Virtual Gromov-Witten Invariants and the Quantum Cohomology Rings of General Type Projective Hypersurfaces

TL;DR

This paper introduces a new way to characterize the generalized mirror transformation for quantum cohomology rings of general type projective hypersurfaces, enabling explicit calculations of Gromov-Witten invariants.

Contribution

It provides a novel characterization of the mirror transformation that simplifies the explicit determination of Gromov-Witten invariants for these hypersurfaces.

Findings

Rederived the mirror transformation for degrees up to 3.

Explicitly determined the transformation for degrees 4 and 5 when the first Chern class equals -H.

Enhanced understanding of quantum cohomology rings of general type hypersurfaces.

Abstract

In this paper, we propose another characterization of the generalized mirror transformation on the quantum cohomology rings of general type projective hypersurfaces. This characterics is useful for explicit determination of the form of the generalized mirror transformation. As applications, we rederive the generalized mirror transformation up to $d=3$ rational Gromov-Witten invariants obtained in our previous article, and determine explicitly the the generalized mirror transformation for the $d=4, 5$ rational Gromov-Witten invariants in the case when the first Chern class of the hypersurface equals $-H$ (i.e., $k-N=1$).