General algebraicity criteria for the real cycle class map

Determine general additional algebraicity conditions that characterize when an integral singular cohomology class in H^k(X(R), Z) lies 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 X and all degrees k, in analogy with the Hodge-type criteria used in the complex case.

Background

The paper studies the real cycle class map c_R from I-cohomology to integral singular cohomology and its surjectivity properties in low dimensions. While c_R becomes an isomorphism after inverting 2 and is an isomorphism for cellular varieties, outside these settings determining whether a given integral cohomology class is algebraic (i.e., lies in the image of c_R) is challenging.

In contrast to the complex case where Hodge-type conditions guide algebraicity, the authors emphasize that no general additional condition is currently known for the real case. Establishing such a criterion would provide a systematic tool to decide algebraicity of classes for arbitrary smooth real varieties and degrees.