8.4. An alternative approach [01C8]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

8.4. An alternative approach

We now give a more explicit description of the solution to $\MA(\varphi)=\mu$ , when $\mu$ is a finite sum of Dirac masses at divisorial points. Let $\omega$ be a form as in §4 (not necessarily normalized), and assume that $\omega$ satisfies the orthogonality property.

Lemma 8.5.

Let $S=\{x_{1},...,x_{N}\}\subset X^{\mathrm{div}}$ be a finite set of divisorial points, and set for $t=(t_{1},...,t_{N})\in\mathbf{R}^{N}$

(8.4)

\varphi_{S,t}:=\sup\left\{\varphi\mid\varphi\in\PSH(X,\omega),\,\varphi(x_{i})\leq t_{i}\text{ for }i=1,...,N\right\}~.

Then $\varphi_{S,t}$ is a continuous $\omega$ -psh function, and $\MA(\varphi_{S,t})$ is supported in $S$ .

Proof.

Let $\mathcal{X}$ be an SNC model such that all $x_{i}$ appear as vertices of $\Delta_{\mathcal{X}}$ . By Theorem 2.10 there exists a constant $M>0$ such that $\sup_{X}\varphi\leq M$ for all $\varphi\in\PSH(X,\omega)$ such that $\varphi(x_{1})\leq t_{1}$ . Since adding a constant $c$ to the $t_{i}$ only replaces $\varphi_{S,t}$ with $\varphi_{S,t}+c$ , we may thus assume $t_{i}\leq-1$ and $\varphi\leq-1$ as soon as $\varphi\in\PSH(X,\omega)$ satisfies $\varphi(x_{1})\leq t_{1}$ . Now let $f_{\mathcal{X}}\in\mathcal{D}(X)_{\mathbf{R}}$ be the unique function that is linear on the faces of $\Delta_{\mathcal{X}}$ , takes value $t_{i}$ at $x_{i}$ for each $i$ , $0$ at any other vertex of $\Delta_{\mathcal{X}}$ , and such that $f_{\mathcal{X}}=f_{\mathcal{X}}\circ p_{\mathcal{X}}$ . Since each $\varphi\in\PSH(X,\omega)$ is convex on the faces of $\Delta_{\mathcal{X}}$ and satisfies $\varphi\leq\varphi\circ p_{\mathcal{X}}$ , we have $\varphi(x_{i})\leq t_{i}$ for all $i$ iff $\varphi\leq f_{\mathcal{X}}$ , hence $\varphi_{S,t}=P_{\omega}(f_{\mathcal{X}})$ . This already shows that $\varphi_{S,t}$ is continuous and $\omega$ -psh, and the orthogonality property further shows that $\MA(\varphi_{S,t})$ is supported in $\{f_{S,t}=f_{\mathcal{X}}\}$ for each SNC model $\mathcal{X}$ as above. We thus see that $\supp\MA(\varphi_{S,t})\subset\bigcap_{\mathcal{X}}\{f_{\mathcal{X}}<0\}$ .

We claim that the latter intersection is in fact equal to $\{x_{1},...,x_{N}\}$ , which will conclude the proof of the lemma. For each model $\mathcal{X}$ and each $x\in X$ we may consider the center (or reduction) $c_{\mathcal{X}}(x)\in\mathcal{X}_{0}$ . Let $E_{i}\in\Div_{0}(\mathcal{X})$ be the component of $\mathcal{X}_{0}$ with generic point $c_{\mathcal{X}}(x_{i})$ , and let $\varphi_{\mathcal{X},i}$ be the model function determined by $E_{i}$ . For each $x\in X$ we have $f_{\mathcal{X}}(x)=f_{\mathcal{X}}(p_{\mathcal{X}}(x))=\sum_{i}t_{i}\varphi_{\mathcal{X},i}(x)$ , hence

\bigcap_{\mathcal{X}}\{f_{\mathcal{X}}<0\}=\bigcup_{i}\bigcap_{\mathcal{X}}\left\{x\in X\mid c_{\mathcal{X}}(x)\in\overline{c_{\mathcal{X}}(x_{i})}\right\}=\{x_{1},...,x_{N}\}.

