Algebraic cycles and topology of real Enriques surfaces

TL;DR

The paper develops an integral equivariant (co)homology framework for real Enriques surfaces and uses it to relate the topology of Y({\mathbb R}) to algebraic cycles on Y. It proves that if all components of Y({\mathbb R}) are orientable, every class in $H_1(Y({\mathbb R}), {\mathbb Z}/2)$ is representable by a real algebraic curve; otherwise the algebraic part has codimension at most one. Through edge maps in the Hochschild–Serre spectral sequence, the authors characterize when a real Enriques surface is GM or $\mathbb Z$-GM, with precise criteria that depend on orientability and the two halves of the real locus. They also compute the Brauer group Br$(Y)$ in terms of integral equivariant data, obtaining explicit formulas based on orientability and half-distribution. The results illuminate how orientability, halves, and covering structures (notably the K3-cover) govern the interplay between algebraic cycles, equivariant topology, and the Brauer group on real Enriques surfaces.

Abstract

For a real Enriques surface Y we prove that every homology class in H_1(Y(R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Proposition 3.1: Poincaré duality
• Proposition 3.2