How can I get a cohomology of hypersurfaces by using their equation?

Question

While studying about complex projective hypersurfaces, I attempts to find a cohomology of this hypersurface : $$X_n=\{(x_0:x_1:x_2:x_3) \in \mathbb{C}\mathbb{P}^3~|~x_0^n+x_1^n+x_2^n+x_3^n=0\}$$

I think that its cohomology only depend on $n$ but I can't prove it. Is there any way to calculate its cohomology by using its equation?

Answer 1

See this blog post. It's true in general that the cohomology of a smooth hypersurface of degree $n$ in $\mathbb{CP}^d$ only depends on $d$ and $n$.