A real viewpoint on the intersection of complex quadrics and its topology

TL;DR

This paper explores the topological relationship between complex projective sets defined by quadrics and their real counterparts, using spectral sequences to compute homology and providing explicit examples for intersecting quadrics.

Contribution

It introduces a method to relate the homology of complex quadrics to their real representations via spectral sequences and demonstrates explicit computations for intersecting quadrics.

Findings

Homology of real sets can be computed efficiently using spectral sequences.

The (Z_2)-cohomology of R is a free H*(C)-module with two generators.

Explicit homology computations are provided for intersections of two complex quadrics.

Abstract

We study the relation between a complex projective set C in CP^n and the set R in RP^(2n+1) defined by viewing each equation of C as a pair of real equations. Once C is presented by quadratic equations, we can apply a spectral sequence to efficiently compute the homology of R; using the fact that the (Z_2)-cohomology of R is a free H*(C)-module with two generators we can in principle reconstruct the homology of C. Explicit computations for the intersection of two complex quadrics are presented.