Cohomology of intersection of hyperplanes

Question

let $X = H_1 \cap ... \cap H_d$ be a compact submanifold of $\mathbb{P}_N$ where the $H_i$ are hyperplanes. I want to compute $H^q(X, \mathcal{O}_{\mathbb{P}_N}(m)|X)$.

I am pretty unexperienced in this area, so I would appreciate any help regarding the excercise (although I do not want a solution to this excercise, just an approach to it).

Thanks in advance!

Answer 1

Let the hyperplane $H_i, i= 1,...,d,$ be defined by the homogeneous polynomial $f_i$ of degree $k_i \in \mathbb N$. Then $X$ is the zero set of the section $f:=(f_1,...,f_d) \in H^0(\mathbb P^n,\mathscr O(k_1) \oplus ... \oplus \mathscr O(k_d))$. As a consequence one has the short exact sequence of sheaves

$$0 \longrightarrow O(-k_1) \oplus ... \oplus \mathscr O(-k_d) \overset{f} \longrightarrow \mathscr O_{\mathbb P^n} \longrightarrow \mathscr O_X \longrightarrow 0.$$

After tensoring by $\mathscr O_{\mathbb P^n}(m)$ and taking the corresponding long exact cohomology sequence one can derive the cohomology on $X$ from the well-known cohomology on projective space. (I do not fill in the details because you asked just for the approach.)