Higher-dimensional analogues of Knebusch’s parity criterion

Ascertain whether algebraicity criteria analogous to Knebusch’s parity condition for the image of W(X) → H^0(X(R), Z) on real curves extend to higher dimensions, specifically to criteria that determine when classes in H^k(X(R), Z) lie in the image of the real cycle class map c_R: H^k(X, I^k) → H^k(X(R), Z) for smooth real varieties of dimension at least two.

Background

For real curves, Knebusch provided a parity condition describing which elements of H0(X(R), Z) come from the Witt group W(X), and the paper relates this to the degree-zero case of the real cycle class map. This suggests a potential role for such conditions as a "real Hodge condition" governing algebraicity.

The authors note that Knebusch’s parity arises from Witt group and purity considerations, leaving open whether comparable conditions exist in higher dimensions to characterize the image of c_R. Clarifying this would significantly advance real integral Hodge-theoretic understanding beyond curves.

References

One might wonder if this result of Knebusch should be considered as a "real Hodge condition". However, we observe that it arises by Witt group considerations and purity. Hence it is not clear if similar conditions should be expected in large dimensions.

— A few computations about the real cycle class map in low dimensions  (2405.14348 - Hornbostel, 2024) in Remark 4.2, Section 4 (Curves)

We therefore leave open the following questions. Fix $d\geq 4$. Does there exist a smooth projective variety $X$ of dimension $d$ over $R$ such that $\evd_{nr,X}$ is not surjective?

— Separating components of real algebraic varieties by cohomological invariants  (2610.18562 - Benoist et al., 13 Oct 2026) in Question 2.1(i), labeled \ref{qnotsurj}\ref{qi}

This leaves open the following question. Fix $d\geq 5$. Does there exist a smooth projective variety $X$ of dimension $d$ over~$R$ with the property that the cokernel of the signature morphism ${W_{nr}(X)\to H0(X(R),Z)}$ is not killed by $2{d-1}$?

— Separating components of real algebraic varieties by cohomological invariants  (2610.18562 - Benoist et al., 13 Oct 2026) in Question 2.2, labeled \ref{qsignnotsurj}