If (U_1,𝜓₁) and (U_2,𝜓₂) are two complex charts, then the transition function is 𝜓₁₂:𝜓₁(U_1∩U_2)→𝜓₂(U_1∩U_2), given by 𝜓₁₂=𝜓₂∘𝜓₁⁻¹. We say M is a complex manifold if it has an atlas of complex charts (U,𝜓), such that all the transition functions are holomorphic, as maps from ℂᵐ to itself.