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 H^0(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.