# Nef line bundles over complex analytic spaces

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?

• Have you looked in Demailly's "Complex Analytic and Differential Geometry"? – M.G. Mar 22 '17 at 18:17
• Yes, I have. But Demailly does not treat of vector bundles over complex analytic spaces. – Armando j18eos Mar 22 '17 at 18:24
• 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? – Horstenson Mar 30 '17 at 17:30
• @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. – Armando j18eos Mar 31 '17 at 8:04
• 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. – Horstenson Mar 31 '17 at 14:20