# Equivariant unitary bordism and equivariant cohomology Chern numbers

# Equivariant unitary bordism and equivariant cohomology Chern numbers

By using the universal toric genus and the Kronecker pairing of bordism and cobordism, we show 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 the book [Moment maps, cobordisms, and Hamiltonian group actions. Appendix J by Maxim Braverman. Mathematical Surveys and Monographs, {\bf 98}. American Mathematical Society, Providence, RI, 2002], where G is a torus. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. As a further application, we also obtain a satisfactory solution of [Question (A), \S1.1, Appendix H] of the above book on unitary Hamiltonian G-manifolds. This is a joint work with Wei Wang.