Invariant Polynomials on complexified bundles with connection

[lavinia]

I would like to know if the following correct.

Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle.

The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle.

From this it seems that the Pontryagin forms derived from the invariant polynomials in the curvature forms are just are the same as for the real curvature forms.

So for instance for an oriented real 2 plane bundle with a connection that is compatible with a metric on the bundle, the first Pontryagin class would be the square of the curvature 2 form and the first Chern polynomial would be zero.

 

[jvicens]

I can confirm that the statements made in the forum post are correct. When extending a connection on a real vector bundle to its complexification, the curvature forms of the complexification will indeed be the same as the curvature 2 forms of the real bundle. This is because the complexification essentially adds a new dimension to the bundle, but does not change the underlying structure of the bundle. Therefore, the Pontryagin forms derived from the invariant polynomials in the curvature forms will also be the same for the real and complex bundles. In the specific example given, the first Pontryagin class would indeed be the square of the curvature 2 form, and the first Chern polynomial would be zero. This is a well-known result in differential geometry and topology.