Quadrics, orthogonal actions and involutions in complex projective spaces. (Q1419564)

The authors consider the canonical action of the special orthogonal group \(\text{SO}(n+ 1,\mathbb{R})\) on \(\mathbb{C} P^n\), the complex projective \(n\)-space, in relationship with the study of the complement of an affine algebraic hypersurface \(V\subset \mathbb{C}^{n+1}\), in particular, for \(V\) a homogeneous quadric with an isolated singularity at the origin, in the projectivized situation. The complement of a non-singular hyperquadric \(Q\) in \(\mathbb{C} P^n\) is diffeomorphic to the total space of the tangent bundle of the real projective \(n\)-space \(\mathbb{R} P^n\), \(\mathbb{C} P^n-Q\cong T(\mathbb{R} P^n)\). This is the starting point for investigations of the geometry of \(\mathbb{C} P^n\). \(\mathbb{C} P^n\) is the double mapping cylinder of a double fibration, and this construction leads to foliations of \(\mathbb{C} P^n -\Pi\) and \(\mathbb{C} P^n-Q\) where \(\Pi\cong\mathbb{R} P^n\) is the fixed point set of the conjugation \(j:\mathbb{C} P^n\to\mathbb{C} P^n\), \(j(z_0,z_1,\dots, z_n)= (\overline z_0,\overline z_n)\). Then the action of \(\text{SO}(n+1,\mathbb{R})\) on \(\mathbb{C} P^n\) is studied, as a group action of hyperpolar (or cohomogeneity 1) isometries, with its orbit space. This leads to a proof of the (classical) theorem that \(\mathbb{C} P^2\) modulo conjugation \(j\) is the sphere \(S^4\). Finally, the action of \(\text{SO}(3,\mathbb{R})\) on \(\mathbb{C} P^3\) is discussed as well as an action of \(\text{SO}(4,\mathbb{R})\) on \(S^7\).