On the Homology of Unions of Certain Non-Degenerate Quadrics in General Position

TL;DR

This paper analyzes the homology of unions of non-degenerate quadratic hypersurfaces in complex space, providing explicit computations and generators relevant to complex analysis and quantum field theory.

Contribution

It introduces a method to decompose and compute the homology of unions of quadratic hypersurfaces and constructs explicit generators for specific arrangements.

Findings

Homology groups decomposed into sums of intersection homologies.

Explicit generators constructed via CW-complexes.

Intersection indices with Borel-Moore homology computed.

Abstract

Motivated by problems arising in the complex analysis of perturbative quantum field theory, we investigate the homology of finite unions of certain non-degenerate quadratic affine hypersurfaces of complex dimension $n$ in general position. The homology of such unions and the homology of $\mathbb{C}^{n+1}$ relative to such unions is decomposed into a direct sum of homology groups of the various possible intersections of these hypersurfaces. This allows us to compute the homology groups up to isomorphism. Furthermore, we compute an explicit set of generators for a specific arrangement of these hypersurfaces by constructing a sufficiently large CW-subcomplex of the union in question. The intersection indices of all generators with the Borel-Moore homology class of $(i\cdot\mathbb{R})^{n+1}$ in the complement of the union of surfaces are computed.