∎

As a consequence of this result, for any divisorial point $x\in X^{\mathrm{div}}$ then

(8.5)

\varphi_{x}:=\sup\left\{\varphi\mid\varphi\in\PSH(X,\omega),\,\varphi(x)\leq 0\right\}

solves $\MA(\varphi_{x})=\{\omega\}^{n}\,\delta_{x}$ , since the two measures have the same mass. More generally we have:

Proposition 8.6.

Let $S=\{x_{1},...,x_{N}\}\subset X^{\mathrm{div}}$ be a finite set of divisorial points and let $\mu$ be a positive Radon measure of mass $\{\omega\}^{n}$ with support contained in $\{x_{1},...,x_{N}\}$ . Then there exists $t\in\mathbf{R}^{N}$ such that the function $\varphi_{S,t}$ defined by (8.4) solves $\MA(\varphi_{S,t})=\mu$ .

Proof.

By Theorem A’, we can choose $\varphi$ be a continuous $\omega$ -psh function satisfying $\MA(\varphi)=\mu$ . Set $t_{i}=\varphi(x_{i})$ for $i=1,...,N$ . We claim that $\varphi_{S,t}=\varphi$ , which will conclude the proof. On the one hand we have $\varphi\leq\varphi_{S,t}$ by (8.4), since $\varphi$ is $\omega$ -psh and satisfies $\varphi(x_{i})\leq t_{i}$ . On the other hand we have $\varphi_{S,t}=\varphi$ on the support of $\MA(\varphi)$ , hence $\varphi_{S,t}\leq\varphi$ by Lemma 8.4. ∎

Remark 8.7.

Consider the setting of Theorem A, i.e. $\{\omega\}$ is the class of an (ample) line bundle $L$ on $X$ . The strategy proposed in the preliminary work [KT00] to solve Monge-Ampère equations mostly deals with the case of a Dirac mass $\mu$ at a divisorial point $x\in X^{\mathrm{div}}$ . The authors introduce the envelope (8.5), and assume by contradiction that $\MA(\varphi_{x})$ is not supported at $x$ . They define a limit functional $F$ obtained by looking at the asymptotics of ball volumes in the space of sections of $mL$ as $m\to\infty$ , and indicate that $F$ should satisfy $F(\varphi_{x}+\varepsilon f)=F(\varphi_{x})+\varepsilon\int f\MA(\varphi_{x})+O(\varepsilon^{2})$ for each $f\in C^{0}(X)$ . Comparing with [BB10] in the complex case, $F$ is likely to coincide with $E_{\omega}\circ P_{\omega}$ , so that a version of the differentiability property (Theorem 7.2) would also be a key ingredient in the approach proposed in [KT00].

Remark 8.8.

We do not know whether the function $\varphi_{x}$ in (8.5) is necessarily a model function. This is the case on a toric variety, see Proposition 9.1 below, but we suspect the answer is no in general.

Pick an SNC model $\mathcal{X}$ , an extension $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ of $L$ , let $\omega$ be the curvature form of the model metric defined by $\mathcal{L}$ . Let also $E$ be a component of $\mathcal{X}$ corresponding to the divisorial point $x=x_{E}$ . We have $\varphi_{x}=P_{\omega}(-f_{E})$ up to a constant. On the other hand, by [BFJ11, Theorem 8.5],

P_{\omega}(-f_{E})=\lim_{m}\frac{1}{m}\log|\mathfrak{a}_{m}|

where $\mathfrak{a}_{m}$ denotes the base-ideal of $m\mathcal{L}^{\prime}$ with $\mathcal{L}^{\prime}:=\mathcal{L}-E$ . As a consequence, $\varphi_{x}$ is indeed a model function as soon as the graded $S$ -algebra $\bigoplus_{m\geq 0}H^{0}(\mathcal{X},m\mathcal{L}^{\prime})$ is finitely generated. Building on Nakayama’s counterexample to the existence of Zariski decompositions [Nak04], it is reasonable to expect this algebra not to be finitely generated in general, and to subsequently prove that $\varphi_{x}$ is not a model function.