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Bando-Siu's paper Stable sheaves and Einstein-Hermitian metrics

Theorem 2 Let $(E,h)$ be a holomorphic vector bundle with a Hermitian metric $h$ defined on a Kähler manifold $(Y,\omega)$ outside a closed subset with locally finite Hausdorff measure of real co-dimension 4. Assume that its curvature tensor $F$ is locally square integrable on $Y$, then

$E$ extends to the whole space $Y$ as a reflexive sheaf $\mathcal{E}$

My question is that Is the extension theorem right for Hermitian manifold?

Thank you!

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  • $\begingroup$ You may assume $\omega$ is Gauduchon metric $\endgroup$
    – user21574
    Commented May 19, 2017 at 2:45
  • $\begingroup$ Moverover, you must impose some positivity notion on initial hermitian metric such that when you run the Hermitian Yang Mills flow , the solution remain positive. $\endgroup$
    – user21574
    Commented May 19, 2017 at 2:52
  • $\begingroup$ Than you for your comments. I wonder why I should assume $\omega$ to be Gauduchon. $\endgroup$
    – Faith
    Commented May 19, 2017 at 3:09
  • $\begingroup$ OK, It seems you need to read a lot of information on non-Kahler geometry. Anyway there is a thesis a person claimed that he has proven your question. There is a mistake on it. $\endgroup$
    – user21574
    Commented May 19, 2017 at 3:11
  • $\begingroup$ In the definition of degree of sheaf when it is equipped with Gauduchon metric then it is well defined since it is independent of a choice of metric $\endgroup$
    – user21574
    Commented May 19, 2017 at 3:15

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