Intuition around covering maps and pull backs to degenerate forms

Question

I apologise in advance for asking what I'm sure is a very nebulous question. The specific set up is - I have $T^*\mathbb{C}\equiv\mathbb{C}^2$ with coordinates $(z,w)$ and the usual (for me) sympletic form $\omega=Re(dz\wedge d\bar{w})$ and I am looking at the pull back under a coordinate change $(z,w)=(u^2,w)$ under which the symplectic form pulls back to something which is degenerate on the hypersurface $Re(u)=\pm Im(u)$. Looking at the big picture, I find it unsurprising that the two form become non-degenerate, as $u^2$ is a branched cover but I can't see intuitively why it is along this hyper surface. I am sure this has an easy answer but any help would be appreciated.

Answer 1

The answer was silly in the end. The hyperplane I described is the one exactly mapped to the imaginary axis by squaring. As the symplectic form is the real part of $dz\wedge d \bar{w}$, of course when we pull back we get a degenerate form