Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space
Masao Jinzenji, Ken Kuwata
TL;DR
This work extends the elliptic virtual structure-constant framework to hypersurfaces and complete intersections in weighted projective spaces with a single Kähler class, enabling the computation of genus $1$ Gromov–Witten invariants via residue-integral graph methods. It introduces and analyzes four graph types, their integrands, and a residue procedure, and ties genus $1$ data to genus $0 mirror maps through conjectured generating functions, providing extensive numerical tests across Fano, Calabi–Yau, and complete-intersection examples, with BCOV consistency in CY cases. The results broaden the applicability of BCOV-type invariants to weighted settings and validate the formalism through multiple cross-checks, including agreements with known results and vanishing predictions for certain K3 scenarios. The work thus offers a computational framework for genus $1$ invariants on a wide class of complete intersections, enriching the landscape of mirror symmetry and enumerative geometry in singular ambient spaces.
Abstract
In this paper, we generalize our formalism of the elliptic virtual structure constants to hypersurfaces and complete intersections within certain weighted projective spaces possessing a single Kähler class.