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The cohomology rings of real toric spaces and smooth real toric varieties
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 3 / June 2022
- Published online by Cambridge University Press:
- 29 June 2021, pp. 720-737
- Print publication:
- June 2022
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We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra (dga) we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are
$\hbox{M}$-varieties.
Symmetric Products of Equivariantly Formal Spaces
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 2 / 01 June 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 272-281
- Print publication:
- 01 June 2018
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- Article
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Let
$X$ be a 
$\text{CW}$ complex with a continuous action of a topological group 
$G$. We show that if $X$ is equivariantly formal for singular cohomology with coefficients in some field 
$\Bbbk $, then so are all symmetric products of $X$ and in fact all its 
$\Gamma $-products. In particular, symmetric products of quasi-projective 
$\text{M}$-varieties are again $\text{M}$-varieties. This generalizes a result by Biswas and D’Mello about symmetric products of $\text{M}$-curves. We also discuss several related questions.
