Symmetry defect of $n$-dimensional complete intersections in ${\mathbb{C}}^{2n-1}$

Abstract

Let $X,Y\subset {\mathbb{C}}^{2n-1}$ be $n$-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point $x\in X$ to a point $y\in Y$. This set is defined as the image of the map $\hat{I}¦(x,y)=\frac{x+y}{2}.$ Under geometric conditions on $X$ and $Y$, we prove that the symmetry defect of $X$ and $Y$, which is the bifurcation set $B(X,Y)$ of the mapping $\hat{I}¦$, is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set $B(X,Y)$ and we present an estimate for its degree. Moreover, for any two $n$-dimensional strong complete intersections $X,Y\subset {\mathbb{C}}^{2n-1}$ (including the case $X=Y$) we introduce a generic symmetry defect set $\tilde{B}(X,Y)$ of $X$ and $Y$, which is defined up to homeomorphism.