Holomorphic bundles and Chern connections

A holomorphic vector bundle has holomorphic transition matrices. In a holomorphic local frame its bar-partial operator differentiates the smooth coefficient functions componentwise, and its square is zero. Conversely, a smooth complex vector bundle with a smooth Leibniz bar-partial operator whose square vanishes admits corresponding holomorphic local frames; this is an integrability theorem, invoked rather than proved here. The base is already a complex manifold. Merely choosing an arbitrary connection on a smooth bundle does not automatically give an integrable holomorphic structure.

Choose column coefficients s and a smooth positive Hermitian matrix H so the squared norm is s-star H s, where star denotes conjugate transpose. Write the connection as d plus a matrix of one-forms acting on the column. Requiring its zero-one part to be ordinary bar-partial in this frame leaves a one-zero matrix. Metric compatibility then forces the displayed expression. This fixes a concrete convention and avoids guessing whether a transpose or a sign has been absorbed into a source's definition of curvature.

The one-zero part of the Chern connection has square zero, as does its zero-one part. Its curvature is therefore of type one-one. For real tangent arguments it is skew-Hermitian relative to the metric, which makes multiplication by i appropriate when discussing Hermitian curvature forms. Vanishing zero-two curvature expresses holomorphic integrability, while vanishing of the entire curvature is the much stronger flatness condition. A nonzero one-one curvature is entirely compatible with a holomorphic bundle.

For a line bundle write its frame norm squared as the positive function exponential minus phi, with phi smooth and real. The connection is minus partial phi. Applying bar-partial reverses the order of partial and bar-partial, giving the plus sign in the displayed curvature. We represent the real first Chern class by i times this curvature divided by two pi. On the dual bundle the metric weight inverts and the curvature changes sign. These conventions will control positivity and the distinction between tangent and canonical first Chern classes later.

For a line frame multiplied by a nowhere-zero holomorphic function g, the metric weight multiplies by the squared modulus of g and the connection matrix gains the logarithmic derivative of g. That added one-form is holomorphic and locally closed, leaving curvature unchanged. The statement is on the overlap where g is nonzero; extending a meromorphic frame through its zeros would require treating divisors and possibly currents, not the smooth frame calculation used here.

A unitary frame instead normalizes length using a smooth real factor and need not be holomorphic. Its connection one-form is purely imaginary on real vectors, with the same curvature. In a quadratic weight on the plane, multiplying that connection by i gives a real rotational one-form. The vector-field instrument can visualize its Euclidean dual when the source parameter is zero. It does not compute a bundle transition, a non-Abelian connection, or a Chern class; those features remain explicit analytic calculations.