Varying a Kahler metric in a neighborhood of a point I would like to know if the following statement (or a more general version of it) is contained in some book or article:
Statement. Let $(U,g)$ be a complex manifold with a Kahler metric $g$ and let $x\in U$ be a point. Let $U_x\subset U$ be a neighborhood of $x$ and $h$ be a Kahler metric on $U_x$. Then there is a smaller neighborhood $U_x'\subset U_x$ of $x$, and a metric $g'$ on $U$, such that $g'=h$ in $U_x'$ and $g'=g$ in $U\setminus U_x$.
I can prove this statement using regularized maximum, but would be grateful for a reference.
 A: This is not literally the answer that the OP wanted (a reference to the literature, which I am not aware of), but following the comments above let me write down the simple gluing argument.
Let $\widetilde{\max}$ be a regularized maximum function, and fix a small coordinate ball $B$ around $x$ (contained in $U_x$), with local coordinates $z=(z_1,\dots,z_n)$, where the Kähler forms of $g$ and $h$ can be written as $\omega_g=dd^c u$, $\omega_h=dd^c v$ for some smooth functions $u,v$ defined in some neighborhood of $\overline{B}$.
Fix a smooth nonnegative cutoff function $\chi$ identically $1$ near $x$ and compactly supported in $B$. We can then find $\delta>0$ small enough so that
$$\tilde{u}(z)=u(z)+\delta\chi(z)\log|z|,$$
is strictly plurisubharmonic on $B$, and of course it is smooth on $B$ minus $x$. We then choose $C>0$ large enough so that $v-C<u$ in a neighborhood of $\partial B$ (where $u=\tilde{u}$).
Then take $$w=\widetilde{\max}(v-C,\tilde{u}),$$
which by construction is smooth and strictly plurisubharmonic on $B$, it equals $v-C$ near $x$ and it equals $u$ near $\partial B$. Then we can define $\omega_{g'}=dd^c w$ on $B$ and $\omega_{g'}=\omega_g$ on $U\backslash B$, and we obtained the desired Kähler form.
