The real cycle class isomorphism for linear schemes
Jan Hennig
TL;DR
This work extends the real cycle class isomorphism from cellular to linear schemes by establishing intermediate bounds for when $H^i(X,\underline{I}^j(\mathcal{L}))$ maps isomorphically to $H^i_{\text{sing}}(X(\mathbb{R}),\mathbb{Z}(\mathcal{L}))$, using Rost–Schmid cohomology and localization techniques. It introduces a refined inductive analysis of linear schemes via $n$-J- and $n$-T-linearity, proving that for smooth $n$-J-linear $X$ the map is an isomorphism for $j \ge i+n$, with corollaries for stratifications by $\mathbb{A}^n_k \times \mathbb{G}_m^{d'}$ giving $j \ge i+d$. The paper also establishes the sharpness of Lerbet's conjectured bound on the cokernel exponent, providing explicit exponents like $2^{d-c}$ in key examples and thereby clarifying the obstruction to bijectivity in broader cases. Collectively, these results broaden the class of schemes for which the real cycle class map is bijective, quantify the failure in non-bijective cases, and validate sharp asymptotic bounds through concrete examples.
Abstract
The real cycle class map $H^i(X,\underline{I}^j(\mathcal{L})) \rightarrow H^i_\text{sing}(X(\mathbb{R}),\mathbb{Z}(\mathcal{L}))$ is an isomorphism for $j\geq \dim(X)+1$ for any scheme $X$ over $\mathbb{R}$ by a result of Jacobson. It is also known to be an isomorphism for $j\geq i$, the earliest possible case, if $X$ is cellular due to Hornbostel-Wendt-Xie-Zibrowius. This paper generalizes their result to linear schemes, providing (precise) intermediate bounds on the range, where the real cycle class map is an isomorphism. Moreover, we show that Lerbet's conjectured upper bound for the exponent of the cokernel of $H^i(X,\underline{I}^i(\mathcal{L})) \rightarrow H^i_\text{sing}(X(\mathbb{R}),\mathbb{Z}(\mathcal{L}))$ cannot be improved. This is part of the author's PhD thesis.