Complex manifolds and holomorphic coordinate changes

A complex n-manifold is a Hausdorff second-countable space locally modeled on open subsets of complex n-space, with coordinate transitions biholomorphic. Biholomorphic means holomorphic with a holomorphic inverse on the relevant overlap. Passing to the maximal compatible atlas removes dependence on the initial chart list. The underlying real manifold has dimension twice n because holomorphic maps are smooth. Hausdorffness and countability remain part of the definition; local analytic formulas alone do not rule out global topological pathologies.

A holomorphic map whose complex derivative is invertible at a point has a holomorphic local inverse near that point. An arbitrary holomorphic map need not be a coordinate change: a power map can have a vanishing derivative, and an exponential can fail global injectivity despite having no critical point. Conversely, a real diffeomorphism can fail the complex-linearity condition on its derivative. Checking inverse domains and the derivative addresses distinct local and global requirements rather than replacing one by the other.

A holomorphic tangent vector transforms using the complex Jacobian of the coordinate change. A one-form transforms dually: the differentials of the new coordinates are the Jacobian times the old differentials, while coefficient columns of a fixed covector transform by the inverse transpose. This duality preserves evaluation of a form on a vector. The canonical bundle, formed from top holomorphic exterior powers of the cotangent bundle, therefore has transition factors determined by Jacobian determinants. Its later curvature will record the first Chern class with a sign opposite to the tangent determinant bundle.

A holomorphic function on the manifold is one whose expression in each holomorphic chart is holomorphic. It suffices to check an atlas because composition with biholomorphic transitions preserves the condition. The same reasoning makes holomorphic maps between manifolds well defined. These definitions require no Hermitian metric and no compactness. A metric can measure a tangent vector's length, but the complex atlas already specifies which functions and sections count as holomorphic before any such measurement is chosen.

On the projective line, the chart with homogeneous first coordinate nonzero uses the ratio z, and the other chart uses its reciprocal w. Their overlap excludes zero in either chart. The reciprocal is holomorphic there, although it has a pole at the missing point when viewed as one affine formula. The second chart supplies a regular coordinate near that point at infinity. A pole of a coordinate expression is thus not automatically a singularity of the underlying manifold.

Local models also clarify the limits of a real matrix diagram. In one complex dimension, multiplication by i is represented by a fixed real rotation J. A real derivative is complex linear exactly when it commutes with J. A shear can have positive determinant and distort a grid smoothly while failing this commutator test. The instrument can test that linear distinction, but it neither constructs a nonlinear complex atlas nor verifies holomorphic compatibility on every overlap of a manifold.