On the integral Hodge conjecture for real varieties, II

TL;DR

This paper proves the real integral Hodge conjecture for 1-cycles on certain uniruled threefolds and conic bundles, while also showing counterexamples over non-archimedean real closed fields and analyzing related theorems.

Contribution

It establishes the conjecture for specific classes of uniruled threefolds and explores its limitations over non-archimedean fields, providing new insights into algebraic cycles.

Findings

Proves the conjecture for various uniruled threefolds and conic bundles.

Shows counterexamples over non-archimedean real closed fields.

Analyzes the validity of Bröcker's EPT theorem for different surfaces.

Abstract

We establish the real integral Hodge conjecture for 1-cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for 1-cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Br\"ocker's EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some K3 surfaces.