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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.
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.
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