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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.

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    $\begingroup$ This statement is definitely "folklore" in the field, and it probably doesn't appear in exactly this form in any reference (I may be wrong of course). The proof is exactly as you hinted, using the regularized maximum (I guess you know the proof, since you only asked for a reference), which I believe was introduced by Demailly for this purpose. $\endgroup$
    – YangMills
    Commented Oct 13, 2017 at 14:03
  • $\begingroup$ Thanks for this comment! My detailed proof takes a half page... I wonder how would you do in an article if you would need to use this statement. Would you just say that is a folklore result or would you give a proof? $\endgroup$
    – aglearner
    Commented Oct 13, 2017 at 17:06
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    $\begingroup$ The proof is easy. Let $\widetilde{\max}$ be a regularized maximum, choose a small coordinate ball $B$ around $x$ (contained in $U_x$), where we can write $\omega_g=dd^c u, \omega_h=dd^c v$, and fix a cutoff function $\chi$ identically $1$ near $x$ and compactly supported in $B$. Choose $\delta>0$ small enough so that $\tilde{u}(z)=u(z)+\delta\chi(z)\log|z|$ is strictly psh on $B$, then choose $C>0$ large so that $v-C < u$ in a neighborhood of $\partial B$ (where $u=\tilde{u}$). Then $w=\widetilde{\max}(v-C,\tilde{u})$ is smooth and strictly psh in $B$. $\endgroup$
    – YangMills
    Commented Oct 14, 2017 at 1:49
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    $\begingroup$ By construction $w$ equals $v-C$ near $x$ and it equals $u$ near $\partial B$. So you can define the Kahler form of $g'$ to be equal to $dd^c w$ in $B$ and to $\omega_g$ on $U\backslash B$. This satisfies your requirements. $\endgroup$
    – YangMills
    Commented Oct 14, 2017 at 1:50
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    $\begingroup$ You can just use this argument in your article. Or yours (which is probably the same), and just say that it's folklore but you include a proof for the reader's convenience, or something like that. $\endgroup$
    – YangMills
    Commented Oct 14, 2017 at 1:52

1 Answer 1

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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.

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