A real analogue of the Hodge conjecture
Samuel Lerbet
TL;DR
The paper develops a real-analytic analogue of the Hodge conjecture using Chow–Witt theory, defining the real cycle class map from Chow–Witt groups to the cohomology of the real locus with twist by a line bundle. It establishes a robust algebraic toolkit based on Milnor–Witt $K$-theory, Rost–Schmid complexes, and $\mathbb{A}^1$-homotopy theory to study the image and exponent of these real cycle class maps in codimensions $c\in\{0,d-2,d-1,d\}$. A key result shows surjectivity in top degree and yields descending exponent bounds that translate into concrete torsion constraints for the cokernels in low codimensions, with explicit descriptions for curves and surfaces. The paper then proposes a refined $\mathbf{I}$-Hodge conjecture (involving exponents $2^{d-c}$) and proves it in several important cases (curves, surfaces, and many threefolds under vanishing hypotheses), while outlining a program to connect these invariants with Steenrod operations and Pardon's spectral sequence. Overall, the work clarifies how Chow–Witt theory captures real-algebraic cycle information and opens a path toward a principled, exponent-based analogue of the Hodge conjecture in real algebraic geometry.
Abstract
Given a smooth scheme $X$ over the field $\mathbb{R}$ of real numbers and a line bundle $\mathcal{L}$ on $X$ with associated topological line bundle $L=\mathcal{L}(\mathbb{R})$, we study the real cycle class map $\widetildeγ_{\mathbb{R}}:\widetilde{\mathrm{CH}}^c(X,\mathcal{L})\rightarrow\mathrm{H}^c(X(\mathbb{R}),\mathbb{Z}(L))$ from the $c$-th Chow-Witt group of $X$ to the $c$-th cohomology group of its real locus $X(\mathbb{R})$ 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 an analogue of the Hodge conjecture in terms of the exponents of the cokernel of $\widetildeγ_{\mathbb{R}}$ that is corroborated by the results obtained in those codimensions.