Equivariant Chow cohomology of nonsimplicial toric varieties

For a toric variety X Σ determined by a polyhedral fan $\sum \subseteq {\text{ N}}$ , Payne shows that the equivariant Chow cohomology is the Sym(N)-algebra C⁰(Σ) of integral piecewise polynomial functions on Σ. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2012-08, Vol.364 (8), p.4041-4051
Main Author: SCHENCK, HAL
Format: Article
Language:English
Subjects:
Geometry
Hyperplanes
Mathematical functions
Mathematical rings
Mathematical theorems
Polynomials
Tangents
Vector fields
Vertices
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Summary:For a toric variety X Σ determined by a polyhedral fan $\sum \subseteq {\text{ N}}$ , Payne shows that the equivariant Chow cohomology is the Sym(N)-algebra C⁰(Σ) of integral piecewise polynomial functions on Σ. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf C⁰(Σ) on ℙ ℚ (N), showing that the Chern classes depend on subtle geometry of Σ and giving criteria for the splitting of C⁰(Σ) as a sum of line bundles. For certain fans associated to the reflection arrangement A n , we describe a connection between C⁰(Σ) and logarithmic vector fields tangent to A n .
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2012-05409-2