Abstract
AbstractLet M be a compact complex manifold containing an irreducible curve C such that M — C is Kähler; in this paper we study the link between some cohomological properties of C and the obstructions to the existence of a Kähler metric on the whole of M. In particular we get that, if M is not Kähler, then C is a \documentclass{article}\pagestyle{empty}\begin{document}$ \left({\partial + \bar \partial} \right) $\end{document} –exact current, or there exists a positive current S of bidimension (1, 1) such that \documentclass{article}\pagestyle{empty}\begin{document}$ \partial \bar \partial S = 0,\,\chi _C S = 0 $\end{document} and S + C is \documentclass{article}\pagestyle{empty}\begin{document}$ \left({\partial + \bar \partial} \right) $\end{document} –exact. If C is a smooth rational curve, more precise results are given in connection with the normal bundle NC|M.
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