I want to show that $T^n$ is diffeomorphic to the quotient of $T^{n+1}$ in $\mathbb CP^n$. I set it up in the following manner:
Take $S^n \times 1 \in \mathbb C^{n+1}-\{0\}$, and denote the quotient map by $Q: \mathbb C^{n+1}-\{0\} \to \mathbb CP^n$. The idea is to show that $Q$ is a proper immersion when restricted to $S^n \times 1$. It seems extremely messy to compute the Jacobian of the quotient map since for the $n-$torus it is most convenient to use real charts while for $\mathbb CP^n$ it is most convenient to use complex charts. Then it is really hard to compute the Jacobian. What is the right strategy? Also, how to show that it is proper?
Update: I realize that I can probably use the chart $\phi:(0,1) \times \ldots \times (0,1) \to S^n \times 1$ such that $\phi(\theta_1, \ldots, \theta_n) = (e^{i\theta_1}, \ldots, e^{i\theta_n}, 1)$ for the $n-$ torus. But still, I am mapping from a real domain to complex coordinates. How should this work?