Given two nonsingular projective algebraic varieties $X,Y \subset {{\mathbf {P}}^n}$, $Y \subset {{\mathbf {P}}^n}$ meeting transversely, it is classical that one may express the Chern classes of their intersection $X \cap Y$ in terms of the Chern classes of $X$ and $Y$ and the Kähler form (hyperplane class) of ${{\mathbf {P}}^n}$. This depends on global considerations. However, by putting a hermitian connection on the tangent bundle of $X$, we may interpret the Chern classes of $X$ as invariant polynomials in the curvature form of the connection. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) $X$, $Y \subset {{\mathbf {P}}^n}$, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of $X \cap Y$ and those of $X$ and $Y$ is rather complicated. However, when we average integrals of Chern forms of $X \cap gY$ over all elements $g$ of the group of motions of ${{\mathbf {P}}^n}$, these can be expressed in a universal fashion in terms of integrals of Chern forms of $X$ and $Y$. This is, then, the kinematic formula for the unitary group.