Equivariant Elliptic Cohomology, Toric Varieties, and Derived Equivalences

In this article we study the equivariant elliptic cohomology of complex toric varieties. We prove a reconstruction theorem showing that equivariant elliptic cohomology encodes considerable non-trivial information on the equivariant 1-skeleton of a toric variety X (although it is not a complete invariant of its GKM graphs). We obtain a complete characterization of smooth and proper toric surfaces with isomorphic equivariant elliptic cohomology. Contrary to ordinary cohomology and K-theory, elliptic cohomology is expected not to be a derived invariant of algebraic varieties. We verify this prediction by showing that elliptic cohomology distinguishes derived equivalent varieties. More precisely, we show that there exist pairs of equivariantly derived equivalent toric varieties with non-isomorphic equivariant elliptic cohomology.