Exercise 4, § 4

In $P_n(\mathbf{C})$ let $H_n$ be the "quadric" defined by the equation

Show that every point of $H_n$ has an open neighbourhood homeomorphic to $\mathbf{C}^{n-1}$, that $H_n$ is connected, and that the intersection of $H_n$ with the complement of any complex projective hyperplane is connected (to prove this last assertion, reduce to the case $n = 2$).

Show that $H_2$ is homeomorphic to $S_2$, and $H_3$ to $S_2 \times S_2$ (use the parametric representation of the quadric by means of its rectilinear generators).