Vanishing Theorems for Real Algebraic Cycles

TL;DR

This paper proves a vanishing theorem for reduced Lawson homology of real algebraic varieties, extending known conjectures and establishing dualities and computations linking homology, cohomology, and equivariant theories.

Contribution

It establishes the analogue of the Friedlander-Mazur conjecture for real varieties' reduced Lawson homology and connects it with equivariant dualities and Bredon cohomology computations.

Findings

Vanishing of reduced Lawson homology in degrees above the variety's dimension.

Characterization of motivic cohomology via homotopy groups of averaged cycles.

Equivariant Poincare duality linking real morphic cohomology and Lawson homology.

Abstract

We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare duality between equivariant Friedlander-Walker real morphic cohomology and dos Santos' real Lawson homology. We use this together with an equivariant extension of the mod-2 Beilinson-Lichtenbaum conjecture to compute some real Lawson homology groups in terms of Bredon cohomology.