemenschneider sheaf $\mathscr {J}(X,\omega _X)$ on the regular scheme X equals $\omega _X$ [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Definition A.1]. We conclude from [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Proposition 7.14(c)], and formula (4.3) above that

$$ \begin{align*} \tau({\mathcal O}_{\mathscr X},0)|_X&=\tau(\omega_{\mathscr X},0)\otimes_{{\mathcal O}_{\mathscr X}}{\mathcal O}_{\mathscr X}(-K_{\mathscr X})|_X =\mathscr{J}(X,\omega_X)\otimes_{{\mathcal O}_X}{\mathcal O}_X(-K_X)={\mathcal O}_X. \end{align*} $$

It follows that the ideal sheaf $\tau ({\mathcal O}_{\mathscr X},0)$ is vertical.

Definition 4.3. Let D be a divisor on ${\mathscr X}$ with linear series $|D|\neq 0$ and $\lambda \in {\mathbb Q}_{>0}$ . Define the perturbation friendly test ideal of the linear series $|D|$ to be

(4.5) $$ \begin{align} \tau({\mathcal O}_{\mathscr X},\lambda\cdot |D|):=\sum_{E\in |D|}\tau({\mathcal O}_{\mathscr X},\lambda\cdot E). \end{align} $$

Thanks to the Noetherian assumption, one can pick a finite number of elements $D_1,\dots , D_r$ in $|D|$ such that $\sum _{i=1}^{r}\tau ({\mathcal O}_{\mathscr X},\lambda \cdot D_i)$ agrees with (4.5).

Lemma 4.4. If D is a divisor on ${\mathscr X}$ with linear system $|D|\neq \emptyset $ , then

(4.6) $$ \begin{align} \tau({\mathcal O}_{\mathscr X}, |D|) = \mathfrak{b}_{|D|}\cdot\tau({\mathcal O}_{\mathscr X}, 0) \end{align} $$

where denotes the base ideal of the linear system $|D|$ .

Proof. Our proof follows Hacon, Lamarche, and Schwede [Reference Hacon, Lamarche and Schwede13, Lemma 7.5(b)]. Using (4.4) and (4.5), one gets

$$\begin{align*}\tau({\mathcal O}_{\mathscr X}, |D|)=\sum_{E\in |D|} \tau({\mathcal O}_{\mathscr X}, E)=\sum_{E\in |D|}\tau({\mathcal O}_{\mathscr X}, 0)\otimes {\mathcal O}_{\mathscr X}(-E)=\mathfrak{b}_{|D|}\cdot \tau({\mathcal O}_{\mathscr X},0), \end{align*}$$

which shows (4.6).

Definition 4.5. Let D be a $\mathbb {Q}$ -divisor with Iitaka dimension $\kappa (D)\geq 0$ , so $mD$ is a divisor such that $|mD|\neq \emptyset $ for all large and sufficiently divisible $m\in \mathbb {N}$ , and $\lambda \in {\mathbb Q}_{>0}$ . Define the asymptotic perturbation friendly test ideal of $|D|$ by

(4.7)

One can show effective global generation and subadditivity also for the asymptotic perturbation friendly global test ideals.

Theorem 4.7. Let D be a $\mathbb {Q}$ -divisor on ${\mathscr X}$ with Iitaka dimension $\kappa (D)\geq 0$ .

(i) Let ${\mathscr H}={\mathcal O}_{\mathscr X}(H)$ be a globally generated ample line bundle, E be a divisor, and $\lambda \in {\mathbb Q}_{>0}$ . Let $n=\dim {\mathscr X}\otimes _Rk$ . If $E-K_{\mathscr X}-\lambda D$ is big and nef, then

(4.9) $$ \begin{align} \tau({\mathcal O}_{\mathscr X},\lambda\cdot ||D||)\otimes{\mathcal O}_{\mathscr X}(nH+E) \end{align} $$

is globally generated by a sub linear series of $H^0({\mathscr X}, {\mathcal O}_{\mathscr X}(nH+E))$ for all $n\geq {\mathrm {dim}({\mathscr X}\otimes _Rk)}$ .

