The real cycle class isomorphism for linear schemes

TL;DR

This paper extends the understanding of the real cycle class map's isomorphism range for linear schemes over real numbers, establishing precise bounds and confirming conjectured limitations on cokernel exponents.

Contribution

It generalizes known isomorphism results from cellular schemes to linear schemes, providing exact bounds and verifying Lerbet's conjecture on cokernel exponents.

Findings

Real cycle class map is an isomorphism for linear schemes within specific bounds.

Established intermediate bounds for the isomorphism range.

Confirmed Lerbet's conjecture on the cokernel exponent cannot be improved.

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.