Strong McKay Correspondence, String-theoretic Hodge Numbers and Mirror Symmetry

TL;DR

The paper develops a coherent framework linking the McKay correspondence, E-polynomials, and string-theoretic Hodge numbers to study mirror symmetry for singular Calabi–Yau varieties. It defines string cohomology $H^*_{st}(X)$ and the string-theoretic invariants $h^{p,q}_{st}(X)$ for GV-varieties, proves the strong McKay correspondence in low dimensions and for abelian quotient singularities, and provides computational tools via toric geometry to determine these invariants. It then derives mirror-duality formulas for Calabi–Yau complete intersections in toric Fano varieties, illustrating the reciprocity with explicit Greene–Plesser constructions and Dwork-type families, and confirms the symmetry of string-theoretic Hodge data in the Greene–Plesser setting. By connecting orbifold Hodge theory, crepant desingularizations, and quantum cohomology limits, the work extends mirror symmetry to singular spaces and clarifies when and how crepant resolutions govern the limiting cohomology.

Abstract

In the revised version of the paper, we correct misprints and add some new statements.

Theorems & Definitions (63)

• Conjecture 1.1
• Remark 1.2
• Conjecture 1.3
• Proposition 3.1
• Definition 3.2
• Remark 3.3
• Proposition 3.4
• Definition 3.5
• Proposition 3.6
• Proposition 3.7