Guillemin–Ginzburg–Karshon's conjecture on equivariant cohomology Chern numbers

Let $G$ be a torus, and let $\Omega_*^{U,G}$ denote the equivariant geometric unitary bordism ring of closed unitary $G$-manifolds. An integral equivariant cohomology Chern number is a characteristic number obtained by integrating an integral equivariant cohomology Chern class over an equivariant bordism class.

Guillemin–Ginzburg–Karshon's conjecture. An element $\beta\in\Omega_*^{U,G}$ is zero if and only if all integral equivariant cohomology Chern numbers of $\beta$ vanish.

This conjecture removes the isolated-fixed-point hypothesis from their partial result, which established the equivalence for closed unitary $G$-manifolds with only isolated fixed points. It asks whether these characteristic numbers form a complete system of invariants for equivariant geometric unitary bordism when $G$ is a torus.