Existence of a line without real points in quadric intersection

TL;DR

This paper investigates the topology of intersections of three quadrics in Euclidean 6-space, establishing the existence of a complex projective line without real points under certain conditions to classify manifold types.

Contribution

It proves the existence of a line without real points in the complex projectivisation of quadrics under specific assumptions, aiding in classifying manifold topologies.

Findings

Existence of a complex projective line without real points established

Conditions under which such lines exist are identified

Progress towards classifying manifold homeomorphism types

Abstract

The topology of the intersection of three quadrics in Euclidean 6-space is studied using Kollar results. This needs an existence of a line without real points in the complex projectivisation of quadrics. We establish the existence of such a line under some assumptions. It is a step to obtain a list of all possible homeomorfism types of manifolds under consideration.