Let $(M,I)$ be a complex manifold with a complex structure $I$, i.e. an endomorphism $I$ of the tangent bundle such that $I^2 = -Id$ and such that the subbundle $T^{1,0}$ of eigenvectors of $I \otimes \mathbb{C}$ with eigenvalue $i$ in $TM \otimes \mathbb{C}$ is involutive.

How to construct a torsion free connection $\nabla$ such that $\nabla(I)=0$?


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