ON THE QUANTUM COHOMOLOGY RINGS OF GENERAL TYPE PROJECTIVE HYPERSURFACES AND GENERALIZED MIRROR TRANSFORMATION

Abstract

In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with nonpositive first Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi–Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three-point Gromov–Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in good correspondence with the terms that appear in the generalized mirror transformation.