I've been trying to google around this topic without success, apologies in advance if I missed an obvious resource.
I'm trying to understand what manifold (compact or not) underlies the complex Lie groups $SU(p,q)$. For example, the manifold of $SU(2,0)$ (i.e. $SU(2)$) is given by $x^2+y^2+z^2+w^2=1$, so it's the three-sphere, or the manifold of $SU(1,1)$ is given by $x^2+y^2-z^2-w^2=1$, so it's a four dimensional hyperbolae.
EDITED QUESTION: Is there a general rule to write the system of equations that define the manifold?
The only general rule is to use the definition.
To define $SU(p,q)$, fix $A$ to be a $(p+q) \times (p+q)$ complex Hermitian matrix of signature $(p,q)$. Then you define $SU(p,q)$ to be all $(p+q) \times (p+q)$ complex matrices $M$ satisfying the following two equations: $$M^*AM=A \quad\text{and}\quad \text{det}(M)=1 $$ If you treat the entries of $M$ as variables, and equate matrix entries in the first equation, then you get $(p+q)^2$ polynomial equations in $(p+q)^2$ variables. The second equation is one more polynomial equation in those same variables. Thus, the underlying manifold of $SU(p,q)$ is defined by a system of $(p+q)^2 + 1$ polynomial equations in the $(p+q)^2$ entries of $M$.