Quantum Cohomology Rings of Toric Manifolds

Abstract

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

23 pages, Latex

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