Let $\mathbb CP^3$ be projective space. Consider polynomial $f$ of degree $k$ satisfying $f(a \mathbb v)=a^k f(\mathbb v)$ for complex number $a$ and vector $v \in \mathbb C^4$. For example $f=xy+z^2-t^2$. Then equation $f=0$ is well defined on projective space. Let $M:=\{f=0\}$ be surface in projective space. It is 2-dimensional complex manifold. Which different manifolds (topologically) can be obtain this way ?
Next we can continue and define $M=\{f=0; g=0\}$ in projective space $\mathbb CP^4$. For example $f=x^2+y^2; g=zt+u^2$ for five variables $x,y,z,t,u$. Which different manifolds (topologically) can be obtain this way ?
My next goal is to intersect such complex manifold (treated as real 4-manifold) with real hyperplane to see what 3-manifolds can be obtained this way. Does this make sense? I started from Brieskorn manifolds and come to this point.
Generically these are smooth complete intersections, and their topology is well-understood, although most of them have no simpler description than this (e.g. K3 surfaces, which occur as quartic hypersurfaces in $\mathbb{CP}^3$). Their Chern classes and cohomology are known and not hard to calculate, and only depend on the degrees of the defining equations. See, for example, this blog post for the case of hypersurfaces (a single equation).