Symmetry Defect of n- Dimensional Complete Intersections in \(\mathbb C^{2n-1}\)

Let \(X, Y \subset \mathbb {C}^{2n-1}\) $X,Y\subset {\mathbb{C}}^{2n-1}$ be n-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point \(x \in X\) $x\in X$ to a point \(y \in Y\) $y\in Y$ . This set is defined as the image of the map \(\Phi (x,y)=\frac{x+y}{2}.\) $\mathrm{\Phi }(x,y)=\frac{x+y}{2}.$ 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 \(\Phi \) $\mathrm{\Phi }$ , 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 \(X,Y\subset \mathbb {C}^{2n-1}\) $X,Y\subset {\mathbb{C}}^{2n-1}$ (including the case \(X=Y\) $X=Y$ ) we introduce a generic symmetry defect set \(\tilde{B}(X,Y)\) $\tilde{B}(X,Y)$ of X and Y, which is defined up to homeomorphism. The set \(\tilde{B}(X,Y)\) $\tilde{B}(X,Y)$ is an algebraic variety. Finally we show that in the real case if X, Y are compact, then the set \(\tilde{B}(X,Y)\) $\tilde{B}(X,Y)$ is a hypersurface and it has only Thom-Boardman singularities. In particular if X is compact, then \(\tilde{B}(X)\) $\tilde{B}(X)$ is a hypersurface, which has only Thom-Boardman singularities.