Curvature of the Chern connection - Imported Knowlpedia Content

Definition

Let $E\to X$ be a holomorphic vector bundle over a complex manifold, equipped with a Hermitian metric $h$ . If $\nabla$ is its Chern connection, the Chern curvature is the curvature

F_\nabla=\nabla^2\in\Omega^2(X;\operatorname{End}E).

It has pure type $(1,1)$ : $F_\nabla^{2,0}=F_\nabla^{0,2}=0$ . In a holomorphic frame with metric matrix $H$ , using the convention $\nabla=d+H^{-1}\partial H$ , one has

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

Thus it is determined jointly by the holomorphic structure and the Hermitian metric.

Type and local formula

The vanishing of the $(0,2)$ -part expresses the integrability of the holomorphic structure, while metric compatibility eliminates the conjugate $(2,0)$ -part. Thus the Chern connection packages the holomorphic and Hermitian data into an $\operatorname{End}E$ -valued $(1,1)$ -form. The matrix formula and its transformation law are developed in Kobayashi, Chapter I, §5.

For a holomorphic line bundle with a local holomorphic frame $e$ and $H=h(e,e)$ , the formula reduces to

F_\nabla=\bar\partial\partial\log H.

Changing the curvature or Hermitian-linearity convention may insert a minus sign.

Geometric consequences

Invariant polynomials applied to $F_\nabla$ produce closed differential forms representing the Chern classes of $E$ . In particular, under the displayed convention,

\frac{\sqrt{-1}}{2\pi}\operatorname{tr}(F_\nabla)

represents the first Chern class in de Rham cohomology. Contraction of $F_\nabla$ with a Hermitian form on the base gives the mean-curvature endomorphism used in the Hermitian Yang–Mills equation.

Examples and non-examples

The trivial holomorphic bundle with its constant standard metric has $H=I$ , hence $F_\nabla=0$ . A nonconstant Hermitian metric on the same holomorphic line bundle can have nonzero Chern curvature. The curvature of an arbitrary connection on a complex vector bundle is not Chern curvature unless that connection is the Chern connection for specified holomorphic and Hermitian structures; in particular, it need not have type $(1,1)$ .

References

Shoshichi Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton University Press, 1987. Publisher record. Relevant: Chapter I, §5, curvature of Hermitian holomorphic vector bundles.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: §4.2, Chern connections and curvature.