Reduction of beta-integrable 2-Segre structures

TL;DR

This paper demonstrates that locally beta-integrable (2,n)-Segre structures can be reduced to torsion-free S^1*GL(n,R)-structures via holomorphic sections of a twistor bundle, linking geometric reductions to complex algebraic varieties.

Contribution

It introduces a method to reduce beta-integrable (2,n)-Segre structures to torsion-free structures using twistor bundle sections, and characterizes reductions on homogeneous structures via smooth quadrics.

Findings

Reductions correspond to holomorphic sections of a twistor bundle.

On the homogeneous Grassmannian, reductions relate to smooth quadrics in CP^{n+1}.

Every beta-integrable (2,n)-Segre structure admits such a reduction.

Abstract

We show that locally every beta-integrable (2,n)-Segre structure can be reduced to a torsion-free S^1*GL(n,R)-structure. This is done by observing that such reductions correspond to sections with holomorphic image of a certain `twistor bundle'. For the homogeneous (2,n)-Segre structure on the oriented 2-plane Grassmannian, the reductions are shown to be in one-to-one correspondence with the smooth quadrics in CP^{n+1} without real points.