Let F be smooth complex bundle with flat connection D. Let H be hermitian metric, D not nec. compatible with H. FACT: There is another connection DD, commpatible with H, such that (1,0) form of (D-DD) and (0,1) form of (D-DD) are adjoint concerning H. Question: Assume DD^2 has zero (0,2)-part. When are the holomorphic structures given by by DD^(0,1) and D^(0,1) isomorphic ?
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