Chern connection - Imported Knowlpedia Content

Definition

Let $E\to X$ be a holomorphic vector bundle over a complex manifold, with holomorphic transition maps and a Hermitian metric $h$ . The Chern connection is the unique connection $\nabla$ that is a Hermitian connection and whose $(0,1)$ -part equals the bundle Dolbeault operator:

\nabla^{0,1}=\bar\partial_E.

Equivalently, a smooth section is holomorphic exactly when its $(0,1)$ -covariant derivative vanishes. Both the holomorphic structure and the metric are essential: neither one alone determines this connection.

Local formula

In a local holomorphic frame, let $H=(h(e_j,e_k))$ be the Hermitian metric matrix, using the convention that $h$ is linear in its second argument. The connection matrix of the Chern connection is

A=H^{-1}\partial H.

Thus its $(0,1)$ -part vanishes in that frame, while metric compatibility determines its $(1,0)$ -part. This formula also proves uniqueness locally.

Curvature

The curvature is

F_\nabla=\bar\partial\!\left(H^{-1}\partial H\right),

in the same convention, and has type $(1,1)$ . For a holomorphic line bundle with local holomorphic frame $e$ and $h(e,e)=H$ , one has $A=\partial\log H$ and $F_\nabla=\bar\partial\partial\log H$ . These formulas underlie the differential-geometric construction of Chern forms.

Conventions and scope

Changing whether the Hermitian form is linear in its first or second argument, or changing the sign convention for curvature, alters the displayed local formulas but not the invariant characterization. The construction applies to every Hermitian holomorphic vector bundle; no Kähler metric on the base is required Kobayashi, Chapter I, §4.

References

S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton University Press, 1987. DOI record. Relevant: Chapter I, §4, existence, uniqueness, and curvature of the Chern connection.
D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §4.2, Hermitian holomorphic bundles and their Chern connections.