Equivariant Chow cohomology of nonsimplicial toric varieties

TL;DR

This paper investigates the equivariant Chow cohomology of nonsimplicial toric varieties, revealing how Chern classes depend on fan geometry and establishing links to logarithmic vector fields for specific arrangements.

Contribution

It introduces a spectral sequence analysis of the associated sheaf, providing new criteria for sheaf splitting and connecting cohomology to geometric properties of fans.

Findings

Chern classes depend on subtle fan geometry

Criteria for sheaf splitting as sum of line bundles

Connection between cohomology and logarithmic vector fields

Abstract

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.