Invariance property of orbifold elliptic genus for multi-fans

TL;DR

This paper explores the invariance of the orbifold elliptic genus for multi-fans, extending concepts from toric varieties to orbifold torus spaces and establishing functorial properties under birational transformations.

Contribution

It introduces orbifold elliptic classes and genera for multi-fans and demonstrates their functorial behavior under birational morphisms, generalizing existing McKay correspondence results.

Findings

Orbifold elliptic genus is invariant under certain birational transformations.

Defines orbifold elliptic class for multi-fans.

Establishes a topological analogue of McKay correspondence.

Abstract

Multi-fan is an analogous notion of fan. As a fan is associated to a toric variety a multi-fan is associated to a torus orbifold. Orbifold elliptic class and orbifold elliptic genus are defined for a triple of a multi-fan, a set of generating integral vectors of one dimensional cones and a $\mathbb{Q}$ divisor. They are shown to behave functorially with respect to birational morphisms between these triples. The result may be considered as a combinatorial or topological analogue of the main result of Borisov and Libgober, McKay correspondence for elliptic genera, Ann. of Math., 161 (2005).