6.1. Monge-Ampère equations on normal Kähler spaces [02F6]

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6.1. Monge-Ampère equations on normal Kähler spaces

Let $\Omega$ be a smooth Kähler metric on $V$ . A classical result of P. Lelong states that if $U$ is relatively compact in $V$ , then $U^{reg}$ is of finite volume with respect to the smooth volume form $\Omega_{reg}^{n}$ .

This has been generalized by E.Bedford and A.Taylor in [BT], where the authors study Monge-Ampère measures for locally bounded psh functions. Since these measures do not charge proper analytic subsets, we obtain:

Proposition 6.1.

Let $\Omega$ be a semi-Kähler current with $L_{loc}^{\infty}$ potentials on $V$ . The Monge-Ampère measure $\Omega_{reg}^{n}$ is well defined on $V_{reg}$ and satisfies $\int_{U^{reg}}\Omega_{reg}^{n}<\infty,\text{ for all relatively compact subset }U\subset V.$

For any resolution $\pi:X\to V$ , the Monge-Ampère measure $(\pi^{*}\Omega)^{n}$ is well defined on $X$ and satisfies $\pi_{*}(\pi^{*}\Omega)^{n}=j_{*}\Omega_{reg}^{n}$ . Moreover if $\bar{\pi}:\bar{X}\to V$ is a resolution dominating $\pi$ (i.e. $\bar{\pi}=\pi\circ\psi$ for some bimeromorphic proper holomorphic map $\psi:\bar{X}^{\prime}\to X^{\prime}$ ), then $\psi_{*}(\bar{\pi}^{*}\Omega)^{n}=(\pi^{*}\Omega)^{n}$ .

The measure $\pi_{*}(\pi^{*}\Omega)^{n}$ is thus well defined on $V$ and independent of the choice of resolution. We will call it the Monge-Ampère measure of $\Omega$ and denote it by $\Omega^{n}$ . The mass of this measure only depends on the cohomology class of $\Omega$ , as follows again from [BT]:

Lemma 6.2.

Assume $V$ is compact. Let $\Omega_{1},\Omega_{2}$ two semi Kähler currents with $L_{loc}^{\infty}$ potentiel on $V$ . If they are cohomologous, i.e. $\Omega_{1}=\Omega_{2}+dd^{c}\varphi$ for some $\varphi\in L^{\infty}(X)$ , then $\int_{V}\Omega_{1}^{n}=\int_{V}\Omega_{2}^{n}$ .

We can now reformulate some of our previous results.

Theorem 6.3.

Let $V$ be a $n$ -dimensional compact normal Kähler space and $\Omega$ be a smooth Kähler form on $V$ . Then for every $f\in L^{p}(V,\Omega^{n})$ , $p>1$ , such that $\int_{V}f\Omega^{n}=\int_{X}\Omega^{n}$ , there is a unique $\varphi\in{\mathcal{C}}^{0}(V)$ such that

(\Omega+dd^{c}\varphi)^{n}=f\Omega^{n}\text{ and }\sup_{V}\varphi=-1.

Proof.

Let $\pi:X\to V$ be a resolution of $V$ . We may define a semipositive big smooth form on $X$ by $\omega=\pi^{*}\Omega$ . By Theorem 2.1 and Proposition 3.1 we can solve uniquely $(\omega+dd^{c}\bar{\varphi})^{n}=f\circ\pi\omega^{n}$ where $\bar{\varphi}$ is a continuous function on $X$ such that $\omega+dd^{c}\bar{\varphi}$ is semipositive. Let $F$ be a fiber of $\pi$ and $i:F\to X$ the inclusion map. $F$ is connected by Zariski’s main theorem. Furthermore $i^{*}\omega+dd^{c}i^{*}\bar{\varphi}$ is semipositive on $F$ . Since $i^{*}\omega=0$ , it follows that $i^{*}\varphi$ is a continuous psh function on $F$ . Hence $i^{*}\bar{\varphi}$ is constant. This implies that $\bar{\varphi}=\varphi\circ\pi$ where $\varphi$ is a continuous function on $V$ . We do have $(\Omega+dd^{c}\varphi)^{n}=f\Omega^{n}$ . ∎