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On the integral cohomology ring of toric orbifolds and singular toric varieties

Anthony Bahri, Soumen Sarkar and Jongbaek Song

Algebraic & Geometric Topology 17 (2017) 3779–3810
Abstract

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.

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Keywords
toric orbifold, quasitoric orbifold, toric variety, lens space, equivariant cohomology, Stanley–Reisner ring, piecewise polynomial
Mathematical Subject Classification 2010
Primary: 14M25, 55N91, 57R18
Secondary: 13F55, 52B11
References
Publication
Received: 20 December 2016
Revised: 22 March 2017
Accepted: 4 April 2017
Published: 4 October 2017
Authors
Anthony Bahri
Department of Mathematics
Rider University
Lawrenceville, NJ
United States
Soumen Sarkar
Department of Mathematics
Indian Institute of Technology Madras
Chennai
India
Jongbaek Song
Department of Mathematical Sciences
KAIST
Daejeon
South Korea