(ii) For $q,r\in {\mathbb N}_{>0}$ , we have

$$\begin{align*}\tau\bigl({\mathcal O}_{\mathscr X},qr\cdot \|D\|\bigr)\subset \tau\bigl({\mathcal O}_{\mathscr X},r\cdot \|D\|\bigr)^q. \end{align*}$$

Proof. (i) follows from [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and Witaszek4, Remark 8.36]. For the convenience of the reader we give a proof following [Reference Hacon, Lamarche and Schwede13, Lemma 7.5(e)]. For sufficiently divisible $m\in {\mathbb N}$ we conclude from (4.8) that

$$ \begin{align*} \tau({\mathcal O}_{\mathscr X},\lambda\cdot ||D||)\otimes{\mathcal O}_{\mathscr X}(nH+E) &=\tau\Bigl({\mathscr O}_{\mathscr X},\frac{\lambda}{m}\cdot |mD|\Bigr)\otimes{\mathcal O}_{\mathscr X}(nH+E)\\ &=\sum_{i=1}^{r}\tau\Bigl({\mathcal O}_{\mathscr X}, \frac{\lambda}{m}\cdot D_i\Bigr)\otimes{\mathcal O}_{\mathscr X}(nH+E) \end{align*} $$

where we pick a finite number of elements $D_1,\dots , D_r$ in $|mD|$ as in Definition 4.3. From Remark 4.2(iv) we conclude that

$$\begin{align*}\tau\Bigl({\mathcal O}_{\mathscr X}, \frac{\lambda}{m}\cdot D_i\Bigr)\otimes{\mathcal O}_{\mathscr X}(nH+E) \end{align*}$$

is globally generated by $\mathbf B^0({\mathscr X},\frac {\lambda }{m}\cdot D_i, {\mathcal O}_{\mathscr X}(nH+E))\subset H^0({\mathscr X},{\mathcal O}_{\mathscr X}(nH+E))$ for all $n\geq \mathrm {dim}({\mathscr X}\otimes _Rk)$ . This finishes the proof of (i).

(ii) By homogeneity of asymptotic test ideals seen in Lemma 4.6(iv), we may assume $r=1$ . For sufficiently divisible m, we deduce from (4.8) that

$$\begin{align*}\tau\bigl({\mathscr O}_{\mathscr X},q\cdot ||D||\bigr)=\tau\Bigl({\mathscr O}_{\mathscr X},\frac{q}{m}\cdot |mD|\Bigr). \end{align*}$$

For any effective divisor $D_m \sim mD$ , using subadditivity in Remark (4.2)(iii) for the ${\mathbb Q}$ -divisor $\frac {1}{m}D_m$ , we get

$$\begin{align*}\tau\Bigl({\mathscr O}_{\mathscr X},\frac{q}{m}\cdot D_m\Bigr) = \tau\Bigl({\mathscr O}_{\mathscr X},q \cdot \frac{1}{m} D_m\Bigr) \subset \tau\Bigl({\mathscr O}_{\mathscr X}, \frac{1}{m} D_m\Bigr)^q \end{align*}$$

and hence

$$\begin{align*}\tau\bigl({\mathscr O}_{\mathscr X},q\cdot ||D||\bigr) \subset\tau\Bigl({\mathscr O}_{\mathscr X},\frac{1}{m}\cdot |mD|\Bigr)^q \subset\tau\bigl({\mathcal O}_{\mathscr X},\|D\|\bigr)^q \end{align*}$$

proving the claim.

5 Continuity of the envelope of the zero function

Let K be a complete discretely valued field of mixed characteristic $(0,p)$ . Let X be a n-dimensional smooth projective variety over K and L an ample line bundle on X. In the approach of Boucksom–Favre–Jonsson [Reference Boucksom, Favre and Jonsson5], [Reference Boucksom, Favre and Jonsson6] to pluripotential theory on $X^{\mathrm {an}}$ , a model ${\mathscr L}$ of L on the model ${\mathscr X}$ induces a curvature form $\theta $ on $X^{\mathrm {an}}$ and we denote by $P_\theta (u)$ the $\theta $ -psh envelope of a continuous real function u on $X^{\mathrm {an}}$ , see [Reference Gubler, Jell, Künnemann and Martin10, Subsections 2.4–2.6] for details. We follow the strategy from [Reference Boucksom, Favre and Jonsson6] and [Reference Gubler, Jell, Künnemann and Martin10] to show continuity of the envelope of the zero function $P_\theta (0)$ .

Proof. Consider the graded sequence $({\mathfrak a}_{m})_{m>0}$ of base ideals

$$\begin{align*}{\mathfrak a}_m=H^0({\mathscr X},{\mathscr L}^{m})\otimes {\mathscr L}^{-m}\subset {\mathcal O}_{\mathscr X} \end{align*}$$

associated with ${\mathscr L}$ . Write ${\mathscr L}={\mathcal O}_{\mathscr X}(D)$ for some divisor D on ${\mathscr X}$ . Let

denote the associated asymptotic perturbation friendly test ideal of exponent m in mixed characteristic. Note that we denote base ideals by $\mathfrak {a}$ and reserve $\mathfrak {b}$ for test ideals, to keep the same notations as in [Reference Boucksom, Favre and Jonsson6] and [Reference Gubler, Jell, Künnemann and Martin10]. Motivated by Lemma 4.6 and Remark 4.2(v), we consider also the coherent ideals

These ideals have the following properties:

1. (a) We have ${\mathfrak a}_m^{\prime } \subset {\mathfrak b}_m$ and these coherent ideal sheaves are vertical for m sufficiently large and divisible.

