Abstract

Given a smooth variety $X$ over the field $ℝ$ of real numbers and a line bundle $\mathcal{ℒ}$ on $X$ with associated topological line bundle $L=\mathcal{ℒ}(ℝ)$, we study the quadratic real cycle class map ${\tilde{\gamma }}_{ℝ}^c:{\widetilde{\mathrm{CH}<!--FUNCTION\; APPLICATION-->}}^c(X,\mathcal{ℒ})\to {H<!--FUNCTION\; APPLICATION-->}^c(X(ℝ),\mathbb{Z}(L))$ from the $c$-th Chow–Witt group of $X$ to the $c$-th cohomology group of its real locus $X(ℝ)$ with coefficients in the local system $\mathbb{Z}(L)$ associated with $L$. We focus on the cases $c\in \{0,d-2,d-1,d\}$ where $d$ is the dimension of $X$, and we formulate a precise conjecture on the image of ${\tilde{\gamma }}_{ℝ}$ in terms of the exponents of its cokernel that is corroborated by the results obtained in those codimensions.

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