  We compute the measure with multiplicity of the set of complex planes
intersecting a compact domain in a complex space form. The result is given in
terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two
different versions for the Gauss-Bonnet-Chern formula in complex space forms.
One of them gives the Gauss curvature integral in terms of the Euler
characteristic, and some hermitian intrinsic volumes. The other one, which is
shorter, involves the measure of complex hyperplanes meeting the domain. As a
tool, we obtain variation formulas in integral geometry of complex space forms.
