Complex slices on a real variety
Oleg Viro
TL;DR
The paper develops a framework for studying how complex subvarieties of a real algebraic variety intersect the real locus, introducing and exploiting the notion of slices: real subvarieties obtained as transverse intersections with complex subvarieties in the complexification $X_{ ext C}$. It shows that sliced subvarieties have even codimension, carry a natural complex structure on their transverse bundle, and realize integer cohomology classes; codimension-two slices are precisely bases of real pencils of hypersurfaces. A central result is an upper bound for the linking number between a real projective curve bounding in its complexification and a codimension-two slice, derived via cap-linking and the Thom-class formalism. The work unites transversality, complex conjugation, coorientations, and integer (co)homology to yield geometric invariants and inequalities with clear implications for real algebraic geometry and the topology of real slices.
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.