the Chern connection
/ CHERN, also 'churn' /
If you put a notion of length on the fibres of a holomorphic vector bundle — a Hermitian metric — you would like a way to differentiate sections that is compatible both with that length (so lengths behave under parallel transport) and with the complex structure (so the differentiation respects holomorphy). On a real Riemannian manifold the analogous demand uniquely picks out the Levi-Civita connection. The Chern connection is the exact complex-geometry counterpart: the unique connection compatible with both the Hermitian metric and the holomorphic structure.
Precisely, let E be a holomorphic vector bundle with a Hermitian metric h. A connection nabla on E splits into a (1,0)-part and a (0,1)-part. The Chern connection is the unique connection that (i) is compatible with h, meaning d of the inner product equals the inner product of the covariantly differentiated arguments, and (ii) has (0,1)-part equal to the operator partial-bar of the holomorphic structure. Conditions (i) and (ii) overdetermine the connection just enough to make it unique. In a local holomorphic frame with metric matrix H, the connection 1-form is theta = H^{-1} partial H, and the curvature is the (1,1)-form F = partial-bar(H^{-1} partial H). For a line bundle with local metric e^{-phi} (a single positive function), the curvature is simply F = partial partial-bar phi, the fundamental local formula of the subject.
Why it matters: the curvature of the Chern connection represents, up to a universal constant, the first Chern class c_1(E) (Chern-Weil theory), so curvature literally computes a topological invariant. For line bundles, positivity of this curvature (i)F > 0 as a (1,1)-form is the precise meaning of a 'positive line bundle', the hypothesis of the Kodaira vanishing and embedding theorems. A common confusion: the Chern connection generally has nonzero torsion when viewed as a connection on the underlying real tangent bundle, and it coincides with the Levi-Civita connection only in the Kähler case — that coincidence is in fact one of several equivalent definitions of Kähler.
Take O(-1) over CP^1 with the metric induced from the standard inner product on C^2: in the chart with coordinate z the metric is h = 1 + |z|^2, so phi = log(1 + |z|^2). The Chern curvature is F = partial partial-bar phi = dz ^ d(z-bar) / (1 + |z|^2)^2. Integrating (i/2pi)F over CP^1 gives -1, which is exactly the degree c_1(O(-1)) = -1.
Curvature integrates to the Chern number: (i/2pi) integral of F over CP^1 gives c_1(O(-1)) = -1.
The Chern connection equals the Levi-Civita connection only when the metric is Kähler. In general the two differ (the Chern connection has torsion on the real tangent bundle), so do not assume the curvatures coincide off the Kähler case.