The Quantum Cohomology of Homogeneous Varieties

The notion of quantum cohomology was first proposed by Witten [Va, Wi], based on topological field theory. Its mathematical theory was only established recently by Y. Ruan and the second named author [RT, Ru], where they proved the existence of the quantum rings on semi-positive symplectic manifolds. (Fano manifolds are particular semi-positive manifolds.) In this note, we will provide a purely algebro-geometric proof to the existence of quantum cohomology rings for a special class of manifolds already treated there, namely homogeneous manifolds. By using algebro-geometric approach, we can prove the existence of quantum cohomology of homogeneous varieties defined over any algebraically closed field. This should be useful in enumerative geometry. We believe the approach here can be applied to a larger class of algebraic varieties, such as toric varieties.