Every compact Kähler manifold has a canonical $spin^c$ structure. Moreover, the associated Dirac operator is isomorphic to $\overline{\partial} + \overline{\partial}^*$, acting on $\Omega^{(0,\bullet)}$ twisted by an appropiate line bundle. When the Kähler condition is relaxed to Hermitian, what can one say about the operator $\overline{\partial} + \overline{\partial}^*$?
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3$\begingroup$ The operator still has the same principal symbol of the $spin^c$ Dirac operator, so it still produces the same index. However, its kernel could be different, so it no longer computes Dolbeault cohomology. $\endgroup$ – Sebastian Goette Feb 1 '16 at 17:56