Vanishing theorems for real algebraic cycles (Q2892852)

Let \(X\) be a quasiprojective real algebraic variety. Denote by \({\mathcal Z}_q(X_{\mathbb C})\) the topological group of algebraic cycles. The Galois group \(G=\text{Gal}({\mathbb C}/{\mathbb R})\) acts on that group. Following \textit{J.-H. Teh} [Indiana Univ. Math. J. 59, No. 1, 327--384 (2010; Zbl 1206.14021)] there is considered the quotient of the invariant cycles NEWLINE\[NEWLINE{\mathcal R}_q(X)=\frac{{\mathcal Z}_q(X_{\mathbb C})^G}{{\mathcal Z}_q(X_{\mathbb C})^{\text{av}}}\,,NEWLINE\]NEWLINE where \({\mathcal Z}_q(X_{\mathbb C})^{\text{av}}\) consists of the cycles of the form \(a+\bar a\). The reduced (real) Lawson homology is defined as the homotopy groups NEWLINE\[NEWLINERL_qH_n(X)=\pi_{n-q}(R_q(X))\,.NEWLINE\]NEWLINE The main result of the paper is vanishing \(RL_qH_n(X)=0\) for \(n>\dim(X)\). That is a version of Friedlander-Mazur conjecture \textit{E. M. Friedlander} and \textit{B. Mazur} [Mem. Am. Math. Soc. 529, 110 p. (1994; Zbl 0841.14019)] for real varieties.NEWLINENEWLINEThe proof is based on \textit{V. Voevodsky} work on Milnor conjecture [Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)]. In fact the real morphic cohomology \(L^qH{\mathbb R}^{q-k,k}(X;{\mathbb Z}/2)\), which is dual to mod 2 version of real Lawson homology NEWLINE\[NEWLINEL_qH{\mathbb R}_{q-k,k}(X)=\pi_k{\mathcal Z}_q(X_{\mathbb C})^G,NEWLINE\]NEWLINE is isomorphic to Voevodsky's motivic cohomology \({\mathbb H}_{\mathcal M}^{2q-k,q}(X;{\mathbb Z}/2)\). The Milnor conjecture states that this group is isomorphic to étale cohomology \(H^{2q-k}_{et}(X;\mu_2^{\otimes q})\). The former group is related to the Bredon and Borel cohomologies. The key technical lemma is to show the natural maps connecting these groups commute.