A New Cohomology Theory for Orbifold

TL;DR

The paper defines orbifold cohomology groups $H^*_{orb}(X)$ for almost complex orbifolds and orbifold Dolbeault cohomology $H^{p,q}_{orb}(X)$ for complex orbifolds, motivated by orbifold string theory. It constructs a rigorous orbifold cup product $\cup_{orb}$ using multi-sector geometry, evaluation maps, and obstruction bundles, proving that the total cohomology carries a graded ring structure with a unit and satisfies Poincaré duality; the Dolbeault theory inherits a compatible bi-grading and duality. The associativity of $\cup_{orb}$ is established via a detailed analysis of the moduli space of ghost maps and gluing constructions on the boundary of ${\cal M}_4$, with explicit decompositions and compatibility with the untwisted sector. The work is illustrated by examples (Kummer surfaces, Borcea–Voisin threefolds, and weighted projective spaces) that highlight degree shifting, rational degrees in twisted sectors, and connections to the center of the group algebra in global quotients. Together, these results lay the classical foundation for orbifold quantum cohomology and point toward further connections with crepant resolutions, mirror symmetry, and birational geometry.

Abstract

Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are computed.