Halving spaces and lower bounds in real enumerative geometry

TL;DR

This paper introduces the concept of halving spaces, a topological framework with group actions, to establish lower bounds in real enumerative geometry, extending previous theories to rational cohomology.

Contribution

It develops the theory of halving spaces, generalizes cohomological results to rational coefficients, and applies these to real and quaternionic flag manifolds for enumerative bounds.

Findings

Real even and quaternionic flag manifolds are circle spaces.

New lower bounds for real and quaternionic Schubert problems.

Generalization of Borel and Haefliger's results to rational cohomology.

Abstract

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $\Gamma$ with additional cohomological properties. For $\Gamma=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $\Gamma=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.