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If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of $\omega$.

My question is what's the analog for degenerated Kähler metrics? For example, if $\omega$ belongs to a semi-ample class, we know that $\omega$ must degenerate along some exceptional sub-varieties. What's the Chern-Weil theory for this case? Any reference for this? Thanks.

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    $\begingroup$ You may see Chern classes for parabolic structures, for example, plarabolic vector bundle, see arxiv.org/pdf/math/0612144.pdf $\endgroup$ – user21574 Jun 2 '17 at 9:49
  • $\begingroup$ For example, if you have a divisor $D$ on $X$ and you want to intorduce Chern class , then it is better to introduce parabolic Chern class. This idea become important in Hitchin Kobayashi correspondence for framed vector bundle $\endgroup$ – user21574 Jun 2 '17 at 10:01
  • $\begingroup$ See this thesis math.unice.fr/~carlos/documents/… $\endgroup$ – user21574 Jun 2 '17 at 10:23
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    $\begingroup$ Chern classes are defined for any complex vector bundle on any real manifold; we don't need Kaehler structures at all. Just take the Chern forms of any Hermitian connection. Milnor and Stasheff explain the topological perspective, without picking connections at all, in terms of classifying spaces of bundles. If neither of these answers is what you are looking for, perhaps you could clarify the question. $\endgroup$ – Ben McKay Jul 2 '17 at 9:14
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For curves, you can look at section 3 in https://eudml.org/doc/182935

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    $\begingroup$ Hi Julien! Welcome!!! $\endgroup$ – diverietti Jul 2 '17 at 11:21
  • $\begingroup$ Hi! didn't see your message :) $\endgroup$ – Julien Grivaux Aug 15 '17 at 10:04

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