  This paper shows that the integral equivariant cohomology Chern numbers
completely determine the equivariant geometric unitary bordism classes of
closed unitary $G$-manifolds, which gives an affirmative answer to the
conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$,
Appendix H], where $G$ is a torus. As a further application, we also obtain a
satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary
Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal
toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of
bordism and cobordism. Our approach heavily exploits Quillen's geometric
interpretation of homotopic unitary cobordism theory. Moreover, this method can
also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism
and can still derive the classical result of tom Dieck.
