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The Kähler identities give commutation relations between the Lefschetz operator ($\alpha \mapsto \omega \wedge \alpha$), the differential operators $\partial, \bar{\partial}$ and its adjoints $\Lambda,\partial^*, \bar{\partial}^*$. (see Math World page).

Usually, these identities are used to show that the Laplacians $\Delta_d,\Delta_{\partial},\Delta_{\bar{\partial}}$ agree up to a constant multiple and this is of great importance for Hodge theory on compact Kähler manifolds.

Does anyone know another interesting application of these identities? Are they used to study non-compact Kähler manifolds?

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    $\begingroup$ This may be unsatisfactory, but I hate to see this very interesting question unnoted. The Kähler condition posses restrictions to the topology, for compact as well as for non-compact manifolds. The easiest ones follow by Hodge symmetry (e.g. the second betti number must be even and therewith one can easily show, that there is no compact Kähler manifold with fundamental group $SL_2(\mathbb{Z})$). But there are other(/stronger) implications as well. I recently found an excellent book on the topic called "Fundamental groups of compact Kähler manifolds" by several authors, e.g. J. Amoros. $\endgroup$
    – Ben
    May 6, 2012 at 11:41
  • $\begingroup$ This is not unsatisfactory at all! I'll check this book out, thanks for the advice. $\endgroup$
    – Lucas Kaufmann
    May 6, 2012 at 23:17
  • $\begingroup$ Another application of the Kähler identities I forgot to mention is the Hard Lefschetz Theorem, which can be prooved using the fact that the Lefschetz operator commutes with the laplacians. But this is still in the compact world. $\endgroup$
    – Lucas Kaufmann
    May 6, 2012 at 23:20

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