Complex slices on a real variety

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.

18 pages