We show that the integral equivariant cohomology Chern numbers completely
determine the equivariant geometric unitary bordism classes of closed unitary
–manifolds,
which gives an affirmative answer to a conjecture posed by Guillemin, Ginzburg and Karshon
(Moment maps, cobordisms, and Hamiltonian group actions, Remark H.5 in Appendix H.3),
where
is a torus. As a further application, we also obtain a satisfactory
solution of their Question (A) (Appendix H.1.1) on unitary Hamiltonian
–manifolds.
Our key ingredients in the proof are the universal toric genus defined by Buchstaber,
Panov and 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
–equivariant
unoriented bordism and can still derive the classical result of tom Dieck.
Keywords
equivariant unitary bordism, Hamiltonian bordism,
equivariant cohomology Chern number
