8. Continuity of the envelope [039V]

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8. Continuity of the envelope

We present consequences of our application of test ideals in the last subsection under the assumption that we have resolution of singularities. As in Section 7 let $K$ be a complete discretely valued field of positive characteristic $p>0$ . Let $X$ be a smooth projective variety over $K$ of dimension $n$ . Consider $\theta\in\mathcal{Z}^{1,1}(X)$ with ample de Rham class $\{\theta\}\in N^{1}(X)$ .

Definition 8.1.

We say that $X$ is of geometric origin from a $d$ -dimensional family over a field $k$ if there exist a normal $d$ -dimensional variety $B$ over $k$ , a point $b\in B^{(1)}$ of codimension one and a projective variety $Y$ over $K^{\prime}=k(B)$ such that

(i)

there exists an isomorphism $\widehat{\mathcal{O}}_{B,b}\stackrel{{\scriptstyle\sim}}{{\to}}K^{\circ}$ of rings where $\widehat{\mathcal{O}}_{B,b}$ denotes the completion of the discrete valuation ring $R\coloneqq{\mathcal{O}}_{B,b}$ ,
(ii)

an isomorphism $Y\otimes_{K^{\prime}}K\simeq X$ over $K$ with $K^{\prime}\to K$ induced by (i).

Usually, we read these isomorphisms as identifications. Moreover, if $L$ is a line bundle (resp. if $\theta$ is a closed $(1,1)$ -form) on $X$ , we say that $(X,L)$ (resp. $(X,\theta)$ ) is of geometric origin from a $d$ -dimensional family over a field $k$ if the above conditions are satisfied and if we can also find a line bundle $L^{\prime}$ on $Y$ inducing $L$ by the base change $K/K^{\prime}$ (resp. a line bundle $\mathscr{L}_{R}$ on an $R$ -model $\mathscr{X}_{R}$ of $Y$ inducing $\theta$ by the base change ${K^{\circ}}/R$ ).

We can now formulate our main result about the continuity of the envelope:

Theorem 8.2.

Let $X$ be a smooth $n$ -dimensional projective variety over $K$ of geometric origin from a $d$ -dimensional family over a perfect field $k$ . Assume that resolution of singularities holds over $k$ in dimension $d+n$ . If $\theta$ is a closed $(1,1)$ -form on $X$ with ample de Rham class $\{\theta\}$ and if $u\in C^{0}({X^{{\mathrm{an}}}})$ , then ${P}_{\theta}(u)$ is a uniform limit of $\theta$ -psh model functions and thus ${P}_{\theta}(u)$ is continuous on ${X^{{\mathrm{an}}}}$ .

Using resolution of singularities in dimension three over a perfect field proven by Cossart–Piltant (see Theorem 6.3), we get the following application:

Corollary 8.3.

Let $X$ be a smooth projective surface over $K$ of geometric origin from a $1$ -dimensional family over a perfect field $k$ . Then the conclusion of Theorem 8.2 holds unconditionally.

We will prove Theorem 8.2 in several steps. First, we prove it in a completely geometric situation:

Lemma 8.4.

If we assume additionally that $(X,\theta)$ is of geometric origin from a $d$ -dimensional family over a perfect field $k$ , then Theorem 8.2 holds.

Proof.

Recall that the space of model functions ${\mathscr{D}}(X)$ is dense in $C^{0}(X^{\mathrm{an}})$ for the topology of uniform convergence [Gub98, Thm. 7.12]. Hence we may assume that $u\in{\mathscr{D}}(X)$ by Proposition 2.9(v). Observe that by Proposition 2.9(vii), we may replace $(\theta,u)$ by a suitable multiple. Hence we may assume without loss of generality that the model function $u$ is defined by a vertical divisor on a ${K^{\circ}}$ -model $\mathscr{X}^{\prime}$ . It is clear that we can choose $\mathscr{X}^{\prime}$ dominating the geometric model $\mathscr{X}=\mathscr{X}_{R}\otimes_{R}{K^{\circ}}$ of $\theta$ from Definition 8.1. It follows from Proposition 5.2(a) that we may assume $\mathscr{X}=\mathscr{X}^{\prime}$ . By Proposition 2.9(iv) we get

(8.1)

{P}_{\theta}(u)={P}_{\theta+dd^{c}u}(0)+u.

By construction, the class $\theta+dd^{c}u$ is induced by a line bundle on $\mathscr{X}_{R}$ and hence Corollary 7.4 yields that ${P}_{\theta+dd^{c}u}(0)$ is a uniform limit of $(\theta+dd^{c}u)$ -psh model functions $\varphi_{i}$ . Then $P_{\theta}(u)$ is the uniform limit of the sequence of $\theta$ -psh functions $\varphi_{i}+u$ by (8.1). ∎

In the lemma above we have proven Theorem 8.2 under the additional assumption that the $(1,1)$ -form $\theta$ is defined geometrically. In the next lemma, we relax this assumption a bit only assuming that the de Rham class of $\theta$ is defined geometrically.

Lemma 8.5.

