Equivariant elliptic cohomology, toric varieties, and derived equivalences

TL;DR

This paper investigates the equivariant elliptic cohomology of complex toric varieties, revealing its partial encoding of the 1-skeleton and demonstrating that it is not a derived invariant, unlike ordinary cohomology.

Contribution

It provides a partial reconstruction theorem for equivariant elliptic cohomology and shows that it can distinguish non-isomorphic derived equivalent toric varieties, highlighting its richer structure.

Findings

Elliptic cohomology encodes significant information about the equivariant 1-skeleton.

It is not a derived invariant of algebraic varieties.

Existence of non-isomorphic equivariantly derived equivalent toric varieties with different elliptic cohomology.

Abstract

In this article we study the equivariant elliptic cohomology of complex toric varieties. We prove a partial reconstruction theorem showing that equivariant elliptic cohomology encodes considerable non-trivial information on the equivariant 1-skeleton of a toric variety X (although it stops short of being a complete invariant of its GKM graphs). Elliptic cohomology is supposed to encode higher categorical geometric data, and proposals have been made linking elliptic cocycles to categorified bundles. In particular, contrary to ordinary cohomology and K-theory, elliptic cohomology is expected not to be a derived invariant of algebraic varieties. Our second main result is to verify this prediction by showing that there exist pairs of equivariantly derived equivalent toric varieties with non-isomorphic equivariant elliptic cohomology.