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.

Virtual Gromov–witten Invariants and the Quantum Cohomology Rings of General Type Projective Hypersurfaces

“…Recently, some works on the quantum cohomology ring of the general type projective hypersurface have appeared [13], [5], [10]. In [13], Lian, Liu and Yau generalized their mirror principle to the case of the general type projective hypersurface and proposed a theoretical recipe to construct the generating function of a certain type of Gromov-Witten invariants including gravitational descendants from the hypergeometric data.…”

J Functions, Non-Nef Toric Varieties and Equivariant Local Mirror Symmetry of Curves

“…From elementary combinatorial considerations, the sum of contributions of all the star graphs can be rewritten by introducing the expression, 15. In other words, summing up graphs with one edge and star graphs is closely related to taking the inverse of the mirror map in the mirror computation.…”

Gauss–manin System and the Virtual Structure Constants

“…With this idea, we speculated that we can generalize the formula (2.24) to the N − k < 0 case. In [9] and [8], we gave some numerical evidence of this generalization up to some lower degree of rational curves.…”

“…In [9], we argued that this formula must have deep connection with toric compactification of the moduli space of rational curves in P N −1 . With this idea, we speculated that we can generalize the formula (2.24) to the N − k < 0 case.…”