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Is there an Ohsawa-Takagushi $L^2$-Extension theorem for Kahler manifolds? For projective varieties Siu-Paun proved:

Let $\pi \colon X \to \mathbb D$ be a projective family of $n$-folds and $X_0$ be the central fiber. Let $L \to X$ be a line bundle with singular hermitian metric $e^{-\kappa}$ such that $\sqrt[]{-1}\partial\bar\partial \kappa \geq 0$. Take $s\in H^0(X_0,K_{X_0}^m\otimes L)$ such that $$\int_{X_0}|s|^2\omega^{-n(m-1)}e^{-\kappa}<\infty.$$ Then there is a section $\sigma\in H^0(X,K_{X}^m\otimes L)$ such that $\sigma|_{X_0}=s\wedge (d\pi)^m$ and $$\int_{X}|\sigma|^2\omega^{-(n+1)(m-1)}e^{-\kappa}<\infty.$$

Does such a theorem work for K\"ahler manifolds? If not, where is the difficulty in proof of such theorems for Kahler manifolds?

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    $\begingroup$ You should have a look at that paper by Junyan Cao: arxiv.org/pdf/1404.6937v1.pdf $\endgroup$
    – Henri
    Commented Jan 8, 2017 at 20:21
  • $\begingroup$ @Henri He has assumed that central fiber is smooth, which central fiber $X_0$ may not be smooth and can have very bad singularities. $\endgroup$
    – pickasa
    Commented Jan 8, 2017 at 21:12
  • $\begingroup$ Well, you make that assumption too. In any case, if the central fiber is singular, then nothing is known even if the family is projective and the central fiber has snc singularities. $\endgroup$
    – Henri
    Commented Jan 8, 2017 at 23:01
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    $\begingroup$ If this (or something very similar) were true, then I believe one could show invariance of plurigenera for Kahler manifolds, which is open. (This was probably implicit in the question based on the reference to the papers of Siu and Paun, but I thought I would mention it just in case!) $\endgroup$
    – msteve
    Commented Jan 8, 2017 at 23:28
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    $\begingroup$ In fact if you assume hermitian metric of central fiber is smooth and singular hermitian metric $L\to X$ has at worst algebraic singularities or is Kawamata log terminal singularities i.e, $\mathcal I(h_L|_{X_0})=\mathcal O_{X_0}$, then you can have Ohsawa-Takegushi-Manivel theorem by recent result of Roufi or Li Yi, and on its recent generalized version of Demailly see this video college-de-france.fr/site/claire-voisin/… $\endgroup$
    – user21574
    Commented Jan 11, 2017 at 20:56

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