If we assume additionally that the de Rham class $\{\theta\}$ is induced by an ample line bundle $L$ on $X$ such that $(X,L)$ is of geometric origin from a $d$ -dimensional family over the perfect field $k$ , then Theorem 8.2 holds.

Proof.

By Proposition 2.9(vi), we may assume that $\theta\in\mathcal{Z}^{1,1}(X)_{\mathbb{Q}}$ . By Proposition 2.9(vii), we may replace $L$ by a positive tensor power and $\theta$ by the corresponding multiple, and so we may assume that $L$ is very ample. In the notation of Definition 8.1, the assumption that $(X,L)$ is of geometric origin means that $L$ is the pull-back of a line bundle $L^{\prime}$ on the projective variety $Y$ over $K^{\prime}=k(B)$ . It follows easily from [Har77, Prop. III.9.3] and [EGAIV, Prop. 2.7.1(xii)] that $L^{\prime}$ is very ample. Then $L^{\prime}$ extends to a very ample line bundle on a projective $R$ -model of $Y$ for the discrete valuation ring $R=\mathcal{O}_{B,b}$ from Definition 8.1. By base change to ${K^{\circ}}$ , we conclude that there is a closed $(1,1)$ -form $\theta^{\prime}$ on ${X^{{\mathrm{an}}}}$ with de Rham class $\{\theta^{\prime}\}=\{\theta\}$ such that $(X,\theta^{\prime})$ is of geometric origin from a $d$ -dimensional family over $k$ .

By the $dd^{c}$ -lemma in [BFJ16a, Thm. 4.3] (see also the second author’s thesis [Jel16, Thm. 4.2.7] for generalizations) and using the rationality assumption on $\theta$ from the beginning of the proof, there is $v\in{\mathscr{D}}(X)$ such that $\theta^{\prime}=\theta+dd^{c}v$ . It follows from Proposition 2.9(iv) that

\displaystyle{P}_{\theta}(u)-v={P}_{\theta+dd^{c}v}(u-v)={P}_{\theta^{\prime}}(u-v).

By Lemma 8.4, the function ${P}_{\theta^{\prime}}(u-v)$ is a uniform limit of $\theta^{\prime}$ -psh functions. Adding $v$ , we get the claim for ${P}_{\theta}(u)$ . ∎

To prove Theorem 8.2 in full generality, the idea is to reduce to the above geometric situation by a similar trick as in [BFJ15, Appendix A].

Proof of Theorem 8.2.

We note first that by Proposition 2.9 (viii) the property that ${P}_{\theta}(u)$ is a uniform limit of $\theta$ -psh model functions is equivalent to the property that it is a continuous function. Let $K^{\prime}/K$ be a finite normal extension and denote by $q\colon X^{\prime}\coloneqq X\otimes_{K}{K^{\prime}}\to X$ the natural projection. Let $\theta^{\prime}=q^{*}\theta\in\mathcal{Z}^{1,1}(X^{\prime})$ . Then by Lemma 2.11 we have that

q^{*}{P}_{\theta}(u)={P}_{\theta^{\prime}}(q^{*}u).

It follows from [Ber90, Prop. 1.3.5] that ${X^{{\mathrm{an}}}}$ is as a topological space equal to the quotient of $(X^{\prime})^{\rm an}$ by the automorphism group of $K^{\prime}/K$ . We conclude that ${P}_{\theta}(u)$ is continuous if and only if ${P}_{\theta^{\prime}}(q^{*}u)$ is continuous.

Hence we can replace $K$ by a finite normal extension. Adapting the same argument as in [BFJ15, Lemma A.7] to characteristic $p$ , there exists a finite normal extension $K^{\prime}/K$ and a function field $F$ of transcendence degree $d$ over $k$ with $K^{\prime}$ as completion as in Definiton 8.1 such that $X^{\prime}\coloneqq X\otimes_{K}K^{\prime}$ is the base change of a projective variety $Y$ over $F$ with $N^{1}(Y/F)_{\mathbb{Q}}\to N^{1}(X^{\prime}/K^{\prime})_{\mathbb{Q}}$ surjective. Replacing $K^{\prime}$ by $K$ , we can assume that there is $Y$ as above with a surjective map

(8.2)

N^{1}(Y/F)_{\mathbb{Q}}\to N^{1}(X/K)_{\mathbb{Q}}

induced by the natural projection $X\to Y$ . To prove continuity of ${P}_{\theta}(u)$ , we may assume that the de Rham class $\{\theta\}$ is in $N^{1}(X)_{\mathbb{Q}}$ by using an approximation argument based on Proposition 2.9(vi). We conclude from surjectivity in (8.2) that there is a non-zero $m\in N$ such that $\{m\theta\}$ is induced by a line bundle $L$ with $(X,L)$ of geometric origin from a $d$ -dimensional family over $k$ (in fact from $Y$ ). By Proposition 2.9(vii), we have ${P}_{\theta}(u)=\frac{1}{m}{P}_{m\theta}(mu)$ and hence continuity follows from Lemma 8.5. ∎