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I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.

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Can you clarify the question? The Dolbeault cohomology is defined with coefficients in any flat unitary holomorphic bundle or more generally with coefficients in any holomorphic Higgs bundle. I don't see what positivity has to do with the definition. Are you thinking about a harmonic metric somehow? –  Tony Pantev Aug 9 '13 at 12:28
Hassan is probably thinking about the relation to geometric quantization, where it matters (or at least historically mattered) that the index of the Dolbeault operator equals the dimension of holomorphic sections = naive quantum states only when the bundle is sufficiently positive.… –  Urs Schreiber Aug 9 '13 at 12:36
Urs Schreiber@ Yes, exactly –  Hassan Jolany 海桑乔朗丽 Aug 9 '13 at 13:16
@Hassan, okay, but then the question remains: what extly are you asking for? Texts on Dolbeault cohomology will automatically treat the general case, so you can pick any reference on Dolbeault that you want! :-) Or do you rather want a reference which discusses what the non-positivity of the prequantum bundle means for the interpretation of Dolbeault-geometric quantization? –  Urs Schreiber Aug 9 '13 at 14:22
Urs Schreiber@ I looking for second parts of your comment. Just a reference for non-positivity case –  Hassan Jolany 海桑乔朗丽 Aug 9 '13 at 15:52

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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 non-positive case more detailed.

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