Equivariant Chow cohomology of nonsimplicial toric varieties

Schenck, Hal

doi:10.1090/S0002-9947-2012-05409-2

Equivariant Chow cohomology of nonsimplicial toric varieties
HTML articles powered by AMS MathViewer

by Hal Schenck PDF

Trans. Amer. Math. Soc. 364 (2012), 4041-4051 Request permission

Abstract:

For a toric variety $X_\Sigma$ determined by a polyhedral fan $\Sigma \subseteq N$, Payne shows that the equivariant Chow cohomology is the $\mathrm {Sym}(N)$-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 $\mathrm {A_n}$.

References

Emili Bifet, Corrado De Concini, and Claudio Procesi, Cohomology of regular embeddings, Adv. Math. 82 (1990), no. 1, 1–34. MR 1057441, DOI 10.1016/0001-8708(90)90082-X
Louis J. Billera, The algebra of continuous piecewise polynomials, Adv. Math. 76 (1989), no. 2, 170–183. MR 1013666, DOI 10.1016/0001-8708(89)90047-9
Louis J. Billera and Lauren L. Rose, A dimension series for multivariate splines, Discrete Comput. Geom. 6 (1991), no. 2, 107–128. MR 1083627, DOI 10.1007/BF02574678
Michel Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry, Parameter spaces (Warsaw, 1994) Banach Center Publ., vol. 36, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 25–44. MR 1481477
M. Brion, Equivariant Chow groups for torus actions, Transform. Groups 2 (1997), no. 3, 225–267. MR 1466694, DOI 10.1007/BF01234659
Michel Brion and Michèle Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math. 482 (1997), 67–92. MR 1427657, DOI 10.1515/crll.1997.482.67
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Terry McDonald and Hal Schenck, Piecewise polynomials on polyhedral complexes, Adv. in Appl. Math. 42 (2009), no. 1, 82–93. MR 2475315, DOI 10.1016/j.aam.2008.06.001
Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), no. 1, 29–41. MR 2199564, DOI 10.4310/MRL.2006.v13.n1.a3
Hal Schenck, A spectral sequence for splines, Adv. in Appl. Math. 19 (1997), no. 2, 183–199. MR 1459497, DOI 10.1006/aama.1997.0534
Hiroaki Terao, Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula, Invent. Math. 63 (1981), no. 1, 159–179. MR 608532, DOI 10.1007/BF01389197
Hiroaki Terao, Free arrangements of hyperplanes and unitary reflection groups, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 8, 389–392. MR 596011