If $a>0$ and $b\in \mathbb{R}$, let $X\subset \mathbb{C}^2$ be the set $$X=\{(z,w)\in \mathbb{C}^2\;\}\{\;|1-zw|^2=a, \quad |z|^2-|w|^2=b\;\}.$$
Is $X$ a closed surface? This is what I expect because $X$ is defined by two equations. Also, how can I compute the homology of $X$ (if $X$ is a surface, I'd like to determine which one it is. Probably not a sphere.)?
Maybe this is obvious. Thanks.