QUANTUM COHOMOLOGIES OF SYMMETRIC PRODUCTS

In this paper we investigate the quantum cohomologies of symmetric products of Kahler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov-Witten invariant and relative Gromov-Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line P-1, which is the k-dimensional complex projective space P-k.