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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Equivariant Chow cohomology of nonsimplicial toric varieties


Author: Hal Schenck
Journal: Trans. Amer. Math. Soc. 364 (2012), 4041-4051
MSC (2010): Primary 14M25; Secondary 14F43, 13D02, 52B99
Published electronically: February 17, 2012
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Abstract: For a toric variety $ X_\Sigma $ determined by a polyhedral fan $ \Sigma \subseteq N$, Payne shows that the equivariant Chow cohomology is the $ \textup {Sym}(N)$-algebra $ C^0(\Sigma )$ of integral piecewise polynomial functions on $ \Sigma $. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf $ \mathcal {C}^0(\Sigma )$ on $ \mathbb{P}_{\mathbb{Q}}(N)$, showing that the Chern classes depend on subtle geometry of $ \Sigma $ and giving criteria for the splitting of $ \mathcal {C}^0(\Sigma )$ as a sum of line bundles. For certain fans associated to the reflection arrangement $ \mathrm {A_n}$, we describe a connection between $ C^0(\Sigma )$ and logarithmic vector fields tangent to $ \mathrm {A_n}$.


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Additional Information

Hal Schenck
Affiliation: Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, Illinois 61801
Email: schenck@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05409-2
PII: S 0002-9947(2012)05409-2
Keywords: Chow ring, toric variety, piecewise polynomial function.
Received by editor(s): March 1, 2010
Received by editor(s) in revised form: June 29, 2010
Published electronically: February 17, 2012
Additional Notes: The author was supported by NSF 0707667
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.