I'm a beginner in complex geometry and I want to ask a question about the definition which I did not find explicit formula.
What's the meaning of 'a connection preserve the complex structure'? $$ \nabla J=0. $$ Does it mean that for local coordinate $\left(U ; x^i, y^i\right)$, for any two $v_1$ and $v_2$ in {$x^i, y^i $}, $$ \nabla_{v_i} J(v_j)=0. $$