The kinematic formula in complex integral geometry

Abstract

Given two nonsingular projective algebraic varieties X, Y c P meeting transversely, it is classical that one may express the Chern classes of their intersec- tion X n Y in terms of the Chern classes of X and Y and the Kahler form (hyperplane class) of P. 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 connec- tion. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) X, Y c P, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of X n Y and those of X and Y is rather complicated. However, when we average integrals of Chern forms of X n g Y over all elements g of the group of motions of P, these can be expressed in a universal fashion in terms of integrals of Chem forms of X and Y. This is, then, the kinematic formula for the unitary group.