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The cohomology rings of smooth toric varieties and quotients of moment-angle complexes

Matthias Franz

Geometry & Topology 25 (2021) 2109–2144

Partial quotients of moment-angle complexes are topological analogues of smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring of such a partial quotient in terms of a torsion product involving the corresponding Stanley–Reisner ring. We show that their formula gives the correct cup product if 2 is invertible in the chosen coefficient ring, but not in general. We rectify this by defining an explicit deformation of the canonical multiplication on the torsion product.

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toric variety, partial quotient, moment-angle complex, cohomology ring
Mathematical Subject Classification
Primary: 14M25, 57S12
Secondary: 14F45, 55N91
Received: 7 February 2020
Revised: 24 April 2020
Accepted: 23 May 2020
Published: 12 July 2021
Proposed: Frances Kirwan
Seconded: Dan Abramovich, Anna Wienhard
Matthias Franz
Department of Mathematics
University of Western Ontario
London, ON