I have read that the surface $K3$ in $\mathbb{CP}^3$ defined by the equation $x^4+y^4+z^4+w^4=0$ is a spin manifold whose $\hat{A}$-genus is non-vanishing. However, not knowing algebraic geometry, I am not sure where to look in order to better understand some basic properties of this surface and was wondering whether anyone could point me to some specific references (notes, book chapters) that might address the following.
- Why is $K3$ a smooth manifold?
- Does $K3$ separate $\mathbb{CP}^3$ into two parts - that is, whether there exist two compact submanifolds $A$ and $B$ with common boundary $K3$ such that $A\cup B=\mathbb{CP}^3$?
Many thanks!
Edit: As pointed in the comments, the answer to question 2 is no because of dimensional considerations. So I would like to ask the following modified question:
- Does there exist a (5-dimensional) real submanifold $N$ of $\mathbb{CP}^3$ that is spin such that we can find compact submanifolds $A$ and $B$ of $N$ with common boundary $K3$, with $A\cup B=N$?