2. (b) We have ${\mathfrak b}_{ml} \subset {\mathfrak b}_m^l$ for all $l,m\in {\mathbb N}_{>0}$ .

3. (c) There exists some $m_0 \geq 0$ such that $\mathfrak {b}_m\otimes {\mathscr A}^{\otimes m_0}\otimes {\mathscr L}^{\otimes m} $ is globally generated for all integers $m>0$ .

Indeed these properties can be seen as follows: by definition of the asymptotic test ideals, we have $\tau ({\mathcal O}_{\mathscr X}, |mD|) \subset {\mathfrak b}_m$ . For sufficiently large and divisible m, we have

$$\begin{align*}{\mathfrak a}_m^{\prime}={\mathfrak a}_m\cdot\tau({\mathcal O}_{\mathscr X},0) = \tau({\mathcal O}_{\mathscr X}, |mD|) \subset {\mathfrak b}_m \end{align*}$$

by Lemma 4.4, and the base ideal ${\mathfrak a}_m$ is vertical as L is ample. Since ${\mathfrak a}_m^{\prime }={\mathfrak a}_m\cdot \tau ({\mathcal O}_{\mathscr X},0)$ , Remark 4.2(v) yields that ${\mathfrak a}_m^{\prime }$ is vertical. This proves (a).

Property (b) follows from Theorem 4.7(ii). Property (c) is shown as follows. Choose a divisor H on ${\mathscr X}$ such that ${\mathcal O}_{\mathscr X}(H)$ is ample and globally generated. We have $n=\mathrm {dim}\,{\mathscr X}\otimes _Rk$ . As ${\mathscr A}$ is ample, we can choose $m_0\in {\mathbb N}$ such that the line bundle

(5.1) $$ \begin{align} {\mathscr A}^{\otimes m_0}\otimes{\mathcal O}_{\mathscr X}\bigl(-K_{\mathscr X}-(n+1)H\bigr) \end{align} $$

is globally generated. Given $m\in {\mathbb N}$ we apply Theorem 4.7(i) to . Since $E-mD-K_{{\mathscr X}}=H$ is big and nef, we get that

(5.2) $$ \begin{align} \mathfrak{b}_m\otimes{\mathcal O}_{\mathscr X}(mD+H+K_{\mathscr X}+n H) \end{align} $$

is globally generated. Taking the tensor product of (5.1) and (5.2) we see that $\mathfrak {b}_m\otimes {\mathscr A}^{\otimes m_0}\otimes {\mathscr L}^{\otimes m}$ is globally generated.

Finally (a), (b) and (c) imply our claim by the strategy of the proof of [Reference Boucksom, Favre and Jonsson6, Theorem 8.5]. For convenience of the reader, we give here some details. For $m \gg 0$ , let

which is a super-additive sequence of $\theta $ -psh functions as shown at the beginning of the proof of loc. cit. Then Step 1 of the quoted proof shows that $P_\theta (0)= \sup_{m} \varphi _m = \lim _m \varphi _m$ on the quasi-monomial points of ${X^{\mathrm {an}}}$ . Using [Reference Gubler, Jell, Künnemann and Martin10, Proposition 2.10], this holds pointwise on the whole ${X^{\mathrm {an}}}$ .

We also consider the functions

It follows from the above that the sequence $\varphi _m^{\prime }$ also converges pointwise to $P_\theta (0)$ on ${X^{\mathrm {an}}}$ .

Then Step 2 of loc. cit. works again in our setting using (a)–(c) above as follows. For m sufficiently large as above and for all $l \in {\mathbb N}_{>0}$ , we have seen in (a) and (b) that ${\mathfrak a}_{ml}^{\prime} \subset {\mathfrak b}_{ml} \subset {\mathfrak b}_m^l$ for all $l \in {\mathbb N}_{>0}$ and hence

$$\begin{align*}\frac{1}{m} \log |{\mathfrak b}_m| \geq \sup_l \frac{1}{ml} \log|{\mathfrak a}_{ml}^{\prime}|=\sup_{l} \varphi_{ml}^{\prime} \end{align*}$$

on ${X^{\mathrm {an}}}$ . We conclude that

(5.3) $$ \begin{align} \frac{1}{m} \log |{\mathfrak b}_m| \geq \lim_m \varphi_m^{\prime} =P_\theta(0) \end{align} $$

on ${X^{\mathrm {an}}}$ . The remaining part of the proof of Step 2 is literally the same as in loc. cit. and even simpler as we have Step 1 on ${X^{\mathrm {an}}}$ and not only on the quasi-monomial points of ${X^{\mathrm {an}}}$ . Note that we use the global generation property (c) there.

