Equivariant Semi-topological Invariants, Atiyah's KR-theory, and Real Algebraic Cycles

TL;DR

This paper develops spectral sequences linking real morphic cohomology and semi-topological K-theory, proves their compatibility with existing theories, and applies them to compute real K-theory and verify conjectures for real varieties.

Contribution

It introduces a new spectral sequence connecting real morphic cohomology with semi-topological K-theory and proves its compatibility with established theories, advancing understanding of real algebraic cycles.

Findings

Established an Atiyah-Hirzebruch type spectral sequence for real varieties.

Proved compatibility with Dugger's spectral sequence for KR-theory.

Provided new proofs and computations for real semi-topological K-theory and related conjectures.

Abstract

We establish an Atiyah-Hirzebruch type spectral sequence relating real morphic cohomology and real semi-topological K-theory and prove it to be compatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah's KR-theory constructed by Dugger. An equivariant and a real version of Suslin's conjecture on morphic cohomology are formulated, proved to come from the complex version of Suslin conjecture and verified for certain real varieties. In conjunction with the spectral sequences constructed here this allows the computation of the real semi-topological K-theory of some real varieties. As another application of this spectral sequence we give an alternate proof of the Lichtenbaum-Quillen conjecture over $\R$, extending an earlier proof of Karoubi and Weibel.