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Let $L$ be a line bundle over a compact complex manifold $X$ with a Hermitian metric $\omega$: $L$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

In order to define the nef line bundles over complex analytic spaces:

are there references about Hermitian metrics, differential forms, Chern connections in the complex analytic space framework?

Any answer, comment, advice will be appreciated.

Thanks in advance.

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  • $\begingroup$ Have you looked in Demailly's "Complex Analytic and Differential Geometry"? $\endgroup$
    – M.G.
    Mar 22, 2017 at 18:17
  • $\begingroup$ Yes, I have. But Demailly does not treat of vector bundles over complex analytic spaces. $\endgroup$
    – Armando j18eos
    Mar 22, 2017 at 18:24
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    $\begingroup$ You won't find anything about Hermitian metrics (do those even make sense for complex analytic spaces?) or chern connections but I think you can find differential forms in Grauert/Remmert - Coherent analytic sheaves and in Fischer - Complex analytic geometry. Other than that, can't you use some equivalent definition from algebraic geometry to define it for complex analytic spaces? $\endgroup$
    – Horstenson
    Mar 30, 2017 at 17:30
  • $\begingroup$ @Horstenson After a couple of days and research on Internet: I started as you suggest; I am consulting Fischer, Grauert - Remmert but Theory of Stein Spaces, and Gunning - Rossi Analytic Functions of Several Complex Variables. However, thank you for your advices. $\endgroup$
    – Armando j18eos
    Mar 31, 2017 at 8:04
  • $\begingroup$ Ok, good luck. From what I've seen so far complex analytic spaces are not heavily presented in the literature, especially in english. There's more in french and german. $\endgroup$
    – Horstenson
    Mar 31, 2017 at 14:20

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The standard reference about differential forms on complex analytic spaces is V. Ancona and B. Gaveau, Differential Forms on Singular Varieties: de Rham and Hodge Theory Simplified, Chapman and Hall, 2006. It doesn't seem to treat Chern classes, but there is some discussion of Hermitian metrics (appearing on resolutions, I think).

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