Let $X$ be a smooth complex projective variety with an ample line bundle $L$, and let $D\subset X$ be a smooth divisor. Suppose in an analytic neighborhood $U$ of $D$ there is a Kahler form $\omega$ such that the class of $\omega|_D$ in $H^{1,1}(D)$ equals $c_1(L)$. How to prove that there exists an even smaller analytic neighborhood $U'\subset U$ of $D$ such that $\omega$ can be extended from $U'$ to a Kahler form on whole $X$?
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9$\begingroup$ If $D$ be simple normal crossing divisor and take $D=\sum_{j=1}^n D_j$ where $D_j=\{\sigma_j=0\}$ , then $\sqrt{-1}\partial\bar\partial (-\log (|\sigma_j|^2))$ extends to a smooth real (1,1)-form on the whole $X$ $\endgroup$– user21574Commented May 14, 2017 at 17:33
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10$\begingroup$ See numdam.org/article/AFST_2014_6_23_4_893_0.pdf in Kahler current sense and not as Kahler form $\endgroup$– user21574Commented May 14, 2017 at 18:16
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5$\begingroup$ Hassan, thanks for the link! It is not far from what I am looking for $\endgroup$– aglearnerCommented May 14, 2017 at 18:52
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3$\begingroup$ You may see Skoda-El Mir extension theorem also which is related to your question but in Kahler current sense and not smooth Kahler form $\endgroup$– user21574Commented May 14, 2017 at 20:08
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2$\begingroup$ See Theorem 4.1 of arxiv.org/pdf/math/0609617.pdf $\endgroup$– user21574Commented May 15, 2017 at 2:00
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