6. Applications to regularity [05BJ]

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6. Applications to regularity

In this section we use the connection of the non-archimedean Monge-Ampère operator to the real one to transfer two known regularity results for the solutions of the real Monge-Ampère equation to the non-archimedean case. Again $K$ denotes a non-archimedean non-trivially valued field.

Definition 6.1.

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open subset and $k\in\mathbb{N}$ . We write $C^{k}(\Omega)$ for the space of real valued, $k$ times continuously differentiable functions on $\Omega$ . Furthermore we denote by $L_{loc}^{1}(\Omega)$ the space of locally integrable functions on $\Omega$ i.e. functions $f:\Omega\rightarrow\mathbb{R}$ such that the restriction of $f$ to any compact subset of $\Omega$ is integrable. Let $f,g\in L_{loc}^{1}(\Omega)$ and $\beta\in\mathbb{N}^{n}$ . We say that $g$ is the $\beta$ -th weak derivative of $f$ if for any test function $\varphi\in C^{\infty}(\Omega)$ with compact support we have

\int_{\Omega}fD^{\beta}\varphi\;\boldsymbol{dx}=(-1)^{|\beta|}\int_{\Omega}g\varphi\;\boldsymbol{dx}

where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\mathbb{R}^{n}$ . We denote by $W^{k,1}_{loc}(\Omega)$ the space of locally integrable functions on $\Omega$ whose weak derivatives exist up to order $k$ .

Proposition 6.2.

Let $X$ be an $n$ -dimensional proper variety over $K$ and $\overline{L}$ a line bundle with a fixed formal metric. Let $\mu$ be a positive Borel measure on $X^{\textup{an}}$ and $\varphi$ a continuous function on $X^{\textup{an}}$ such that the metric on $\overline{L}\otimes\overline{\mathcal{O}}^{\varphi}$ is semipositive and solving the equation

c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})^{n}=\mu.

Let $\tau$ be an $n$ -dimensional open face of some skeleton $\Delta$ associated to a strongly nondegenerate strictly polystable formal model $\mathfrak{X}$ of $X^{\textup{an}}$ . Suppose that $\mathfrak{X}$ is algebraic, $\overline{L}$ has a model on $\mathfrak{X}$ and $\lambda\cdot\boldsymbol{dx}\leq\mu\leq\Lambda\cdot\boldsymbol{dx}$ on $\tau$ for some $\lambda,\Lambda>0$ where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\tau$ . Assume that $\varphi=\varphi\circ p_{\mathfrak{X}}$ . Then $\varphi\in W^{2,1}_{loc}(\tau)$ .

Proof.

By Corollary B.4 $\varphi$ is convex on every closed face of $\Delta$ . Note that the metric on $\overline{L}$ is trivial on $p_{\mathfrak{X}}^{-1}(\tau)$ . Hence we can apply Corollary 5.7 to get

\mu=c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})^{n}=\Deg(S)\cdot n!\cdot\MA(\varphi)

on $\tau$ where $S$ is the stratum of $\tilde{\mathfrak{X}}$ corresponding to $\tau$ . Now the claim follows from the corresponding fact in the real case [Moo15, Theorem 1.2]. ∎

Remark 6.3.

The condition $\varphi=\varphi\circ p_{\mathfrak{X}}$ is not automatic as shown by a counterexample of Burgos and Sombra, see [GJKM19, Appendix A].

Proposition 6.4.

Let $X$ be a smooth projective curve over $K$ and $\overline{L}$ a line bundle with a fixed formal metric. Let $\mu$ be a positive Borel measure on $X^{\textup{an}}$ and $\varphi$ a continuous function on $X^{\textup{an}}$ such that the metric on $\overline{L}\otimes\overline{\mathcal{O}}^{\varphi}$ is semipositive and solving the equation

c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})=\mu.

If $\tau$ is an open face of the skeleton $\Delta$ of a strictly semistable algebraic model $\mathscr{X}$ of $X^{\textup{an}}$ on which $\overline{L}$ has an algebraic model, $\mu$ is supported on $\Delta$ and $\mu=f\cdot\boldsymbol{dx}$ on $\tau$ for some positive function $f\in C^{k}(\tau)$ where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\tau$ then $\varphi\in C^{k+2}(\tau)$ .

Proof.

By [GJKM19, Proposition 1.2] we have $\varphi=\varphi\circ p_{\mathfrak{X}}$ . As in the previous result $\varphi$ is convex on $\tau$ and

\mu=c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})=\Deg(S)\cdot\MA(\varphi)

on $\tau$ . But a solution to the archimedean Monge-Ampère problem is given by a second antiderivative of $f$ and the solution is unique up to addition of a linear function. Hence $\varphi\in C^{k+2}(\tau)$ and $\Deg(S)\cdot\varphi^{\prime\prime}=f$ . ∎