Timeline for Does $F_{A}^{0,2}=0$ for a connection $A$ on $TM$ almost complex give a complex structure?

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Jan 18, 2023 at 18:16	comment	added	Robert Bryant		@PaulCusson: An Hermitian connection $A$ on the tangent bundle that is without torsion wil always have $F_A^{0,2}=0$. This is a consequence of the first Bianchi identity.
Jan 18, 2023 at 17:05	comment	added	Paul Cusson		Another question just to make sure I understood your comment. Are you saying that all we need is a Hermitian connection without torsion to imply integrability, or are you also assuming that $F_{A}^{0,2}=0$ as a required condition?
Dec 10, 2022 at 19:11	vote	accept	Paul Cusson
Dec 10, 2022 at 14:42	history	edited	Robert Bryant	CC BY-SA 4.0	Added an actual counterexample in dimension 4.
Dec 10, 2022 at 14:35	comment	added	Robert Bryant		Well, an Hermitian connection on the tangent bundle that was without torsion would already imply that the almost complex structure was integrable. Anyway, I have a better example that I will add to my above answer.
Dec 10, 2022 at 14:08	comment	added	Paul Cusson		Thank you, this clarifies a lot for me. In that case, would finding a connection with curvature of type $(1,1)$ without torsion of type $(0,2)$ be enough, or is this simply an obstruction?
Dec 10, 2022 at 13:07	history	answered	Robert Bryant	CC BY-SA 4.0