I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
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The reference for Dolbeault Cohomology related to index theory I know are e.g. Ballmann "lectures on Kähler manifolds" or Huybrechts "complex geometry". You may also find something in Berline, Getzler, Vergne "Heat kernels..." Unfortunately I don't know if they treat the nonpositive case more detailed. 

