Semi-Kähler currents [02EW]

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Semi-Kähler currents

Definition 5.7.

A semi-Kähler, resp. Kähler, resp. smooth Kähler, potential on $V$ is a family $(U_{i},\varphi_{i})_{i\in I}$ where $(U_{i})$ is an open covering of $V$ and $\varphi_{i}$ a psh function, resp. a strongly psh function, resp. a ${\mathcal{C}}^{\infty}$ -smooth strongly psh function, on $U_{i}$ such that $\varphi_{i}-\varphi_{j}$ is pluriharmonic on $U_{i}\cap U_{j}$ .

Define an equivalence relation on semi-kähler potentials requiring that $(U_{i},\varphi_{i})\sim(V_{j},\psi_{j})$ iff $\varphi_{i}-\psi_{j}$ is pluriharmonic on $U_{i}\cap V_{j}$ .

Definition 5.8.

A smooth Kähler metric $\Omega$ on $V$ is a $\sim$ -equivalence class of smooth Kähler potentials. A semi-Kähler (resp. Kähler) current on $V$ is a $\sim$ -equivalence class of semi-Kähler (resp. Kähler) potentials.

A semi-Kähler current $\Omega=(U_{i},\varphi_{i})_{i\in I}\mod\sim$ is said to have $L_{loc}^{\infty}$ (resp. ${\mathcal{C}}^{0}$ , resp. Hölder continuous) potentials iff each $\varphi_{i}$ is $L_{loc}^{\infty}$ (resp. ${\mathcal{C}}^{0}$ , resp. Hölder continuous).

We will on occasion drop the requirement that the local potentials of $\Omega$ are psh, replacing it by the requirement that they are locally the sum of a smooth and a psh function. The current $\Omega$ will then be called a quasi positive closed current on $V$ .

If it has locally bounded potentials, $\Omega$ is fully determined by the closed $(1,1)$ form $\Omega_{reg}$ on $V_{reg}$ defined on $U_{i}$ by $\Omega_{reg}=dd^{c}\varphi_{i}$ .

Let $\Omega$ be a smooth Kähler metric on $V$ with Kähler potential $(U_{i},\varphi_{i})$ . An upper semi-continuous function $\varphi:X\to\mathbb{R}\cup{-\infty}$ is said to be $\Omega$ -psh iff $\forall i$ $\varphi_{i}+\varphi$ is psh on $U_{i}$ . The semi-Kähler current whose potential is $(U_{i},\varphi+\varphi_{i})$ is denoted by $\Omega+dd^{c}\varphi$ .

Example 5.9.

Let $V=\mathbb{C}^{2}/{\pm 1}$ . Let $(x,y)$ be the usual affine coordinates on $\mathbb{C}^{2}$ , $(u,v,w)$ those on $\mathbb{C}^{3}$ . The formulas $u=x^{2},\ v=y^{2},\ w=xy$ realize $V$ as the closed subscheme of $\mathbb{C}^{3}$ whose equation is $uv-w^{2}=0$ . We have two ‘natural’Kähler metrics on $V$ , the first one is smooth with potential $\varphi^{1}=|u|^{2}+|v|^{2}+|w|^{2}$ , induced by the euclidean Kähler metric of $\mathbb{C}^{3}$ , the second one is the Kähler current whose potential is $\varphi^{2}=|u|+|v|$ . On $V^{reg}$ it is the quotient of the euclidean metric restricted to $\mathbb{C}^{2}-\{0\}$ . Near $0$ , $dd^{c}\varphi^{2}\gg dd^{c}\varphi^{1}$ .

The metric $dd^{c}\varphi^{2}$ is an example of an orbifold Kähler metric on $V$ . The results of [Y] extend without major modifications to Kähler orbifolds. For instance, in each Kähler class of a nodal K3 surface there is a unique Ricci flat orbifold metric.