Complex slices on a real variety

TL;DR

This paper studies special real algebraic varieties called complex slices, characterizes them via pencils of hypersurfaces, and establishes an upper bound for the linking number of certain real curves.

Contribution

It introduces the concept of complex slices on real varieties, characterizes them through pencils of hypersurfaces, and provides bounds on linking numbers in this context.

Findings

Complex slices are special real algebraic varieties cooriented and representing integer cohomology classes.

A codimension 2 projective variety is a slice iff it is a base of a pencil of real hypersurfaces.

An upper bound is established for the linking number of real projective curves bounding in their complexification.

Abstract

Let $X$ be a real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$. Complex slices are real algebraic varieties of a very special kind. They are cooriented, realize an integer cohomology class. A codimension 2 projective variety is a slice, iff it is a base of pencil of real algebraic hypersurfaces. We prove an upper bound for the linking number of a real projective curve bounding in its complexification with a slice of codimension two.