Quantum Cohomology Rings of Toric Manifolds

Abstract

We compute the quantum cohomology ring $H^*_φ({\bf P}, {\bf C})$ of an arbitrary $d$-dimensional smooth projective toric manifold ${\bf P}_Σ$ associated with a fan $Σ$. The multiplicative structure of $H^*_φ({\bf P}_Σ, {\bf C})$ depends on the choice of an element $avarphi$ in the ordinary cohomology group $H^2({\bf P}_Σ, {\bf C})$. There are many properties of the quantum cohomology rings $H^*_φ({\bf P}_Σ, {\bf C})$ which are supposed to be valid for quantum cohomology rings of Kähler manifolds