A Poincar\'e-Hopf type formula for Chern character numbers

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincar'e-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincar'e-Hopf index formula to the case of complex vector fields (to appear in Mathematische Zeitschrift)