EQUIVARIANT CHOW COHOMOLOGY OF TORIC VARIETIES

By Sam Payne. 2005

98 citations recorded by the source corpus.

We show that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which lo- calization holds in equivariant Chow cohomology with integer coe!cients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(!) is a smooth, complete complex toric variety then the follow- ing rings are canonically isomorphic: the equivariant singular cohomology ring H ! T (X), the equivariant Chow cohomology ring A ! (X), the Stanley-Riesner ring SR(!), and the ring of integral piecewise polynomial functions PP ! (!). If X is simplicial but not smooth then H ! (X) may have torsion and the natural map from SR(!) takes monomial generators to piecewise linear functions with ra- tional, but not necessarily integral, coe"cients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A ! (X)Q to H ! (X)Q and from H ! T (X) to PP ! (!), but these maps are far

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0228da365f97fb2d8b7fe47a6fff4f58fd6cec0e

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Citations in the corpus

Operational K-Theory (2013, result): “The results in this section are closely analogous to those proved by Kimura for operational Chow theory in [Ki], and extended to the equivariant setting in [EG1, Pa1], but the proofs for operational K - theory are substantially diﬀerent.”
Moduli of toric vector bundles (2007, result): “framing. The remaining problem of understanding which families of ﬁltrations come from ﬂat families of toric vector bundles motivated recent work on the equivariant Chow cohomology of toric varieties [Pay06a], which clariﬁed what it means to ﬁx the Chern classes in Klyachko’s classiﬁcation, as follo”
Logarithmic Gromov–Witten theory with expansions (2019, background): “The rational Chow cohomology of Artin fans is the ring of piecewise polynomials [52, 53, 64, 67].”
On the structure of instability in moduli theory (2014, background): “ny φ: [k] →[n] in C, the restriction homomorphismH∗(Θn) →H∗(Θk) induced by the morphism Θk →Θn agrees with the restriction of polynomial functions along the corresponding inclusion (R≥0)k⊂(R≥0)n. See [P1] for a veriﬁcation of these properties in the case of equivariant Chow cohomology. ON THE STRUCT”
Toric vector bundles, branched covers of fans, and the resolution property (2006, background): “e i-th equivariant Chern class cT i (E) corresponds to the piecewise-polynomial function Ψi E, where Ψi E|σ = ei(u(σ)), the i-th elementary symmetric function in the multiset of linear functions u(σ) [Pay06]. We write cT (E) for the equivariant total Chern class cT(E) = 1+cT 1 (E) +···+cT r (E). Pro”
Stellahedral geometry of matroids (2022, unknown): “(cid:2) Let [ E ] ∈ 𝐾 𝑇 ( 𝑋 Σ ) be an equivariant K -class, localizing to [ E ] 𝜎 = 𝑘 𝜎 𝑖 = 1 𝑎 𝜎,𝑖 𝑇 𝑚 𝜎,𝑖 at a torus-ﬁxed point corresponding to a maximal cone 𝜎 ∈ Σ ( 𝑛 ) .”
Tropical intersection theory from toric varieties (2009, background): “Payne [17] gave a combinatorial description of A∗T (X) even in the case that X is not complete.”