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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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    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.2012-05-06

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