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 \(\Phi (x,y)=\frac{x+y}{2}.\) 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 \(\Phi \), 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. The set \(\tilde{B}(X,Y)\) is an algebraic variety. Finally we show that in the real case if X, Y are compact, then the set \(\tilde{B}(X,Y)\) is a hypersurface and it has only Thom-Boardman singularities. In particular if X is compact, then \(\tilde{B}(X)\) is a hypersurface, which has only Thom-Boardman singularities.