Please re-state...has zero measure when you integrate something on it.
I am reading a paper by Claude LeBrun, which I have been discussing with several of my classmates but we cannot seem to explain a fact he uses: A codimension 2 submanifold B of the complex plane CP_2 has zero measure when you integrate something on it. I believe this is general statement, but really what we are concerned with is determining the first Chern class, so we are integrating functions of curvature 2-forms.
I have been trying to explain why this works using some theorems of exterior algebra and Hodge duality (since a∧*b=<a,b>du for measure du) but I haven't really gotten anywhere. Any insight would be appreciated.