6 Resolution of singularities

Let $K^\circ $ be a complete discrete valuation ring with field of fractions K and . Let X be a smooth projective variety over K of dimension n.

It is essential to show that projective models are dominated by SNC-models. In order to prove this we are going to use the following assumption.

7 Continuity of the envelope

Let K be a complete discretely valued field of mixed characteristic $(0,p)$ and . Let L be an ample line bundle on a regular projective variety X of dimension n over K. Let $\theta $ be a closed $(1,1)$ -form on X with ample de Rham class $\{\theta \}$ induced by a model ${\mathscr L}$ of L on a model ${\mathscr X}$ of X.

Proof. Using that u is a uniform limit of model functions on X, we may assume that u is itself a model function by [Reference Gubler, Jell, Künnemann and Martin10, Proposition 2.9(v)]. The model function u is defined by a vertical ${\mathbb Q}$ -divisor on a proper model ${\mathscr X}$ of X over S. By [Reference Gubler, Jell, Künnemann and Martin10, Proposition 2.9(7)], we may replace $(\theta ,u)$ for the proof by $(m\theta ,m u)$ for some $m\in {\mathbb N}$ . Hence we may assume without loss of generality that u is actually defined by a vertical divisor on ${\mathscr X}$ . If we apply [Reference Lütkebohmert16, Lemma 2.2] to ${\mathscr X}$ and a projective model of X as in Remark 6.2, then ${\mathscr X}$ is dominated by a projective model of X. Hence we can assume without loss of generality that ${\mathscr X}$ is a projective model. Then we choose a resolution of singularities ${\mathscr X}'\to {\mathscr X}$ as in Lemma 6.4 such that some power $L^{\otimes m}$ of L extends to an ample line bundle ${\mathscr A}$ on the regular projective model ${\mathscr X}'$ of X. As before we may assume that $m=1$ . By [Reference Gubler, Jell, Künnemann and Martin10, Proposition 2.9(4)], we get

$$\begin{align*}P_\theta(u)=P_{\theta+dd^cu}(0)+u. \end{align*}$$

By construction the class $\theta +dd^cu$ is induced by a line bundle ${\mathscr L}$ on ${\mathscr X}'$ whose restriction to X is isomorphic to L and Theorem 5.1 shows that $P_{\theta +dd^cu}(0)$ is a uniform limit of $(\theta +dd^cu)$ -psh model functions. This finishes our proof.

8 The Monge–Ampère equation

Let K be a complete discretely valued field with valuation ring R of mixed characteristic $(0,p)$ . In this section, we consider a projective regular variety X over K. We assume that resolution of singularities holds for projective models of X and embedded resolution of singularities holds for regular projective models of X. It follows that every projective model of X is dominated by a projective SNC-model. By Lemma 6.4 and Remark 6.5, any ample line bundle on X extends to an ample line bundle on a suitable dominating SNC model after possibly passing to a positive tensor power. As explained in [Reference Gubler, Jell, Künnemann and Martin10, Section 9], these assumptions are enough to set up a pluripotential theory for $\theta $ -psh functions with respect to a closed $(1,1)$ -form $\theta $ on ${X^{\mathrm {an}}}$ with ample de Rham class $\{\theta \}$ . All the results of [Reference Boucksom, Favre and Jonsson6, Sections 1–7] hold. If we assume continuity of the envelope for ample line bundles on X, then monotone regularization [Reference Boucksom, Favre and Jonsson6, Theorem 8.7] holds as well in our setting which is crucial to extend the Monge–Ampère measure $\mathrm {MA}_\theta (\varphi )$ from $\theta $ -psh model functions to bounded $\theta $ -psh functions on ${X^{\mathrm {an}}}$ . Then the results from [Reference Boucksom, Favre and Jonsson6, Sections 4–6] hold in our setting by the same arguments.

Remark 8.2. If L is an ample line bundle on X and if $\theta $ is induced by a model $({\mathscr X},{\mathscr L})$ of $(X,L)$ , then there is a bijective correspondence [Reference Boucksom, Favre and Jonsson5, Section 2.6]

(8.1) $$ \begin{align} \{\theta\mbox{-psh functions}\}\longleftrightarrow\{\mbox{semipositive metrics of } L\},\,\,\, \varphi\longmapsto {\|\hspace{1ex}\|}_{\mathscr L} \cdot e^{-\varphi}. \end{align} $$

It follows that Theorems 5.1 and 7.1 imply Theorem 1.1 and that Theorem 8.1 implies Theorem 1.2.