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We examine the integral cohomology rings of certain families of 2n–dimensional orbifolds X that are equipped with a well-behaved action of the n–dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs (Q,λ), where Q is a simple convex n–polytope and λ a labeling of its facets, and from n–dimensional fans Σ. In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on (Q,λ) or on Σ which ensure that the integral cohomology groups H∗(X) of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring H∗(X) as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.
Keywords: toric orbifold, quasitoric orbifold, toric variety, lens space, equivariant cohomology, Stanley–Reisner ring, piecewise polynomial
Bahri, Anthony 1 ; Sarkar, Soumen 2 ; Song, Jongbaek 3
@article{10_2140_agt_2017_17_3779,
author = {Bahri, Anthony and Sarkar, Soumen and Song, Jongbaek},
title = {On the integral cohomology ring of toric orbifolds and singular toric varieties},
journal = {Algebraic and Geometric Topology},
pages = {3779--3810},
publisher = {mathdoc},
volume = {17},
number = {6},
year = {2017},
doi = {10.2140/agt.2017.17.3779},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3779/}
}
TY - JOUR AU - Bahri, Anthony AU - Sarkar, Soumen AU - Song, Jongbaek TI - On the integral cohomology ring of toric orbifolds and singular toric varieties JO - Algebraic and Geometric Topology PY - 2017 SP - 3779 EP - 3810 VL - 17 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3779/ DO - 10.2140/agt.2017.17.3779 ID - 10_2140_agt_2017_17_3779 ER -
%0 Journal Article %A Bahri, Anthony %A Sarkar, Soumen %A Song, Jongbaek %T On the integral cohomology ring of toric orbifolds and singular toric varieties %J Algebraic and Geometric Topology %D 2017 %P 3779-3810 %V 17 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3779/ %R 10.2140/agt.2017.17.3779 %F 10_2140_agt_2017_17_3779
Bahri, Anthony; Sarkar, Soumen; Song, Jongbaek. On the integral cohomology ring of toric orbifolds and singular toric varieties. Algebraic and Geometric Topology, Tome 17 (2017) no. 6, pp. 3779-3810. doi: 10.2140/agt.2017.17.3779
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